What is the difference in assumptions between the one-mean t-test and the one-mean z-test?
The one mean z-test - ? is known
The one mean t-test - ? unknown
What are the two ways to create a narrower confidence interval?
(1) decrease ?
(2) increase n (sample size)
What are the two ways to create a wider confidence interval?
(1) increase ?
(2) decrease n (sample size)
Formula for confidence interval, sigma known
What does ? signify?
The area in both tails. Usually .10 or less
A sample of 36 soda cans was taken and the average number of ounces was found to be 12.4 oz. (assume ?=2). Find a 95% confidence interval for �, the average number of oz in the can. Interpret your results.
(11.747, 13.053)
We are 95% confident that the average number of ounces in a soda can is between 11.8 ounces and 13.1 ounces.
What determines "z" in the confidence interval formula?
Changing the confidence level.
What is the relationship between confidence (high or low) and the length of a confidence interval?
Higher confidence means a wider interval. Lower confidence means a narrower interval.
A sample of 25 typists has an average typing speed of 85 wpm. Assume �=19. Find a 99% confidence interval for the average speed of typists. Interpret your results.
(75.2, 94.8)
We are 99% confident the average typing speed for typists is between 75.2 words per minute and 94.8 words per minute.
What calculator function is used to find a confidence interval when � is known?
STAT TESTS Z-interval
Formula to calculate E (margin of error)
What is the relationship between the margin of error and the length of the confidence interval?
The margin of error is half the length of the confidence interval.
Relate the precision with which x-bar estimates � with the size of the margin of error.
The more precision with which x-bar estimates �, the smaller the margin of error.
An AP poll found that 38% of parents said they were unlikely to give permission for their kids to be vaccinated at school (sample of 1003 adults). The margin of sampling error is +- 3.1 percentage points for all adults. What is the confidence interval? Wh
(34.9, 41.1)
Length of confidence interval: 6.2
Formula to find the sample size required for a 1-? confidence interval.
The FAA estimated with 90% confidence that the mean flight time from Albuquerque NM to Dallas TX to be between 99 minutes and 107.8 minutes. Assume n = 9 and ? = 8. (1) Find the margin of error. (2) Find how many times larger must a sample size be to halv
(1) 4.4
(2) 4 (36/9)
What are the properties of the t-distribution?
a) centered at 0 (like the standard normal distribution)
b) symmetric/bell-shaped
c) wider than the normal distribution
d) area under curve = 1
e) uses degrees of freedom (n-1) to find t-values
Which would result in a wider confidence interval? 90% confidence level or 95% confidence level?
95% confidence level would result in a wider confidence interval. Increasing the confidence level increases the length of the confidence interval.
Which would result in a wider confidence interval?
n=100 or n=400
n=100 would result in a wider confidence interval. Decreasing the sample size increases the length of the confidence interval.
What is the appropriate z-value for a 95% confidence level?
1.96
What is the appropriate z-value for a 90% confidence level?
1.645
What is the appropriate z-value for a 99% confidence level?
2.576
What is the appropriate z-value for an 80% confidence level?
1.282
What is the appropriate z-value for an 85% confidence level?
1.44
(Hint: invnorm (.075))
What is the appropriate z-value for a 50% confidence level?
.674
(Hint: invnorm (.25))
Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let � denote the average alcohol content for the population of all bottles of the brand under study. Suppo
(1) The 90% confidence interval would have been narrower because decreasing the confidence level from 95% to 90% will decrease the confidence interval. The value of Z?/2 for a 90% confidence level (1.28) is smaller than the Z?/2 value for the 95% confiden
Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, 95 of the resulting interval will include �. Is this statement correct? Why or why not?
** This statement is correct. In the long run, after computing the corresponding 95% confidence interval many times, 95 of the resulting confidence intervals will include �.
Two intervals (114.4, 115.6) and (114.1, 115.9) are confidence intervals for �=true average resonance frequency (in hertz) for all tennis rackets of a certain type. (1) What is the value of the sample mean resonance frequency? (2) The confidence level for
(1) x-bar = 115 hertz
(2) the 90% confidence interval is (114.4, 115.6) and the 99% confidence interval is (114.1, 115.9). The confidence interval is larger for the 99% confidence level than for the 90% confidence level.
Five hundred randomly selected working adults living in Calgary, Canada were asked how long, in minutes, their typical daily commute was. The resulting sample mean commute time was 28.5 minutes. Construct and interpret a 90% confidence interval for the me
(26.72, 30.28)
We are 90% confident that the average daily commute time for working adults living in Calgary, Canada, will be between 26.72 minutes and 30.28 minutes.
According to Bride's Magazine, getting married these days can be expensive when all costs are included. A simple random sample of 20 recent U.S. weddings yielded data on wedding costs in dollars (sum of data is $526,538). (1) use the data to obtain a poin
(1) $26,326.9
(2) No. It is unlikely that a sample mean (x-bar) will exactly equal the population mean, �. Some sampling error is to be anticipated.
Consumer Reports provides information on new automobile models - including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of 35 new models yields data on fuel tank capacity, in gallons (sum of data is 66
(1) x-bar = 18.997
(2) (17.81, 20.18) We can be 95.44% confident that the mean fuel tank capacity of all 2003 automobile models is somewhere between 17.81 and 20.18 gallons.
(3) obtain a normal probability plot of the data.
(4) No, because the sample size
Find the confidence level and ? for a 90% confidence interval.
Confidence level = .90
? = .10
Find the confidence level and ? for a 99% confidence interval.
Confidence level = .99
? = .01
What are the assumptions required for using the z-interval procedure?
(1) simple random sample
(2) normal population or large sample (�30)
(3) sigma (?) known
How important is the normality assumption for the z-interval procedure?
The z-interval procedure works well when the variable is normally distributed and reasonably well if the variable is not normally distributed and the sample size is small or moderate, provided the variable is not too far from being normally distributed.
What is meant by saying that a statistical procedure is "robust"?
A statistical procedure that works reasonably well even when one of its assumptions is violated (or moderately violated) is called a robust procedure relative to that assumption.
Assume that the population standard deviation is known. Is it reasonable to use the z-interval procedure to obtain a confidence interval for the population mean under each of the following circumstances: (1) the sample data contains no outliers, the varia
(1) Reasonable, because of the roughly normal distribution, sample size need not be greater than 30 and outliers do not exist that might call into question the normality assumption.
(2) Not reasonable, because sample size is too small.
(3) Reasonable beca
Of 95% and 99% confidence levels, which will result in the confidence interval's giving a more precise estimate of �?
The 95% confidence level because decreasing the confidence level improves the precision.
Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 20
n = 36
? = 3
confidence level = 95%
(19.0, 21.0)
Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 30
n = 25
? = 4
confidence level = 90%
(28.7, 31.3)
Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 50
n = 16
? = 5
confidence level = 99%
(46.8, 53.2)
A random sample of 18 venture-capital investments in the fiber optics business sector yielded the following data, in millions of dollars (sum of the data is $113.97 million). (1) determine a 95% confidence interval for the mean amount, �, of all venture-c
(1) $5.389 million to $7.274 million
(2) We can be 95% confident that the mean amount of all venture-capital investments in the fiber optics business sector is somewhere between $5.389 million and $7.274 million.
Given: safety limit set for cadmium in dry vegetables at 0.5 ppm. Random sample of 12 Bp mushrooms, data obtained sums to 6.31 ppm.
Find and interpret a 99% confidence interval for the mean cadmium level of Bp mushrooms. Assume a population standard devia
0.251 ppm to 0.801 ppm
We can be 99% confident that the mean cadmium level of all Bp mushrooms is somewhere between 0.251 ppm and 0.801 ppm.
