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AP STATISTICS – CHAPTER 11 NOTES GOAL: To find confidence intervals and significance tests for the mean of a single population and for comparing the means of two populations. 11.1: INFERENCE FOR THE MEAN OF A POPULATION NOTES:

1. In practice, σ is unknown and we must estimate it from the data. 2. This need to estimate σ changes some details of tests and confidence interval, but not

their interpretation. 3. Conditions for inferences about a mean:

• Data consists of a SRS of size n from the population of interest. • Observations from the population have a normal distribution with mean µ and

standard deviation σ . 4. When the standard deviation of a statistic is estimated from the data, the result is called

the standard error of the statistic. The standard error of the sample mean x sn

is .

5. When we know the value of σ , we work with the one-sample z statistic. (Chapter 10)

6. When we do not know the value of σ , we must substitute the standard error sn

of x

for its standard deviation σn

.

7. This statistic does not have a normal distribution. It has a t distribution.

Draw as SRS of size n from a population that has the normal distribution with mean

µ σand standard deviation . The one-sample t statistic t xsn

=− µ has the t

distribution with n – 1 degree of freedom.

NOTES:

1. The t-statistic has the same interpretation as any standardized statistic: measures how far x is from its mean µ in standard deviation units.

2. There is a different t distribution for each sample size. Therefore, we specify a particular t distribution by giving its degree of freedom. (page 618)

USING THE t TABLE OR CALCULATOR TO FIND CRITICAL VALUES FOR THE t DISTRIBUTION:

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3. Use P-values of critical values from the t distribution with n – 1 degrees of freedom in place of the normal values.

4. The One-Sample t Procedure (p.622) follows the same pattern as the One-Sample z Procedure.

CONSTRUCTING A CONFIDENCE INTERVAL FOR A POPULATION MEAN USING t PROCEDURES: ESTIMATE ± ∗t SEestimate , where SE stands for “Standard Error”. NOTES:

1. Keep in mind the three C’s – conclusion, connection, and context. 2. Confidence levels and P-values from the t procedure are not very sensitive to lack of

normality. They are more sensitive to violation of the random sampling condition. 3. For a sample size less than 15, use the t procedure if the data are close to normal. 4. For a sample size at least 15, use the t procedure except when outliers are present or the

distribution is strongly skewed. 5. For large samples (roughly greater than or equal to 40), use the t procedure even if the

distribution is clearly skewed. 6. Power of the t test: measures the ability to detect deviations from the null hypothesis.

Recall: A matched pairs design, matches subjects in pairs and each treatment is given to one subject in each pair. MATCHED PAIRS t PROCEDURES: Comparative studies are more convincing than single-sample investigations. To compare the responses to the two treatments in a matched pairs design, apply the one-sample t procedures to the observed differences. That is: the parameter x x1 2 1 2− = −µ µ in a matched pairs t procedures is the mean difference in the responses of the two treatments within matched pairs subjects in the entire population. 11.2: COMPARING TWO MEANS Definition: Two-sample problems involve the comparison of two populations or two treatments. Goal: Given two separate samples from each population or each treatment, compare the responses to two treatments or compare the characteristics of two populations. A two-sample problem:

• Randomly divides subjects into two groups and exposes each group to a different n • Comparing random samples separately selected from two populations • Nor matching of the units; samples can be of different size

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• Both populations are normally distributed • Samples are independent (matching violates independence) • Mean and standard deviations of the two populations are unknown • The mean x x1 2 1 2− = −µ µ (unbiased estimator) • The variance of the differences is the sum of the variances; the standard deviation is the

square root of the sum. • Two-sample z statistics vs. two-sample t statistics

NOTES:

1. The two-sample t procedures err on the safe side, since it reports higher P-values and lower confidence than are actually true.

2. Sample size strongly influences the P-values of a test. 3. When two population distributions have different shapes, larger samples are

needed. Therefore, when planning a two-sample study you should usually choose equal sample sizes.