When we say that a sample statistic is a " point estimate, " we mean that it is
our best guess of a population parameter.
Simply put, " bias " means that our sample statistic is not a good estimate of the corresponding population parameter. Which of the following situations might lead to bias?
taking a nonrepresentative sample
b. adding 3 to each measurement
c. cluster sampling
If " bias " refers to whether or not our estimate is a good predictor of a population parameter, " efficiency " refers to
how far our estimate is likely to deviate from the population parameter.
Having the square root of N in the denominator of the formula for the standard error of a mean means that
as N gets larger, the standard error will get smaller.
From a sample of 200 private college students, you found that the average number of hours of study time each week is 25 with a standard deviation of 6. A point estimate of the average study time for all private college students would be
25
Two sample statistics are unbiased estimators. They are
Means and Proportions
An estimator is unbiased if the mean of its sampling distribution is equal to
The population value
The more efficient the estimate, the more the sampling distribution
is clustered around the mean.
What are Inferential Statistics
Tools that are used to make a "best guess" about the value of a population parameter.
We are using statistics from a sample to estimate parameters in a population.
An always-present concern about inferential statistics:
We can never be 100% sure that our statistics generated from a sample reflects "what's really so" in the entire population.
It could be that we happened to pick an unusual sample!
Margin of error is
a statistical concept (and calculation!) that we use to account for the fact that our statistics might be different than the "true" population parameter!
Two Main Uses of Inferential Statistics
Setting confidence intervals
Performing hypothesis tests
Setting confidence intervals
It's an statistical estimate of how confident we are that a sample mean that we calculate is close to the true population mean (that most likely we will never know).
An important tool for honest reporting of scientific results.
Performing hypothesis tests
Unlike the confidence interval, we begin with a guess about what number the true population value is (based upon a theory).
The statistic tells us the likelihood that my guess (hypothesis) is right or wrong, given the data I collect on my sample.
Basic Concepts of Confidence Intervals
Point estimate
Lower limit
Upper limit
Margin of error
Point estimate
The middle of the confidence interval.
The sample mean (statistic), which is our "best guess" of the population mean (parameter).
Lower limit
The lowest value on the confidence interval (left side)
Upper limit
The highest value on the confidence interval (right side)
Margin of error
The distance between the lower limit and confidence interval (which = the distance between the confidence interval and the upper limit)
Margin of Error
determines width of confidence interval. The narrower the confidence interval is, the more confident we are about our results.
Margin of error accounts for sampling error
Sampling error (p. 216)
is exactly how far off our sample statistic is from the population parameter (something we usually don't know!).
What Determines How Big (Wide) the Margin of Error Is (p. 231)
Depends upon equation,
Sample size (N)
The percent confidence interval that we want (Z)
Variation in the sample (s)
Sample size (N)
Larger sample size= smaller margin of error.
This is because we have more confidence that most often, a larger sample is more representative of the population than a smaller sample.
The percent confidence interval that we want (Z)
Bigger z= bigger margin of error= more confidence that the true population mean is somewhere in that confidence interval.
Variation in the sample (s)
More variation in the variable= larger margin of error.
This is because more dispersion (variety) of answers means that not as many people in the sample have scores that are close to the point estimate (the sample mean).
IQ has a normal distribution in a sample of students.
The 95% confidence interval= (100, 105)
Find the point estimate:
It is right in the middle of the confidence interval! (100+105)/2= 102.5
Find the margin of error:
It is the distance between the point estimate and the lower or upper limit (either one). 105 - 102.5 = 2.5
Question: Did McCain have a significantly
higher % of the suburban women vote
compared to Obama?
Margin of error= 3.1%
Answer to question:
CI for McCain: 44 + 3.1= (40.9, 47.1)
CI for Obama: 38 + 3.1= (34.9, 41.1)
Answer is no, McCain did not have a significantly higher % of the suburban women vote. Even though McCain had a higher vote % (44% vs. 38%
Sampling Concepts
Probability Samples
Non-probability Samples
Stratified Sample
Cluster Sample
These are important because the sampling methods you choose can largely influence sampling error!
