HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 10.2 Hypothesis Testing for Population Means ( Known)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Use the rejection region to draw a conclusion. o Use the p-value to draw a conclusion.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Means ( Known) Test Statistic for a Hypothesis Test for a Population Mean ( Known) When the population standard deviation is known, the sample taken is a simple random sample, and either the sample size is at least 30 or the population distribution is approximately normal, the test statistic for a hypothesis test for a population mean is given by
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Means ( Known) Test Statistic for a Hypothesis Test for a Population Mean ( Known) (cont.) where x̄ is the sample mean, is the presumed value of the population mean from the null hypothesis, is the population standard deviation, and n is the sample size.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rejection Regions
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rejection Regions Decision Rule for Rejection Regions Reject the null hypothesis, H 0, if the test statistic calculated from the sample data falls within the rejection region.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rejection Regions Rejection Regions for Hypothesis Tests for Population Means ( Known)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) The state education department is considering introducing new initiatives to boost the reading levels of fourth graders. The mean reading level of fourth graders in the state over the last 5 years was a Lexile reader measure of 800 L. (A Lexile reader measure is a measure of the complexity of the language that a reader is able to comprehend.) The developers of a new program claim that their techniques will raise the mean reading level of fourth graders by more than 50 L.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) To assess the impact of their initiative, the developers were given permission to implement their ideas in the classrooms. At the end of the pilot study, a simple random sample of 1000 fourth graders had a mean reading level of 856 L. It is assumed that the population standard deviation is 98 L. Using a 0.05 level of significance, should the findings of the study convince the education department of the validity of the developers’ claim?
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) Solution Step 1:State the null and alternative hypotheses. The developers want to show that their classroom techniques will raise the fourth graders’ mean reading level to more than 850 L. This is written mathematically as > 850, and since it is the research hypothesis, it will be H a. The mathematical opposite is ≤ 850. Thus, we have the following hypotheses.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) Step 2:Determine which distribution to use for the test statistic, and state the level of significance. Note that the hypotheses are statements about the population mean, is known, the sample is a simple random sample, and the sample size is at least 30. Thus, we will use a normal distribution, which means we need to use the z-test statistic.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) In addition to determining which distribution to use for the test statistic, we need to state the level of significance. The problem states that = Step 3:Gather data and calculate the necessary sample statistics. At the end of the pilot study, a simple random sample of 1000 fourth graders had a mean reading level of 856 L. The population standard deviation is 98 L.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) Thus, the test statistic is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) Step 4:Draw a conclusion and interpret the decision. Remember that we determine the type of test based on the alternative hypothesis. In this case, the alternative hypothesis contains “>,” which indicates that this is a right ‑ tailed test. To determine the rejection region, we need a z-value so that 0.05 of the area under the standard normal curve is to its right. If 0.05 of the area is to the right then 1 0.05 = 0.95 is the area to the left.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) If we look up in the body of the cumulative z-table, the corresponding critical z-value is Alternately, we can look up c = 0.95 in the table of critical z-values for rejection regions. Either way, the rejection region is z ≥
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right ‑ Tailed, Known) (cont.) The z-value of 1.94 falls in the rejection region. So the conclusion is to reject the null hypothesis. Thus the evidence collected suggests that the education department can be 95% sure of the validity of the developers’ claim that the mean Lexile reader measure of fourth graders will increase by more than 50 points.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. p-Values p-value A p-value is the probability of obtaining a sample statistic as extreme or more extreme than the one observed in the data, when the null hypothesis, H 0, is assumed to be true.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.11: Calculating the p-Value for a z-Test Statistic for a Left-Tailed Test Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data have been collected and the test statistic was calculated to be z = 1.34.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.11: Calculating the p-Value for a z-Test Statistic for a Left-Tailed Test (cont.) Solution The alternative hypothesis tells us that this is a left- tailed test. Therefore, the p-value for this situation is the probability that z is less than or equal to 1.34, written p-value = P(z −1.34). Use a table or appropriate technology to find the area under the standard normal curve to the left of z = Thus, the p-value ≈
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.11: Calculating the p-Value for a z-Test Statistic for a Left-Tailed Test (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.12: Calculating the p-Value for a z-Test Statistic for a Right-Tailed Test Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data have been collected and the test statistic was calculated to be z = 2.78.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.12: Calculating the p-Value for a z-Test Statistic for a Right-Tailed Test (cont.) Solution The alternative hypothesis tells us that this is a right- tailed test. Therefore, the p-value for this situation is the probability that z is greater than or equal to 2.78, written p-value = P(z 2.78). Use a table or appropriate technology to find the area under the standard normal curve to the right of z = Thus, the p-value ≈
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.12: Calculating the p-Value for a z-Test Statistic for a Right-Tailed Test (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.13: Calculating the p-Value for a z-Test Statistic for a Two-Tailed Test Calculate the p-value for a hypothesis test with the following hypotheses. Assume that data have been collected and the test statistic was calculated to be z = 2.15.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.13: Calculating the p-Value for a z-Test Statistic for a Two-Tailed Test (cont.) Solution The alternative hypothesis tells us that this is a two- tailed test. Thus, the p-value for this situation is the probability that z is either less than or equal to 2.15 or greater than or equal to 2.15, which is written mathematically as Use a table or appropriate technology to find the area under the standard normal curve to the left of z 1 = −2.15. The area to the left of z 1 = −2.15 is
