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CHAPTER 7 HYPOTHESIS TESTING WITH ONE SAMPLE Hypothesis testing in this chapter test the value of a population parameter (population mean (μ), or population proportion (p)) against some specified value k. Hypothesis testing is a very important tool that helps us to decide whether to accept or reject a claim about the population parameter. In this chapter we will illustrate how to: Use the z-test to test means of large samples or known δ. Use the t-test to test means of small samples and unknown δ. Use the z-test to test populations’ proportions. Hypothesis Testing for the Mean (Large Samples or δ Known) When the population standard deviation δ is known or the sample size is large (n 30), the z-test is appropriate to test the null hypothesis (H0) against the alternative hypothesis (Ha). Excel’s function for the z-test is given below. ZTEST (array, x, sigma): This returns the P-value for a right tailed test (H0: μ k versus Ha: μ > k). Note that the Excel documentation for the z-test should be ignored. It is mistakenly says that z-test gives the result of two-tailed test. The Excel’s function ZTEST (array, x), i.e., if there is no sigma value is given then Excel calculates the sample standard deviation (s) from the sample data and it uses it instead of sigma (δ). Recall that the formula for finding the test statistics is z = (xbar- μ)/ (δ/n), where xbar is the sample mean, μ is the hypothesis mean, δ is the standard deviation, and n is the sample size. z-test can also be used to apply a left-tailed test (H0: μ k versus Ha: μ < k) or a two-tailed test (H0: μ= k versus Ha: μ k). To apply a left tailed test, for the case μ < k, simply apply a right-tailed test and then subtract the result from 1. To apply a two-tailed test for the case μ k, either double the P- value from a right tailed test (μ>k) or double the P-value from a left tailed test (μ<k). 143

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CHAPTER 7

HYPOTHESIS TESTING WITH ONE SAMPLE

Hypothesis testing in this chapter test the value of a population parameter (population mean (μ), or population proportion (p)) against some specified value k. Hypothesis testing is a very important tool that helps us to decide whether to accept or reject a claim about the population parameter. In this chapter we will illustrate how to:

Use the z-test to test means of large samples or known δ. Use the t-test to test means of small samples and unknown δ. Use the z-test to test populations’ proportions.

Hypothesis Testing for the Mean (Large Samples or δ Known) When the population standard deviation δ is known or the sample size is large (n ≥ 30), the z-test is appropriate to test the null hypothesis (H0) against the alternative hypothesis (Ha). Excel’s function for the z-test is given below. ZTEST (array, x, sigma): This returns the P-value for a right tailed test (H0: μ ≤ k versus Ha: μ > k). Note that the Excel documentation for the z-test should be ignored. It is mistakenly says that z-test gives the result of two-tailed test. The Excel’s function ZTEST (array, x), i.e., if there is no sigma value is given then Excel calculates the sample standard deviation (s) from the sample data and it uses it instead of sigma (δ). Recall that the formula for finding the test statistics is z = (xbar- μ)/ (δ/√n), where xbar is the sample mean, μ is the hypothesis mean, δ is the standard deviation, and n is the sample size. z-test can also be used to apply a left-tailed test (H0: μ ≥ k versus Ha: μ < k) or a two-tailed test (H0: μ= k versus Ha: μ ≠ k). To apply a left tailed test, for the case μ < k, simply apply a right-tailed test and then subtract the result from 1. To apply a two-tailed test for the case μ ≠ k, either double the P-value from a right tailed test (μ>k) or double the P-value from a left tailed test (μ<k).

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The predefined Excel’s function ZTEST can be called as follows. Select Insert Function. Select Statistical category, and scroll through the list of functions and highlight ZTEST [see Figure (7.1)] and click OK. A dialog box should be opened [see Figure (7.2)], and then it has to be completed. Note that sigma (δ) is optional. If this box left blank, Excel will compute the standard deviation for the data in the specified array and use s instead of δ in the computation of z. In the dialog box X means the k value in the hypothesis test. Once the P-value is computed, the user can then compare it with α, the level of significance. If P-value ≤ α, the null hypothesis (H0) will be rejected. But if P-value > α, the null hypothesis will not be rejected.

