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AP Stats - Chapter 23

Page history last edited by Mrs. Gallagher 16 years ago

 

Chapter 23: Inferences About Means

 

 

Key Vocabulary:

t-distribution a family of distributions indexed by their degrees of freedom, the t models are unimodal, symmetric, and bell shaped, but generally have fatter tails and a narrower center than the Normal Model.  As the defress of freedom increase, t distributions approch the normal Model.

t-table

degrees of freedom-a whole family of related distributions that depend on a parameter

one-sample t-interval-hypothesis test that finds the population mean

one-sample t-test- hypothesis test that finds the true mean.

Calculator Skills:

T-Interval- first go to STATPLOT to create a histogram of the data, then go to the STATTESTS menu and select (8) Tinterval. Inpt: Data, List: whatever list you plugged the data in (L1).

T-test- go to the STAT TESTS menu and choose 2: T-TEST.

tcdf (leftend, rightend, df) go to distribution and number five is tcdf. to find a t-value greater than 1.645, you must put in a degree of freedom. so if you use twelve degrees of freedom it would be tcdf(1.645,99,12)

 

 

 

1. What is the standard deviation of the sample mean ?

t=(measured value-mean)/ (standard error of the measured value)

 

the symbol for the standard deviation of the sample is "s"

 

2. What is the standard error of the sample mean ?

     s / square root(n)

 

3. Describe the similarities between a standard normal distribution and a t distribution.

 they are both unimodal, symmetric, and bell shaped

 

4. Describe the differences between a standard normal distribution and a t distribution.

t-models with few degrees of freedom have fatter tails than normal models. If a t-model has infinite degrees of freedom then it is exactly a Normal model. However, then you would need an infinite number of data values and that isn't practical.

 

5. How do you calculate the degrees of freedom for a t distribution?

 In order to calculate the degrees of freedom, you simply do (n-1), n being the sample size.

6. What happens to the t distribution as the degrees of freedom increase?

 

As the degrees of freedom increase, the t-distribution looks more and more like the Normal Model. (In fact, the t-model with infinite degrees of freedom is exactly normal (but we never have an infinite number of data values))

 

7. How would you construct a level C confidence interval for  if  is unknown?

 

 

8. The z-Table gives the area under the standard normal curve to the left of z. What does the t-Table give?

 

9. Samples from normal distributions have very few outliers. If your data contains outliers, what does this suggest?

 

It suggests that the normal model will not work.  If there are outliers in the data, you might want to set them aside and after they are singled out, you should look at them carefully.  Sometimes, it is obvious when a data value is wrong and the justifiaction for removing themare clear.  When there is no justification, you might wnat to run the analysis both with and without the outliers and note any differences.

 

10. If the size of the SRS is less than 15, when can we use t procedures on the data?

 For very small samples less than 15, the data should follow a normal model pretty closely. I fthere are outliers or skewness, you should not use these methods.

 

11. If the size of the SRS is between 15 and 40, when can we use t procedures on the data?

the t methods will work well  as long as he dara are unimodal and symmetric.  MAke a histogram

 

 

12. If the size of the SRS is at least 40, when can we use t procedures on the data?

 

t procedures are safe to use even when data are skewed. if there are outliers, remove them and report results without them. if there are more than one modes, consider each one separately.

 

IMPORTANT:

A correct way to interpret a mean's __% confidence level.

Based on this sample , we can say, with __% confidence, that the mean ______ is between __ and __.

 

 

 

 

 

 

 

 

 

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