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.13: Calculating the p-Value for a z-Test Statistic for a Two-Tailed Test (cont.) Since the standard normal curve is symmetric about its mean, 0, the area to the right of z 2 = 2.15 is also Thus, the p-value is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.13: Calculating the p-Value for a z-Test Statistic for a Two-Tailed Test (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator As we read in Example 10.10, the state education department is considering introducing new initiatives to boost the reading levels of fourth graders. The mean reading level of fourth graders in the state over the last 5 years was a Lexile reader measure of 800 L. (A Lexile reader measure is a measure of the complexity of the language that a reader is able to comprehend.) The developers of a new program claim that their techniques will raise the mean reading level of fourth graders by more than 50 L.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) To assess the impact of their initiative, the developers were given permission to implement their ideas in the classrooms. At the end of the pilot study, a simple random sample of 1000 fourth graders had a mean reading level of 856 L. It is assumed that the population standard deviation is 98 L. Calculate the p-value for this hypothesis test.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) Solution First, recall from Example that the hypotheses are as follows. The alternative hypothesis tells us that this is a right- tailed test. Thus, in terms of the sampling distribution of sample means for a sample size of n = 1000, the
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) p-value for this situation is the probability of getting a sample mean greater than or equal to 856, written Press and then to get the DISTR (distributions) menu. Choose option 2:normalcdf(. Remember that you must enter the lower and upper bounds of the area under the curve as well as the mean and standard deviation of the normal distribution. For this example, the lower bound of the area is the sample mean, x̄ = 856.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) Since we want the area to the right, the upper bound of the area is infinity, so we must enter a really large value of x̄. Thus, we will enter 1û99 for the upper bound. The mean of the sampling distribution is the hypothesized value of the population mean, The standard deviation of the sampling distribution is the population standard deviation divided by the square root of the sample size,
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) Enter n ormalcdf(856,1û99, 850, 98/ð (1000)) and press.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) The p-value returned is approximately , as shown in the screenshot.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.14: Calculating the p-Value for a z-Test Statistic Using a TI-83/84 Plus Calculator (cont.) Note that by solving for the p-value in one step on the calculator, we eliminate any possible rounding errors. Thus, this method will always yield the most accurate p-value. Later in this chapter, we will see that the calculator can perform the entire hypothesis test in one step, and the output includes this p-value.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. p-Values Conclusions Using p-Values If p-value ≤ , then reject the null hypothesis. If p-value > , then fail to reject the null hypothesis.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.15: Determining the Conclusion to a Hypothesis Test Using the p-Value For a certain hypothesis test, the p-value is calculated to be p-value = a.If the stated level of significance is 0.05, what is the conclusion to the hypothesis test? b.If the level of confidence is 99%, what is the conclusion to the hypothesis test? Solution a.The level of significance is = Next, note that < Thus, p-value ≤ , so we reject the null hypothesis.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.15: Determining the Conclusion to a Hypothesis Test Using the p-Value (cont.) b.The level of confidence is 99%, so the level of significance is calculated as follows. Next, note that > Thus, p-value > , so we fail to reject the null hypothesis.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) A researcher claims that the mean age of women in California at the time of a first marriage is higher than 26.5 years. Surveying a simple random sample of 213 newlywed women in California, the researcher found a mean age of 27.0 years. Assuming that the population standard deviation is 2.3 years and using a 95% level of confidence, determine if there is sufficient evidence to support the researcher’s claim.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Solution Step 1:State the null and alternative hypotheses. The researcher’s claim is investigating the mean age at first marriage for women in California. Therefore, the research hypothesis, H a, is that the mean age is greater than 26.5, > The logical opposite is ≤ Thus, the null and alternative hypotheses are stated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Step 2:Determine which distribution to use for the test statistic, and state the level of significance. Note that the hypotheses are statements about the population mean, the population standard deviation is known, the sample is a simple random sample, and the sample size is at least 30. Thus, we will use a normal distribution and calculate the z-test statistic.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) We will draw a conclusion by computing the p-value for the calculated test statistic and comparing the value to . For this hypothesis test, the level of confidence is 95%, so the level of significance is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Step 3:Gather data and calculate the necessary sample statistics. From the information given, we know that the presumed value of the population mean is = 26.5, the sample mean is x̄ = 27.0, the population standard deviation is = 2.3, and the sample size is n = 213.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Thus, the test statistic is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Step 4:Draw a conclusion and interpret the decision. The alternative hypothesis tells us that we have a right-tailed test. Therefore, the p-value for this test statistic is the probability of obtaining a test statistic greater than or equal to z = 3.17, written as To find the p-value, we need to find the area under the standard normal curve to the right of z = 3.17.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, Known) (cont.) Using a normal distribution table or appropriate technology, we find that the area is p-value ≈ Comparing this p-value to the level of significance, we see that < 0.05, so p-value ≤ . Thus, the conclusion is to reject the null hypothesis. Therefore, we can say with 95% confidence that there is sufficient evidence to support the researcher’s claim that the mean age at first marriage for women in California is higher than 26.5 years.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) A recent study showed that the mean number of children for women in Europe is 1.5. A global watch group claims that German women have a mean fertility rate that is different from the mean for all of Europe. To test its claim, the group surveyed a simple random sample of 128 German women and found that they had a mean fertility rate of 1.4 children. The population standard deviation is assumed to be 0.8. Is there sufficient evidence to support the claim made by the global watch group at the 90% level of confidence?
