Skip to main content
Statistics LibreTexts

9.E: Two-Sample Problems (Exercises)

  • Page ID
    1113
  • These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.

    9.1: Comparison of Two Population Means: Large, Independent Samples

    Basic

    Q9.1.1

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(90\%\) confidence, \[n_1=45, \bar{x_1}=27, s_1=2\\ n_2=60, \bar{x_2}=22, s_2=3\]
    2. \(99\%\) confidence, \[n_1=30, \bar{x_1}=-112, s_1=9\\ n_2=40, \bar{x_2}=-98, s_2=4\]

    Q9.1.2

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(95\%\) confidence, \[n_1=110, \bar{x_1}=77, s_1=15\\ n_2=85, \bar{x_2}=79, s_2=21\]
    2. \(90\%\) confidence, \[n_1=65, \bar{x_1}=-83, s_1=12\\ n_2=65, \bar{x_2}=-74, s_2=8\]

    Q9.1.3

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(99.5\%\) confidence, \[n_1=130, \bar{x_1}=27.2, s_1=2.5\\ n_2=155, \bar{x_2}=38.8, s_2=4.6\]
    2. \(95\%\) confidence, \[n_1=68, \bar{x_1}=215.5, s_1=12.3\\ n_2=84, \bar{x_2}=287.8, s_2=14.1\]

    Q9.1.4

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(99.9\%\) confidence, \[n_1=275, \bar{x_1}=70.2, s_1=1.5\\ n_2=325, \bar{x_2}=63.4, s_2=1.1\]
    2. \(90\%\) confidence, \[n_1=120, \bar{x_1}=35.5, s_1=0.75\\ n_2=146, \bar{x_2}=29.6, s_2=0.80\]

    Q9.1.5

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the \(p\)-value of the test as well.

    1. Test \(H_0:\mu _1-\mu _2=3\; vs\; H_a:\mu _1-\mu _2\neq 3\; @\; \alpha =0.05\) \[n_1=35, \bar{x_1}=25, s_1=1\\ n_2=45, \bar{x_2}=19, s_2=2\]
    2. Test \(H_0:\mu _1-\mu _2=-25\; vs\; H_a:\mu _1-\mu _2<-25\; @\; \alpha =0.10\) \[n_1=85, \bar{x_1}=188, s_1=15\\ n_2=62, \bar{x_2}=215, s_2=19\]

    Q9.1.6

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the \(p\)-value of the test as well.

    1. Test \(H_0:\mu _1-\mu _2=45\; vs\; H_a:\mu _1-\mu _2>45\; @\; \alpha =0.001\) \[n_1=200, \bar{x_1}=1312, s_1=35\\ n_2=225, \bar{x_2}=1256, s_2=28\]
    2. Test \(H_0:\mu _1-\mu _2=-12\; vs\; H_a:\mu _1-\mu _2\neq -12\; @\; \alpha =0.10\) \[n_1=35, \bar{x_1}=121, s_1=6\\ n_2=40, \bar{x_2}=135 s_2=7\]

    Q9.1.7

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the \(p\)-value of the test as well.

    1. Test \(H_0:\mu _1-\mu _2=0\; vs\; H_a:\mu _1-\mu _2\neq 0\; @\; \alpha =0.01\) \[n_1=125, \bar{x_1}=-46, s_1=10\\ n_2=90, \bar{x_2}=-50, s_2=13\]
    2. Test \(H_0:\mu _1-\mu _2=20\; vs\; H_a:\mu _1-\mu _2>20\; @\; \alpha =0.05\) \[n_1=40, \bar{x_1}=142, s_1=11\\ n_2=40, \bar{x_2}=118 s_2=10\]

    Q9.1.8

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach. Compute the \(p\)-value of the test as well.