Probability Samples
In this sampling strategy, every person in the population of interest has a chance of being selected for the sample.
Example of probability sample:
You're studying exercise frequency among CSUF college students. You get your participants by recruiting them all day at the Titan Student Union.
Since most students hang around there at least sometimes, this is likely to be a probability sample.
Non-probability Samples (p. 214)
Not everyone in the population of interest has an equal chance of being selected for the sample.
Non- probability Example:
You're studying exercise frequency among Cal State Fullerton students. You get your participants by recruiting them at the gym.
Since nearly all students in the gym are working out, students who do not work out would not be represented in this sample!
Probability vs. Non-probability Samples
In reality, we often use non-probability samples because we do not have potential access to the entire population of interest.
Non-probability samples are often
ok for doing hypothesis testing of a theoretical model (e.g., factors associated with exercise frequency)
if you are trying to estimate a confidence interval for a characteristic in the population (e.g., mean IQ in the US population), you should use
probability samples (not non-probability samples)!
Stratified Sampling
The population is divided into groups (strata), based upon some variable that's important.
Then you randomly pick a specific number of participants from each group to be in your sample.
That way, each group represents a fixed % of the total sample. The %
Example of Stratified Sampling
These people were stratified by gender and then 2 participants per group
were randomly selected to be in the sample.
Cluster Sampling
Is possible when your population of interest naturally occurs in "clusters" (e.g., churches, companies, schools, etc.).
Instead of sampling individuals, we randomly select clusters of individuals
Cluster Example:
There are 100+ schools in the Orange County School District, and I randomly select 3 schools and survey all of the students in those three schools.
The two main types of inferential statistics are
setting confidence intervals and testing hypothesis.
The basic components of the confidence interval are
the point estimate, margin of error, lower limit and upper limit. You can find the point estimate and the margin of error if you have the lower and upper limit.
Probability samples are those which are based upon
the entire population of interest, whereas non-probability samples are those that are based only on a subset of the population.
Two types of probability sampling strategies: stratified sampling and cluster sampling.
In hypothesis testing, the __________ is the critical assumption, the assumption that is actually tested.
null hypothesis
Which assumption must be true in order to justify the use of hypothesis testing?
random sampling
If we reject the null hypothesis of " no difference " at the 0.05 level,
the odds are 20 to 1 in our favor that we have made a correct decision.
Given the same alpha level or p -level, the one-tailed test
makes it more likely that the H 0 will be rejected.
one-tailed test of significance could be used whenever
the researcher can predict a direction for the difference.
A sample of people attending a professional football game averages 13.7 years of formal education while the surrounding community averages 12.1 years. The difference is significant at the 0.05 level. What could we conclude?
The sample is significantly more educated than the community as a whole.
A researcher is interested in the effect that neighborhood crime-watch efforts have on the crime rate in the inner city, but he is unwilling to predict the direction of the difference. The appropriate test of a hypothesis would be
two tailed
Do sex education classes and free clinics that offer counseling for teenagers reduce the number of pregnancies among teenagers? The appropriate test of a hypothesis would be
one tailed test
When reading output from SPSS, in order to find the significance of a onetailed test for a difference in means, one needs to
multiply the significance of the two-tailed test by 0.5.
When H a is that the mean is " greater than " some value, you have a __________; when H a is that the mean is " not equal to " some value, you have a ___________.
one tailed test, two tailed test
Alpha Level represents...
The percent to which we can be certain a result is not due to chance! (Significance level!)
If P value is < than a
we reject the null
If P is > than a
we do not reject the null
Fail to reject the null hypothesis" simply means that
the evidence in favor of rejection was not strong enough
The central problem in the case of two-sample hypothesis tests is to determine
if two populations differ significantly on the trait in question.
