1. Chapter 11: Inference for Distributions
The Practice of Statistics
(Yates)

2. 11.1 Inference for the Mean of a Population
Confidence intervals and tests of significance for the mean of a normal population
Mean is based on sample mean
Population standard deviation is unknown so estimate using sample data
Standard Error of the sample mean:

3. Assumptions for Inference about a Mean
Data are a simple random sample (SRS) on size n from the population
Observations from the population have a normal distribution with mean and standard deviation both of which are unknown

4. One-Sample t Statistic and t-Distribution
The one-sample t statistic
has the t distribution with n – 1 degrees of freedom

5. Facts about the t Distributions
Density curves – similar in shape to standard normal curve
Symmetric about zero
Bell-shaped
Spread – somewhat greater than the standard normal distribution
More probability in tails and less in the center
Due to substituting the estimate for population standard deviation
As the degrees of freedom k increase the t(k) density curve approaches the N(0, 1) curve

6. One-Sample t Procedures
To test the hypothesis
Compute the one-sample t statistic
Against one of the following

7. Matched Pairs t Procedures
Matched pairs design
Subjects are matched in pairs
Each treatment given to one subject in each pair
Also used in before-and-after observations on the same subjects

8. Robustness of t Procedures
A confidence interval or significance test is called robust if the confidence level or P-value does not change very much when the assumptions of the procedure are violated
t procedures are strongly influenced by outliers
Skewness affects the t procedures
Make a plot to check for outliers and skewness
When population is not normal, larger sample sizes improve accuracy (central limit theorem)

9. Using t Procedures
SRS assumption is more important than assumption of Normal distribution
Sample size less than 15: use t procedures only if data is nearly normal, without outliers
Sample size at least 15: use t procedures except if there are outliers or strong skewness
Large samples, : t procedures can be used even in clearly skewed distributions

10. The Power of the t Test
Power: measures the ability of test to detect deviations from the null hypothesis
Power of one-sample t test: probability that test will reject null hypothesis when the mean has alternative value
To calculate
assume fixed level of significance
Usually

11. 11.2 Comparing Two Means Two –Sample Problems Goal of inference is
to compare the responses to two treatments
to compare the characteristics of two populations
Have a separate sample from each treatment or each population

12. Assumptions for Comparing Two Means
Two SRSs from two distinct populations
The samples are independent
One sample has no influence on the other
Measure the same variable for both samples
Both populations are normally distributed
Means and standard deviations of the populations are unknown

13. Two-Sample z Statistic
Normal distribution of the statistic

14. Two-Sample t Procedures
The population standard deviations are unknown

15. Two-Sample Problems
Two-Sample t statistic is used with t critical values in inference
Option 1: use procedures based on the statistic t with critical values from a t distribution with degrees of freedom computed from the data
Option 2: use procedures based on the statistic t with critical values from a t distribution with degrees of freedom equal to the smaller of