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.) By-Hand 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.