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8.E: Testing Hypotheses (Exercises)

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.

8.1: The Elements of Hypothesis Testing

Q8.1.1

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number \(\mu _0\) and write \(H_0:\mu =\mu _0\) and the appropriate analogous expression for \(H_a\).)

  1. The average July temperature in a region historically has been \(74.5^{\circ}F\). Perhaps it is higher now.
  2. The average weight of a female airline passenger with luggage was \(145\) pounds ten years ago. The FAA believes it to be higher now.
  3. The average stipend for doctoral students in a particular discipline at a state university is \(\$14,756\). The department chairman believes that the national average is higher.
  4. The average room rate in hotels in a certain region is \(\$82.53\). A travel agent believes that the average in a particular resort area is different.
  5. The average farm size in a predominately rural state was \(69.4\) acres. The secretary of agriculture of that state asserts that it is less today.

Q1.1.2

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number \(\mu _0\) and write \(H_0:\mu =\mu _0\) and the appropriate analogous expression for \(H_a\).)

  1. The average time workers spent commuting to work in Verona five years ago was \(38.2\) minutes. The Verona Chamber of Commerce asserts that the average is less now.
  2. The mean salary for all men in a certain profession is \(\$58,291\). A special interest group thinks that the mean salary for women in the same profession is different.
  3. The accepted figure for the caffeine content of an \(8\)-ounce cup of coffee is \(133\) mg. A dietitian believes that the average for coffee served in a local restaurants is higher.
  4. The average yield per acre for all types of corn in a recent year was \(161.9\) bushels. An economist believes that the average yield per acre is different this year.
  5. An industry association asserts that the average age of all self-described fly fishermen is \(42.8\) years. A sociologist suspects that it is higher.

Q1.1.3

Describe the two types of errors that can be made in a test of hypotheses.

Q1.1.4

Under what circumstance is a test of hypotheses certain to yield a correct decision?

Answers

  1.  
    1. \(H_0:\mu =74.5\; vs\; H_a:\mu >74.5\)
    2. \(H_0:\mu =145\; vs\; H_a:\mu >145\)
    3. \(H_0:\mu =14756\; vs\; H_a:\mu >14756\)
    4. \(H_0:\mu =82.53\; vs\; H_a:\mu \neq 82.53\)
    5. \(H_0:\mu =69.4\; vs\; H_a:\mu <69.4\)
  2.  
  3. A Type I error is made when a true \(H_0\) is rejected. A Type II error is made when a false \(H_0\) is not rejected.

8.2: Large Sample Tests for a Population Mean

Basic

  1. Find the rejection region (for the standardized test statistic) for each hypothesis test.
    1. \(H_0:\mu =27\; vs\; H_a:\mu <27\; @\; \alpha =0.05\)
    2. \(H_0:\mu =52\; vs\; H_a:\mu \neq 52\; @\; \alpha =0.05\)
    3. \(H_0:\mu =-105\; vs\; H_a:\mu >-105\; @\; \alpha =0.10\)
    4. \(H_0:\mu =78.8\; vs\; H_a:\mu \neq 78.8\; @\; \alpha =0.10\)
  2. Find the rejection region (for the standardized test statistic) for each hypothesis test.
    1. \(H_0:\mu =17\; vs\; H_a:\mu <17\; @\; \alpha =0.01\)
    2. \(H_0:\mu =880\; vs\; H_a:\mu \neq 880\; @\; \alpha =0.01\)
    3. \(H_0:\mu =-12\; vs\; H_a:\mu >-12\; @\; \alpha =0.05\)
    4. \(H_0:\mu =21.1\; vs\; H_a:\mu \neq 21.1\; @\; \alpha =0.05\)
  3. Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.
    1. \(H_0:\mu =141\; vs\; H_a:\mu <141\; @\; \alpha =0.20\)
    2. \(H_0:\mu =-54\; vs\; H_a:\mu <-54\; @\; \alpha =0.05\)
    3. \(H_0:\mu =98.6\; vs\; H_a:\mu \neq 98.6\; @\; \alpha =0.05\)
    4. \(H_0:\mu =3.8\; vs\; H_a:\mu >3.8\; @\; \alpha =0.001\)
  4. Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.
    1. \(H_0:\mu =-62\; vs\; H_a:\mu \neq -62\; @\; \alpha =0.005\)
    2. \(H_0:\mu =73\; vs\; H_a:\mu >73\; @\; \alpha =0.001\)
    3. \(H_0:\mu =1124\; vs\; H_a:\mu <1124\; @\; \alpha =0.001\)
    4. \(H_0:\mu =0.12\; vs\; H_a:\mu \neq 0.12\; @\; \alpha =0.001\)
  5. Compute the value of the test statistic for the indicated test, based on the information given.
    1. Testing \(H_0:\mu =72.2\; vs\; H_a:\mu >72.2,\; \sigma \; \text{unknown}\; n=55,\; \bar{x}=75.1,\; s=9.25\)
    2. Testing \(H_0:\mu =58\; vs\; H_a:\mu >58,\; \sigma =1.22\; n=40,\; \bar{x}=58.5,\; s=1.29\)
    3. Testing \(H_0:\mu =-19.5\; vs\; H_a:\mu <-19.5,\; \sigma \; \text{unknown}\; n=30,\; \bar{x}=-23.2,\; s=9.55\)
    4. Testing \(H_0:\mu =805\; vs\; H_a:\mu \neq 805,\; \sigma =37.5\; n=75,\; \bar{x}=818,\; s=36.2\)
  6. Compute the value of the test statistic for the indicated test, based on the information given.
    1. Testing \(H_0:\mu =342\; vs\; H_a:\mu <342,\; \sigma =11.2\; n=40,\; \bar{x}=339,\; s=10.3\)
    2. Testing \(H_0:\mu =105\; vs\; H_a:\mu >105,\; \sigma =5.3\; n=80,\; \bar{x}=107,\; s=5.1\)
    3. Testing \(H_0:\mu =-13.5\; vs\; H_a:\mu \neq -13.5,\; \sigma \; \text{unknown}\; n=32,\; \bar{x}=-13.8,\; s=1.5\)
    4. Testing \(H_0:\mu =28\; vs\; H_a:\mu \neq 28,\; \sigma \; \text{unknown}\; n=68,\; \bar{x}=27.8,\; s=1.3\)
  7. Perform the indicated test of hypotheses, based on the information given.
    1. Test \(H_0:\mu =212\; vs\; H_a:\mu <212\; @\; \alpha =0.10,\; \sigma \; \text{unknown}\; n=36,\; \bar{x}=211.2,\; s=2.2\)
    2. Test \(H_0:\mu =-18\; vs\; H_a:\mu >-18\; @\; \alpha =0.05,\; \sigma =3.3\; n=44,\; \bar{x}=-17.2,\; s=3.1\)
    3. Test \(H_0:\mu =24\; vs\; H_a:\mu \neq 24\; @\; \alpha =0.02,\; \sigma \; \text{unknown}\; n=50,\; \bar{x}=22.8,\; s=1.9\)
  8. Perform the indicated test of hypotheses, based on the information given.
    1. Test \(H_0:\mu =105\; vs\; H_a:\mu >105\; @\; \alpha =0.05,\; \sigma \; \text{unknown}\; n=30,\; \bar{x}=108,\; s=7.2\)
    2. Test \(H_0:\mu =21.6\; vs\; H_a:\mu <21.6\; @\; \alpha =0.01,\; \sigma \; \text{unknown}\; n=78,\; \bar{x}=20.5,\; s=3.9\)
    3. Test \(H_0:\mu =-375\; vs\; H_a:\mu \neq -375\; @\; \alpha =0.01,\; \sigma =18.5\; n=31,\; \bar{x}=-388,\; s=18.0\)