According to an article, the mean duration of imprisonment for 32 patients with chronic PTSD was 33.4 months. Assuming that ? = 42 months, determine a 95% confidence interval for the mean duration of imprisonment, �, of all East German political prisoners
18.8 to 48.0 months
We can be 95% confident that the mean duration of imprisonment, �, of all East German political prisoners with chronic PTSD is somewhere between 18.8 and 48.0 months.
For the same value of ?, are t-values greater than or less than z-values?
For the same value of ?, t-values will be greater than z-values.
Find the area to the right of t=1.771 with df=13.
.05
Find the area to the right of t=2 with df = 30.
Between 0.05 and 0.025
Compare t-curves to the standard normal curve as the number of degrees of freedom becomes larger.
As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve.
Formula for confidence interval for one population mean when ? is unknown.
Standardized version of x-bar
...
Studentized version of x-bar
...
How does the distributions of the standard and studentized versions of x-bar differ?
The studentized version has more spread (wider).
Assumptions for the One-Mean t-Interval Procedure
(1) simple random sample
(2) normal population or large sample
(3) ? unknown
Formula for the confidence interval for � (t-interval procedure)
...
When is a confidence interval exact? When is a confidence interval approximately correct?
The confidence interval is exact for normal populations and is approximately correct for large samples from non-normal populations.
The publication Amusement Business provides figures on the cost for a family of four to spend the day at one of America's Amusement parks. A random sample of 25 families of four that attended amusement parks yielded the following costs, rounded to the nea
(182.29, 204.35)
We are 95% confident that the average cost for a family of four to spend the day at an American amusement park will be between $182.29 and $204.35.
null hypothesis
The hypothesis to be tested
alternative hypothesis
The alternative to the null hypothesis
hypothesis test
The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.
two-tailed test
If the primary concern is deciding whether a population mean, �, is different from a specified value �?, we express the alternative hypothesis as:
H?: � � �?
left-tailed test
If the primary concern is deciding whether a population mean, �, is less than a specified value �?, we express the alternative hypothesis as:
H?: � < �?
right-tailed test
If the primary concern is deciding whether a population mean, �, is greater than a specified value �?, we express the alternative hypothesis as:
H?: � > �?
one-tailed test
A hypothesis test that is either left-tailed or right-tailed.
A snack food company produces a 454 g bag of pretzels and insists that the mean net weight of the bags is 454 g. As part of its program, the quality assurance department periodically performs a hypothesis test to decide whether the packaging machine is wo
(1) H?: � = 454 g
(2) H?: � � 454 g
(3) two tailed
What mathematical signs are allowed in the null hypothesis?
0
What mathematical signs are allowed in the alternative hypothesis?
�, <, >
The mean charitable contribution per household in the U.S. in 2000 is $1623. A researcher claims that the level of giving has changed since then. State the null and the alternative hypotheses.
H?: � = $1623
H?: � � $1623
Federal law requires that a jar of peanut butter labeled 32 oz. must contain at least 32 oz. A consumer advocate feels that a certain manufacturer is shorting customers by underfilling jars so that the mean content is less than 32 oz. State the null and a
H?: � � 32
H?: � < 32
left tailed test
A confidence interval for a population mean has a margin of error of 3.4 Determine the length of the confidence interval.
6.8
A confidence interval for a population mean has a margin of error of 3.4. If the sample mean is 52.8, obtain the confidence interval.
(49.4, 56.2)
True or False. The length of a confidence interval can be determined if you know only the margin of error.
True
True or False. The margin of error can be determined if you know only the length of the confidence interval.
True
True or False: The confidence interval can be obtained if you know only the margin of error.
False
True or False: The confidence interval can be obtained if you know only the margin of error and the sample mean.
True
The method for computing the sample size required to obtain a confidence interval with a specified confidence level and margin of error - the number resulting from the formula should be rounded up to the nearest whole number. (1) Why do you want a whole n
(1) The sample size cannot be a fraction.
(2) The result (n) is the smallest value that will provide the required margin of error. If the number were rounded down, the sample size would not be sufficient to ensure the required margin of error.
Infants treated for pulmonary hypertension, called the PH group, were compared with those not so treated, called the control group. One of the characteristics measured was head circumference. The mean head circumference of the 10 infants in the PH group w
(1) (33.108, 35.292)
(2) E = 1.1 cm
(3) We can be 90% confident that the error made in estimating � by x-bar is at most 1.1 cm.
(4) 68
Explain the difference in the formulas for the standardized and the studentized version of x-bar.
The denominator of the standardized version of x-bar (z-score) uses the population standard deviation, ?, whereas the denominator of the studentized version of x-bar (t-score) uses the sample standard deviation, s.
A variable has a mean of 100 and a standard deviation of 16. 4 observations of this variable have a mean of 108 and a sample standard deviation of 12. Determine the observed value of the:
(1) standardized version of x-bar
(2) studentized value of x-bar
(1) 1
(2) 1.33
Two t-curves have degrees of freedom, 12 and 20, respectively. Which one more closely resembles the standard normal curve? Explain your answer.
The df=20 because as the number of degrees of freedom increases, t-curves look increasingly like the standard normal curve.
According to Scarborough Research, more than 85% of working adults commute by car. Of all U.S. cities, Washington, D.C. and New York City have the longest commute times. A sample of 30 commuters in the Washington, D.C. area yielded the following commute t
(1) (24.855, 31.085)
(2) We are 90% confident that the average commute time of all commuters in Washington, D.C. is between 24.855 minutes and 31.085 minutes.
A data set gives the additional sleep in hours obtained by a sample of 10 patients using laevohysocyamine hydrobromide (with xbar=2.33 hr, s=2.002 hr). (1) Obtain and interpret a 95% confidence interval for the additional sleep that would be obtained on a
(1) 0.90 hr to 3.76 hr
We can be 95% confident that the additional sleep that would be obtained on average for all people using the drug is somewhere between 0.90 hr and 3.76 hr.
(2) It appears so, because, based on the confidence interval, we can be 95%
Explain the meaning of the term hypothesis as used in inferential statistics.
A hypothesis is a statement that something is true.
Given: safety limit set for cadmium in dry vegetables at 0.5 ppm. A hypothesis test is to be performed to decide whether the mean cadmium level in Bp mushrooms is greater than the government's recommended limit. (1) Determine the null hypotheses, (2) Dete
(1) null hypothesis H?: � = .5 ppm
(2) alternative hypothesis H?: � > .5 ppm
(3) right tailed test
The recommended dietary allowance (RDA) of iron for adult females under the age of 51 is 18 milligrams (mg) per day. A hypothesis test is to be performed to decide whether adult females under the age of 51 are, on average, getting less than the RDA of 18
(1) null hypothesis H?: � = 18 mg
(2) alternative hypothesis H?: � < 18 mg
(3) left tailed test
According to the Bureau of Crime Statistics and Research of Australia, the mean length of imprisonment for motor-vehicle theft offenders in Australia is 16.7 months. You want to perform a hypothesis test to decide whether the mean length of imprisonment f
(1) H?: � = 16.7 months
(2) H?: � � 16.7 months
(3) two tailed test
test statistic
the z-score (or t-score) that determines if an average is unusual or not
Calculation formula for the test statistic (z-score or t-score)
...
What does the z or t test statistic tell us?