Figure (7.1)

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Figure (7.2)

Example 7.1:

The data below represent the miles per gallon gasoline consumption (highway) for a random sample of 44 makes and models of passenger cars.

30 35 24 27 25 35 31 33 30 27 24 18 15 29 28 32 32 24 25 37 26 26 30 49 28 25 27 51 24 28 27 26 31

13 27 31 29 24 17 29 17 29 32 18

In testing the hypothesis that the population mean mile per gallon gasoline consumption for such cars is greater than 25 mpg, find the P-value.

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Solution: Since is δ is not known and n ≥ 30, we use s instead of δ in calculating test statistic.

- Open a new worksheet and enter the miles per gallon in column A. - In cell C1, type Null Hypothesis: μ ≤ 25, and in cell C2, type

Alternative Hypothesis: μ > 25. - In cell C3, type P-value for the right-tailed test. - In cell F3, enter the command = ZTEST (A2:A45, 25) and click

Enter. This returns a value 0.004392 as a P-value for the right-tailed test.

- In cell C5, select Tools Data Analysis Descriptive statistics. Select Summary Statistics, and click OK.

- In cell C23, type z = (Sample mean – Hyp. mean)/(Standard Error), where

Standard error is equal to δ/√n. If δ is unknown and the sample size is large (n ≥ 30), s is used instead of δ.

- In cell F23, type = (D7- 25)/D8, and click the Enter key. This returns 2.620363894 as a value of the test statistic z defined above.

- In cell C24, type P-value = P (z>2.62036894). - In cell F24, type = 1-NORMSDIST (2.620363814), and click the

Enter key. This return 0.004391799, which is another way of computing the P-value besides using the z-test [see Figure (7.3)].

Figure (7.3)

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Now, a worksheet template will be created to test claims about populations’ means for large samples or known δ based on the P-value method (P-value is used to make decisions) for left-tailed, right-tailed, or two-tailed test.8

- Open a new worksheet, and in cell A1, type Hypothesis testing about a population mean (large sample or sigma known).

- In cell A3, type input Data. - In cells A4, A5, A6, A7, and A8, type Sample_mean, Hyp_mean, n,

Std_Dev, and Alpha, respectively. - Highlight cells A4: A8, and click on the Align right icon. - In cell A10, type Standard Error. - In cell A11, type test statistic, z. - In cell A13, type Testing for a right-tailed test. - In cells A14 and A15 type P-value, and Conclusion, respectively. - In cell A 17, type Testing for a left-tailed test. - In cells A18 and A19 type P-value and Conclusion, respectively. - In cell A21, type testing for a two-tailed test. - In cells A22, and A23, type P-value and Conclusion, respectively. - In cell B10, type = Std_Dev/ SQRT (n), and click the Enter key. - In cell B11, type = (Sample_mean –Hyp_mean) / (Standard Error),

and click Enter. - In cell B14, type = 1- NORMSDIST (z) and click the Enter key. - In cell B15, type = if (P-value > Alpha, “Do no reject H0”, “Reject

H0”) and click Enter. - In cell B18, type = NORMSDIST (z), and click the Enter key. - In cell B19, type = if (P-value > Alpha, “Do no reject H0”, “Reject

H0”) and click the Enter key. - In cell B22, type = if (z > 0, 2* (1-NORMSDIST (z)),

2*NORMSDIST (z)), and click the Enter key. - In cell B23, type = if (P-value > Alpha, “Do no reject H0”, “Reject

H0”) and click the Enter key.