When testing for the significance of the difference between two means, which is the proper assumption?
that samples are independent as well as random
When random samples are drawn so that the selection of a case for one sample has no effect on the selection of cases for another sample, the samples are
independent
If the Levene test in SPSS output reports an outcome with significance less than our alpha (say, 0.05), we should probably
not assume that the two samples ' variances are equal.
SPSS reports significance levels for two-tailed tests by default; to find the significance of one-tailed tests,
divide the two-tailed significance by 2.
When looking at a difference in proportions between two populations, to be wrong in rejecting the null hypothesis means that
there is not a significant difference between the groups.
Samples from two high schools are being tested for the difference in their levels of prejudice. One sample contains 39 respondents, and the other contains 47 respondents. The appropriate sampling distribution is
the t-distribution.
When solving the formula for finding Z with sample proportions in the twosample case, we must first estimate
the population proportion.
When testing the significance of the difference between two sample proportions, the null hypothesis is
P u1= P u
From a university population, random samples of 145 men and 237 women have been asked if they have ever cheated in a college class. Eight percent of the men and 6 percent of the women say that they have. What is the appropriate test to assess the signific
Test for the significance of the difference between two sample proportions, large samples.
For all tests of hypotheses, the probability of rejecting the null hypothesis is a function of
the size of the observed difference.
In twins studies, in which one twin is assigned to one group and one to another, the selection of subjects is
non independent
Why do the degrees of freedom matter when performing t-tests?
The shape of the t-distribution differs for different degrees of freedom.
Our null hypothesis about paired means or proportions is usually that there is no difference between
the scores in the population
In the section ex., we tested for difference in means between two non-independent groups. What could we have done if the variable of interest were categorial instead of interval/ratio?
We could have converted the outcome into a dichotomy and assigned dummy coding scores.
The higher the alpha level
the greater the probability of rejecting the null hypothesis.
Four tests of significance were conducted on the same set of results: for test 1: alpha � 0.05, two-tailed test; for test 2: alpha � 0.10, one-tailed test; for test 3: alpha � 0.01, two-tailed test; for test 4: alpha � 0.01, one-tailed test. Which test is
Test 2
The value of all test statistics is directly proportional to
sample size
The larger the sample size
the more likely we are to reject the null hypothesis.
A difference between samples that is shown to be statistically significant may also be
a. practically insignificant.
b. theoretically insignificant.
c. sociologically insignificant.
d. all of the above.
D
If a difference between samples is not statistically significant, it is almost certainly ___________. On the other hand, a statistically significant difference is not necessarily ___________.
unimportant; important
The Levene test can be helpful in
deciding which estimator of the standard error of the mean difference is most appropriate.
A One Sample Z-Test is used to...
Compare a sample mean to the population mean
What are the 7 Steps to Follow when computing an inferential statistic?
State the Null and Alternative Hypotheses
Set the level of risk (Alpha Level)
Select the appropriate test statistic
Compute the test statistic value
Determine the value needed to reject the null
Compare your obtained value and critical value
State Results
An Alpha Level is...
A value selected by the researcher (typically .05) which indicates the percent to which he/she wants to be certain that the result was not due to chance.
Follow up question! If a value is Significant at the .05 level... what does this mean?
This means that the researcher can be at least 95% certain that the result was NOT due to chance
One Sample Z-Test
Used to compare a sample's mean to the population mean on one variable.
Helped us to identify if a sample's result was significantly different from "The Norm"
Can sometimes be impractical because we may not know the Normative values for a population
A Z-Test only lets us
work with one sample and we have to compare it to the population
Wont work for every study
Independent Samples T-Test!
(sometimes called independent means t-test)
Compares two samples on one variable
Independent T-Test Tools Needed:
______ Identified Significance Level
______ Computed T-Statistic
______ Degrees Of Freedom
Significance Level for comparing two samples
Just like with ANY inferential Statistic
First Identify your ALPHA level
Commonly used one????