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Solution Step 1:State the null and alternative hypotheses. The watch group is investigating whether the mean fertility rate for German women is different from the mean for all of Europe. Thus, they need to find evidence that the mean fertility rate is not equal to 1.5.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) So the research hypothesis, H a, is that the mean does not equal 1.5, ≠ 1.5. The logical opposite is = 1.5. Thus, the null and alternative hypotheses are stated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Step 2:Determine which distribution to use for the test statistic, and state the level of significance. Note that the hypotheses are statements about the population mean, is known, the sample is a simple random sample, and the sample size is at least 30. Thus, we will use a normal distribution and calculate the z-test statistic.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) We will draw a conclusion by computing the p-value for the calculated test statistic and comparing the value to . For this hypothesis test, the level of confidence is 90%, so the level of significance is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Step 3:Gather data and calculate the necessary sample statistics. From the information given, we know that the presumed value of the population mean is = 1.5, the sample mean is the population standard deviation is = 0.8, and the sample size is n = 128.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Thus, the test statistic is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Step 4:Draw a conclusion and interpret the decision. The alternative hypothesis tells us that we have a two-tailed test. Therefore, the p-value for this test statistic is the probability of obtaining a test statistic that is either less than or equal to z 1 = −1.41 or greater than or equal to z 2 = 1.41, which is written mathematically as To find the p-value, we need to find the sum of the areas under the standard normal curve to the left of z 1 = −1.41 and to the right of z 2 = 1.41.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.)
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) By looking up z 1 = −1.41 in the cumulative normal distribution table, we find that the area to the left is equal to Since the standard normal curve is symmetric about its mean, 0, the area to the right of z 2 = 1.41 is also Thus the p-value is calculated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, Known) (cont.) Comparing the p-value to the level of significance, we see that > 0.10, so p-value > . Thus, the conclusion is to fail to reject the null hypothesis. This means that, at a 90% level of confidence, the evidence does not support the watch group’s claim that the fertility rate of German women is different from the mean for all of Europe.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) A recent study showed that the mean number of children for women in Europe is 1.5. A global watch group claims that German women have a mean fertility rate that is different from the mean for all of Europe. To test its claim, the group surveyed a simple random sample of 128 German women and found that they had a mean fertility rate of 1.4 children. The population standard deviation is assumed to be 0.8. Is there sufficient evidence to support the claim made by the global watch group at the 90% level of confidence? Use a TI-83/84 Plus calculator to answer the question.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Solution Even when working with technology, Steps 1 and 2 are the same, so we will just reiterate the important information for these two steps from Example Step 1:State the null and alternative hypotheses. The hypotheses are stated as follows.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Step 2:Determine which distribution to use for the test statistic, and state the level of significance. Since is known, the sample is a simple random sample, and the sample size is at least 30, we use the normal distribution and z-test statistic for this hypothesis test for the population mean. The level of significance is = 0.10.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Step 3:Gather data and calculate the necessary sample statistics. This is where we begin to use a TI-83/84 Plus calculator. Let’s start by writing down the information from the problem, as we will need to enter these values into the calculator. We know that the presumed value of the population mean is = 1.5, the sample mean is x̄ = 1.4, the population standard deviation is = 0.8, and the sample size is n = 128.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Press, scroll to TESTS, and choose option 1:Z- Test. Since we know the sample statistics, choose Stats instead of Data. For À, enter the value from the null hypothesis, thus enter 1.5 for À. Enter the rest of the given values, as shown in the following screenshot on the left. Choose the alternative hypothesis øÀ. Highlight Calculate and press.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) The output screen, shown on the right, displays the alternative hypothesis, calculates the z-test statistic and the p-value, and then reiterates the sample mean and sample size that were entered.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Step 4:Draw a conclusion and interpret the decision. The p-value given by the calculator is approximately It is not identical to the one we found when using the table because there were several intermediate steps when we found the p-value by hand that reduced the accuracy of the p-value. However, the conclusion is the same since the two p-values are extremely close.
HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.18: Performing a Hypothesis Test for a Population Mean Using a TI-83/84 Plus Calculator (Two-Tailed, Known) (cont.) Since p-value > , the conclusion is to fail to reject the null hypothesis. This means that at a 90% level of confidence, the evidence does not support the watch group’s claim that the fertility rate of German women is different from the mean for all of Europe.