After you finished editing, and retyping the worksheet, it should have the cell contents as shown in Figure (7.4). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic z should be replaced by B11. To display the formula view [see Figure (7.5)], instead of the output [see Figure (7.6)], that will be used in testing claims about populations means for large samples or known δ by using the P-value method, hold down <Ctrl> key and tap the <~> key. Note

8 Goffinet/Koehler/Merchant, Microsoft Excel Manual, McGraw Hill, 2007.

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that the cells B10, B11, B14, B15, B18, B19, B22, and B23 in the template created display # Div /0!; but they will change when the cells B4:B8 are filled with their data values.

Figure (7.4)

Figure (7.5)

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Figure (7.6)

The template created, based on the P-value method is saved under the name “Hypothesis Testing_Population Mean_Large Samples_Pvalue_Template.xls” [see Figure (7.6)]. This template can be used to test claims about the population means for large samples or known population standard deviation (δ) using the P-value method. Example 7.2: The average salary for public community college instructor a specific year was reported to be \$41,375. A random sample of 50 public community college instructors in a particular State had a mean \$43, 860 and a standard deviation of \$6125. Is there enough evidence at α = 0.05 to conclude that the mean salary differs from \$41,375. Solution: The null and alternative hypotheses are given below. Null Hypothesis (H0): μ= \$ 41,375 Alternative Hypothesis (Ha): μ ≠ \$41,375 (Claim)

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- Open the template that was created under the name “Hypothesis Testing_Population Mean_Large Samples _Pvalue_Template.xls”, and enter the input data: Sample_mean = 43,860, Hyp_mean = 41,375, Std_Dev = 6125, Alpha = 0.05 and n = 50.

- Activate the other cells and save this worksheet as Example (7.2). - The worksheet that shows the results is shown in Figure (7.7). Thus, it

is clear from cell B23 (since the test is a two tailed-test) that the null hypothesis (H0) is rejected. That means there is enough evidence to conclude that the mean salary differs from \$ 41,375.

Note that the template that was created can be used to test a claim about a population mean if δ is known or the sample size is large (n ≥ 30). If δ is unknown and the sample size is large (n ≥ 30), sample standard deviation s should be used instead of the population standard deviation δ. If a set of data is given, the sample mean (xbar), sample standard deviation (s), and standard error can be obtained from the summary statistic. P-value can be computed as explained in the template or by using z-test as it was explained in Example 7.1.

Figure (7.7)

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Example 7.3: The data below represent the miles per gallon gasoline consumption (highway) for a random sample of 44 makes and models.

30 35 24 27 25 35 31 33 30 27 24 18 15 29 28 32 32 24 25 37 26 26 30 49 28 25 27 51 24 28 27 26 31

13 27 31 29 24 17 29 17 29 32 18

Test the hypothesis that the mean mile per gallon gasoline consumption for such cars is greater than 25 mpg.

a. Do we know δ for the mpg consumption? If not, use the value of s for the value of δ. Can we use the normal distribution for the hypothesis test?

b. State Null and Alternative Hypothesis, and type them on the worksheet?

c. Use z-test with sigma (δ) omitted. d. Find the P-value and compare it with Alpha (α) = 0.05. Do we reject

the Null Hypothesis? e. Use the Descriptive statistics dialog box to generate Summary

statistics for the data and place the results on the worksheet. Solution:

a. δ is not known, the sample standard deviation (s) will be used instead of δ. Yes we can use normal distribution test since n ≥ 30.

b. Null Hypothesis (H0): μ≤ 25; Alternative Hypothesis (Ha): μ>25 (Claim) - Open the file “Hypothesis Testing _Population Mean_Large

Sample_P-value_Template.xls”. - In cells A25 and A26, type Null hypothesis and Alternative

hypothesis. c. Using z-test to find P-value

- Enter the data values in column C. - In cell A28, type P-value and in cell B28, type = ZTEST (C2:C45,

25) and click the Enter key. This returns the P-value for the right tailed test as another way of finding P-value using Microsoft Excel.

d. Use Descriptive statistics to compute Summary statistics and type the result starting at the cell D2.

