Mann – Introductory Statistics, Fifth Edition, Students Solutions Manual 187
9.1 a. The null hypothesis is a claim about a population parameter that is assumed to be true until it is declared false.
b. An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.
c. The critical point(s) divides the whole area under a distribution curve into rejection and non- rejection regions.
d. The significance level, denoted by α, is the probability of making a Type I error, that is, the probability of rejecting the null hypothesis when it is actually true.
e. The nonrejection region is the area where the null hypothesis is not rejected.
f. The rejection region is the area where the null hypothesis is rejected.
g. A hypothesis test is a two-tailed test if the rejection regions are in both tails of the distribution curve; it is a left-tailed test if the rejection region is in the left tail; and it is a right-tailed test if the rejection region is in the right tail.
h. Type I error: A type I error occurs when a true null hypothesis is rejected. The probability of committing a Type I error, denoted by α is: α = P(H0 is rejected | H0 is true)
Type II error: A Type II error occurs when a false null hypothesis is not rejected. The probability of committing a Type II error, denoted by β, is: β = P(H0 is not rejected |H0 is false)
9.3 A hypothesis test is a two-tailed test if the sign in the alternative hypothesis is "≠ "; it is a left-tailed test if the sign in the alternative hypothesis is " < " (less than); and it is a right-tailed test if the sign in the alternative hypothesis is " > " (greater than). Table 9.3 on page 405 of the text describes these relationships.
9.5 a. Left-tailed test b. Right-tailed test c. Two-tailed test
9.7 a. Type II error b. Type I error
9.9 a. H0: μ = 20 hours; H1: μ ≠ 20 hours; a two-tailed test
b. H0: μ = 10 hours; H1: μ > 10 hours; a right-tailed test
c. H0: μ = 3 years; H1: μ ≠ 3 years; a two-tailed test
d. H0: μ = $1000; H1: μ $1000; a left-tailed test
e. H0: μ = 12 minutes; H1: μ 12 minutes; a right-tailed test
9.11 For a two-tailed test, the p–value is twice the area in the tail of the sampling distribution curve beyond the observed value of the sample test statistic.
For a one-tailed test, the p–value is the area in the tail of the sampling distribution curve beyond the observed value of the sample test statistic.
9.13 a. Step 1: H0: µ = 46; H1: µ≠ 46; A two-tailed test.
Step 2: Since n > 30, use the normal distribution.
Step 3: = = 9.7 / = 1.53370467
z = ( – μ) / = (49.60–46) / 1.53370467 = 2.35
From the normal distribution table, area to the right of z = 2.35 is .5 – .4906 = .0094 approximately. Hence, p–value = 2(.0094) = .0188
b. Step 1: H0: µ = 26; H1: µ < 26; A left-tailed test.
Step 2: Since n > 30, use the normal distribution.
Step 3: = = 4.3 / = .74853392
z = ( – μ) / = (24.3–26) / .74853392 = –2.27
From the normal distribution table, area to the left of z = –2.27 is .5 –.4884 = .0116 approximately. Hence, p–value = .0116
c. Step 1: H0: µ = 18; H1: µ > 18; A right-tailed test.
Step 2: Since n > 30, use the normal distribution.
Step 3: = = 7.8 / = 1.05175179
z = ( – μ) / = (20.50 – 18) / 1.05175179 = 2.38
From the normal distribution table, area to the right of z = 2.38 is .5 – .4913 = .0087 approximately. Hence, p –value = .0087
TI-83: If you have raw data, first enter your data in a list. Otherwise just select Stat, highlight TESTS, highlight 1: Z Test, and press the ENTER key. If you have raw data choose Data and press the ENTER key, and then using the down arrow to move to enter the requested values and the list name, but be sure to keep Freq set at 1. On the other hand if you have summary statistics, instead of Data highlight Stats and press the ENTER key. Using the down arrow, scroll down to start entering the requested values. In this example we do not have raw data, so will follow the latter method. We select Stat, scroll down enter for part a, µ0 = 46, σ = s = 9.7, 49.60, n = 40, for µA highlighting µ0, highlighting Calculate, and pressing the ENTER key. The results are shown below where p is the p value we seek.
