Chapter 9: Hypothesis Testing GBS221, Class April 15, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College

Transcription

1 Chapter Objectives 1. Learn how to formulate and test hypotheses about a population mean and a population proportion. 2. Be able to use an Excel worksheet to conduct hypothesis tests about population means and proportions. 3. Understand the types of errors possible when conducting a hypothesis test. 4. Be able to determine the probability of making various errors in hypothesis tests. 5. Know how to compute and interpret p-values. 6. Know the definition of the following terms: null hypothesis, alternative hypothesis, type I error, type II error, critical value, level of significance, one-tailed test, two-tailed test, p-value 1. Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. The null hypothesis, denoted by H 0, is a tentative assumption about a population parameter. The alternative hypothesis, denoted by H a. is the opposite of what is stated in the null. Hypothesis testing is similar to a criminal trial. The hypotheses are: o H 0 : The defendant is innocent o H a : The defendant is guilty. Testing the Validity of a Claim o Manufacturers' claims are usually given the benefit of the doubt and stated as the null hypothesis. o The conclusion that the claim is false comes from sample data that contradict the null hypothesis. Testing in Decision-Making Situations o A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. o Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier. Summary of Forms for Null and Alternative Hypotheses: Population Mean o The equality part of the hypotheses always appears in the null hypothesis. o In general, a hypothesis test about the value of a population mean must take one of the following three forms (where H 0 is the hypothesized value of the population mean). H 0 : µ > µ 0 H 0 : µ < µ 0 H 0 : µ = µ 0 H a : µ < µ 0 H a : µ > µ 0 H a : µ!= µ 0 2. Using Excel for Hypothesis Tests Excel does not provide built-in routines for the hypothesis tests presented in this chapter. To handle these situations, we present Excel worksheets that we designed to use as templates

2 for testing hypotheses about a population mean and a population proportion. The worksheets are easy to use and can be modified to handle any sample data. For the population mean when σ is known (Data File: Hyp Sigma Known.xlsx) o We illustrate using the MaxFlight golf ball distance example in Section 9.3. The data are in column A of an Excel worksheet. The population standard deviation σ = 12 is assumed known and the level of significance is α =.05. The following steps can be used to test the hypothesis H 0 : μ = 295 versus H a : μ!= 295. (!= means not equal) o Refer to Figure 9.8 as we describe the procedure. The worksheet in the background shows the cell formulas used to compute the results shown in the foreground worksheet. o The data are entered into cells A2:A51. The following steps are necessary to use the template for this data set. Step 1. Enter the data range A2:A51 into the =COUNT cell formula in cell D4 Step 2. Enter the data range A2:A51 into the =AVERAGE cell formula in cell D5 Step 3. Enter the population standard deviation σ = 12 into cell D6 Step 4. Enter the hypothesized value for the population mean 295 into cell D8 o The remaining cell formulas automatically provide the standard error, the value of the test statistic z, and three p-values. Because the alternative hypothesis ( μ 0!= 295) indicates a two-tailed test, the p-value (Two Tail) in cell D15 is used to make the rejection decision. o With p-value =.1255 = α >.05, the null hypothesis cannot be rejected. The p-values in cells D13 or D14 would be used if the hypotheses involved a one-tailed test. For the population mean when σ is unknown (Data File: Hyp Sigma Unknown.xlsx) o We illustrate using the Heathrow Airport rating example in Section 9.4. The data are in column A of an Excel worksheet. The population standard deviation σ is unknown and will be estimated by the sample standard deviation s. The level of significance is α =.05. The following steps can be used to test the hypothesis H 0 : μ < 7 versus H a : μ > 7. o Refer to Figure 9.9 as we describe the procedure. The background worksheet shows the cell formulas used to compute the results shown in the foreground version of the worksheet. The data are entered into cells A2:A61. The following steps are necessary to use the template for this data set. Step 1. Enter the data range A2:A61 into the =COUNT cell formula in cell D4 Step 2. Enter the data range A2:A61 into the =AVERAGE cell formula in cell D5 Step 3. Enter the data range A2:A61 into the =STDEV cell formula in cell D6 Step 4. Enter the hypothesized value for the population mean 7 into cell D8 o The remaining cell formulas automatically provide the standard error, the value of the test statistic t, the number of degrees of freedom, and three p-values. Because the alternative hypothesis (μ > 7) indicates an upper tail test, the p-value (Upper Tail) in cell D15 is used to make the decision. With p-value =.0353 < α =.05, the null hypothesis is rejected. The p-values in cells D14 or D16 would be used if the hypotheses involved a lower tail test or a two-tailed test.