The z or t test statistic tells us how far x-bar is from � in standard deviations (i.e. the number of standard deviations from the mean). It is the statistic used as a basis for deciding whether the null hypothesis should be rejected.
rejection region
The set of values for the test statistic that lead us to reject H? (tail or tails of the distribution)
non-rejection region
The set of values for the test statistic that lead us not to reject H?
critical values
the boundaries for the rejection/non-rejection regions
Type I Error
Rejecting the null hypothesis when it is in fact true
Type II Error
Not rejecting the null hypothesis when it is in fact false.
Type I Error probability
the probability of a Type I error, denoted ?, also called the significance level of the hypothesis test
significance level
the probability of making a Type I error, that is, of rejecting a true null hypothesis (denoted ?)
Type II Error probability
the probability of a Type II error, denoted ? - a Type II error occurs if the test statistic falls in the non-rejection region when in fact the null hypothesis is false.
What is the relationship between Type I and Type II Error probabilities?
For a fixed sample size, the smaller we specify the significance level, ?, the larger will be the probability, ?, of not rejecting a false null hypothesis.
Possible conclusions for a hypothesis test
If the null hypotheses is rejected, we conclude that the alternative hypothesis is true. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis.
Steps for Hypothesis Tests for One Population Mean when ? is known (6)
(1) State the null and alternative hypotheses
(2) Decide on a value for ? (significance level)
(3) Compute the test statistic Z
(4) Find the critical values
(5) Conclusion
(6) Interpretation
Calculation of critical values (for hypothesis test for one population when ? is known)
+/- Z (?/2) - two tailed test
- Z ? - left tailed test
Z? - right tailed test
Suppose a CEO of a company wants to determine whether the average amount of wasted time during an 8-hour day for employees at the company is less than 120 minutes. A random sample of 10 employees gave these results:
108, 131, 112, 113, 117, 113, 130, 105,
Conclusion: Do not reject H?
Interpretation: There is not enough evidence to conclude that the average amount of wasted time at the company is less than 120 minutes.
True or False: If it is important not to reject a true null hypothesis, the hypothesis test should be performed at a small significance level.
True. The significance level ? of a hypothesis test is the probability of making a Type I error (rejecting a true null hypothesis). If this is important, the lower the probability of making such an error the better; thus you should use a small significanc
True or False: For a fixed sample size, decreasing the significance level of a hypothesis test results in an increase in the probability of making a Type II error.
True. For a fixed sample size, the smaller you specify the significance level ?, the larger will be the probability ?, of not rejecting a false null hypothesis.
Define test statistic
The statistic used as a basis for deciding whether the null hypothesis should be rejected
Define rejection region
The set of values for the test statistic that leads to rejection of the null hypothesis.
Define non-rejection region
The set of values for the test statistic that leads to non-rejection of the null hypothesis.
Define critical values
The values of the test statistic that separate the rejection and non-rejection regions. A critical value is considered part of the rejection region.
Define significance level
The probability of making a Type I error, that is, of rejecting a true null hypothesis.
Identify the two types of incorrect decisions in a hypothesis test. For each incorrect decision, what symbol is used to represent the probability of making that type of error?
(1) Type I - rejecting a true null hypothesis (symbol ?)
(2) Type II - not rejecting a false null hypothesis (symbol ?)
Would it be appropriate to use a t-interval for a sample size of 15? Explain.
It would not be appropriate because the assumption of a normal population or a large sample is not met. We know nothing of the population and the sample is small.
What is the appropriate t-value for the following confidence level and sample size: confidence level 95%, n=17
2.120
What is the appropriate t-value for the following confidence level and sample size: confidence level 90%, n=12
1.796
What is the appropriate t-value for the following confidence level and sample size: confidence level 99%, n=24
2.807
The following data are airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: 57, 54, 55, 51, 56, 48, 52, 51, 59, 59 . (1) Compute and interpret a 90% confidence interval for the mean airborne time for fligh
(1) (52.07, 56.33)
(2) Recommend an arrival time of 10:57 a.m., so that 0% of the flights would be late.
A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured i
246
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the
The alternative hypothesis was chosen because the mean strength of welds should be greater than 100, thus the alternative hypothesis H?: � > 100. The primary concern of the research is to decide whether the population mean is greater than the specified va
Does this pair comply with the rules for setting up hypotheses? If not, explain why.
H?: � = 15; H?: � = 15
Does not comply. H? must be stated as � 15, <15, or > 15.
Does this pair comply with the rules for setting up hypotheses? If not explain why.
H?: � = 10; H?: � > 12
Does not comply. H? must use the same number as H?, the null hypothesis.
Does this pair comply with the rules for setting up hypotheses? If not explain why.
H?: � = 123; H?: � < 123
Complies
Does this pair comply with the rules for setting up hypotheses? If not explain why.
H?: � = 123; H?: � = 125
Does not comply. H? must use the same number as H? and cannot contain the equal sign.
Does this pair comply with the rules for setting up hypotheses? If not explain why.
H?: � = 50; H?: � � 50
Complies
Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170. Let � represent the true mean blood chole
H?: � = 170 (the null hypothesis is the hypothesis to be tested)
H?: � < 170
Using the test statistic formula for the z-score, determine the required critical value(s) for a right-tailed test with ? = 0.05.
1.645
Using the test statistic formula for the z-score, determine the required critical value(s) for a left-tailed test with ? = 0.05.
-1.645
Using the test statistic formula for the z-score, determine the required critical value(s) for a two-tailed test with ? = 0.05.
-1.96 and +1.96
Decide in the following situations whether the z-test is an appropriate method for conducting
the hypothesis test for a population mean: (a) no outliers, distribution highly skewed, sample size 24, (b) no outliers, mildly skewed, sample size 70
(a) not appropriate
(b) appropriate
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at 0.5 parts per million (ppm). A ran
Given: significance level 0.05
? = 0.37
H?: � = 0.5 ppm
H?: � > 0.5 ppm
Test statistic: z=0.24
Critical value: 1.645
Conclusion: 0.24 is in the non-rejection region
Interpretation: There is not enough evidence to conclude that the mean level of cadmium in
Given:
n=45
x-bar = 14.68
? = 4.2
? = .01
H?: � = 18
H?: � < 18
Reject or do not reject null hypothesis?
Test statistic: z = -5.3
Critical value: -2.326
Conclusion: -5.3 is in the rejection region
Interpretation: There is sufficient evidence to conclude that ... the alternative hypothesis is true.
Describe the meaning of P-value of a hypothesis test
To obtain the P-value of a hypothesis test, we assume that the null hypothesis is true and compute the probability of observing a value of the test statistic as extreme as or more extreme than that observed. By extreme we mean "far from what we would expe
If you have a p-value of 0.0168 and a z-score of +/- 2.39, interpret the meaning of these values in context.
The probability of getting a z-score more extreme than +/- 2.39 is 0.0168.
On the calculator (TI-84), how do you find the area to the left of a particular z-score?
NORMALCDF (-1000, Z-score, 0, 1)
On the calculator (TI-84), how do you find the area to the right of a particular z-score?
NORMALCDF (Z-score, 1000, 0, 1)
What p-value(s) describe weak or no evidence against the null hypothesis?
P-value > .10
What p-value(s) describe moderate evidence against the null hypothesis?
.05 < P � .10
What p-value(s) describe strong evidence against the null hypothesis?
.01 < P � .05
What p-value(s) describe very strong evidence against the null hypothesis?
P � .01
How does a large test statistic relate to the area in the tail?
A large test statistic means that there is a smaller area in the tail.
How does a small test statistic relate to the area in the tail?
A small test statistic means that there is a larger area in the tail.