    1. Test \(H_0:\mu _1-\mu _2=13\; vs\; H_a:\mu _1-\mu _2<13\; @\; \alpha =0.01\) \[n_1=35, \bar{x_1}=100, s_1=2\\ n_2=35, \bar{x_2}=88, s_2=2\]
    2. Test \(H_0:\mu _1-\mu _2=-10\; vs\; H_a:\mu _1-\mu _2\neq -10\; @\; \alpha =0.10\) \[n_1=146, \bar{x_1}=62, s_1=4\\ n_2=120, \bar{x_2}=73 s_2=7\]

    Q9.1.9

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach.

    1. Test \(H_0:\mu _1-\mu _2=57\; vs\; H_a:\mu _1-\mu _2<57\; @\; \alpha =0.10\) \[n_1=117, \bar{x_1}=1309, s_1=42\\ n_2=133, \bar{x_2}=1258, s_2=37\]
    2. Test \(H_0:\mu _1-\mu _2=-1.5\; vs\; H_a:\mu _1-\mu _2\neq -1.5\; @\; \alpha =0.20\) \[n_1=65, \bar{x_1}=16.9, s_1=1.3\\ n_2=57, \bar{x_2}=18.6 s_2=1.1\]

    Q9.1.10

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach.

    1. Test \(H_0:\mu _1-\mu _2=-10.5\; vs\; H_a:\mu _1-\mu _2>-10.5\; @\; \alpha =0.01\) \[n_1=64, \bar{x_1}=85.6, s_1=2.4\\ n_2=50, \bar{x_2}=95.3, s_2=3.1\]
    2. Test \(H_0:\mu _1-\mu _2=110\; vs\; H_a:\mu _1-\mu _2\neq 110\; @\; \alpha =0.02\) \[n_1=176, \bar{x_1}=1918, s_1=68\\ n_2=241, \bar{x_2}=1782 s_2=146\]

    Q9.1.11

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach.

    1. Test \(H_0:\mu _1-\mu _2=50\; vs\; H_a:\mu _1-\mu _2>50\; @\; \alpha =0.005\) \[n_1=72, \bar{x_1}=272, s_1=26\\ n_2=103, \bar{x_2}=213, s_2=14\]
    2. Test \(H_0:\mu _1-\mu _2=7.5\; vs\; H_a:\mu _1-\mu _2\neq 7.5\; @\; \alpha =0.10\) \[n_1=52, \bar{x_1}=94.3, s_1=2.6\\ n_2=38, \bar{x_2}=88.6 s_2=8.0\]

    Q9.1.12

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach.

    1. Test \(H_0:\mu _1-\mu _2=23\; vs\; H_a:\mu _1-\mu _2<23\; @\; \alpha =0.20\) \[n_1=314, \bar{x_1}=198, s_1=12.2\\ n_2=220, \bar{x_2}=176, s_2=11.5\]
    2. Test \(H_0:\mu _1-\mu _2=4.4\; vs\; H_a:\mu _1-\mu _2\neq 4.4\; @\; \alpha =0.05\) \[n_1=32, \bar{x_1}=40.3, s_1=0.5\\ n_2=30, \bar{x_2}=35.5 s_2=0.7\]

    Applications

    Q9.1.13

    In order to investigate the relationship between mean job tenure in years among workers who have a bachelor’s degree or higher and those who do not, random samples of each type of worker were taken, with the following results.

    n \(\bar{x}\) s
    Bachelor’s degree or higher 155 5.2 1.3
    No degree 210 5.0 1.5
    1. Construct the \(99\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(1\%\) level of significance, the claim that mean job tenure among those with higher education is greater than among those without, against the default that there is no difference in the means.
    3. Compute the observed significance of the test.

    Q9.1.14

    Records of \(40\) used passenger cars and \(40\) used pickup trucks (none used commercially) were randomly selected to investigate whether there was any difference in the mean time in years that they were kept by the original owner before being sold. For cars the mean was \(5.3\) years with standard deviation \(2.2\) years. For pickup trucks the mean was \(7.1\) years with standard deviation \(3.0\) years.