3 o This template can be used to make hypothesis testing computations for other applications. For instance, to conduct a hypothesis test for a new data set, enter the new sample data into column A of the worksheet and modify the formulas in cells D4, D5, and D6 to correspond to the new data range. Enter the hypothesized value for the population mean into cell D8 to obtain the results. If the new sample data have already been summarized, the new sample data do not have to be entered into the worksheet. In this case, enter the sample size into cell D4, the sample mean into cell D5, the sample standard deviation into cell D6, and the hypothesized value for the population mean into cell D8 to obtain the results. For population proportion (Data File: Hypothesis p.xlsx) o We illustrate using the Pine Creek golf course survey data presented in Section 9.5. The data of Male or Female golfer are in column A of an Excel worksheet. Refer to Figure 9.10 as we describe the procedure. The background worksheet shows the cell formulas used to compute the results shown in the foreground worksheet. The data are entered into cells A2:A401. The following steps can be used to test the hypothesis H 0 : p <.20 versus H a : p >.20. Step 1. Enter the data range A2:A401 into the =COUNTA cell formula in cell D3 Step 2. Enter Female as the response of interest in cell D4 Step 3. Enter the data range A2:A401 into the =COUNTIF cell formula in cell D5 Step 4. Enter the hypothesized value for the population proportion.20 into cell D8 o The remaining cell formulas automatically provide the standard error, the value of the test statistic z, and three p-values. Because the alternative hypothesis (p 0 >.20) indicates an upper tail test, the p-value (Upper Tail) in cell D14 is used to make the decision. o With p-value =.0062 < α =.05, the null hypothesis is rejected. The p-values in cells D13 or D15 would be used if the hypothesis involved a lower tail test or a two-tailed test. o This template can be used to make hypothesis testing computations for other applications. For instance, to conduct a hypothesis test for a new data set, enter the new sample data into column A of the worksheet. Modify the formulas in cells D3 and D5 to correspond to the new data range. Enter the response of interest into cell D4 and the hypothesized value for the population proportion into cell D8 to obtain the results. If the new sample data have already been summarized, the new sample data do not have to be entered into the worksheet. In this case, enter the sample size into cell D3, the sample proportion into cell D6, and the hypothesized value for the population proportion into cell D8 to obtain the results. 3. and 4. Errors The null and alternative hypotheses are competing statements about the population. Either the null hypothesis H 0 is true or the alternative hypothesis Ha is true, but not both. Ideally the hypothesis testing procedure should lead to the acceptance of H 0 when H 0 is true and the

4 rejection of H 0 when H a is true. Unfortunately, the correct conclusions are not always possible. Because hypothesis tests are based on sample information, we must allow for the possibility of errors. The first row of Table 9.1 shows what can happen if the conclusion is to accept H 0. If H 0 is true, this conclusion is correct. However, if Ha is true, we make a Type II error; that is, we accept H 0 when it is false. The second row of Table 9.1 shows what can happen if the conclusion is to reject H 0. If H 0 is true, we make a Type I error; that is, we reject H 0 when it is true. However, if Ha is true, rejecting H 0 is correct. The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. In practice, the person responsible for the hypothesis test specifies the level of significance. By selecting α, that person is controlling the probability of making a Type I error. If the cost of making a Type I error is high, small values of α are preferred. If the cost of making a Type I error is not too high, larger values of α are typically used. Applications of hypothesis testing that only control for the Type I error are called significance tests. Many applications of hypothesis testing are of this type. 5. p-values A p-value is a probability that provides a measure of the evidence against the null hypothesis provided by the sample. Smaller p-values indicate more evidence against H 0. The p-value is used to determine whether the null hypothesis should be rejected. A small p-value indicates the value of the test statistic is unusual given the assumption that H0 is true. The value of the test statistic is used to compute the p-value. The method used depends on whether the test is a lower tail, an upper tail, or a two-tailed test. For a lower tail test, the p-value is the probability of obtaining a value for the test statistic as small as or smaller than that provided by the sample. Thus, to compute the p-value for the lower tail test in the σ known case, we must find the area under the standard normal curve to the left of the test statistic. After computing the p-value, we must then decide whether it is small enough to reject the null hypothesis; the decision involves comparing the p-value to the level of significance. Rejection rule using p-values o Reject H 0 if p-value < α Critical Value Approach o The critical value approach requires that we first determine a value for the test statistic called the critical value. For a lower tail test, the critical value serves as a benchmark for determining whether the value of the test statistic is small enough to

5 KEY TERMS reject the null hypothesis. It is the value of the test statistic that corresponds to an area of α (the level of significance) in the lower tail of the sampling distribution of the test statistic. In other words, the critical value is the largest value of the test statistic that will result in the rejection of the null hypothesis. o In the σ known case, the sampling distribution for the test statistic z is a standard normal distribution. o Rejection Rule Reject H 0 if z < -z α where -z α is the critical value; that is, the z value that provides an area of α in the lower tail of the standard normal distribution. Null hypothesis The hypothesis tentatively assumed true in the hypothesis testing procedure. Alternative hypothesis The hypothesis concluded to be true if the null hypothesis is rejected. Type I error The error of rejecting H 0 when it is true. Type II error The error of accepting H 0 when it is false. Level of significance The probability of making a Type I error when the null hypothesis is true as an equality. One-tailed test A hypothesis test in which rejection of the null hypothesis occurs for values of the test statistic in one tail of its sampling distribution. Test statistic A statistic whose value helps determine whether a null hypothesis should be rejected. p-value A probability that provides a measure of the evidence against the null hypothesis provided by the sample. Smaller p-values indicate more evidence against H 0. For a lower tail test, the p-value is the probability of obtaining a value for the test statistic as small as or smaller than that provided by the sample. For an upper tail test, the p-value is the probability of obtaining a value for the test statistic as large as or larger than that provided by the sample. For a two-tailed test, the p-value is the probability of obtaining a value for the test statistic at least as unlikely as or more unlikely than that provided by the sample. Critical value A value that is compared with the test statistic to determine whether H 0 should be rejected. Two-tailed test A hypothesis test in which rejection of the null hypothesis occurs for values of the test statistic in either tail of its sampling distribution.

6 KEY FORMULAS