2 methods for determining whether to reject or not reject the null hypothesis
(1) compare the test statistic to the critical values; where the test statistic falls (rejection region or non-rejection region)
(2) compare the p-value to ?
If the p-value is low, H? must go!
If the p-value � ?, reject H?
If the p-value > ?, do not rejec
A hot tub manufacturer advertises that with its heating equipment a temperature of 100 degrees F can be achieved in at most 15 minutes. A random sample of 20 tubs is selected and the time needed to reach 100 degrees is determined for each tub. The sample
(1) H?: � � 15
H?: � > 15
(2) ? = 0.01
(3) Test statistic: t = 4.47
(4) P-value = .000013 (using t-test on calculator)
(5) Compare p-value to ?
.000013 � 0.01
Reject H?
(6) There is enough evidence to suggest the average time for a hot tub to reach 100 de
An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg decides to carry out a fuel efficiency test. Six non-professional drivers are selected and each one drives a car from Phoenix to Los Angeles. The resulting fuel effi
(1) H?: � � 30 mpg
H?: � < 30 mpg
(2) ? = 0.05
(3) test statistic t = -1.16
(4) p-value = .1493 (t-test on calculator)
(5) compare p-value to ?
.1493 � 0.05 ? NO
Do not reject H?
(6) There is not enough evidence to conclude that the average fuel efficienc
For which of the following p-values would the null hypothesis be rejected at a level of ? = 0.05:
(a) .001
(b) .021
(c) .078
(d) .047
(e) .148
(a) reject
(b) reject
(c) do not reject
(d) reject
(e) do not reject
State two reasons why including the p-value is prudent when you are reporting the results of a hypothesis test.
(1) it allows you to assess significance at any desired level
(2) it permits you to evaluate the strength of the evidence against the null hypothesis
What is the p-value of a hypothesis test?
The probability of observing a value of the test statistic as extreme or more extreme than that observed. By extreme we mean "far from what we would expect to observe if the null hypothesis is true.
When does the p-value provide evidence against the null hypothesis?
When the p-value is less than or equal to the significance level, ?
True or False: The p-value is the smallest significance level for which the observed sample data result in rejection of the null hypothesis.
True
The p-value for a hypothesis test is 0.083. For each of the following significance levels, decide whether the null hypothesis should be rejected:
(1) ? = 0.05
(2) ? = 0.10
(3) ? = 0.06
(1) Do not reject (0.083 > 0.05)
(2) Reject (0.083 � 0.10)
(3) Do not reject (0.083 > 0.06)
Determine the strength of the evidence against the null hypothesis:
(a) p = 0.06
(b) p = 0.35
(c) p = 0.027
(d) p = 0.004
(1) moderate
(2) weak or none
(3) strong
(4) very strong
Determine the p-value.
Z = 2.03 right-tailed test
0.0212
Determine the p-value.
Z = -0.31 right-tailed test
0.6217
Determine the p-value.
Z = -0.74 left-tailed test
0.2296
Determine the p-value.
Z = 1.16 left-tailed test
0.8770
Determine the p-value.
Z = -1.66 two-tailed test
0.0970
Determine the p-value.
Z = 0.52 two-tailed test
0.6030
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at 0.5 parts per million (ppm). A ran
(1) H?: � = 0.5 ppm
H?: � > 0.5 ppm
(2) ? = 0.05
(3) test statistic: Z = 0.24
(4) p-value =0.4044 (calculator t-test)
(5) conclusion: compare p-value to ?:
Is 0.4044 � 0.05? No. Do not reject.
(6) There is not sufficient evidence to conclude that the aver
According to Communications Industry Forecast & Report, the average person watched 4.66 hours of television per day in 2002. A random sample of 20 people gave the number of hours of television watched per day for last year (x-bar = 4.835 hours, s = 2.291
(1) H?: � = 4.66 hours
H?: � � 4.66 hours
(2) ? = 0.10
(3) Test statistic t = 3.416
(4) p-value: p=0.7364 (from calculator t-test)
(5) Conclusion: compare p-value to ?
Is 0.7364 � 0.10? No. Do not reject.
(6) There is not sufficient evidence that the aver
In the following problem, decide whether applying the t-test to perform a hypothesis test for the population mean appears reasonable.
The Florida State Center for Health Statistics reported that, for cardiovascular hospitalizations, the mean age of women
The t-test is not reasonable because the sample is small (20) and a histogram of the data shows that the data is not normally distributed.
For the following:
Right tailed test, n=20, t=2.235,
(1) Estimate the p-value from the Table
(2) Based on your estimate from part (1), state at which significance levels the null hypothesis can be rejected, at which significance levels it cannot be reject
(1) 0.01 < P < 0.025
(2) We can reject H? at any significance level of 0.025 or larger, and we cannot reject H? at any significance level of 0.01 or smaller. For significance levels between 0.01 and 0.025, the table is not sufficiently detailed to help us
For the following:
Left-tailed test, n=10, t = -3.381,
(1) Estimate the p-value from the Table
(2) Based on your estimate from part (1), state at which significance levels the null hypothesis can be rejected, at which significance levels it cannot be reje
(1) P < 0.005
(2) We can reject H? at any significance level of 0.005 or larger. For significance levels smaller than 0.005, the table is not sufficiently detailed to help us to decide whether to reject H?.
For the following:
Two-tailed test, n = 17, and t = -2.733,
(1) Estimate the p-value from the Table
(2) Based on your estimate from part (1), state at which significance levels the null hypothesis can be rejected, at which significance levels it cannot be
(1) 0.01 < P < 0.02
(2) We can reject H? at any significance level of 0.02 or larger, and we cannot reject H? at any significance level of 0.01 or smaller. For significance levels between 0.01 and 0.02, the table is not sufficiently detailed to help us to
Given:
x-bar = 20
s = 4
n = 32
H?: � = 22, H?: � < 22
? = 0.05
(1) Use the one-mean t-test to perform the required hypothesis test about the mean, �, of the population from which the sample was drawn.
(2) Find (or estimate) the P-value and determine the s
(1) t = -2.82; critical value = -1.696
P-value = 0.004 (P < 0.05) Reject H?.
(2) Very strong (P < 0.01)
Given:
x-bar = 24
s = 4
n = 15
H?: � = 22, H?: � > 22
? = 0.05
(1) Use the one-mean t-test to perform the required hypothesis test about the mean, �, of the population from which the sample was drawn.
(2) Find (or estimate) the P-value and determine the s
(1) t = 1.94; critical value = 1.761
p-value = 0.037 (0.025 < P < 0.05) Reject H?.
(2) Strong (0.01 < P < 0.05)
Given:
x-bar = 23
s = 4
n = 24
H?: � = 22, H?: � � 22
? = 0.05
(1) Use the one-mean t-test to perform the required hypothesis test about the mean, �, of the population from which the sample was drawn.
(2) Find (or estimate) the P-value and determine the s
(1) t = 1.22; critical values = +/- 2.069
p-value = .233 (P > 0.20) Do not reject H?.
(2) Weak or none (P > 0.10)
What calculator function is sued to find a confidence interval when ? is unknown?
STATS - TESTS - T-Interval
What value does 1 - ? give you?
The confidence level
For a fixed confidence level, how does increasing the sample size affect precision?
Increasing the sample size improves the precision.
What factor determines the precision with which x-bar estimates �?