Z − Testµ46
z = 2.347257645
p = .0189121347
29.6
n = 40
MINITAB: Select the following Stat, Basic Statistics, and 1-Sample Z to generate a pop-up menu. In the pop-up menu, click beside the words Summarized data as long as you do indeed have summary data , beside the words Sample size enter your n, beside the word Mean enter your , beside the word standard deviation enter s or if known σ and beside the words Test mean type in µ0. Next click on Options to generate another pop-up menu where you make sure it says the correct Alternative hypothesis. By default µ1 is set to “not equal to”, if your alternative is something else, like greater than or less than, then use the down arrow to highlight the left tailed or right tailed µ1. Once the correct µ1 is in the box, then click on OK. This returns you to the original pop-up menu where you also click on OK and the resulting confidence interval appears in the Session section. The pop-up menus as well as the resulting confidence interval are shown below. The results are shown below where p is the p value we seek.
Excel: Open up KADDSTAT and select KADD, Hypothesis Testing, One Sample, Population mean using Z. In the pop-up menu in the first empty box enter µ0 and make sure not equal are highlighted if you are doing a two tailed test. If you have summary statistics and do not have raw data, like in this question, then click beside user input other wise click beside the other choice and enter your range of cells. After clicking beside user input if you, in the boxes enter the n, , σ if known or s if σ is unknown. In the box beside cell: type in the address on your worksheet where you want the answer to appear. Finally click on OK. For part a, the results and the pop-up menu are shown below.
Caution: For parts b and c, when using technology be sure to select the correct tail for your one tailed test instead the “NOT EQUAL”to used with the two tail test in these instructions which is only appropriate for part a.
9.15 a. Step 1: H0: µ = 72; Hl: µ > 72; A right-tailed test.
Step 2: Since n > 30, use the normal distribution.
Step 3: = = 6 / = 1.0
z = ( – μ) / = (74.07 – 72) / 1.0 = 2.07
The area under the standard normal curve to the right of z = 2.07 is .5 – .4808 = .0192.
Hence p–value = .0192
b. For α = .01, do not reject H0, since p–value > .01.
c. For α = .025, reject H0, since p–value < .025.
9.17 H0: µ = 16.3 weeks; H1: µ >16.3 weeks; A right-tailed test.
= = 4.2 / = .21
Test statistic: z = ( – μ) / = (16.9 – 16.3) / .21 = 2.86
The area under the standard normal curve to the right of z = 2.86 is .5 – .4979 = .0021 approximately. Hence, p–value = .0021
For α = .02, reject H0, since p–value < .02.
9.19 H0: µ ≥ 14 hours; H1 : µ < 14 hours; A left-tailed test.
= = 3.0 / = .21213203
Test statistic: z = ( – μ) / = (13.75 – 14) / .21213203 = –1.18
The area under the standard normal curve to the left of z = –1.18 is .5 – .3810 = .1190
Hence, p–value = .1190.
For α = .05, do not reject H0, since p–value > .05.
9.21 a. H0: µ = 10 minutes; H1: µ < 10 minutes; A left-tailed test.
= = 3.75 / = .375
z = ( – μ) / = (9.25 – 10) / .375 = –2.00
The area under the standard normal curve to the left of z = –2.00 is .5 – .4772 = .0228.
Hence, p–value = .0228
b. Do not reject H0 for α = .02, since p–value > .02.
Reject H0 for α = .05, since p–value < .05.
9.23 a. H0: µ = 32 ounces; H1: µ ≠ 32 ounces; A two-tailed test.
= .15 / = .02535463
z = ( – μ) / (31.90 – 32) / .02535463 = –3.94
The area under the standard normal curve to the left of z = –3.94 is approximately .5 –.5 = 0.
Hence, p–value = 2(.00) = .00 approximately
b. For α = .01, reject H0, since p–value < .01.
For α = .05, reject H0, since p–value < .05.
Thus in either case, the inspector will stop the machine.
9.25 The level of significance in a test of hypothesis is the probability of making a Type I error. It is the area under the probability distribution curve where we reject H0.
9.27 The critical value of z separates the rejection region from the nonrejection region and is found from a table such as the standard normal distribution table. The observed value of z is the value calculated for a sample statistic such as .
9.29 a. The rejection region lies to the left of z = –2.58 and to the right of z = 2.58.
The nonrejection region lies between z = –2.58 and z = 2.58.
b. The rejection region lies to the left of z = –2.58.
The nonrejection region lies to the right of z = –2.58.
c. The rejection region lies to the right of z = 1.96.
The nonrejection region lies to the left of z = 1.96.
Note: To find the critical value or values of z to create the rejection and nonrejection regions one method is to follow the procedures outlined in Chapter 6 of the text or in problems like 6.53 in Chapter 6 of the solutions manual.