    1. Construct the \(95\%\) confidence interval for the difference in the means based on these data.
    2. Test the hypothesis that there is a difference in the means against the null hypothesis that there is no difference. Use the \(1\%\) level of significance.
    3. Compute the observed significance of the test in part (b).

    Q9.1.15

    In previous years the average number of patients per hour at a hospital emergency room on weekends exceeded the average on weekdays by \(6.3\) visits per hour. A hospital administrator believes that the current weekend mean exceeds the weekday mean by fewer than \(6.3\) hours.

    1. Construct the \(99\%\) confidence interval for the difference in the population means based on the following data, derived from a study in which \(30\) weekend and \(30\) weekday one-hour periods were randomly selected and the number of new patients in each recorded.
    n \(\bar{x}\) s
    Weekends 30 13.8 3.1
    Weekdays 30 8.6 2.7
    1. Test at the \(5\%\) level of significance whether the current weekend mean exceeds the weekday mean by fewer than \(6.3\) patients per hour.
    2. Compute the observed significance of the test.

    Q9.1.16

    A sociologist surveys \(50\) randomly selected citizens in each of two countries to compare the mean number of hours of volunteer work done by adults in each. Among the \(50\) inhabitants of Lilliput, the mean hours of volunteer work per year was \(52\), with standard deviation \(11.8\). Among the \(50\) inhabitants of Blefuscu, the mean number of hours of volunteer work per year was \(37\), with standard deviation \(7.2\).

    1. Construct the \(99\%\) confidence interval for the difference in mean number of hours volunteered by all residents of Lilliput and the mean number of hours volunteered by all residents of Blefuscu.
    2. Test, at the \(1\%\) level of significance, the claim that the mean number of hours volunteered by all residents of Lilliput is more than ten hours greater than the mean number of hours volunteered by all residents of Blefuscu.
    3. Compute the observed significance of the test in part (b).

    Q9.1.17

    A university administrator asserted that upperclassmen spend more time studying than underclassmen.

    1. Test this claim against the default that the average number of hours of study per week by the two groups is the same, using the following information based on random samples from each group of students. Test at the \(1\%\) level of significance.
    n \(\bar{x}\) s
    Upperclassmen 35 15.6 2.9
    Underclassmen 35 12.3 4.1
    1. Compute the observed significance of the test.

    Q9.1.18

    An kinesiologist claims that the resting heart rate of men aged \(18\) to \(25\) who exercise regularly is more than five beats per minute less than that of men who do not exercise regularly. Men in each category were selected at random and their resting heart rates were measured, with the results shown.

    n \(\bar{x}\) s
    Regular exercise 40 63 1.0
    No regular exercise 30 71 1.2
    1. Perform the relevant test of hypotheses at the \(1\%\) level of significance.
    2. Compute the observed significance of the test.

    Q9.1.19

    Children in two elementary school classrooms were given two versions of the same test, but with the order of questions arranged from easier to more difficult in Version \(A\) and in reverse order in Version \(B\). Randomly selected students from each class were given Version \(A\) and the rest Version \(B\). The results are shown in the table.

    n \(\bar{x}\) s
    Version A 31 83 4.6
    Version B 32 78 4.3
    1. Construct the \(90\%\) confidence interval for the difference in the means of the populations of all children taking Version \(A\) of such a test and of all children taking Version \(B\) of such a test.
    2. Test at the \(1\%\) level of significance the hypothesis that the \(A\) version of the test is easier than the \(B\) version (even though the questions are the same).
    3. Compute the observed significance of the test.