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e. Now, from the Summary statistic obtained, use Sample_mean (Mean) = 27.8409, Std_Dev (standard deviation) = 7.1915, n= 44, Alpha = 0.05 and Hyp_mean=25 as input data in the cells B2:B8 in the worksheet template. It is clear that P-value = 0.0044 ≤ α = 0.05, so the null hypothesis (H0) is rejected [see Figure (7.8)].

Figure (7.8)

We can test claims about population mean for large samples or known δ by using the critical-value method (critical values are used to make decisions) as follows.

For a left-tailed test, find the z-score (critical z-value, z0) that corresponds to an area α.

For a right-tailed test, find the z-score (critical z-value, z0) that corresponds to an area (1- α).

For a two tailed test, find the z-scores (critical z-values, -z0, + z0) that correspond to ½ α and 1- ½ α.

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Then we need to find the test statistic z. If the test statistic z is in the critical region, we reject the Null hypothesis (H0). Similarly, as we did before a worksheet template will be created to test claims about populations’ means for large samples or known δ using the critical-value method as follows.

- Open a new worksheet, and in cell A1, type Hypothesis testing about a population mean (large sample or sigma known) using critical values.

- In cell A3, type input data. - In cells A4, A5, A6, A7, and A8, type Sample_mean, Hyp_mean, n,

Std_Dev, and Alpha, respectively. - Highlight cells A4:A8, and click on Align right icon. - In cell A10, type Standard Error. - In cell A11, type Test Statistic, z. - In cell A13, type Testing for a right-tailed test. - In cells A14 and A15 type Crt_zValue, and Conclusion. - In cell A17, type Testing for a left-tailed test. - In cells A18, and A19, type Crt_zValue, and Conclusion. - In cell A21, type Testing for a two-tailed test. - In cells A22, and A23, type AbsCrt_zValue, and Conclusion. - In all B10, type = Std_Dev/ SQRT (n), and click the Enter key. - In cell B11, type = (Sample_mean-Hyp_mean) / (Standard Error), and

click the Enter key. - In cell B14, type = NORMS INV (Alpha), and click the Enter key. - In cell B15, type = if (z > Crt_zValue, “Reject H0”, “Do Not Reject

H0” ), and click the Enter key. - In B18, type = NORMSINV (Alpha), and click the Enter key. - In B19, type if (z < Crt_zValue, “Reject H0”, “Do Not Reject H0”),

and click the Enter key. - In cell B22, type = ABS (NORMSINV (Alpha/2)), and click the Enter

key. - In cell B23, if (OR (z < -AbsCrt_zValue, z > AbsCrt_zValue), “Reject

H0”, “Do Not Reject H0”), and click the Enter key.

After you finished editing, and retyping the worksheet, it should have the cell contents as shown in Figure (7.9). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic z should be replaced by B11. To display the formula view [see Figure (7.10)] instead of the output [see Figure (7.11)], that will be used in testing claims about populations’ means for large samples or known δ by using critical-value method, hold down <Ctrl> key and tap the <~> key.

153

Note that the cells B10, B11, B14, B15, B18, B19, B22, and B23 in the template created display # Div /0!; but they will change when the cells B4:B8 are filled with their data values.

Figure (7.9)

Figure (7.10)

154

Figure (7.11)

The template created, based on the critical value method was saved under the name “Hypothesis Testing_Population Mean_Large Samples_Criticalvalue_Template.xls” [see Figure (7.11)]. This template can be used to test claims about populations’ means for large samples or known standard deviation (δ) by using the critical value method. Example 7.4: The Average salary for public community college instructor for a specific year was reported to be \$ 41,375. A random sample of 50 public community college instructors in particular state had a mean of \$43,860 and a standard deviation of \$6125. Is there enough evidence at α = 0.05 to conclude that the mean salary differs form \$ 41,375? Use critical-value method.