The length of the confidence interval, determined by the margin of error.
point estimate
The value of a statistic used to estimate a parameter
A tarantula has two body parts. The anterior part of the body is covered above by a shell, or carapace. A simple random sample of 15 of adult male Brazilian giant tawny red tarantulas provided the following data on carapace length, in millimeters (mm): 15
(17.52, 19.34)
We can be 95.44% confident that the average carapace length of all adult male Brazilian giant tawny red tarantulas is between 17.52 mm and 19.34 mm.
Explain why the margin of error determines the precision with which a sample mean estimates a population mean.
The length of a confidence interval, and thus the precision with which x-bar estimates �, is determined by the margin of error.
A confidence interval for a population mean has length 20. (a) Determine the margin of error. (b) If the sample mean is 60, obtain the confidence interval.
(a) 10
(b) (50, 70)
For the confidence interval (5.289, 7.274), obtain the margin of error by:
(a) taking half the length of the confidence interval
(b) using the formula for E
margin of error = .94
For the confidence interval (18.8, 48): (a) Determine the margin of error E. (b) For a 95% confidence level, explain the meaning of E in this context in terms of the accuracy of the measurement.
(a) margin of error = 14.6
(b) We are 95% confident that the maximum error made in using x-bar to estimate � is 14.6.
Professor Thomas Stanley of Georgia State University has surveyed millionaires since 1973. Among other information, Professor Stanley obtains estimates for the mean age, �, of all U.S. millionaires. Suppose that one year's study involved a simple random s
(a) 172
(b) We used s in place of ? because ? was unknown. We can do this because the sample of size 36 is large enough to provide an estimate of ? and the variation is not likely to change much from one year to the next.
Suppose that a simple random sample is taken from a normal population having a standard deviation of 10 for the purpose of obtaining a 95% confidence interval for the mean of the population. (a) If the sample size is 4, obtain the margin of error. (b) Rep
(a) 9.80
(b) 4.90
(c) It appears that quadrupling the sample size will halve the margin of error. Therefore, increasing n from 16 to 64 will decrease the margin of error from 4.90 to 2.45.
Explain why there is more variation in the possible values of the studentized version of x-bar than in the possible values of the standardized version of x-bar.
The variation in the possible values of the standardized version of x-bar is due only to the variation in x-bar, while the variation in the studentized version results not only from the variation in x-bar, but also from the variation in the sample standar
For a t-curve with df=6, use Table IV to find each t-value: (a) t (sub 0.10), (b) t (sub 0.025), (c) t (sub 0.001)
(a) 1.440
(b) 2.447
(c) 3.143
For a t-curve with df=21, find each t-value: (a) The t-value having area 0.10 to its right (b) t (sub 0.01), (c) The t-value having area 0.025 to its left (Hint: a t-curve is symmetric about 0), (d) The two t-values that divide the area under the curve in
(a) 1.323
(b) 2.518
(c) -2.080
(d) +/- 1.721
A simple random sample of size 100 is taken from a population with unknown standard deviation. A normal probability plot of the data displays significant curvature but no outliers. Can you reasonably apply the t-interval procedure? Explain your answer.
It is reasonable to use the t-interval procedure since the sample size is large and for large degrees of freedom (99), the t-distribution is very similar to the standard normal distribution. Another way of expressing this is that the sampling distribution
With the following information, use the one-mean t-interval procedure to find a confidence interval for the mean of the population from with the sample was drawn:
x-bar = 20
n = 36
s = 3
confidence level = 95%
(18.98, 21.02)
With the following information, use the one-mean t-interval procedure to find a confidence interval for the mean of the population from with the sample was drawn:
x-bar = 30
n = 25
s = 4
confidence level = 90%
(28.63, 31.37)
With the following information, use the one-mean t-interval procedure to find a confidence interval for the mean of the population from with the sample was drawn:
x-bar =50
n = 16
s = 5
confidence level = 99%
(46.32, 53.68)
A variable of a population has a mean of 266 and a standard deviation of 16. Ten observations of this variable have a mean of 262.1 and a sample standard deviation of 20.4. Obtain the observed value of the (a) standardized version of x-bar, (b) studentize
(a) -0.77
(b) -0.605
Explain the difference between a point estimate of a parameter and a confidence-interval estimate of a parameter.
A point estimate of a parameter consists of a single value with no indication of the accuracy of the estimate. A confidence interval consists of an interval of numbers obtained from a point estimate of the parameter together with a percentage that specifi
Must the variable under consideration be normally distributed for you to use the z-interval procedure or t-interval procedure? Explain your answer.
No. The z-interval procedure can be used almost anytime with large samples because the sampling distribution of x-bar is approximately normal for large n. The same is true for the t-interval procedure because when n is large, the t-distribution is very si
Suppose that you have obtained a sample with the intent of performing a particular statistical-inference procedure. What should you do before applying the procedure to the sample data? Why?
Before applying a particular statistical inference procedure, we should look at graphical displays of the sample data to see if there appear to be any violations of the conditions required for the use of the procedure.
A confidence interval for a population mean has a margin of error of 10.7. (a) Obtain the length of the confidence interval. (b) If the mean of the sample is 75.2, determine the confidence interval.
(a) 21.4
(b) (64.5, 85.9)
Dr. Thomas Stanley of Georgia State University has surveyed millionaires since 1973. Among other information, Stanley obtains estimates for the mean age, �, of all U.S. millionaires. Suppose that 36 U.S. millionaires are randomly selected (mean = 58.53).
(54.3, 62.8)
Abigail Camp Dimon found the mean shell length of 461 randomly selected specimens of N. trivittata to be 11.9 mm. (a) assuming that ? = 2.5 mm, obtain a 90% confidence interval for the mean length, �, of all N. trivittata. (b) interpret your answer from p
(a) (11.71, 12.09)
(b) We can be 90% confident that the mean length of N. trivittata is somewhere between 11.71 and 12.09 mm.
(c) Since the sample size is very large, the distribution of sample means will be approximately normal regardless of the shape of
For a t-curve with df = 18, obtain the t-values of the following:
(a) The t-value having area 0.025 to its right
(b) t (sub 0.05)
(c) The t-value having area 0.10 to its left
(d) The two t-values that divide the area under the curve into a middle 0.99 are
(a) 2.101
(b) 1.734
(c) -1.330
(d) +/- 2.878
In a Singapore edition of Business Times, diamond pricing was explored. The price of a diamond is based on the diamond's weight, color, and clarity. A simple random sample of 18 one-half carat diamonds had the following prices, in dollars: 1676, 1995, 144
($1880.07, $2049.37)
We can be 90% confident that the mean diamond price for one-half carat diamonds is between $1880.07 and $2049.37.
True or False (and give a reason for your answer): If a 95% confidence interval for a population mean, �, is from 33.8 to 39.0, the mean of the population must lie somewhere between 33.8 and 39.0
False. We are 95% confident that the mean lies in the interval from 33.8 to 39.0, but about 5% of the time, the procedure will produce an interval that does not contain the population mean. Therefore, we cannot say that the mean must lie in the interval.
If you obtained one thousand 95% confidence intervals for a population mean, �, roughly how many of the intervals would actually contain �?
950
Suppose that you intend to find a 95% confidence interval for a population mean by applying the one-mean z-interval procedure to a sample of size 100. (a) What would happen to the precision of the estimate if you used a sample of size 50 instead but kept
(a) Reducing the sample size from 100 to 50 will reduce the precision of the estimate (result in a longer confidence interval).
(b) Reducing the confidence level from .95 to .90 while maintaining the sample size will increase the precision of the estimate
Suppose that you plan to apply the one-mean z-interval procedure to obtain a 90% confidence interval for a population mean, �. You know that ? = 12 and that you are going to use a sample of size 9. (a) What will be your margin of error? (b) What else do y
(a) 6.58
(b) To obtain the confidence interval, you also need to know x-bar.
