1. AP STATISTICS LESSON 11 – 1 (DAY 2) The t Confidence Intervals and Tests

2. ESSENTIAL QUESTION: How are t confidence intervals and significance test constructed? Objectives: To construct t confidence intervals. To construct t significance tests. To use the C table to retrieve t* values and P- values.

3. The t Confidence Intervals and Tests To analyze samples from normal populations with unknown σ, just replace the standard deviation σ/√ n of x by its standard error s/√n. The z procedures then become one-sample t procedures. Use P-values or critical values from the t distribution with n – 1 degrees of freedom in place of the normal values.

4. The One-Sample t Procedures Draw an SRS of size n from a population having unknown mean μ. A level C confidence interval for μ is x ± t* s/√n Where t* is the upper (1-C)/2 critical value for the t(n – 1) distribution. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases. The test the hypothesis H 0 : μ = μ 0 based on an SRS of size n, computed the one-sample t statistic t = (x – μ o )/(s/√ n )

5. The One-sample t Procedure (continued…) In terms of a variable T having then t(n – 1) distribution, the P-value for a test of H o against These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases. H a : μ > μ o is P( T ≥ t) H a : μ < μ o is P( T ≤ t) H a : μ ≠ μ o is 2P( T ≥ |t|)

6. Example 11.2 Page 622 Auto Pollution Minitab stemplot of the data (page 623) The one-sample t confidence interval has the form: estimate ± t* SE estimate (where SE stands for “standard error”)

7. Example 11.3 Page 624 Significance Test For μ When σ is Unknown Step 1. Identify the population of interest and the parameter you want to draw conclusions about. State null and alternative hypotheses in words and symbols. Step 2. Choose the appropriate inference procedure. Step 3. If the conditions are met, carry out the inference procedure. Step 4. Interpret your results in the context of the problem.

8. Example 11.3 Page 624 Significance Test For μ When σ is Unknown (continued…). Notice the linkage between the P-value computed in Step 3 and the conclusion drawn in Step 4. If you keep in mind the three C’s – conclusion, connection, context – then you will include all the important elements in your interpretation of results. The procedures used depend on conditions that are often reasonable but not always easy to check: random sampling and a normal population. Fortunately, we will see that confidence levels, and P-values from the t procedures are not very sensitive to lack of normality. Violation of the random sampling condition is more serious.

9. The Output for the One- sample t Test of Example 11.3 Minitab – Page 627