Why is sampling often preferable to conducting a census for the purpose of obtaining information about a? population?
Sampling is less costly and can be done more quickly than a census.
?
Symbol for mean
Dotplot
Constructed by drawing a horizontal axis that displays all possible observations. Then each observation is recorded by placing a dot over the appropriate value on the horizontal axis.
You cannot know the sampling distribution of the sample mean? exactly, by what distribution can you often approximate? it?
Normal distribution
What is another name for the standard deviation of "x bar"?
Standard error of the mean
Why is "x bar" called something else?
The standard deviation of x bar determines the amount of sampling error to be expected when a population mean is estimated by a sample mean.
What symbol represents the standard error of mean?
Sigma Subscript X
How do we calculate the standard deviation of X bar?
Standard Deviation / Square Root of N
Three Standard Deviation Rule
States that almost all the observations in any data set lie within three standard deviations to either side of the mean.
How do you calculate the upper and lower limits?
?x + or - 3 (?x)
When dealing with the lower limit, what happens when it falls below zero?
Just becomes zero because you cannot have a negative amount of items
How does "n" affect graphs of distribution of x bar?
The larger n gets, the skinnier the graph becomes
The Central Line Theorem
Variable x bar is normally distributed regardless of the distribution of the variable under consideration.
How large does a sample size have to be to be considered large?
n = 30
What is the distribution of X Bar?
Normal with mean mu and stand dev as sigma / square root n
The mean and standard deviation of x bar do not depend on what?
Normal Distribution
How do we calculate z-scores?
(Xbar - Mean) / ( Standard deviation)
How do you find probability or percentage of samples within a specific interval?
1. Find your intervals (Mean +/- x)
2. Find z scores of each interval using Z-Score equation
3. Use chart to find the areas of both z-scores
4. Subtract areas
5. Multiply by 100
only if percentage asked
What is the value of a statistic used to estimate a parameter?
Point estimate of the parameter
When estimating an unknown? parameter, what does the margin of error? indicate?
The accuracy of an estimate for the value of the unknown parameter
Point Estimate is the same as what?
X Bar
Point Estimate Equation if given confidence interval
XBAR - number of intervals * standard deviation
Margin of Error
Confidence Interval / 2
The endpoints of the confidence interval is expressed as
point estimate +/- margin of error
Confidence Level
Confidence Interval in decimal form
How do we calculate ??
(1 - Confidence level)
How does sample size and margin of error correlate if confidence level is held steady?
Negative Correlation
How does confidence level and margin of error correlate if sample size is held steady?
Positive Corelation
How is a confidence interval shown on a number line?
- Confidence above with arrows across whole interval
- Error Margins below splitting Confidence Interval
How do you find your confidence interval?
XBAR +/- Z of ?/2 * (standard deviation / square root of n)
What does Z of ? mean?
Critical value of the confidence level
Margin of Error Equation
E = Z?/2 * (Stand dev / sqrt n)
Z Score of 95% Confidence
1.96
Z Score of 99% Confidence
2.58
If the confidence level gets lower than
it is a more accurate estimate
Why do you need to consider the studentized version of x overbarx to develop a? confidence-interval procedure for a population mean when the population standard deviation is? unknown?
Because the population standard deviation ? is not? known, the sample standard deviation must be used as an estimate for ? in the formula for the standardized version of x overbarx. The result is the studentized version of XBAR
When do you know to use T scores?
When you have sample standard deviation
When do you know to use Z scores?
When you have population sample standard deviation
Significance Level
Probability of making a Type 1 error
What determines the likeliness of rejecting a true null hypothesis?
Significance Level
The lower the significance level, the lower the....
Likeliness of rejecting a true null hypothesis
Type 1 Error
Rejecting a true null hypothesis, denoted by alpha
Type 2 Error
Not rejecting a false null hypothesis, denoted by Beta
Null Hypothesis
Population mean = Specified Value
Alternative Hypothesis
Population mean = different from null hypothesis
If a graph is left tailed...
Ha < Ho
If graph is right tailed...
Ha > Ho
If graph is two tailed....
Ha =x Ho
Correct decision
No rejections for either
Approaches to Hypothesis Testing
- Critical Value
- P- Value
Critical Value Approach
Uses values of test statistics that separate rejections & non rejections
Test Statistics
(XBAR - mean of Ho) / (stand dev / sqrt of n)
P-Value Testing
- Allows assessing significance at any desired level.
- Allows for evaluating the strength of the evidence against null hypothesis
P-Values vs Significance Level
- If P-Value is less than significance level, null hypothesis can be rejected
- If P-Value is greater than significance level, null hypothesis cannot be rejected
Critical Values for left tailed graphs....
-t alpha
Critical Values for right tailed graphs....
t alpha
Critical Values for two tailed graphs....
+/-t (alpha/2)