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Solution:

- Since δ is not known, the sample standard deviation (s) will be used instead of δ.

- The null and alternative hypotheses are as follows. Null Hypothesis (H0): μ= \$ 41, 37; Alternative Hypothesis (Ha): μ ≠ \$41,375 (Claim)

- Open the worksheet template file that was created under the name “Hypothesis Testing_Population Mean_Large Samples_Criticalvalue_Template.xls”, and enter the input data: Sample_mean = 43,860, Hyp_mean = 41,375, Std_Dev = 6125, Alpha = 0.05, and n = 50.

- Save this worksheet as Example (7.4). - From the worksheet shown in Figure (7.12), the null hypothesis (H0)

is rejected. This means there is enough evidence to conclude that the mean salary differs form \$ 41,375.

Figure (7.12)

156

Hypothesis Testing for the Mean (Small Samples and δ Unknown) To test a claim about a population mean using a small sample (n< 30) from a normal, or nearly normal distribution when sigma (δ) is unknown, student’s t-distribution (z-distribution is not appropriate) will be used. The test statistics t is defined as t = (xbar- μ)/(s/√n), where xbar is the sample mean, μ is the hypothesis mean, s is the sample standard deviation, and n is the sample size. This is the same as t = (Sample_mean –Hyp_mean)/ (Standard Error), where Standard Error = s/ SQRT (n), and the degrees of freedom df=n-1. Note that Microsoft Excel has two commands for student’s t-distribution under statistical options of paste function. TDIST (x, degrees of freedom, tails): This returns the area in the tail of student’s t-distribution beyond the specified value of x, for the specified degree of freedom and number of tails (1 or 2). TINV (probability, degree of freedom): This returns the critical t-value, t0, such that the area in the two tails beyond the t0 value equals the specified probability for the specified degrees of freedom. TINV can be used to find the critical t-value, t0, to use in testing claims about populations’ means of small samples (n < 30) and unknown δ by using the critical-value method. Testing claims about populations’ means of small sample ( n < 30) and unknown δ can be done in similar ways as testing claims about populations’ means of large sample ( n ≥ 30 ) or known δ. One method is based on the P-value, where the null hypothesis will be rejected if the P-value ≤ α. The second method is based on the critical values t0 and the critical regions, where the null hypothesis will be rejected if the test statistics, t falls in the critical region. In this section, two worksheet templates will be created; the first worksheet is based on the P-value method, and the second one is based on the critical-value method. These worksheets will be used to test claims about a populations’ means for small samples (n < 30) and unknown δ. In similar way, as for large samples, the first worksheet that has cell contents is shown

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in Figure (7.13). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic t should be replaced by B12. The worksheet template that will be used in testing claims about populations’ means for small samples and unknown δ by using the P-value method is shown in Figure (7.14). To display the formula view (instead of the output), hold down the <Ctrl> key and tap the <~> key. Note that the cells B11, B12, B15, B16, B19, B20, B23, and B24 in the template created display # Div /0!; but they will change when the cells B4:B8 are filled with their data values.

Figure (7.13)

The template created, based on the P-value method is saved under the name “Hypothesis Testing_Population Mean_Small Samples_Pvalue_Template.xls” [see Figure (7.14)]. This template can be used to test claims about populations’ means for small samples and unknown population standard deviation (δ) by using the P-value method.

158

Figure (7.14)

Example 7.5: Employment information services claims that the mean annual pay for full-time female workers over age 25 and without high school diploma is \$17,100. The annual pay for a random sample of 12 full-time female workers without a high school diploma is listed. At α = 0.05, test the claim that the mean salary is \$17,100. Use the P-value method. 16,009 16,790 17,328 18,161 16,631 21,028 16,114 17,176 17,503 19,764 15,316 18,801 Solution:

- The null and alternative hypotheses are given below. Null Hypothesis (H0): μ= \$17100 (Claim); Alternative Hypothesis (Ha): μ ≠ 17,100.