9.31 If H0 is not rejected, the difference between the hypothesized value of μ and the observed value of is "statistically not significant".
9.33 a. .10 b. .02 c. .005
9.35 n = 90, = 15, s = 4, and = = = .42163702
a. Critical value: z = –2.33
Observed value: z = ( – μ) / = (15 – 20) / .42163702 = –11.86
b. Critical value: z = –2.58 and 2.58.
Observed value: z = ( – μ) / = (15 – 20) / .42163702 = –11.86
Note: To find the observed value of z and the critical value of z follow the procedures outlined in Chapter 6 of the text or in problems 6.17 and 6.53 of the solutions manual.
9.37 a. The rejection region lies to the left of z = –2.33.
The nonrejection region lies to the right of z = –2.33.
b. The rejection region lies to the left of z = –2.58 and to the right of z = 2.58.
The nonrejection region lies between z = –2.58 and z = 2.58.
c. The rejection region lies to the right of z = 2.33.
The nonrejection region lies to the left of z = 2.33.
9.39 a. n = 100, = 43, s = 5, and = = = .50
Critical value: z = –1.96
Test statistic: z = (– μ) / =(43 – 45) / .50 = –4.00; Reject H0.
b. n = 100, = 43.8, s = 7, and = = = .70
Critical value: z = –1.96
Test statistic: z = ( – μ) / = (43.8 – 45) / .70 = –1.71; Do not reject H0.
Comparing parts a and b shows that two samples selected from the same population can yield opposite conclusions on the same test of hypothesis.
TI-83: If you have raw data, first enter your data in a list. Otherwise just select Stat, then TESTS, then 1: Z Test, and finally pressing the ENTER key. If you have raw data choose Data and press the ENTER key, and then using the down arrow to move to enter the requested values and the list name, but be sure to keep Freq set at 1. On the other hand if you have summary statistics, instead of Data highlight Stats and press the ENTER key. Using the down arrow, scroll down to start entering the requested values. In this example we do not have raw data, so will follow the latter method. We select Stat, scroll down, and enter for part a, µ0 = 45, σ = s = 5, 43, n = 100. Then we highlight < 45 for µA, highlight Calculate, and press the ENTER key. The results are shown below where p is the p value we seek to compare with α. If we make out comparison in terms of z, we calculate the critical value or values of z first using the methods describe previously. Next we compare the critical value of z with the observed value of z reported below. Since the observed z is less than the critical value of z we do not reject the null hypothesis for part b, but the observed z is more than the critical value of z so we can reject the null hypothesis in part a.
Z − Test / Z − Testµ< 45 / µ< 45
z = − 4 / z = − 1.714285714
p = 3.1686035E −5 / p = .043238098
43 / 43.8
n = 100 / n = 100
MINITAB: Select the following Stat, Basic Statistics, and 1-Sample Z to generate a pop-up menu. In the pop-up menu, click beside the words Summarized data as long as you do indeed have summary data , beside the words Sample size enter your n, beside the word Mean enter your , beside the word standard deviation enter s or if known σ and beside the words Test mean type in µ0. Next click on Options to generate another pop-up menu where you enter in the confidence level or 1− α and then make sure it says the correct Alternative hypothesis. By default µ1 is set to “not equal to”, if your alternative is something else, like greater than or less than, then use the down arrow to highlight a left tailed or right tailed µ1. Once the correct µ1 is in the box, then click on OK. This returns you to the original pop-up menu where you also click on OK and the resulting confidence interval appears in the Session section. The pop-up menus for part a as well as the results of the hypothesis tests for both parts are shown below. Since the observed z is less than the critical value of z we do not reject the null hypothesis for part b, but the observed z is more than the critical value of z so we can reject the null hypothesis for part a.
Excel: Open up KADDSTAT and select KADD, Hypothesis Testing,, One Sample, Population mean using Z. In the pop-up menu in the first empty box enter µ0 and make sure not equal are highlighted if you are doing a two tailed test. If you have summary statistics and do not have raw data, like in this question, then click beside user input other wise click beside the other choice and enter your range of cells. After clicking beside user input if you, in the boxes enter the n, , σ if known or s if σ is unknown. In the box beside cell: type in the address on your worksheet where you want the answer to appear. For part a, the results and the pop-up menu are shown below, while for part b only the results are displayed. Since the observed z is less than the critical value of z we do not reject the null hypothesis for part b, but for part a the observed z is more than the critical value of z so we can reject the null hypothesis.