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 17 is taken from a population. A normal probability plot of the sample data is found to
t-interval procedure
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 50 is taken from a population. A normal probability plot of the sample data is found to
z-interval procedure
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 25 is taken from a population. A normal probability plot of the sample data shows three
z-interval procedure
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 20 is taken from a population. A normal probability plot of the sample data shows three
Neither procedure should be used.
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 128 is taken from a population. A normal probability plot of the sample data shows no ou
z-interval procedure
Decide whether the appropriate method for obtaining the confidence interval is the z-interval procedure, the t-interval procedure, or neither:
A random sample of size 13 is taken from a population. A normal probability plot of the sample data shows no out
Neither procedure should be used.
Following are the arterial blood pressures, in millimeters of mercury (mm Hg), for a random sample of 16 children of diabetic mothers:
(x-bar = 85.99 mm Hg, s = 8.08 mm Hg)
(a) Apply the t-interval procedure to find a 95% confidence interval for the mean
(81.69, 90.29)
We can be 95% confident that the mean arterial blood pressure of all children of diabetic mothers is somewhere between 81.69 and 90.29 mm Hg.
Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations. Discussing the benefits and downsides of the screening process, the article states that, although the rate of false-
(a) Type I
(b) The Type I error would be a false positive (thinking cancer is present when there is none) and would likely result in following up by the radiologists. There would be perhaps unnecessary additional tests which would increase costs.
(c) A Ty
Find the critical value(s) for the following tests:
(a) Right tail test with ? = 0.10
(b) Left tail test with ? = 0.01
(c) Two tail test with ? = 0.05
(a) 1.282
(b) -2.326
(c) +/- 1.96
In a study of computer use, 1000 randomly selected Canadian internet users were asked how much time they spend using the Internet in a typical week. The mean of the 1000 resulting observations was 12.7 hours. Assume the population standard deviation is 5
1. H?: � = 12.5 hours, H?: � > 12.5 hours
2. ? = 0.05
3. test statistic: z = 1.26
4. critical value: 1.645
5. conclusion: Do not reject the null hypothesis.
6. interpretation: There is not enough evidence to conclude that the average time spent using the
A credit bureau analysis of undergraduate student credit records found that the average number of credit cards in an undergraduate's wallet was 4.09. It was also reported that in a random sample of 132 undergraduates, the sample mean number of credit card
1. H?: � = 4.08, H?: � < 4.09
2. ? = 0.05 (not given)
3. Test statistic: z = -14.26
4. Critical value: -1.645
5. Conclusion: Reject the null hypothesis.
6. There is sufficient evidence to conclude that the average number of credit cards that undergraduate
Consider the null and alternative hypotheses:
H?: � = 30 lb (mean has not increased)
H?: � > 30 lb (mean has increased)d
where � is last year's mean cheese consumption for all Americans. Explain what each of the following would mean:
(a) Type I error
(b)
(a) a Type I error would occur if, in fact, � = 30 lb, but the results of the sampling lead to the conclusion that � > 30 lb.
(b) a Type II error would occur if, in fact, � > 30 lb, but the results of the sampling fail to lead to that conclusion
(c) A cor
According to an FBI document, the mean value lost to purse snatching was $332 in 2002. For last year, 12 randomly selected purse-snatching offenses yielded the following values lost, to the nearest dollar: 207, 237, 422, 226, 272, 205, 362, 348, 165, 266,
(a) H?: � = 332, H?: � < 332
? = 0.05
test statistic: t = -1.909
Critical value = -1.782
Conclusion: reject H?
Interpretation: At the 5% significance level, the data provides sufficient evidence to conclude that the mean value lost because of purse snatch
In May, 2002, the average cost of a private room in a nursing home was $168 per day. For August 2003, a random sample of 11 nursing homes yielded the following daily costs, in dollars, for a private room in a nursing home: 73, 159, 199, 182, 192, 208, 181
Test Statistic: W = 48
Reject the null hypothesis.
Assumptions for the Wilcoxon Signed Rank Test
(1) simple random sample
(2) symmetric distribution (triangular, uniform, symmetric bimodal)
Steps for the Wilcoxon Signed Rank Test
(1) Determine null and alternative hypotheses
(2) Determine value for ?
(3) Construct the table to find W, the test statistic
(4) Find critical value(s)
(5) Number line; plot test statistic and critical value(s)
(6) Conclusion: reject H? or do not reject
In the Wilcoxon signed rank test, how do you assign ranks if there is a tie (1) between 2 values of |D|, (2) between 3 values of |D|
(1) If there is a tie between two values of |D|, then average the ranks and assign that average to the two values
(2) If there is a tie between three values of |D|, then assign the middle rank to all three values
In the Wilcoxon signed rank test, if an observation is equal to �?, what should happen to that observation?
The observation is to be tossed out and the sample size reduced accordingly.
Test statistic for z-test
Test statistic for t-test
The National Center for Test Statistics reports that the median birth weight of U.S. babies was 7.4 lb in 2002. A random sample of this year's births provided the following weights, in pounds: 8.6, 8.8, 7.4, 8.2, 5.3, 9.2, 13.8, 5.6, 7.8, 6.0, 5.7, 11.6,
(1) H?: � = 7.4 lb
H?: � � 7.4 lb
(2) ? = 0.05
(3) Test statistic (from created table): W = 57.5
(4) Critical values - Right side 74, Left side 17
(5) Conclusion: Do not reject H?
(6) Interpretation: There is not enough evidence to conclude that the media
For the 2002 baseball season, the median baseball salary was determined to be $800,000. A random sample of 14 salaries was conducted with the following salaries in 2005 (in thousands) as follows: 316, 326, 331, 332, 335, 550, 750, 950, 1300, 3000, 4000, 6
(1) H?: � = 800
H?: � > 800
(2) ? = 0.05
(3) Test Statistic (Construct Table): W = 71
(4) Critical Value = 79
(5) Conclusion: Do not reject H?
(6) Interpretation: At the 5% significance level, there is not sufficient evidence to conclude that the median b
Standardized version of the difference of two means (x-bar? - x-bar?)
Sampling Distribution for x-bar? - x-bar?
Mean �? - �?
Std Deviation: See above
Shape: normal distribution
Technically, what is a nonparametric method? In current statistical practice, how is that term used?
A nonparametric method is an inferential method not concerned with parameters (such as � and ?). Common statistical practice is to refer to most methods that can be applied without assuming normality as nonparametric.
What assumption must be met in order to use the Wilcoxon signed rank test?
a symmetric distribution
In a Wilcoxon signed rank test, an observation equals �? (the value given for the mean in the null hypothesis), that observation should be removed and the sample size reduced by 1. Why does that need to be done?
Because the D-value for such an observation equals 0, a sign cannot be attached to the rank of |D|
The Wilcoxon signed rank test can be used to perform a hypothesis test for a population median, ?, as well as for the population mean. Why is that so?
For a symmetric distribution, the mean and median are equal.
During the late 1800s, Lake Wingra in Madison, Wisconsin, was frozen over an average of 124.9 days per year. A random sample of eight recent years provided the following data on numbers of days that the lake was frozen over: 103, 80, 79, 135, 134, 77, 80,
(1) H?: � = 124.9
H?: � < 124.9
(2) ? = 0.05
(3) Test Statistic (from constructed table): W = 3
(4) Critical value = 6
(5) Conclusion: Reject H?
