Week 8 October Three Mini-Lectures QMM 510 Fall 2014

9-2 Chapter 9 Testing a Mean: Known ML 8.1 Population Variance LO9-5: Find critical values of z or t in tables or by using Excel. The test statistic is compared with a critical value from a table. The critical value is the boundary between two regions (reject H 0, do not reject H 0 ) in the decision rule. The critical value shows the range of values for the test statistic that would be expected by chance if the null hypothesis were true. These z-values can be computed using Excel, MINITAB etc.

9-3 The hypothesized mean  0 that we are testing is a benchmark.The hypothesized mean  0 that we are testing is a benchmark. The value of  0 does not come from a sample.The value of  0 does not come from a sample. The test statistic compares the sample mean with the hypothesized mean  0.The test statistic compares the sample mean with the hypothesized mean  0. Chapter 9 LO9-6: Perform a hypothesis test for a mean with known σ using z. Testing a Mean: Known Population Variance

9-4 Testing the Hypothesis Testing the Hypothesis Step 1: State the hypotheses For example, H 0 :   216 mm H 1 :  > 216 mm Step 2: Specify the decision rule: For example, for  =.05 for the right-tail area, reject H 0 if z calc > 1.645; otherwise do not reject H 0. Chapter 9 Example: Paper Manufacturing Testing a Mean: Known Population Variance

9-5 Step 3: Collect sample data and calculate the test statistic. If H 0 is true, then the test statistic should be near 0 because the sample mean should be near μ 0. The value of the test statistic is. Chapter 9 Step 4: The test statistic falls in the right rejection region, so we reject the null hypothesis H 0 : μ  216 and conclude the alternative hypothesis H 1 : μ > 216 at the 5% level of significance. Testing a Mean: Known Population Variance Testing the Hypothesis Testing the Hypothesis Example: Paper Manufacturing

9-6 Step 5: Take action. Now that we have concluded that the process is producing paper with an average width greater than the specification, it is time to adjust the manufacturing process to bring the average width back to specification. Our course of action could be to readjust the machine settings or it could be time to resharpen the cutting tools. At this point it is the responsibility of the process engineers to determine the best course of action. Chapter 9 Testing a Mean: Known Population Variance Testing the Hypothesis Testing the Hypothesis Example: Paper Manufacturing

9-7 Using the p-Value Approach Using the p-Value Approach The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true.The p-value is the probability of the sample result (or one more extreme) assuming that H 0 is true. The p-value can be obtained using Excel’s cumulative standard normal function =NORM.S.DIST(z).The p-value can be obtained using Excel’s cumulative standard normal function =NORM.S.DIST(z). The p-value can also be obtained from Appendix C-2.The p-value can also be obtained from Appendix C-2. Using the p-value, we reject H 0 if p-value  .Using the p-value, we reject H 0 if p-value  . Chapter 9 Testing a Mean: Known Population Variance

9-8 Two-Tailed Test of Hypothesis Two-Tailed Test of Hypothesis Chapter 9 Example: Paper Manufacturing Testing a Mean: Known Population Variance

9-9 Chapter 9 Testing a Mean: Known Population Variance Two-Tailed Test of Hypothesis Two-Tailed Test of Hypothesis Example: Paper Manufacturing

9-10 Chapter 9 Testing a Mean: Known Population Variance Two-Tailed Test of Hypothesis Two-Tailed Test of Hypothesis Example: Paper Manufacturing

9-11 Chapter 9 Testing a Mean: Known Population Variance Two-Tailed Test of Hypothesis Two-Tailed Test of Hypothesis Example: Paper Manufacturing

9-12 Testing the Hypothesis Testing the Hypothesis Chapter 9 Using the p-Value Approach Using the p-Value Approach p-value = <  = 0.05, so the null hypothesis is rejected. Testing a Mean: Known Population Variance

9-13 Analogy to Confidence Intervals Analogy to Confidence Intervals A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean.A two-tailed hypothesis test at the 5% level of significance (  =.05) is exactly equivalent to asking whether the 95% confidence interval for the mean includes the hypothesized mean. If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis.If the confidence interval includes the hypothesized mean, then we cannot reject the null hypothesis. Chapter 9 Testing a Mean: Known Population Variance

9-14 When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with n – 1 degrees of freedom.When the population standard deviation  is unknown and the population may be assumed normal, the test statistic follows the Student’s t distribution with n – 1 degrees of freedom. Using Student’s t Using Student’s t Chapter 9 Testing a Mean: Unknown ML 8.2 Population Variance LO9-7: Perform a hypothesis test for a mean with unknown σ using t.