Applications

  1. In the past the average length of an outgoing telephone call from a business office has been \(143\) seconds. A manager wishes to check whether that average has decreased after the introduction of policy changes. A sample of \(100\) telephone calls produced a mean of \(133\) seconds, with a standard deviation of \(35\) seconds. Perform the relevant test at the \(1\%\) level of significance.
  2. The government of an impoverished country reports the mean age at death among those who have survived to adulthood as \(66.2\) years. A relief agency examines \(30\) randomly selected deaths and obtains a mean of \(62.3\) years with standard deviation \(8.1\) years. Test whether the agency’s data support the alternative hypothesis, at the \(1\%\) level of significance, that the population mean is less than \(66.2\).
  3. The average household size in a certain region several years ago was \(3.14\) persons. A sociologist wishes to test, at the \(5\%\) level of significance, whether it is different now. Perform the test using the information collected by the sociologist: in a random sample of \(75\) households, the average size was \(2.98\) persons, with sample standard deviation \(0.82\) person.
  4. The recommended daily calorie intake for teenage girls is \(2,200\) calories/day. A nutritionist at a state university believes the average daily caloric intake of girls in that state to be lower. Test that hypothesis, at the \(5\%\) level of significance, against the null hypothesis that the population average is \(2,200\) calories/day using the following sample data: \(n=36,\; \bar{x}=2,150,\; s=203\)
  5. An automobile manufacturer recommends oil change intervals of \(3,000\) miles. To compare actual intervals to the recommendation, the company randomly samples records of \(50\) oil changes at service facilities and obtains sample mean \(3,752\) miles with sample standard deviation \(638\) miles. Determine whether the data provide sufficient evidence, at the \(5\%\) level of significance, that the population mean interval between oil changes exceeds \(3,000\) miles.
  6. A medical laboratory claims that the mean turn-around time for performance of a battery of tests on blood samples is \(1.88\) business days. The manager of a large medical practice believes that the actual mean is larger. A random sample of \(45\) blood samples yielded mean \(2.09\) and sample standard deviation \(0.13\) day. Perform the relevant test at the \(10\%\) level of significance, using these data.
  7. A grocery store chain has as one standard of service that the mean time customers wait in line to begin checking out not exceed \(2\) minutes. To verify the performance of a store the company measures the waiting time in \(30\) instances, obtaining mean time \(2.17\) minutes with standard deviation \(0.46\) minute. Use these data to test the null hypothesis that the mean waiting time is \(2\) minutes versus the alternative that it exceeds \(2\) minutes, at the \(10\%\) level of significance.
  8. A magazine publisher tells potential advertisers that the mean household income of its regular readership is \(\$61,500\). An advertising agency wishes to test this claim against the alternative that the mean is smaller. A sample of \(40\) randomly selected regular readers yields mean income \(\$59,800\) with standard deviation \(\$5,850\). Perform the relevant test at the \(1\%\) level of significance.
  9. Authors of a computer algebra system wish to compare the speed of a new computational algorithm to the currently implemented algorithm. They apply the new algorithm to \(50\) standard problems; it averages \(8.16\) seconds with standard deviation \(0.17\) second. The current algorithm averages \(8.21\) seconds on such problems. Test, at the \(1\%\) level of significance, the alternative hypothesis that the new algorithm has a lower average time than the current algorithm.
  10. A random sample of the starting salaries of \(35\) randomly selected graduates with bachelor’s degrees last year gave sample mean and standard deviation \(\$41,202\) and \(\$7,621\), respectively. Test whether the data provide sufficient evidence, at the \(5\%\) level of significance, to conclude that the mean starting salary of all graduates last year is less than the mean of all graduates two years before, \(\$43,589\).

Additional Exercises

  1. The mean household income in a region served by a chain of clothing stores is \(\$48,750\). In a sample of \(40\) customers taken at various stores the mean income of the customers was \(\$51,505\) with standard deviation \(\$6,852\).
    1. Test at the \(10\%\) level of significance the null hypothesis that the mean household income of customers of the chain is \(\$48,750\) against that alternative that it is different from \(\$48,750\).
    2. The sample mean is greater than \(\$48,750\), suggesting that the actual mean of people who patronize this store is greater than \(\$48,750\). Perform this test, also at the \(10\%\) level of significance. (The computation of the test statistic done in part (a) still applies here.)
  2. The labor charge for repairs at an automobile service center are based on a standard time specified for each type of repair. The time specified for replacement of universal joint in a drive shaft is one hour. The manager reviews a sample of \(30\) such repairs. The average of the actual repair times is \(0.86\) hour with standard deviation \(0.32\) hour.
    1. Test at the \(1\%\) level of significance the null hypothesis that the actual mean time for this repair differs from one hour.
    2. The sample mean is less than one hour, suggesting that the mean actual time for this repair is less than one hour. Perform this test, also at the \(1\%\) level of significance. (The computation of the test statistic done in part (a) still applies here.)