P = .05
The T Statistic:
Lets Break it Down
Essentially what we are working with, is a ratio:
The Difference in Sample Means divided by
The Variability of Groups
The T Statistic pooled = :
variability of groups
it will always be given
Because a T-Test deals with Variance, It operates on a Few different assumptions
The most Important of these is referred to as Homogeneity of Variance
The Assumption states that the variances between the two groups will
be equal
Degrees of Freedom
The rank of a quadratic form"
What?!?!?
Essentially, it is a value which provides an approximation of the Sample Size
How far can our variables vary?
Its sort of a way of keeping score
"This many values must be taken into account
IMPORTANT: The calculation for Degrees of Freedom is
different for each inferential test.
DF stands for
degrees of freedom
Why do DF matter?
Larger sample size = Smaller Critical Value
Effect Size
provides a measure of HOW different two groups are from one another.
Effect Size Values
The value ranges outlined by Jacob Cohen
Small = 0.0 - .20
Medium = .20 - .50
Large = .50 and above
Reporting a T-Test
So For our First Example
T-Statistic = 2.75
Df = 18
Significant at p=.05
So...
t(18)=2.75, p< 0.05
if our t score is larger than the critical value we
reject the null.
smaller we accept the null
Surprise! We followed these Steps! for the T- Test
State the Null and Alternative Hypotheses
Set the level of risk (Alpha Level)
Select the appropriate test statistic
Compute the test statistic value
Determine the value needed to reject the null
Compare your obtained value and critical value
State Results
Step 1: State the Null and Research Hypothesis
H0 = There will be no difference in average time between those who are vegetarian and those who are carnivores
H1 = There will be a difference in average time between those who are vegetarian and those who are carnivores
step 2)
Set Alpha Level!
Lets Stick with .05
Step 3)
Calculate the T-statistic!
X1-X2 / pooledSD
step 4)
Calculate the Degrees of Freedom!
df= (n1- 1) + (n2- 1)
how do you find the critical value?
off that sheet of paper
0.12 < 2.093 = Not Significant
We fail to reject the Null!
Conclusion = "There is no significant difference between vegetarians and carnivores in regard to their average mile time.
Dependent T-Test
Also Called a Paired Samples T-Test
Compares one sample on two means over time
The Same subjects are Tested more than once, and we are comparing those two scores!
We are still comparing two "Groups" however those groups are made up of the same people at different time points.
Dependent T-Test
Both T-Tests are reported the same way...
Requires the T-Value, The Alpha Level and the Degrees Of Freedom
t(df)=tstatistic, p(<)alphalevel
There are two types of T-Test
independent and dependent
Each Has their own T-Test Formula
Independent -
Two samples compared on the same dependent variable
Dependent -
One sample, two time points. Compared on the same dependent variable.
For both indepedent and dependent T-Tests:
Using the obtained T-Statistic, identified alpha level, degrees of freedom, and critical level, we determine weather or not to Reject the Null!
Chi-Square analysis!!!
It allows you to
compare two or more groups on nominal or ordinal data!
Example: Gender, Age, Hair color, Income Level etc...
Using Expected Frequency:
The Expected probability of Two independent events happening at the same time...
What information does a frequency table contain?
Information regarding the "frequency" of responses to the various categories of a variable
A Correlation tells us
about the relationship between variables.
Bi-Variate Correlation = 2 variables
Represented with one number
Range between -1 and 1
The Pearson correlation focuses on
variables which we consider "continuous"
They can assume any value on a continuum
Doesn't work as well for non continuous
Nominal etc...
Coefficient of Determination
A much more precise way to
interpret the correlation coefficient
Correlations measure the similarities between two variables
The Coefficient of Determination
Describes the amount of variance which can be accounted for by the variance in another variable
This is calculated by squaring the Correlation Coefficent,
Bi Variate correlation -
two variables
When we have more than two variables we can use a "Correlation Matrix
For each Pair of Variables, there is a
Pearson's r value (correlation value)
Illustrates several bi-variate correlations for all variables.
Correlation coefficient
Describes the relationship between two different variables
Pearsons r
Represented with a range of numbers
-1 0 1
Correlation coefficient is used to test hypotheses that...