- From the given data, we can use Summary statistic to find the sample mean and the sample standard deviation. Another way to find the sample mean (Sample_mean) and sample standard deviation (Std_Dev) is by using the following predefined Excel’s functions:

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AVERAGE (data range), and STDEV (data range) then press the Enter key.

- Open worksheet template based on the P-value to test hypothesis about population mean of small samples (n < 30) and unknown δ.

- Enter sample data in column D. - Enter the input data: Sample_mean, Hyp_mean = 17,100, n = 12,

Std_Dev, and Alpha = 0.05, in the cells B4:B8. Note that the Sample_mean in cell B4 can be entered by typing the Excel’s command = AVERAGE (D2: D13) and clicking the Enter key. Also, Std_Dev in cell B7, can be entered by typing the Excel’s command = STDEV (D2:D13) and clicking the Enter key.

- By activating other cells and saving this worksheet as Example 7.5, we get the results which are summarized in the following figure [see Figure (7.15)].

- As shown in Figure (7.15), the P-value for the two-tailed test we had is 0.36357 > α = 0.05, so the null hypothesis (H0) is not rejected. This means that there is enough evidence to support the claim that the mean salary is \$17,100.

Figure (7.15)

160

The second worksheet template, which is based on the critical-value method, is created in a similar way as a worksheet template for testing claims about populations’ means for large samples in the previous section. The worksheet, which has cell contents, is shown in Figure (7.16). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic t should be replaced by B12. The worksheet template that will be used in testing claims about populations’ means for small samples and unknown δ, by using the critical-value method is shown in Figure (7.17). To display the formula view (instead of the output), hold down the <Ctrl> key and tap the <~> key. Note that the cells B11, B12, B15, B16, B19, B20, B23, and B24 in the template created display # Div /0!, and #NUM!; but they will change when the cells B4:B8 are filled with their data values.

Figure (7.16)

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Figure (7.17)

The template created, based on the P-value method is saved under the name “Hypothesis Testing_Population Mean_Small Samples_Criticalvalue_Template.xls” [see Figure (7.17)]. This template can be used to test claims about populations’ means for small samples and unknown population standard deviation (δ) by using the critical value method. Example 7.6: Redo Example 7.5 by using critical-value method. Solution:

- Note that the null and alternative hypotheses are as given below. Null Hypothesis (H0): μ= \$17100 (Claim); Alternative Hypothesis (Ha): μ ≠17,100.

- From the given data, we can use Summary statistic to find the sample mean and the sample standard deviation. Another way to find the sample mean and sample standard deviation is by using the following predefined Excel’s functions: AVERAGE (data range), and STDEV (data range) then press the Enter key.

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- Open the worksheet template that was created under the name “Hypothesis Testing_Population Mean_Small Samples_Criticalvalue_Template.xls”.

- Enter sample data in column E. - Enter the input data: Sample_mean, Hyp_mean = 17,100, n = 12,

Std_Dev, and Alpha = 0.05, in the cells B4:B8. Note that the Sample_mean in cell B4 can be entered by typing the Excel’s command = AVERAGE (E2: E13) and clicking the Enter key. Also, Std_Dev in cell B7, can be entered by typing the Excel’s command = STDEV (E2:E13) and clicking the Enter key.

- By activating other cells and saving this worksheet as Example 7.6, we get the results which are summarized in the following figure [see Figure (7.18)].

- As shown in Figure (7.18), the test statistic value t is not in the critical region for the available two-tailed test. Thus, the null hypothesis (H0) is not rejected. This means that there is enough evidence to support the claim that the mean salary is \$17,100.