Large Data Set Exercises

Large Data Set missing from the original

  1. Large \(\text{Data Set 1}\) records the SAT scores of \(1,000\) students. Regarding it as a random sample of all high school students, use it to test the hypothesis that the population mean exceeds \(1,510\), at the \(1\%\) level of significance. (The null hypothesis is that \(\mu =1510\)).
  2. Large \(\text{Data Set 1}\) records the GPAs of \(1,000\) college students. Regarding it as a random sample of all college students, use it to test the hypothesis that the population mean is less than \(2.50\), at the \(10\%\) level of significance. (The null hypothesis is that \(\mu =2.50\)).
  3. Large \(\text{Data Set 1}\) lists the SAT scores of \(1,000\) students.
    1. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean \(\mu\).
    2. Regard the first \(50\) students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean exceeds \(1,510\), at the \(10\%\) level of significance. (The null hypothesis is that \(\mu =1510\)).
    3. Is your conclusion in part (b) in agreement with the true state of nature (which by part (a) you know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?
  4. Large \(\text{Data Set 1}\) lists the GPAs of \(1,000\) students.
    1. Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person was measured. Compute the population mean \(\mu\).
    2. Regard the first \(50\) students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean is less than \(2.50\), at the \(10\%\) level of significance. (The null hypothesis is that \(\mu =2.50\)).
    3. Is your conclusion in part (b) in agreement with the true state of nature (which by part (a) you know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?

Answers

  1.  
    1. \(Z\leq -1.645\)
    2. \(Z\leq -1.645\; or\; Z\geq 1.96\)
    3. \(Z\geq 1.28\)
    4. \(Z\leq -1.645\; or\; Z\geq 1.645\)
  2.  
  3.  
    1. \(Z\leq -0.84\)
    2. \(Z\leq -1.645\)
    3. \(Z\leq -1.96\; or\; Z\geq 1.96\)
    4. \(Z\geq 3.1\)
  4.  
  5.  
    1. \(Z = 2.235\)
    2. \(Z = 2.592\)
    3. \(Z = -2.122\)
    4. \(Z = 3.002\)
  6.  
  7.  
    1. \(Z = -2.18,\; -z_{0.10}=-1.28,\; \text{reject}\; H_0\)
    2. \(Z = 1.61,\; z_{0.05}=1.645,\; \text{do not reject}\; H_0\)
    3. \(Z = -4.47,\; -z_{0.01}=-2.33,\; \text{reject}\; H_0\)
  8.  
  9. \(Z = -2.86,\; -z_{0.01}=-2.33,\; \text{reject}\; H_0\)
  10.  
  11. \(Z = -1.69,\; -z_{0.025}=-1.96,\; \text{do not reject}\; H_0\)
  12.  
  13. \(Z = 8.33,\; z_{0.05}=1.645,\; \text{reject}\; H_0\)
  14.  
  15. \(Z = 2.02,\; z_{0.10}=1.28,\; \text{reject}\; H_0\)
  16.  
  17. \(Z = -2.08,\; -z_{0.01}=-2.33,\; \text{do not reject}\; H_0\)
  18.  
  19.  
    1. \(Z =2.54,\; z_{0.05}=1.645,\; \text{reject}\; H_0\)
    2. \(Z = 2.54,\; z_{0.10}=1.28,\; \text{reject}\; H_0\)
  20.  
  21. \(H_0:\mu =1510\; vs\; H_a:\mu >1510\). Test Statistic: \(Z = 2.7882\). Rejection Region: \([2.33,\infty )\). Decision: Reject \(H_0\).
  22.  
  23.  
    1. \(\mu _0=1528.74\)
    2. \(H_0:\mu =1510\; vs\; H_a:\mu >1510\). Test Statistic: \(Z = -1.41\). Rejection Region: \([1.28,\infty )\). Decision: Fail to reject \(H_0\).
    3. No, it is a Type II error.

8.3: The Observed Significance of a Test

Basic

  1. Compute the observed significance of each test.
    1. Testing \(H_0:\mu =54.7\; vs\; H_a:\mu <54.7,\; \text{test statistic}\; z=-1.72\)
    2. Testing \(H_0:\mu =195\; vs\; H_a:\mu \neq 195,\; \text{test statistic}\; z=-2.07\)
    3. Testing \(H_0:\mu =-45\; vs\; H_a:\mu >-45,\; \text{test statistic}\; z=2.54\)
  2. Compute the observed significance of each test.
    1. Testing \(H_0:\mu =0\; vs\; H_a:\mu \neq 0,\; \text{test statistic}\; z=2.82\)
    2. Testing \(H_0:\mu =18.4\; vs\; H_a:\mu <18.4,\; \text{test statistic}\; z=-1.74\)
    3. Testing \(H_0:\mu =63.85\; vs\; H_a:\mu >63.85,\; \text{test statistic}\; z=1.93\)
  3. Compute the observed significance of each test. (Some of the information given might not be needed.)
    1. Testing \(H_0:\mu =27.5\; vs\; H_a:\mu >27.5,\; n=49,\; \bar{x}=28.9,\; s=3.14,\; \text{test statistic}\; z=3.12\)
    2. Testing \(H_0:\mu =581\; vs\; H_a:\mu <581,\; n=32,\; \bar{x}=560,\; s=47.8,\; \text{test statistic}\; z=-2.49\)
    3. Testing \(H_0:\mu =138.5\; vs\; H_a:\mu \neq 138.5,\; n=44,\; \bar{x}=137.6,\; s=2.45,\; \text{test statistic}\; z=-2.44\)
  4. Compute the observed significance of each test. (Some of the information given might not be needed.)
    1. Testing \(H_0:\mu =-17.9\; vs\; H_a:\mu <-17.9,\; n=34,\; \bar{x}=-18.2,\; s=0.87,\; \text{test statistic}\; z=-2.01\)
    2. Testing \(H_0:\mu =5.5\; vs\; H_a:\mu \neq 5.5,\; n=56,\; \bar{x}=7.4,\; s=4.82,\; \text{test statistic}\; z=2.95\)
    3. Testing \(H_0:\mu =1255\; vs\; H_a:\mu >1255,\; n=152,\; \bar{x}=1257,\; s=7.5,\; \text{test statistic}\; z=3.29\)
  5. Make the decision in each test, based on the information provided.
    1. Testing \(H_0:\mu =82.9\; vs\; H_a:\mu <82.9\; @\; \alpha =0.05\), observed significance \(p=0.038\)
    2. Testing \(H_0:\mu =213.5\; vs\; H_a:\mu \neq 213.5\; @\; \alpha =0.01\), observed significance \(p=0.038\)
  6. Make the decision in each test, based on the information provided.
    1. Testing \(H_0:\mu =31.4\; vs\; H_a:\mu >31.4\; @\; \alpha =0.10\), observed significance \(p=0.062\)
    2. Testing \(H_0:\mu =-75.5\; vs\; H_a:\mu <-75.5\; @\; \alpha =0.05\), observed significance \(p=0.062\)

