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Section 10.2 Hypothesis Testing for Population Means (s Known) With valuable content added by D.R.S., University of Cordele 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. There are two ways to get the answer and we learn BOTH. • The “Critical Value” method, comparing your z Test Statistic to the z Critical Value, which was determined by the chosen α Level of Significance. • The “p-Value method”, in which your z Test Statistic leads to a “p-Value”, an area under the normal curve, which is compared to the chosen α Level of Significance. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Means (s Known) Test Statistic for a Hypothesis Test for a Population Mean (s Known) When (1) the population standard deviation is known, (2) the sample taken is a simple random sample, and (3) either the sample size is at least 30 or the population distribution is approximately normal, THEN COMPUTE: the test statistic for a hypothesis test for a population mean is given by x z s n HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Means (s Known) THEN COMPUTE: the test statistic for a hypothesis test for a population mean is given by x z s n My sample mean is how far away from the H0 Null Hypothesis mean? Gauged by the standard deviation for a sample as defined in the Central Limit Theorem. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Means (s Known) Test Statistic for a Hypothesis Test for a Population Mean (s Known) (cont.) where x ̄ is the sample mean, is the presumed value of the population mean from the null hypothesis, s 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 Alternative Hypothesis, Ha Value Value Value The “rejection region” is the one-tail area of size α. Type of Hypothesis Test Left-tailed test Right-tailed test Two-tailed test The “rejection region” consists of two tails, each of them of size α/2, left and right. And the “Fail to Reject H0” region is the big area of size 1 – α. 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, H0, if the test statistic calculated from the sample data falls within the rejection region. Refer to the next three pictures. They illustrate the Left-Tailed, Right-Tailed, and Two-Tailed cases. Be able to draw the right kind of picture for each Hypothesis Test you perform. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Left-Tailed Hypothesis Test: Ha < μ The critical z value is the negative z value which separates the left tail of area α. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Right-Tailed Hypothesis Test: Ha > μ The critical z value is the positive z value which separates the right tail of area α. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A Two-Tailed Hypothesis Test: Ha ≠ μ The critical z values are the +/- z values which separate the two tails, area α/2 each. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Commonly-occurring critical values • These values for the α Level Of Significance (or c Level Of Confidence) are typical. You should already know how to find them using tables and using invNorm(). It is convenient to have this special table for reference. Area in tail(s), α, alpha (and c too) OneTailed Test TwoTailed Test 0.10 (0.90) 1.28 ±1.645 0.05 (0.95) 1.645 ±1.96 0.02 (0.98) 2.05 ±2.33 0.01 (0.99) 2.33 ±2.575 Example 10.10: Using a Rejection Region in a Hypothesis Test for a Population Mean (Right-Tailed, s 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, s 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, s 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 Ha. The mathematical opposite is ≤ 850. Thus, we have the following hypotheses. H0 : 850 Ha : 850 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, s 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, s 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, s 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 a = 0.05. 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, s Known) (cont.) Thus, the test statistic is calculated as follows. x z s n 856 850 98 1000 1.94 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, s 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, s Known) (cont.) If we look up 0.9500 in the body of the cumulative z-table, the corresponding critical z-value is 1.645. 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 ≥ 1.645. 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, s 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, s 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, H0, is assumed to be true. We’ll give examples of how to find a p-value by hand. But usually you’ll get the p-value for free from the TI-84’s ZTest or TTest feature. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. How to calculate the p value for your z Test Statistic You’ve calculated your z = Test Statistic using the formula. Here’s how to define your p-value: • If it’s a left-tailed test, what’s the area to the left of your z ? • If it’s a right tailed test, what’s the area to the right of your z? • If it’s a two-tailed test, what’s 2x the area to the left/right of your neg/pos z? 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. H0 : 0.15 Ha : 0.15 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 lefttailed 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 = 1.34. Thus, the p-value ≈ 0.0901. normalcdf( ________, ________ ) 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. H0 : 0.43 Ha : 0.43 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 righttailed 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 = 2.78. Thus, the p-value ≈ 0.0027. normalcdf( ________, ________ ) 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. H0 : 0.78 Ha : 0.78 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 twotailed 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 p-value P z 2.15 . Use a table or appropriate technology to find the area under the standard normal curve to the left of z1 = −2.15. The area to the left of z1 = −2.15 is 0.0158. 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 z2 = 2.15 is also 0.0158. Thus, the p-value is calculated as follows. p-value 0.0158 2 0.0316 normalcdf( ________, ________ )*2 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 Enter normalcdf(856,1û99, 850, 98/ð (1000)) and press . But in practice, we’re probably They demonstrate here how to find the p-value using normalcdf and “x” values. Recall that the sample mean was bar_x = 856, it was a right tailed-test, the null hypothesis said the mean score was 850, the population standard deviation was 98, and the sample size was 1000. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. going to use TI-84 Z-Test instead to give us the p-values, so don’t put a lot of emphasis on this slide. 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 0.0264, as shown in the screenshot. But again, Z-Test is a much better way !!!! HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Z-Test for this problem. STAT, TESTS, 1:Z-Test. Since they gave us summary statistics, we pick “Stats” (not “Data”, which is for a problem where we have the individuals’ scores). μ0 comes from H0 σ is the “known” stdev. bar-x from our sample n is our sample size Give the Ha inequality Highlight Calculate and press ENTER. Z-Test for this problem. STAT, TESTS, 1:Z-Test. Here are the results. It reminds you of what you told it was the alternative hypothesis. Have a look and make sure it’s what you wanted. It tells you the Test Statistic. It tells you the p-Value. It reminds you of the sample data, the sample mean and sample size you told it. p-Values Conclusions Using p-Values • If p-value ≤ a, then reject the null hypothesis. • If p-value > a, 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 = 0.0146. 