(6) Interpretation: At the 5% significance level, there is sufficient evidence to conclude that the average nu
In 2002, the median age of U.S. residents was 35.7 years. A random sample of 10 U.S. residents taken this year yielded the following data, in years: 42, 45, 62, 49, 14, 39, 57, 11, 36, 26. At the 1% significance level, do the data provide sufficient evide
(1) H?: � = 35.7
H?: � > 35.7
(2) ? = .01
(3) Test Statistic (from constructed table): W = 33
(4) Critical value: 50
(5) Conclusion: Do not reject H?
(6) Interpretation: At the 1% significance level, there is not sufficient evidence to conclude that the m
Consider the following quantities: �, ?, x-bar, s.
Which are parameters and which are statistics? Which are fixed numbers and which are variables?
Parameters: � and ?
Statistics: x-bar and s (sample standard deviation)
Parameters are fixed numbers and statistics are variables.
Why do you need to know the sampling distribution of the difference between two sample means in order to perform a hypothesis test to compare two population means?
So that you can determine whether the observed difference between the two sample means can be reasonably attributed to sampling error or whether that difference suggests that the null hypothesis of equal population means is false and the alternative hypot
Identify the assumption for using the two-means z-test and the two-means z-interval procedure that renders those procedures generally impractical.
The assumption that ? is known (because population standard deviations are usually unknown).
How do you calculate the test statistic for the w-test?
From a constructed chart using (1) observations, (2) difference between observation and �?, (3) absolute values of values calculated in step 2, (4) ranking of the absolute values, (5) signing ranks according to sign in step 2, (6) w = sum of positive rank
With the following hypothesis test: (a) identify the variable, (b) identify the two populations, (c) determine the null and alternative hypotheses, (d) classify the hypothesis test as two tailed, left tailed, or right tailed:
Samples of adolescent offspri
(a) systolic blood pressure
(b) ODM adolescents and ONM adolescents
(c) H?: �? = �?
H?: �? > �?
* �? = mean systolic bp of ODM adolescents
* �? = mean systolic bp of ONM adolescents
(d) right-tailed
A variable of two populations has a mean of 40 and a standard deviation of 12 for one of the populations and a mean of 40 and a standard deviation of 6 for the other population. For independent samples of sizes 9 and 4 respectively, find the mean and stan
Mean: 0
Standard Deviation: 5
The Kelley Blue Book provides information on retail and trad-in values for used card and trucks. The retail value represents the price a dealer might charge after preparing the vehicle for sale. A 2003 Ford Mustang coupe has a 2006 Kelley Blue Book retail
(1) H? = 12,850
H? < 12,850
(2) ? = 0.10
(3) Test statistic: W? = 13
(4) Critical value = (10)(11)/2 - 41
(5) Since W < 14, reject H?
(6) At the 10% significance level, there is sufficient evidence to conclude that the mean asking price for a 2003 Ford Mu
A certain pen has been designed to that true average writing lifetime under controlled conditions is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data
(a) It is appropriate to reject H?.
(b) It is appropriate to reject H?.
(c) Do not reject H? (weak or no evidence - P-value > 0.10)
A credit bureau analysis of undergraduate student credit records found that the average number of credit cards in an undergraduate's wallet was 4.09. It was also reported that in a random sample of 132 undergraduates, the sample mean number of credit card
(a) H?: � = 4.09
H?: � < 4.09
(b) ? = .05 (not given)
(c) Test statistic: z? = -14.27
(d) P-value: � 0 (less than 0.0001)
(e) Conclusion: Reject H?
(f) Interpretation: There is enough evidence to suggest that the average number of credit cards in an under
Fill in the blanks:
(a) For a two-tail test, the ____(a)_____ is the probability of observing a value of the test statistic t that is at least as large in magnitude as the value actually observed, which is the area under the t-curve that lies outside the
(a) p-value
(b) - |t?|
(c) |t?|
Fill in the blanks:
(Use the Book tables) For a right-tailed test with n=15, ? = 0.01 and a value of the test statistic of t = 3.458, the test statistic lies to the right of ____(a)____. This means the true p-value is (choose one: larger/smaller) ____(b)_
(a) 2.977
(b) smaller
(c) 0.005
What are the 3 assumptions of the t-test?
(1) simple random sample
(2) normal population or a large sample
(3) ? unknown
A hot tub manufacturer advertises that with its heating equipment, a temperature of 100� F can be achieved in at most 15 minutes. A random sample of 25 tubs is selected, and the time necessary to achieve a 100� F temperature is determined for each tub. Th
(a) H?: � � 15 minutes
H?: � > 15 minutes
(b) ? = 0.05
(c) Test statistic: t? = 5.68
(d) p-value: < .0001 (�.00000374)
(e) Conclusion: Reject H?
(f) Summary: There is enough evidence to suggest the average time for a hot tub to reach 100� F is greater tha
Test statistic formula for independent samples (2 population means, equal standard deviations)
(Y-bars should show as x-bars)
Critical value(s) for independent samples (2 population means, equal standard deviations)
Two tailed: +/- t (sub ?/2), df = n? + n? - 2
Left tailed : - t (sub ?), df = n? + n? - 2
Right tailed: t (sub ?), df = n? + n? - 2
Test statistic formula for independent samples (2 population means, standard deviations not equal)
Critical value(s) for independent samples (2 population means, standard deviations not equal)
Two tailed: +/- t (sub ?/2), df = ?
Left tailed : - t (sub ?), df = ?
Right tailed: t (sub ?), df = ?
2-mean confidence interval formula for independent samples (standard deviations equal)
2-mean confidence interval formula for independent samples (standard deviations not equal)
* - refers to:
t (sub ?/2), df = ?
Pooled T-Test Assumptions
(1) simple random sample
(2) normal population or large sample
(3) independent samples
(4) equal population standard deviations
Non-pooled T-Test Assumptions
(1) simple random sample
(2) normal population or large sample
(3) independent samples
(4) unequal population standard deviations
How do you know whether to use the pooled t-test or the non-pooled t-test for 2-mean independent samples?
Compare the standard deviations of the two samples. If one is nearly twice the other, use the non-pooled t-test. If they are fairly close, use the pooled t-test.
Of the 4 pooled t-test assumptions (simple random sample, independent samples, normal population or large samples, equal population standard deviations), how important is each of these?
Simple random samples and independent samples are essential assumptions. Moderate violations of the normality assumption are permissible even for small or moderate size samples. Moderate violations of the equal standard deviations requirement are not seri
The U.S. Bureau of Prisons publishes data in Prison Statistics on the times served by prisoners released from federal institutions for the first time. Independent random samples of released prisoners in the fraud and firearms offense categories yielded th
(1) H?: �? = �?
H?: �? < �?
(2) ? = 0.05
(3) Test Statistic: t = -4.058
(4) Critical Value = -1.734, P-value: = .000369
(5) Conclusion: Reject H?
(6) At the 5% significance level, there is sufficient evidence to conclude that the mean time served for frau
L. Smith and D. Haukos examined the relationship of species richness and diversity to playa area and watershed disturbance. Independent random samples of 126 playa with cropland and 98 playa with grassland in Southern Great Plains yielded the following su
(1) H?: �? = �?
H?: �? � �?
(2) ? = 0.05
(3) Test Statistic: t = -1.98
(4) Critical Value = -1.971, P-value: = 0.0493
(5) Conclusion: Reject H?