Applications

  1. A lawyer believes that a certain judge imposes prison sentences for property crimes that are longer than the state average \(11.7\) months. He randomly selects \(36\) of the judge’s sentences and obtains mean \(13.8\) and standard deviation \(3.9\) months.
    1. Perform the test at the \(1\%\) level of significance using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(1\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).
  2. In a recent year the fuel economy of all passenger vehicles was \(19.8\) mpg. A trade organization sampled \(50\) passenger vehicles for fuel economy and obtained a sample mean of \(20.1\) mpg with standard deviation \(2.45\) mpg. The sample mean \(20.1\) exceeds \(19.8\), but perhaps the increase is only a result of sampling error.
    1. Perform the relevant test of hypotheses at the \(20\%\) level of significance using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(20\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).
  3. The mean score on a \(25\)-point placement exam in mathematics used for the past two years at a large state university is \(14.3\). The placement coordinator wishes to test whether the mean score on a revised version of the exam differs from \(14.3\). She gives the revised exam to \(30\) entering freshmen early in the summer; the mean score is \(14.6\) with standard deviation \(2.4\).
    1. Perform the test at the \(10\%\) level of significance using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(10\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).
  4. The mean increase in word family vocabulary among students in a one-year foreign language course is \(576\) word families. In order to estimate the effect of a new type of class scheduling, an instructor monitors the progress of \(60\) students; the sample mean increase in word family vocabulary of these students is \(542\) word families with sample standard deviation \(18\) word families.
    1. Test at the \(5\%\) level of significance whether the mean increase with the new class scheduling is different from \(576\) word families, using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(5\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).
  5. The mean yield for hard red winter wheat in a certain state is \(44.8\) bu/acre. In a pilot program a modified growing scheme was introduced on \(35\) independent plots. The result was a sample mean yield of \(45.4\) bu/acre with sample standard deviation \(1.6\) bu/acre, an apparent increase in yield.
    1. Test at the \(5\%\) level of significance whether the mean yield under the new scheme is greater than \(44.8\) bu/acre, using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(5\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).
  6. The average amount of time that visitors spent looking at a retail company’s old home page on the world wide web was \(23.6\) seconds. The company commissions a new home page. On its first day in place the mean time spent at the new page by \(7,628\) visitors was \(23.5\) seconds with standard deviation \(5.1\) seconds.
    1. Test at the \(5\%\) level of significance whether the mean visit time for the new page is less than the former mean of \(23.6\) seconds, using the critical value approach.
    2. Compute the observed significance of the test.
    3. Perform the test at the \(5\%\) level of significance using the \(p\)-value approach. You need not repeat the first three steps, already done in part (a).

Answers

  1.  
    1. \(p\text{-value}=0.0427\)
    2. \(p\text{-value}=0.0384\)
    3. \(p\text{-value}=0.0055\)
  2.  
  3.  
    1. \(p\text{-value}=0.0009\)
    2. \(p\text{-value}=0.0064\)
    3. \(p\text{-value}=0.0146\)
  4.  
  5.  
    1. reject \(H_0\)
    2. do not reject \(H_0\)
  6.  
  7.  
    1. \(Z=3.23,\; z_{0.01}=2.33\), reject \(H_0\)
    2. \(p\text{-value}=0.0006\)
    3. reject \(H_0\)
  8.  
  9.  
    1. \(Z=0.68,\; z_{0.05}=1.645\), do not reject \(H_0\)
    2. \(p\text{-value}=0.4966\)
    3. do not reject \(H_0\)
  10.  
  11.  
    1. \(Z=2.22,\; z_{0.05}=1.645\), reject \(H_0\)
    2. \(p\text{-value}=0.0132\)
    3. reject \(H_0\)

