Unit 8 Section 7.3

7.3: Hypothesis Testing for the Mean (σ unknown)  The hypothesis test for a mean when the population standard deviation is unknown is more common in real-life situations.  When the population standard deviation is unknown, we use the t-distribution with (n-1) degrees of freedom.

 Specify the level of significance  Identify the degrees of freedom  Find the critical value using the t- Distribution chart.  Left-tailed: Use the “One Tail” column  t is negative  Right-tailed: Use the “One Tail” column  t is positive  Two-tailed: Use the “Two Tail” column  t is positive and negative Section 7.3 Finding Critical Values in a t- Distribution

Example 1: a)Find the critical t value for α= 0.05 with d.f. = 16 for a right tailed t test. b)Find the critical t value for α= 0.01 with d.f. = 22 for a left tailed t test. c)Find the critical t value for α= 0.10 with d.f. = 18 for a two-tailed t test d)Find the critical t value for α= 0.05 with d.f. = 28 for a right tailed t test Section 7.3

Using t-Test for a Mean  Statistical test for the mean of a population  It can be used when at least one of the following conditions are met:  n is greater than or equal to 30  The population is normally distributed.  Formula : Section 7.3

Using the t-Test for a Mean  Verify that σ is not known, the sample is random, and either the population is normally distributed or n is greater than or equal to 30.  State the hypotheses and identify the claim.  Specify the level of significance  Identify the degrees of freedom.  Determine the critical value(s) from the t table  Determine the rejection region  Find the standardized test statistic  Make a decision  Interpret your results Section 7.3

Example 2: A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses’ salaries has a mean of $23,450 and a standard deviation of $400. Is there enough evidence to support the claim to reject the director’s claim at α = Section 7.3

Example 3: An educator claims that the average salary of substitute teachers in school districts in Camden County, New Jersey, is less than $60 per day. A random sample of 8 school districts is selected, and the daily salary (in dollars) are shown. Is there enough evidence to support the educator’s claim at α = 0.10? Section 7.3

Finding P-values  P-values can obtained from using the t distribution chart  The P-values will be represented by an interval (range of numbers) based on the type of test and the d.f. Section 7.3

Example 4: a)Find the P-value when the t test value is 2.056, the sample size is 11, and the test is right-tailed. b)Find the P-value when the t test value is 2.983, the sample size is 6, and the test is two-tailed. Section 7.3

Example 5: A physician claims that joggers’ maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physician’s claim at α = 0.05? Section 7.3

Homework:  Pg : (3 – 21 ODD) Section 7.3