(6) At the 5% significance level, there is sufficient evidence to conclude that there is a difference in the me
Suppose that you know that a variable is normally distributed on each of two populations. Further suppose that you want to perform a hypothesis test based on independent random samples to compare the two population means. In each case, decide whether you
(1) pooled
(2) nonpooled
(3) pooled
(4) nonpooled
Researchers randomly and independently selected 32 former prisoners diagnosed with chronic PTSD and 20 former prisoners that were diagnosed with PTSD after release from prison but had since recovered (remitted). The ages, in years, at arrest yielded the f
(1) H?: �? = �?
H?: �? � �?
(2) ? = 0.10
(3) Test Statistic: t = 1.79
(4) Critical Value = +/- 1.677, P-value: = 0.0794
(5) Conclusion: Reject H?
(6) At the 10% significance level, there is sufficient evidence to conclude that there is a difference in the
Use the nonpooled t-test and the nonpooled t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval:
x-bar? = 10 x-bar? = 12
s? = 2 s? = 5
n? = 15 n? = 15
(a) two tailed test, ? = 0.05
(b) 95% confidence in
(a) t = -1.44
critical values = +/- 2.101
do not reject H?
(b) (-4.92, 0.92)
Use the nonpooled t-test and the nonpooled t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval:
x-bar? = 20 x-bar? = 18
s? = 4 s? = 5
n? = 10 n? = 15
(a) right-tailed test, ? = 0.05
(b) 90% confidence
(a) t = 1.11
critical value = 1.717
do not reject H?
(b) (-1.10, 5.10)
Use the nonpooled t-test and the nonpooled t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval:
x-bar? = 20 x-bar? = 24
s? = 6 s? = 2
n? = 20 n? = 15
(a) left-tailed test, ? = 0.05
(b) 90% confidence i
(a) t = -2.78
critical value = -1.711
reject H?
(b) (-6.46, -1.54)
Researchers obtained the following data on the number of acute postoperative days in the hospital using the dynamic and static systems:
dynamic:
Dynamic: x-bar? = 7.36, s? = 1.22, n? = 14
Static: x-bar? = 10.50, s? = 4.59, n? = 6
At the 5% significance le
(1) H?: �? = �?
H?: �? < �?
(2) ? = 0.05
(3) Test Statistic: t = -1.645
(4) Critical Value = -2.015, P-value: = 0.0781
(5) Conclusion: Do not reject H?
(6) At the 5% significance level, there is not sufficient evidence to conclude that the mean number of
Researchers randomly and independently selected 32 former prisoners diagnosed with chronic PTSD and 20 former prisoners that were diagnosed with PTSD after release from prison but had since recovered (remitted). The ages, in years, at arrest yielded the f
(.23712, 7.1629) or rounded (.2, 7.2); df = ? = +/- 1
We can be 90% confident that the difference between the mean ages at arrest of East German prisoners with chronic PTSD and remitted PTSD is between 0.2 and 7.2 years.
What does "pooling" refer to, in the context of a hypotheses test for two population means?
Since we cannot use the population standard deviation as a basis for calculating the test statistic in a hypothesis test for two population means (because ? is unknown), we use sample information to estimate ?, the unknown population standard deviation. W
Do children diagnosed with ADHD have smaller brains than children without this condition? Brain scans were completed for 152 children with ADHD and 139 children of similar age without ADHD. Summary values for total cerebral volume (in milliliters) are giv
(1) H?: �? = �?
H?: �? < �?
(2) ? = 0.05
(3) Test Statistic: t = -3.3527
(4) P-value: .0004516
(5) Conclusion: Reject H?
(6) Interpretation: There is sufficient evidence to conclude that the mean brain volume of children with ADHD is smaller than the mean
An article investigated the driving behavior of teenagers by observing their vehicles as they left a high school parking lot and then again at a site approximately 1/2 mile from the school.
For this test, use a 0.01 significance level.
The following measu
(1) H?: �? = �?
H?: �? > �?
(2) ? = 0.01
(3) Test Statistic: t = 2.374
(4) P-value: .0181
(5) Conclusion: Do not reject H?
(6) Interpretation: There is not sufficient evidence to conclude that the male teenage drivers exceed the speed limit by more than d
The logic behind the Wilcoxon Signed-Rank Test
The Wilcoxon signed-rank test is based on the assumption that the variable under consideration has a symmetric distribution - one that can be divided into two pieces that are mirror images of each other - but does not require that its distribution be norm
two-tailed test
If the primary concern is deciding whether a population mean, �, is different from a specified value ?, we express the alternative hypothesis as:
H?: � � �?
A hypothesis test whose alternative hypothesis has this form is called a two-tailed test.
left-tailed test
If the primary concern is deciding whether a population mean, �, is less than a specified value �?, we express the alternative hypothesis as:
H?: � < �?
A hypothesis test whose alternative hypothesis has this form is called a left-tailed test.
right-tailed test
If the primary concern is deciding whether a population mean, �, is greater than a specified value �?, we express the alternative hypothesis as:
H?: � > �?
A hypothesis test whose alternative hypothesis has this form is called a right-tailed test.
Basic logic of hypothesis testing
Take a random sample from the population. If the sample data are consistent with the null hypothesis, do not reject the null hypothesis; if the sample data are inconsistent with the null hypothesis (in the direction of the alternative hypothesis), reject
When to use the one-mean z-interval procedure
(a) For small samples - say, of size less than 15 - the z-interval procedure should be used only when the variable under consideration is normally distributed or very close to being so.
(b) For samples of moderate size - say, between 15 and 30 - the z-int
A fundamental principle of data analysis
Before performing a statistical-inference procedure, examine the sample data. If any of the conditions required for using the procedure appear to be violated, do not apply the procedure. Instead use a different, more appropriate procedure, or, if you are
Relate confidence and precision for a fixed sample size
For a fixed sample size, decreasing the confidence level improves the precision, and vice versa.
Margin of error, precision, and sample size
The length of a confidence interval for a population mean, �, and therefore the precision with which x-bar estimates �, is determined by the margin of error, E. For a fixed confidence level, increasing the sample size improves the precision, and vice vers
Basic properties of t-curves
(1) The total area under a t-curve equals 1.
(2) A t-curve extends indefinitely in both directions, approaching but never touching, the horizontal axis as it does so.
(3) A t-curve is symmetric about 0.
(4) As the number of degrees of freedom becomes larg
Relation between Type I and Type II error probabilities
For a fixed sample size, the smaller we specify the significance level, ?, the larger will be the probability, ?, of not rejecting a false null hypothesis.
Obtaining critical values
Suppose that a hypothesis test is to be performed at the significance level, ?. Then the critical value(s) must be chosen so that, if the null hypothesis is true, the probability is ? that the test statistic will fall in the rejection region.
When to use the one-mean z-test
(1) For small samples - say of size less than 15 - the z-test should be used only when the variable under consideration is normally distributed or very close to being so.
(2) For samples of moderate size - say, between 15 and 30 - the z-test can be used u
Decision criterion for a hypothesis test using the p-value
If the P-value is less than or equal to the specified significance level, reject the null hypothesis; otherwise, do not reject the null hypothesis.
Wilcoxon Signed-Rank Test versus the t-test
Suppose that you want to perform a hypothesis test for a population mean. When deciding between the t-test and the Wilcoxon signed-rank test, follow these guidelines:
(1) If you are reasonably sure that the variable under consideration is normally distrib
Choosing between a pooled and nonpooled t-procedure
Suppose you want to use independent simple random samples to compare the means of two populations. To decide between a pooled t-procedure and a nonpooled t-procedure, follow these guidelines: If you are reasonably sure that the populations have nearly equ