8.4: Small Sample Tests for a Population Mean

Basic

  1. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed.
    1. \(H_0: \mu =27\; vs\; H_a:\mu <27\; @\; \alpha =0.05,\; n=12,\; \sigma =2.2\)
    2. \(H_0: \mu =52\; vs\; H_a:\mu \neq 52\; @\; \alpha =0.05,\; n=6,\; \sigma \; \text{unknown} \)
    3. \(H_0: \mu =-105\; vs\; H_a:\mu >-105\; @\; \alpha =0.10,\; n=24,\; \sigma \; \text{unknown} \)
    4. \(H_0: \mu =78.8\; vs\; H_a:\mu \neq 78.8\; @\; \alpha =0.10,\; n=8,\; \sigma =1.7\)
  2. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed.
    1. \(H_0: \mu =17\; vs\; H_a:\mu <17\; @\; \alpha =0.01,\; n=26,\; \sigma =0.94\)
    2. \(H_0: \mu =880\; vs\; H_a:\mu \neq 880\; @\; \alpha =0.01,\; n=4,\; \sigma \; \text{unknown} \)
    3. \(H_0: \mu =-12\; vs\; H_a:\mu >-12\; @\; \alpha =0.05,\; n=18,\; \sigma =1.1\)
    4. \(H_0: \mu =21.1\; vs\; H_a:\mu \neq 21.1\; @\; \alpha =0.05,\; n=23,\; \sigma \; \text{unknown} \)
  3. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed. Identify the test as left-tailed, right-tailed, or two-tailed.
    1. \(H_0: \mu =141\; vs\; H_a:\mu <141\; @\; \alpha =0.20,\; n=29,\; \sigma \; \text{unknown} \)
    2. \(H_0: \mu =-54\; vs\; H_a:\mu <-54\; @\; \alpha =0.05,\; n=15,\; \sigma =1.9\)
    3. \(H_0: \mu =98.6\; vs\; H_a:\mu \neq 98.6\; @\; \alpha =0.05,\; n=12,\; \sigma \; \text{unknown} \)
    4. \(H_0: \mu =3.8\; vs\; H_a:\mu >3.8\; @\; \alpha =0.001,\; n=27,\; \sigma \; \text{unknown} \)
  4. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed. Identify the test as left-tailed, right-tailed, or two-tailed.
    1. \(H_0: \mu =-62\; vs\; H_a:\mu \neq -62\; @\; \alpha =0.005,\; n=8,\; \sigma \; \text{unknown} \)
    2. \(H_0: \mu =73\; vs\; H_a:\mu >73\; @\; \alpha =0.001,\; n=22,\; \sigma \; \text{unknown} \)
    3. \(H_0: \mu =1124\; vs\; H_a:\mu <1124\; @\; \alpha =0.001,\; n=21,\; \sigma \; \text{unknown} \)
    4. \(H_0: \mu =0.12\; vs\; H_a:\mu \neq 0.12\; @\; \alpha =0.001,\; n=14,\; \sigma =0.026\)
  5. A random sample of size 20 drawn from a normal population yielded the following results: \(\bar{x}=49.2,\; s=1.33\)
    1. Test \(H_0: \mu =50\; vs\; H_a:\mu \neq 50\; @\; \alpha =0.01\).
    2. Estimate the observed significance of the test in part (a) and state a decision based on the \(p\)-value approach to hypothesis testing.
  6. A random sample of size 16 drawn from a normal population yielded the following results: \(\bar{x}=-0.96,\; s=1.07\)
    1. Test \(H_0: \mu =0\; vs\; H_a:\mu <0\; @\; \alpha =0.001\).
    2. Estimate the observed significance of the test in part (a) and state a decision based on the \(p\)-value approach to hypothesis testing.
  7. A random sample of size 8 drawn from a normal population yielded the following results: \(\bar{x}=289,\; s=46\)
    1. Test \(H_0: \mu =250\; vs\; H_a:\mu >250\; @\; \alpha =0.05\).
    2. Estimate the observed significance of the test in part (a) and state a decision based on the \(p\)-value approach to hypothesis testing.
  8. A random sample of size 12 drawn from a normal population yielded the following results: \(\bar{x}=86.2,\; s=0.63\)
    1. Test \(H_0: \mu =85.5\; vs\; H_a:\mu \neq 85.5\; @\; \alpha =0.01\).
    2. Estimate the observed significance of the test in part (a) and state a decision based on the \(p\)-value approach to hypothesis testing.

Applications

  1. Researchers wish to test the efficacy of a program intended to reduce the length of labor in childbirth. The accepted mean labor time in the birth of a first child is \(15.3\) hours. The mean length of the labors of \(13\) first-time mothers in a pilot program was \(8.8\) hours with standard deviation \(3.1\) hours. Assuming a normal distribution of times of labor, test at the \(10\%\) level of significance test whether the mean labor time for all women following this program is less than \(15.3\) hours.
  2. A dairy farm uses the somatic cell count (SCC) report on the milk it provides to a processor as one way to monitor the health of its herd. The mean SCC from five samples of raw milk was \(250,000\) cells per milliliter with standard deviation \(37,500\) cell/ml. Test whether these data provide sufficient evidence, at the \(10\%\) level of significance, to conclude that the mean SCC of all milk produced at the dairy exceeds that in the previous report, \(210,250\) cell/ml. Assume a normal distribution of SCC.
  3. Six coins of the same type are discovered at an archaeological site. If their weights on average are significantly different from \(5.25\) grams then it can be assumed that their provenance is not the site itself. The coins are weighed and have mean \(4.73\) g with sample standard deviation \(0.18\) g. Perform the relevant test at the \(0.1\%\) (\(\text{1/10th of}\; 1\%\)) level of significance, assuming a normal distribution of weights of all such coins.
  4. An economist wishes to determine whether people are driving less than in the past. In one region of the country the number of miles driven per household per year in the past was \(18.59\) thousand miles. A sample of \(15\) households produced a sample mean of \(16.23\) thousand miles for the last year, with sample standard deviation \(4.06\) thousand miles. Assuming a normal distribution of household driving distances per year, perform the relevant test at the \(5\%\) level of significance.
  5. The recommended daily allowance of iron for females aged \(19-50\) is \(18\) mg/day. A careful measurement of the daily iron intake of \(15\) women yielded a mean daily intake of \(16.2\) mg with sample standard deviation \(4.7\) mg.
    1. Assuming that daily iron intake in women is normally distributed, perform the test that the actual mean daily intake for all women is different from \(18\) mg/day, at the \(10\%\) level of significance.
    2. The sample mean is less than \(18\), suggesting that the actual population mean is less than \(18\) mg/day. Perform this test, also at the \(10\%\) level of significance. (The computation of the test statistic done in part (a) still applies here.)
  6. The target temperature for a hot beverage the moment it is dispensed from a vending machine is \(170^{\circ}F\). A sample of ten randomly selected servings from a new machine undergoing a pre-shipment inspection gave mean temperature \(173^{\circ}F\) with sample standard deviation \(6.3^{\circ}F\).
    1. Assuming that temperature is normally distributed, perform the test that the mean temperature of dispensed beverages is different from \(170^{\circ}F\), at the \(10\%\) level of significance.
    2. The sample mean is greater than \(170\), suggesting that the actual population mean is greater than \(170^{\circ}F\). Perform this test, also at the \(10\%\) level of significance. (The computation of the test statistic done in part (a) still applies here.)
  7. The average number of days to complete recovery from a particular type of knee operation is \(123.7\) days. From his experience a physician suspects that use of a topical pain medication might be lengthening the recovery time. He randomly selects the records of seven knee surgery patients who used the topical medication. The times to total recovery were:\[\begin{matrix} 128 & 135 & 121 & 142 & 126 & 151 & 123 \end{matrix}\]
    1. Assuming a normal distribution of recovery times, perform the relevant test of hypotheses at the \(10\%\) level of significance.
    2. Would the decision be the same at the \(5\%\) level of significance? Answer either by constructing a new rejection region (critical value approach) or by estimating the \(p\)-value of the test in part (a) and comparing it to \(\alpha \).
  8. A 24-hour advance prediction of a day’s high temperature is “unbiased” if the long-term average of the error in prediction (true high temperature minus predicted high temperature) is zero. The errors in predictions made by one meteorological station for \(20\) randomly selected days were:\[\begin{matrix} 2 & 0 & -3 & 1 & -2\\ 1 & 0 & -1 & 1 & -1\\ -4 & 1 & 1 & -4 & 0\\ -4 & -3 & -4 & 2 & 2 \end{matrix}\]
    1. Assuming a normal distribution of errors, test the null hypothesis that the predictions are unbiased (the mean of the population of all errors is \(0\)) versus the alternative that it is biased (the population mean is not \(0\)), at the \(1\%\) level of significance.
    2. Would the decision be the same at the \(5\%\) level of significance? The \(10\%\) level of significance? Answer either by constructing new rejection regions (critical value approach) or by estimating the \(p\)-value of the test in part (a) and comparing it to \(\alpha \).
  9. Pasteurized milk may not have a standardized plate count (SPC) above \(20,000\) colony-forming bacteria per milliliter (cfu/ml). The mean SPC for five samples was \(21,500\) cfu/ml with sample standard deviation \(750\) cfu/ml. Test the null hypothesis that the mean SPC for this milk is \(20,000\) versus the alternative that it is greater than \(20,000\), at the \(10\%\) level of significance. Assume that the SPC follows a normal distribution.
  10. One water quality standard for water that is discharged into a particular type of stream or pond is that the average daily water temperature be at most \(18^{\circ}F\). Six samples taken throughout the day gave the data: \[\begin{matrix} 16.8 & 21.5 & 19.1 & 12.8 & 18.0 & 20.7 \end{matrix}\]
    The sample mean exceeds \(\bar{x}=18.15\), but perhaps this is only sampling error. Determine whether the data provide sufficient evidence, at the \(10\%\) level of significance, to conclude that the mean temperature for the entire day exceeds \(18^{\circ}F\).