Examine Relationship between variables rather than the difference
Two variables
Finding DF for a correlation Coefficient is easy!
df = n - 2
One Way Analysis of Variance allows
us to compare 3 or more groups on a single dependent Variable
Interval or dependent variable
One way Anova often referred to as
an F-Test
Analysis of Variance (ANOVA) is used when
more than two group means are being tested simultaneously
Means of groups differ from one another on a particular score
What test statistic would you use for a Anova
F- Test
ANOVA examines the
variance between groups and the variances within groups
These variances are then compared against
each other
ANOVA is Similar to t test...only in this case,
you have more than two groups
Computing the F-Test Statistics: we want the within-group variance
to be small and the between-group variance large to find significance
F Test =
F= mean squares between / mean squares within
Degrees of freedom for anova
Two sets of degrees of freedom
one for between and one for within
DF for between groups (anova)
Number of groups minus one
k - 1
3 groups, so 3 - 1 = 2
DF for within groups (anova)
Total sample size minus number of groups
N - k
30 sample size minus 3 groups, so 30 - 3 = 27
F Test: The Plan (anova)
1)A statement of null and research hypothesis
Null Hypothesis
Research Hypothesis
2)Set the level of risk
3)Select the appropriate test statistic
4)Compute the test statistic value (called the obtained value)
-Compute the between-group sum of squares
-Com
F(2.27) = 8.80, p < .05
F = test statistic
2,27 = df between groups and df within groups
8.80 = obtained value
p < .05 = probability less than 5% that null hypothesis is true
One-Way ANOVA: What does it do?
Compare the variance between 3 or more groups
Specifically
The variance between groups - How different are the three groups from one another? (Means)
The variance within groups - how different are the scores within the group from one another? (Means)
And what is the Formula for the F-Ratio?
F= Means of squares between (Variance between groups) / Means of squares within ( Variance Within groups)
SS=
Sum of Squares
n=
total sample size
k=
# of groups
(anova) The less variance evident within groups, the
higher your F-Ratio will be
Higher is good!
As the variance between groups increases (MSb) and the variance within groups decreases (MSw) the F-Ratio
Increases!
The following are the steps to follow when computing a One Sample ANOVA
Identify your Null and Research Hypothesis
Calculate your F-Ratio
Compare your value to a critical value
Make a conclusion about your hypothesis
To read an F-Table, we need three things
Our Alpha Level _ 0.05
(k-1) df Numerator
(n-k) df Denominator
Anova- looking at the table for F- test
Just like with the T-Test Table, always round up if your exact value is not listed
We need to compare our Obtained Value of 4.44 to the critical value of 3.89...
4.44 > 3.89 Meaning we have a significant result!
do we accept or reject the null
We Can Reject the Null hypothesis and conclude that there is a significant difference in overall sales between the three branches!!!
What is a Tukey?
Tukey's Range test
Works in a similar way to the T-Test
Determines where significant differences lie between the groups used in an ANOVA
This is where you can identify what groups are significantly different from one another.
No Need to learn to do by hand, SPSS does a grea
Always keep in mind: The further away a test statistic is from zero, the p-value does down, and the more likely you will
reject the null hypothesis!
F-ratio is the test statistic for what inferential test? and what is the formula
One Way Anova
Mean Square Between Groups divided by Mean Square Within Groups.
Because the F-ratio is a ratio, it's always a positive number (never negative).
Independent samples t-test:
It's a function of the difference between two group means. A t-score of zero means no difference!
One-way ANOVA:
It's a ratio of between group differences to within group differences. The smaller the F-ratio is, the smaller the group differences are.
Pearson's chi-square:
It measures the extent to which observed frequencies are different from expected frequencies in each cell of a contingency table. The closer the chi-square test statistic is to zero, the more likely you will fail to reject the null hypothesis.
Pearson's correlation coefficient:
It measures the strength and direction of a linear relationship between the independent variable and dependent variable. The further away the correlation test statistic number is from zero (either negative or positive), the stronger the relationship is!