9-15 Testing the Hypothesis Testing the Hypothesis Step 1: State the hypotheses H 0 :  = 142 H 1 :   142 Step 2: Specify the decision rule. For  =.10 for the two-tailed test and with d.f. n – 1 = 24  1 = 23, reject H 0 if t calc > or if t calc <  1.714; otherwise do not reject H 0. Chapter 9 Example: Hot Chocolate Testing a Mean: Unknown Population Variance

9-16 Step 3: Collect sample data and calculate the test statistic. If H 0 is true, then the test statistic should be near 0 because the sample mean should be near μ 0. The value of the test statistic is Chapter 9 Step 4: Since the test statistic lies within the range of chance variation, we cannot reject the null hypothesis H 0 : μ = 142. Testing a Mean: Unknown Population Variance Testing the Hypothesis Testing the Hypothesis Example: Hot Chocolate

9-17 Chapter 9 Using the p-value LO9-8: Use tables or Excel to find the p-value in tests of μ. Testing a Mean: Unknown Population Variance

9-18 Confidence Intervals versus Hypothesis Test A two-tailed hypothesis test at the 10% level of significance (  =.10) is equivalent to a two-sided 90% confidence interval for the mean. If the confidence interval does not include the hypothesized mean, then we reject the null hypothesis. The 90% confidence interval for the mean is given next. Because μ = 142 lies within the 90 percent confidence interval [ , ], we cannot reject the hypothesis H 0 : μ = 142 at α =.10 in a two-tailed test. Chapter 9 Testing a Mean: Unknown Population Variance

9-19 To conduct a hypothesis test, we need to know - the parameter being tested, - the sample statistic, and - the sampling distribution of the sample statistic.To conduct a hypothesis test, we need to know - the parameter being tested, - the sample statistic, and - the sampling distribution of the sample statistic. The sampling distribution tells us which test statistic to use.The sampling distribution tells us which test statistic to use. A sample proportion p estimates the population proportion .A sample proportion p estimates the population proportion . For a large sample, p can be assumed to follow a normal distribution. If so, the test statistic is z.For a large sample, p can be assumed to follow a normal distribution. If so, the test statistic is z. Chapter 9 Testing a Proportion ML 8.3 LO9-9: Perform a hypothesis test for a proportion and find the p-value.

9-20 Chapter 9 Testing a Proportion Note: A rule of thumb to assume normality is if n  0  10 and n(1   0 )  10.

9-21 The value of  0 that we are testing is a benchmark such as past experience, an industry standard, or a product specification. The value of  0 does not come from a sample. Chapter 9 Testing a Proportion Critical Value Critical Value The test statistic is compared with a critical z value from a table. The critical value shows the range of values for the test statistic that would be expected by chance if H 0 were true. Choice of π 0 Choice of π 0

9-22 Step 2: Specify the decision rule. For α =.05 for a left-tail test, reject H 0 if z <  1.645; otherwise do not reject H 0. Figure 9.12 Chapter 9 Testing a Proportion Example: Return Policy Example: Return Policy Step 1: State the hypotheses. For example: H 0 :  .13 H 1 :  <.13 Steps in Testing a Proportion

9-23 Steps in Testing a Proportion Steps in Testing a Proportion Figure 9.12 Chapter 9 Step 3: Collect sample data and calculate the test statistic. If H 0 is true, then the test statistic should be near 0 because the sample mean should be near μ 0. The value of the test statistic is: Step 4: Since the test statistic lies in the left-tail rejection region, we reject the null hypothesis H 0 :  .13. Testing a Proportion

9-24 Calculating the p-Value Calculating the p-Value Chapter 9 Testing a Proportion

9-25 Chapter 9 The Effect of  The Effect of  Testing a Proportion

9-26 Chapter 9 The Effect of  The Effect of  Testing a Proportion

9-27 Small Samples and Non-Normality Small Samples and Non-Normality In the case where n  0 < 10, use MINITAB (or any other appropriate software) to test the hypotheses by finding the exact binomial probability of a sample proportion p. For example: Figure 9.19 Chapter 9 Testing a Proportion