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 a = 0.05. Next, note that 0.0146 < 0.05. Thus, p-value ≤ a, 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. a 1c 1 0.99 0.01 Next, note that 0.0146 > 0.01. Thus, p-value > a, 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, s 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: TI-84 Z-Test solution You still have to write the hypotheses: • Null Hypothesis: ________ Alternative: ________ What is the α Level Of Significance? α = _______ What inputs do you give to the TI-84 Z-Test? • 𝜇0 = _______, 𝜎 = ________ • 𝑥 = _______, 𝑛 = ________ • 𝜇: ≠ or < or > The TI-84 Z-Test output gives us a p-value of ________. Therefore we { reject or fail to reject } H0. Example 10.16: Performing a Hypothesis Test for a Population Mean (Right-Tailed, s 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, Ha, is that the mean age is greater than 26.5, > 26.5. The logical opposite is ≤ 26.5. Thus, the null and alternative hypotheses are stated as follows. H0 : 26.5 Ha : 26.5 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, s 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, s Known) (cont.) We will draw a conclusion by computing the p-value for the calculated test statistic and comparing the value to a. For this hypothesis test, the level of confidence is 95%, so the level of significance is calculated as follows. a 1c 1 0.95 0.05 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, s 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 x ̄ = 27.0, the population standard deviation is s = 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, s Known) (cont.) Thus, the test statistic is calculated as follows. x z s n 27.0 26.5 2.3 213 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, s 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 p-value P z 3.17 . 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, s 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, s Known) (cont.) Using a normal distribution table or appropriate technology, we find that the area is p-value ≈ 0.0008. Comparing this p-value to the level of significance, we see that 0.0008 < 0.05, so p-value ≤ a. 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, s 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: TI-84 Z-Test solution You still have to write the hypotheses: • Null Hypothesis: ________ Alternative: ________ What is the α Level Of Significance? α = _______ What inputs do you give to the TI-84 Z-Test? • 𝜇0 = _______, 𝜎 = ________ • 𝑥 = _______, 𝑛 = ________ • 𝜇: ≠ or < or > The TI-84 Z-Test output gives us a p-value of ________. Therefore we { reject or fail to reject } H0. Example 10.17: Performing a Hypothesis Test for a Population Mean (Two-Tailed, s 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, s Known) (cont.) So the research hypothesis, Ha, 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. H0 : 1.5 Ha : 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, s 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, s 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, s Known) (cont.) We will draw a conclusion by computing the p-value for the calculated test statistic and comparing the value to a. For this hypothesis test, the level of confidence is 90%, so the level of significance is calculated as follows. a 1c 1 0.90 0.10 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, s 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 x 1.4, the population standard deviation is s = 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, s Known) (cont.) Thus, the test statistic is calculated as follows. x z s n 1.4 1.5 0.8 128 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, s 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 z1 = −1.41 or greater than or equal to z2 = 1.41, which is written mathematically as p-value P z 1.41 . To find the p-value, we need to find the sum of the areas under the standard normal curve to the left of z1 = −1.41 and to the right of z2 = 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, s 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, s Known) (cont.) By looking up z1 = −1.41 in the cumulative normal distribution table, we find that the area to the left is equal to 0.0793. Since the standard normal curve is symmetric about its mean, 0, the area to the right of z2 = 1.41 is also 0.0793. Thus the p-value is calculated as follows. p-value 0.0793 2 0.1586 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, s Known) (cont.) Comparing the p-value to the level of significance, we see that 0.1586 > 0.10, so p-value > a. 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, s 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, s 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 10.17. Step 1: State the null and alternative hypotheses. The hypotheses are stated as follows. H0 : 1.5 Ha : 1.5 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, s Known) (cont.) Step 2: Determine which distribution to use for the test statistic, and state the level of significance. Since s 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 a = 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, s 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 x ̄ = 1.4, the population standard deviation is s = 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, s Known) (cont.) Press , scroll to TESTS, and choose option 1:ZTest. 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, s 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, s Known) (cont.) Step 4: Draw a conclusion and interpret the decision. The p-value given by the calculator is approximately 0.1573. 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, s Known) (cont.) Since p-value > a, 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. Excel’s Z.TEST(data array, μ, σ) function • • • • Give it the cells (“range”) of the sample data values. And the μ mean from the null hypothesis. And the σ population standard deviation. It gives you back the one-tailed p-value. • So it calculates a z = test statistic based on the sample. • And then it calculates the area in the tail beyond that z • Either left tail beyond negative z • Or right tail beyond positive z, it knows which one. Excel’s Z.TEST(data array, μ, σ) function • • • • • Give it the cells (“range”) of the sample data values. And the μ mean from the null hypothesis. And the σ population standard deviation. It gives you back the one-tailed p-value. You compare that p-value to the α level of significance that was pre-selected in the design. • If p-value < α, then reject H0. • If p-value > α, then fail to reject H0. Excel’s Z.TEST(data array, μ, σ) if you have a Two-Tailed Situation • Recall the two-tailed situation: Ha has a ≠. • So the rejection regions are two tails of area α/2. • Excel Help says to do this to get the p-value: 2 * MIN(Z.TEST(data , μ, σ), 1 – Z.TEST(data , μ, σ)) (whereas we tell TI-84 it’s two-tailed and TI-84 takes care of doubling it transparently, thank you.) • And then decide as usual: • If p-value < α, then reject H0. • If p-value > α, then fail to reject H0.