Additional Exercises

  1. A calculator has a built-in algorithm for generating a random number according to the standard normal distribution. Twenty-five numbers thus generated have mean \(0.15\) and sample standard deviation \(0.94\). Test the null hypothesis that the mean of all numbers so generated is \(0\) versus the alternative that it is different from \(0\), at the \(20\%\) level of significance. Assume that the numbers do follow a normal distribution.
  2. At every setting a high-speed packing machine delivers a product in amounts that vary from container to container with a normal distribution of standard deviation \(0.12\) ounce. To compare the amount delivered at the current setting to the desired amount \(64.1\) ounce, a quality inspector randomly selects five containers and measures the contents of each, obtaining sample mean \(63.9\) ounces and sample standard deviation \(0.10\) ounce. Test whether the data provide sufficient evidence, at the \(5\%\) level of significance, to conclude that the mean of all containers at the current setting is less than \(64.1\) ounces.
  3. A manufacturing company receives a shipment of \(1,000\) bolts of nominal shear strength \(4,350\) lb. A quality control inspector selects five bolts at random and measures the shear strength of each. The data are:\[\begin{matrix} 4,320 & 4,290 & 4,360 & 4,350 & 4,320 \end{matrix}\]
    1. Assuming a normal distribution of shear strengths, test the null hypothesis that the mean shear strength of all bolts in the shipment is \(4,350\) lb versus the alternative that it is less than \(4,350\) lb, at the \(10\%\) level of significance.
    2. Estimate the \(p\)-value (observed significance) of the test of part (a).
    3. Compare the \(p\)-value found in part (b) to and make a decision based on the \(p\)-value approach. Explain fully.
  4. A literary historian examines a newly discovered document possibly written by Oberon Theseus. The mean average sentence length of the surviving undisputed works of Oberon Theseus is \(48.72\) words. The historian counts words in sentences between five successive \(101\) periods in the document in question to obtain a mean average sentence length of \(39.46\) words with standard deviation \(7.45\) words. (Thus the sample size is five.)
    1. Determine if these data provide sufficient evidence, at the \(1\%\) level of significance, to conclude that the mean average sentence length in the document is less than \(48.72\).
    2. Estimate the \(p\)-value of the test.
    3. Based on the answers to parts (a) and (b), state whether or not it is likely that the document was written by Oberon Theseus.

Answers

  1.  
    1. \(Z\leq -1.645\)
    2. \(T\leq -2.571\; or\; T \geq 2.571\)
    3. \(T \geq 1.319\)
    4. \(Z\leq -1645\; or\; Z \geq 1.645\)
  2.  
  3.  
    1. \(T\leq -0.855\)
    2. \(Z\leq -1.645\)
    3. \(T\leq -2.201\; or\; T \geq 2.201\)
    4. \(T \geq 3.435\)
  4.  
  5.  
    1. \(T=-2.690,\; df=19,\; -t_{0.005}=-2.861,\; \text{do not reject }H_0\)
    2. \(0.01<p-value<0.02,\; \alpha =0.01,\; \text{do not reject }H_0\)
  6.  
  7.  
    1. \(T=2.398,\; df=7,\; t_{0.05}=1.895,\; \text{reject }H_0\)
    2. \(0.01<p-value<0.025,\; \alpha =0.05,\; \text{reject }H_0\)
  8.  
  9. \(T=-7.560,\; df=12,\; -t_{0.10}=-1.356,\; \text{reject }H_0\)
  10.  
  11. \(T=-7.076,\; df=5,\; -t_{0.0005}=-6.869,\; \text{reject }H_0\)
  12.  
  13.  
    1. \(T=-1.483,\; df=14,\; -t_{0.05}=-1.761,\; \text{do not reject }H_0\)
    2. \(T=-1.483,\; df=14,\; -t_{0.10}=-1.345,\; \text{reject }H_0\)
  14.  
  15.  
    1. \(T=2.069,\; df=6,\; t_{0.10}=1.44,\; \text{reject }H_0\)
    2. \(T=2.069,\; df=6,\; t_{0.05}=1.943,\; \text{reject }H_0\)
  16.  
  17. \(T=4.472,\; df=4,\; t_{0.10}=1.533,\; \text{reject }H_0\)
  18.  
  19. \(T=0.798,\; df=24,\; t_{0.10}=1.318,\; \text{do not reject }H_0\)
  20.  
  21.  
    1. \(T=-1.773,\; df=4,\; -t_{0.05}=-2.132,\; \text{do not reject }H_0\)
    2. \(0.05<p-value<0.10\)
    3. \(\alpha =0.05,\; \text{do not reject }H_0\)

8.5: Large Sample Tests for a Population Proportion

Basic

On all exercises for this section you may assume that the sample is sufficiently large for the relevant test to be validly performed.

