1. Section 11.1 Inference for the Mean of a Population AP Statistics March 10, 2009

2. AP Statistics, Section 8.2.1 2

3. 3

4. The t statistic The t statistic is used when we don’t know the standard deviation of the population, and instead we use the sample distribution as an estimation. The t statistic has n-1 degrees of freedom (df).

5. The t statistic In statistical tests of significance, we still have H 0 and H a. We need to provide the mu in the calculation of the t statistic. Looking at the t table is fundamentally different than the z table.

6. AP Statistics, Section 8.2.1 6

7. 7

8. 8 The t statistic The t statistic is bigger than the z statistic. We say that t distribution is a more conservative distribution. There is more area in the tails. The t statistic has n-1 degrees of freedom.

9. AP Statistics, Section 8.2.1 9

10. Example Construct and interpret a 95% confidence interval for the mean amount of (NOX) emitted by light duty engines of this type.

11. Step 1: Parameter: The population of interest is all light duty engines of this type. We want to estimate, the mean amount of the pollutant NOX emitted, for all of these engines.

12. Step 2: Conditions: Since we do not know σ, we should construct a one-sample t-interval for the mean NOX level if the conditions are satisfied. SRS: The data comes from a random sample of 46 engines from the population of all light-duty engines of this type. We are not told the data comes from a SRS so we proceed with caution. Normality: Is the population distribution of NOX emissions Normal? We do not know from the problem statement. Let’s examine the sample data. Independence: We must assume that there are at least (10)(46) = 460 light-duty engines of this type since we are sampling without replacement.

14. Step 3: Calculation: Check that the mean NOX emission reading for the 46 light-duty engines in our sample is x-bar = 1.329 grams per mile. Using a graphing calculator we calculate a t-interval, inputting the degrees of freedom at 45. (Using the table we would have to use a degree of freedom of 40 since 45 is not an option)

15. Step 4: Interpretation: We are 95% confident that the true mean level of nitrogen oxides emitted by this type of light-duty engine is between 1.185 and 1.473 grams/mile.

16. Example: Mr. Young Mopping Let’s suppose that Mr. Young has been told that he should mop by 25 after 1. We collect 12 samples with an average 27.58 minutes after 1 p.m. with a standard deviation of 3.848 minutes. Is this evidence that his true mean is after 1:25?

17. Step 1: Mr. Young Mopping Population of interest:  Mr. Young’s mopping Parameter of interest:  average time of arrival during mopping Hypothesis  H 0 : µ=25  H a : µ>25

18. Step 2: Mr. Young Mopping We are using 1 sample t-test? SRS?  No. Proceed with caution. Normality?  Big sample size (> 40)  Sample is somewhat normal because the sample distribution is single peaked, no obvious outliers. Population size is at least 10 times the sample size?  We assume that Mr. Young has done a lot of mopping

19. AP Statistics, Section 8.2.1 19 Step 3: Mr. Young Mopping Calculate the test statistic, and calculate the p-value.

20. AP Statistics, Section 11.1 20 Exercises 11.1-11.19 odd, 11.21-11.35 odd