Figure (7.18)

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Hypothesis Testing for Population Proportions If np ≥ 5 and nq ≥ 5 for a binomial distribution, then the sampling distribution (phat) is normal with mean p and standard deviation √(pq/n). The test statistics for proportion is given by the formula z= (phat – p)/ √ (pq/n), where phat is the sample proportion, p is the Hypothesized proportion, q=1-p, and n is the sample size. To test hypothesis (claims) about populations’ proportions, two worksheet templates will be created to solve such problems. The first worksheet template is based on the P-value method and the second worksheet template is based on the critical-value method. The first worksheet template has cell contents as shown in Figure (7.19). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic z should be replaced by B9. The worksheet template that will be used in testing claims about populations’ proportions by using the P-value method is shown in Figure (7.20). To display the formula view (instead of the template output), hold down the <Ctrl> key and tap the <~> key. Note that the cells B9, B12, B13, B16, B17, B20, and B21 in the template created display # Div /0!, and #NUM!; but they will change when the cells B4:B7 are filled with their data values.

Figure (7.19)

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Figure (7.20)

The template created, based on the P-value method is saved under the name “Hypothesis Testing_Population Proportions _Pvalue_Template.xls” [see Figure (7.20)]. This template can be used to test claims about populations’ proportions by using the P-value method. The second worksheet template has cell contents as shown in Figure (7.21). Formulas can be entered by replacing cell contents by their addresses. For example, the test statistic z should be replaced by B9. The worksheet template that will be used in testing claims about populations’ proportions by using the critical-value method is shown in Figure (7.22). To display the formula view (instead of the output), hold down the <Ctrl> key and tap the <~> key. Note that the cells B9, B12, B13, B16, B17, B20, and B21 in the template created display #NUM!, and # Div /0!; but they will change when the cells B4:B7 are filled with their data values.

165

Figure (7.21)

Figure (7.22)

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The template created, based on the critical-value method is saved under the name “Hypothesis Testing_Population Proportions _Criticalvalue_Template.xls” [see Figure (7.22)]. This template can be used to test claims about populations’ proportions by using the critical value method. Example 7.7: An educator estimates that the dropout rate for seniors at high school in Ohio is 12%. Last year, 34 seniors form a random sample of 200 Ohio seniors withdraw. At α =0.01, is there enough evidence to support the educator claim. Use P-value method or critical-value method. Solution:

- Note that the null and alternative hypotheses are given below. Null Hypothesis (H0): p= 0.12 (Claim); Alternative Hypothesis (Ha): p ≠ 0.12.

- Note that phat= 34/200 = 0.17, n=200, and α =0.01. - Open either worksheet template based on the P-value method or on

the critical-value method for testing claims about populations’ proportions.

- Enter the input data, and then the results are shown in Figure (2.23) and Figure (2.24).

- It is clear from the two worksheets that the null hypothesis H0 is not rejected. Hence, there is enough evidence to support the educator claim.

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Figure (7.23)

Figure (7.24)

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Example 7.8: A medical researcher estimates that no more than 55% of U.S. adults eat breakfast every day. In a random sample of 250 U.S. Adults, 56.4% say they eat breakfast every day. At α =0.05, is there enough evidence to reject the researcher’s claim. Use either the P-value method or critical-value method. Solution:

- Note that the null and alternative hypotheses are given below. Null Hypothesis H0: p ≤ 0.55(Claim); Alternative Hypothesis Ha: p>0.55.

- Note that phat= 0.564, n=250, and α =0.05. - Open either worksheet template based on the P-value or on the

Critical-value for testing claims about populations’ proportions. - Enter the input data, and then the results are shown in Figure (2.25)

and Figure (2.26). - It is clear from the two worksheets that the null hypothesis (H0) is

not rejected. Hence, there is enough evidence to support the educator claim.