  1. Compute the value of the test statistic for each test using the information given.
    1. Testing \(H_0:p=0.50\; vs\; H_a:p>0.50,\; n=360,\; \hat{p}=0.56\).
    2. Testing \(H_0:p=0.50\; vs\; H_a:p\neq 0.50,\; n=360,\; \hat{p}=0.56\).
    3. Testing \(H_0:p=0.37\; vs\; H_a:p<0.37,\; n=1200,\; \hat{p}=0.35\).
  2. Compute the value of the test statistic for each test using the information given.
    1. Testing \(H_0:p=0.72\; vs\; H_a:p<0.72,\; n=2100,\; \hat{p}=0.71\).
    2. Testing \(H_0:p=0.83\; vs\; H_a:p\neq 0.83,\; n=500,\; \hat{p}=0.86\).
    3. Testing \(H_0:p=0.22\; vs\; H_a:p<0.22,\; n=750,\; \hat{p}=0.18\).
  3. For each part of Exercise 1 construct the rejection region for the test for and make the decision based on your answer to that part of the exercise.
  4. For each part of Exercise 2 construct the rejection region for the test for and make the decision based on your answer to that part of the exercise.
  5. For each part of Exercise 1 compute the observed significance (\(p\)-value) of the test and compare it to in order to make the decision by the \(p\)-value approach to hypothesis testing.
  6. For each part of Exercise 2 compute the observed significance (\(p\)-value) of the test and compare it to in order to make the decision by the \(p\)-value approach to hypothesis testing.
  7. Perform the indicated test of hypotheses using the critical value approach.
    1. Testing \(H_0:p=0.55\; vs\; H_a:p>0.55\; @\; \alpha =0.05,\; n=300,\; \hat{p}=0.60\).
    2. Testing \(H_0:p=0.47\; vs\; H_a:p\neq 0.47\; @\; \alpha =0.01,\; n=9750,\; \hat{p}=0.46\).
  8. Perform the indicated test of hypotheses using the critical value approach.
    1. Testing \(H_0:p=0.15\; vs\; H_a:p\neq 0.15\; @\; \alpha =0.001,\; n=1600,\; \hat{p}=0.18\).
    2. Testing \(H_0:p=0.90\; vs\; H_a:p>0.90\; @\; \alpha =0.01,\; n=1100,\; \hat{p}=0.91\).
  9. Perform the indicated test of hypotheses using the \(p\)-value approach.
    1. Testing \(H_0:p=0.37\; vs\; H_a:p\neq 0.37\; @\; \alpha =0.005,\; n=1300,\; \hat{p}=0.40\).
    2. Testing \(H_0:p=0.94\; vs\; H_a:p>0.94\; @\; \alpha =0.05,\; n=1200,\; \hat{p}=0.96\).
  10. Perform the indicated test of hypotheses using the \(p\)-value approach.
    1. Testing \(H_0:p=0.25\; vs\; H_a:p<0.25\; @\; \alpha =0.10,\; n=850,\; \hat{p}=0.23\).
    2. Testing \(H_0:p=0.33\; vs\; H_a:p\neq 0.33\; @\; \alpha =0.05,\; n=1100,\; \hat{p}=0.30\).

Applications

  1. Five years ago \(3.9\%\) of children in a certain region lived with someone other than a parent. A sociologist wishes to test whether the current proportion is different. Perform the relevant test at the \(5\%\) level of significance using the following data: in a random sample of \(2,759\) children, \(119\) lived with someone other than a parent.
  2. The government of a particular country reports its literacy rate as \(52\%\). A nongovernmental organization believes it to be less. The organization takes a random sample of \(600\) inhabitants and obtains a literacy rate of \(42\%\). Perform the relevant test at the \(0.5\%\) (one-half of \(1\%\)) level of significance.
  3. Two years ago \(72\%\) of household in a certain county regularly participated in recycling household waste. The county government wishes to investigate whether that proportion has increased after an intensive campaign promoting recycling. In a survey of \(900\) households, \(674\) regularly participate in recycling. Perform the relevant test at the \(10\%\) level of significance.
  4. Prior to a special advertising campaign, \(23\%\) of all adults recognized a particular company’s logo. At the close of the campaign the marketing department commissioned a survey in which \(311\) of \(1,200\) randomly selected adults recognized the logo. Determine, at the \(1\%\) level of significance, whether the data provide sufficient evidence to conclude that more than \(23\%\) of all adults now recognize the company’s logo.
  5. A report five years ago stated that \(35.5\%\) of all state-owned bridges in a particular state were “deficient.” An advocacy group took a random sample of \(100\) state-owned bridges in the state and found \(33\) to be currently rated as being “deficient.” Test whether the current proportion of bridges in such condition is \(35.5\%\) versus the alternative that it is different from \(35.5\%\), at the \(10\%\) level of significance.
  6. In the previous year the proportion of deposits in checking accounts at a certain bank that were made electronically was \(45\%\). The bank wishes to determine if the proportion is higher this year. It examined \(20,000\) deposit records and found that \(9,217\) were electronic. Determine, at the \(1\%\) level of significance, whether the data provide sufficient evidence to conclude that more than \(45\%\) of all deposits to checking accounts are now being made electronically.
  7. According to the Federal Poverty Measure \(12\%\) of the U.S. population lives in poverty. The governor of a certain state believes that the proportion there is lower. In a sample of size \(1,550,163\) were impoverished according to the federal measure.
    1. Test whether the true proportion of the state’s population that is impoverished is less than \(12\%\), at the \(5\%\) level of significance.
    2. Compute the observed significance of the test.
  8. An insurance company states that it settles \(85\%\) of all life insurance claims within \(30\) days. A consumer group asks the state insurance commission to investigate. In a sample of \(250\) life insurance claims, \(203\) were settled within \(30\) days.
    1. Test whether the true proportion of all life insurance claims made to this company that are settled within \(30\) days is less than \(85\%\), at the \(5\%\) level of significance.
    2. Compute the observed significance of the test.
  9. A special interest group asserts that \(90\%\) of all smokers began smoking before age \(18\). In a sample of \(850\) smokers, \(687\) began smoking before age \(18\).
    1. Test whether the true proportion of all smokers who began smoking before age \(18\) is less than \(90\%\), at the \(1\%\) level of significance.
    2. Compute the observed significance of the test.
  10. In the past, \(68\%\) of a garage’s business was with former patrons. The owner of the garage samples \(200\) repair invoices and finds that for only \(114\) of them the patron was a repeat customer.
    1. Test whether the true proportion of all current business that is with repeat customers is less than \(68\%\), at the \(1\%\) level of significance.
    2. Compute the observed significance of the test.

