Basics of Hypothesis Testing: Testing a Claim about a Population Mean

1. Lecture Slides Elementary StatisticsTwelfth Edition and the Triola Statistics Series by Mario F. Triola

2. Chapter 8Hypothesis Testing 8-1 Review and Preview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim about a Proportion 8-4 Testing a Claim About a Mean 8-5 Testing a Claim About a Standard Deviation or Variance

3. Key Concept This section presents methods for testing a claim about a population mean. Part 1 deals with the very realistic and commonly used case in which the population standard deviation σ is not known. Part 2 discusses the procedure when σ is known, which is very rare.

4. Part 1 When σ is not known, we use a “t test” that incorporates the Student t distribution.

5. Notation n = sample size = sample mean = population mean

6. Requirements The sample is a simple random sample. Either or both of these conditions is satisfied: The population is normally distributed or n > 30.

7. Test Statistic

8. Running the Test P-values: Use technology or use the Student t distribution in Table A-3 with degrees of freedom df = n – 1. Critical values: Use the Student t distribution with degrees of freedom df = n – 1.

9. Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see Figure 7-5 in Section 7-3). 2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected when s is used to estimate σ. 3. The Student t distribution has a mean of t = 0. 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1. 5. As the sample size n gets larger, the Student t distribution gets closer to the standard normal distribution.

10. Example Listed below are the measured radiation emissions (in W/kg) corresponding to a sample of cell phones. Use a 0.05 level of significance to test the claim that cell phones have a mean radiation level that is less than 1.00 W/kg. The summary statistics are: .

11. Example - Continued Requirement Check: We assume the sample is a simple random sample. The sample size is n = 11, which is not greater than 30, so we must check a normal quantile plot for normality.

12. Example - Continued The points are reasonably close to a straight line and there is no other pattern, so we conclude the data appear to be from a normally distributed population.

13. Example - Continued Step 1: The claim that cell phones have a mean radiation level less than 1.00 W/kg is expressed as μ < 1.00 W/kg. Step 2: The alternative to the original claim is μ ≥ 1.00 W/kg. Step 3: The hypotheses are written as:

14. Example - Continued Step 4: The stated level of significance is α= 0.05. Step 5: Because the claim is about a population mean μ, the statistic most relevant to this test is the sample mean: .

15. Example - Continued Step 6: Calculate the test statistic and then find the P-value or the critical value from Table A-3:

16. Example - Continued Step 7: Critical Value Method: Because the test statistic of t = –0.486 does not fall in the critical region bounded by the critical value of t = –1.812, fail to reject the null hypothesis.

17. Example - Continued Step 7: P-value method: Technology, such as a TI-83/84 Plus calculator can output the P-value of 0.3191. Since the P-value exceeds α = 0.05, we fail to reject the null hypothesis.

18. Example Step 8: Because we fail to reject the null hypothesis, we conclude that there is not sufficient evidence to support the claim that cell phones have a mean radiation level that is less than 1.00 W/kg.

19. Finding P-Values Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the P-value corresponding to the given results. a) In a left-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –2.007. b) In a right-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = 1.222. c) In a two-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –3.456.

20. Example – Confidence Interval Method We can use a confidence interval for testing a claim about μ. For a two-tailed test with a 0.05 significance level, we construct a 95% confidence interval. For a one-tailed test with a 0.05 significance level, we construct a 90% confidence interval.

21. Example – Confidence Interval Method Using the cell phone example, construct a confidence interval that can be used to test the claim that μ < 1.00 W/kg, assuming a 0.05 significance level. Note that a left-tailed hypothesis test with α = 0.05 corresponds to a 90% confidence interval. Using methods described in Section 7.3, we find: 0.707 W/kg < μ < 1.169 W/kg

22. Example – Confidence Interval Method Because the value of μ = 1.00 W/kg is contained in the interval, we cannot reject the null hypothesis that μ = 1.00 W/kg . Based on the sample of 11 values, we do not have sufficient evidence to support the claim that the mean radiation level is less than 1.00 W/kg.

23. Part 2 When σ is known, we use test that involves the standard normal distribution. In reality, it is very rare to test a claim about an unknown population mean while the population standard deviation is somehow known. The procedure is essentially the same as a t test, with the following exception:

24. Test Statistic for Testing a Claim About a Mean (with σKnown) The test statistic is: The P-value can be provided by technology or the standard normal distribution (Table A-2). The critical values can be found using the standard normal distribution (Table A-2).

25. Example If we repeat the cell phone radiation example, with the assumption that σ= 0.480 W/kg, the test statistic is: The example refers to a left-tailed test, so the P-value is the area to the left of z = –0.43, which is 0.3336 (found in Table A-2). Since the P-value is large, we fail to reject the null and reach the same conclusion as before.