Figure (7.25)

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Figure (7.26)

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Microsoft Excel Lab Experiments Lab Experiment 7.1: A researcher reports that the average salary of assistant professors is more than \$42,000. A sample of 28 assistant professors has a mean salary of \$43,260. At α = 0.05, test the claim that assistant professors earn more than \$42,000 a year. The standard deviation of the population is \$5,230. Lab Experiment 7.2: The average salary for public school teachers for a specific year was reported to be \$39,475. A random sample of 50 public school teachers in a particular state had a mean of \$41,540 and a standard deviation of \$5,935. Is there sufficient evidence at α=0.05 level to conclude that the mean salary differs form \$39,475. Use the P-value method. Test statistics: _________________P-value: ________________________ Conclusion in your own words: Lab Experiment 7.3: Open or retrieve the worksheet temperature.xls from the CD-ROM included with this manual. The data in column A of this worksheet represent the temperatures of 50 cities in the United States. Temperatures are given below. 98 94 95 104 100 97 82 93 101 97 100 88 89 98 96 98 96 90 92 93 99 100 97 96 96 97 98 99 92 98 98 107 98 99 112 92 97 103 100 101 100 94 87 101 98 99 82 96 103 71 A meteorologist claims that the average of the highest temperature in the United States is 98 degrees Fahrenheit. A random sample of 50 cities is selected, and the temperatures are recorded as shown above. At α = 0.05, test the meteorologist claim by using either P-value method or critical value method.

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Lab Experiment 7.4:9 An Alabama politician claims that the mean annual salary for engineering managers in Alabama is more than the national mean, \$90,000. The annual salaries (in dollars) for a random sample of 34 engineering managers in Alabama are listed. At α = 0.03, is there enough evidence to support the politician’s claim. Use P-value method or critical-value method. 87,512 89,510 82,639 98,897 104,877 87,592 103,632 105,202 93,672 104,878 101,508 88,302 89,231 105,060 95,974 76,955 101,396 69,838 87,728 98,314 104,348 93,493 74,918 89,736 68,060 96,777 82,723 92,944 86,114 97,062 85,975 84,956 95,905 79,042 Test statistics: ___________________P-value: ______________________ Critical z-value: _________________________ Conclusion in your own words: __________________________________ Lab Experiment 7.5: The average undergraduate cost for tuition fees, and room and board for 2-years institutions last year was \$13,252. The following year, a random sample of 20 two-year institutions had a mean of \$15,560 and a standard deviation of \$3,500. Create the template (based on the P-value method) shown in Figure (7.14) and use this template to decide if there is enough evidence at α = 0.01 level of significance to conclude that the mean cost has increased. Test statistics: __________________P-value: ________________________ Conclusion in your own words: ___________________________________

9 Larson/Farber, Elementary Statistics, Pearson Prentice Hall, 2006.

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Lab Experiment 7.6: A company that manufactures school supplies says the teachers spend a mean of more than \$580 of their own money on the school supplies in a year. A random sample of the amounts (in dollars) that 24 teachers spent on school supplies in a recent year is listed below. At α = 0.05, is there enough evidence to support the company’s claim?

523 581 476 600 721 622 575 320 888 875 910 643 596 587 398 622 585 700 590 710 729 857 630 520

Lab Experiment 7.7:10

In a Gallup poll of 1012 randomly selected adults, 9% said that cloning of humans should be allowed. Use a 0.05 significance level to test the claim that less than 10% of all adults say that cloning in humans should be allowed. Can a newspaper run a headline that “less than 10% of all adults are opposed to cloning of human”? Test statistics: ___________________P-value: _____________________ Conclusion in your own words: __________________________________ Lab Experiment 7.8: The statistical abstract reported that 16% of adults attended a musical play past year. A researcher surveyed 90 people and had found that 22 had attended a musical play in the past year. Create the templates shown in Figures 7.20 & 7.22 (based on the P-value or critical-value methods). At α = 0.05 use one of these templates to test the statistical abstract claim. Test statistics: ________________critical z-values: ___________________ P-value: ___________________________________ Conclusion in your own words: ___________________________________

10 Mario F. Triola, Minitab Manual, Pearson Addison Wesley, 2004.

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