Additional Exercises

  1. A rule of thumb is that for working individuals one-quarter of household income should be spent on housing. A financial advisor believes that the average proportion of income spent on housing is more than \(0.25\). In a sample of \(30\) households, the mean proportion of household income spent on housing was \(0.285\) with a standard deviation of \(0.063\). Perform the relevant test of hypotheses at the \(1\%\) level of significance. Hint: This exercise could have been presented in an earlier section.
  2. Ice cream is legally required to contain at least \(10\%\) milk fat by weight. The manufacturer of an economy ice cream wishes to be close to the legal limit, hence produces its ice cream with a target proportion of \(0.106\) milk fat. A sample of five containers yielded a mean proportion of \(0.094\) milk fat with standard deviation \(0.002\). Test the null hypothesis that the mean proportion of milk fat in all containers is \(0.106\) against the alternative that it is less than \(0.106\), at the \(10\%\) level of significance. Assume that the proportion of milk fat in containers is normally distributed. Hint: This exercise could have been presented in an earlier section.

Large Data Set Exercises

Large Data Sets missing

  1. Large \(\text{Data Sets 4 and 4A}\) list the results of \(500\) tosses of a die. Let \(p\) denote the proportion of all tosses of this die that would result in a five. Use the sample data to test the hypothesis that \(p\) is different from \(1/6\), at the \(20\%\) level of significance.
  2. Large \(\text{Data Set 6}\) records results of a random survey of \(200\) voters in each of two regions, in which they were asked to express whether they prefer Candidate \(A\) for a U.S. Senate seat or prefer some other candidate. Use the full data set (\(400\) observations) to test the hypothesis that the proportion \(p\) of all voters who prefer Candidate \(A\) exceeds \(0.35\). Test at the \(10\%\) level of significance.
  3. Lines \(2\) through \(536\) in Large \(\text{Data Set 11}\) is a sample of \(535\) real estate sales in a certain region in 2008. Those that were foreclosure sales are identified with a \(1\) in the second column. Use these data to test, at the \(10\%\) level of significance, the hypothesis that the proportion \(p\) of all real estate sales in this region in 2008 that were foreclosure sales was less than \(25\%\). (The null hypothesis is \(H_0:p=0.25\)).
  4. Lines \(537\) through \(1106\) in Large \(\text{Data Set 11}\) is a sample of \(570\) real estate sales in a certain region in 2010. Those that were foreclosure sales are identified with a \(1\) in the second column. Use these data to test, at the \(5\%\) level of significance, the hypothesis that the proportion \(p\) of all real estate sales in this region in 2010 that were foreclosure sales was greater than \(23\%\). (The null hypothesis is \(H_0:p=0.25\)). 

Answers

  1.  
    1. \(Z = 2.277\)
    2. \(Z = 2.277\)
    3. \(Z = -1.435\)
  2.  
  3.  
    1. \(Z \geq 1.645\); reject \(H_0\)
    2. \(Z\leq -1.96\; or\; Z \geq 1.96\); reject \(H_0\)
    3. \(Z \leq -1.645\); do not reject \(H_0\)
  4.  
  5.  
    1. \(p-value=0.0116,\; \alpha =0.05\); reject \(H_0\)
    2. \(p-value=0.0232,\; \alpha =0.05\); reject \(H_0\)
    3. \(p-value=0.0749,\; \alpha =0.05\); do not reject \(H_0\)
  6.  
  7.  
    1. \(Z=1.74,\; z_{0.05}=1.645\); reject \(H_0\)
    2. \(Z=-1.98,\; -z_{0.005}=-2.576\); do not reject \(H_0\)
  8.  
  9.  
    1. \(Z=2.24,\; p-value=0.025,\alpha =0.005\); do not reject \(H_0\)
    2. \(Z=2.92,\; p-value=0.0018,\alpha =0.05\); reject \(H_0\)
  10.  
  11. \(Z=1.11,\; z_{0.025}=1.96\); do not reject \(H_0\)
  12.  
  13. \(Z=1.93,\; z_{0.10}=1.28\); reject \(H_0\)
  14.  
  15. \(Z=-0.523,\; \pm z_{0.05}=\pm 1.645\); do not reject \(H_0\)
  16.  
  17.  
    1. \(Z=-1.798,\; -z_{0.05}=-1.645\); reject \(H_0\)
    2. \(p-value=0.0359\)
  18.  
  19.  
    1. \(Z=-8.92,\; -z_{0.01}=-2.33\); reject \(H_0\)
    2. \(p-value\approx 0\)
  20.  
  21. \(Z=3.04,\; z_{0.01}=2.33\); reject \(H_0\)
  22.  
  23. \(H_0:p=1/6\; vs\; H_a:p\neq 1/6\). Test Statistic: \(Z = -0.76\). Rejection Region: \((-\infty ,-1.28]\cup [1.28,\infty )\). Decision: Fail to reject \(H_0\).
  24.  
  25. \(H_0:p=0.25\; vs\; H_a:p<0.25\). Test Statistic: \(Z = -1.17\). Rejection Region: \((-\infty ,-1.28]\). Decision: Fail to reject \(H_0\).

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