    Q9.1.20

    The Municipal Transit Authority wants to know if, on weekdays, more passengers ride the northbound blue line train towards the city center that departs at \(8:15\; a.m.\) or the one that departs at \(8:30\; a.m\). The following sample statistics are assembled by the Transit Authority.

    n \(\bar{x}\) s
    8:15 a.m. train 30 323 41
    8:30 a.m. train 45 356 45
    1. Construct the \(90\%\) confidence interval for the difference in the mean number of daily travelers on the \(8:15\; a.m.\) train and the mean number of daily travelers on the \(8:30\; a.m.\) train.
    2. Test at the \(5\%\) level of significance whether the data provide sufficient evidence to conclude that more passengers ride the \(8:30\; a.m.\) train.
    3. Compute the observed significance of the test.

    Q9.1.21

    In comparing the academic performance of college students who are affiliated with fraternities and those male students who are unaffiliated, a random sample of students was drawn from each of the two populations on a university campus. Summary statistics on the student GPAs are given below.

    n \(\bar{x}\) s
    Fraternity 645 2.90 0.47
    Unaffiliated 450 2.88 0.42

    Test, at the \(5\%\) level of significance, whether the data provide sufficient evidence to conclude that there is a difference in average GPA between the population of fraternity students and the population of unaffiliated male students on this university campus.

    Q9.1.22

    In comparing the academic performance of college students who are affiliated with sororities and those female students who are unaffiliated, a random sample of students was drawn from each of the two populations on a university campus. Summary statistics on the student GPAs are given below.

    n \(\bar{x}\) s
    Sorority 330 3.18 0.37
    Unaffiliated 550 3.12 0.41

    Test, at the \(5\%\) level of significance, whether the data provide sufficient evidence to conclude that there is a difference in average GPA between the population of sorority students and the population of unaffiliated female students on this university campus.

    Q9.1.23

    The owner of a professional football team believes that the league has become more offense oriented since five years ago. To check his belief, \(32\) randomly selected games from one year’s schedule were compared to \(32\) randomly selected games from the schedule five years later. Since more offense produces more points per game, the owner analyzed the following information on points per game (ppg).

    n \(\bar{x}\) s
    ppg previously 32 20.62 4.17
    ppg recently 32 22.05 4.01

    Test, at the \(10\%\) level of significance, whether the data on points per game provide sufficient evidence to conclude that the game has become more offense oriented.

    Q9.1.24

    The owner of a professional football team believes that the league has become more offense oriented since five years ago. To check his belief, \(32\) randomly selected games from one year’s schedule were compared to \(32\) randomly selected games from the schedule five years later. Since more offense produces more offensive yards per game, the owner analyzed the following information on offensive yards per game (oypg).

    n \(\bar{x}\) s
    oypg previously 32 316 40
    oypg recently 32 336 35

    Test, at the \(10\%\) level of significance, whether the data on offensive yards per game provide sufficient evidence to conclude that the game has become more offense oriented.

    Large Data Set Exercises

    Large Data Sets are absent

    1. Large \(\text{Data Sets 1A and 1B}\) list the SAT scores for \(1,000\) randomly selected students. Denote the population of all male students as \(\text{Population 1}\) and the population of all female students as \(\text{Population 2}\).
      1. Restricting attention to just the males, find \(n_1\), \(\bar{x_1}\) and \(s_1\). Restricting attention to just the females, find \(n_2\), \(\bar{x_2}\) and \(s_2\).
      2. Let \(\mu _1\) denote the mean SAT score for all males and \(\mu _2\) the mean SAT score for all females. Use the results of part (a) to construct a \(90\%\) confidence interval for the difference \(\mu _1-\mu _2\).
      3. Test, at the \(5\%\) level of significance, the hypothesis that the mean SAT scores among males exceeds that of females.
    2. Large \(\text{Data Sets 1A and 1B}\) list the SAT scores for \(1,000\) randomly selected students. Denote the population of all male students as \(\text{Population 1}\) and the population of all female students as \(\text{Population 2}\).
      1. Restricting attention to just the males, find \(n_1\), \(\bar{x_1}\) and \(s_1\). Restricting attention to just the females, find \(n_2\), \(\bar{x_2}\) and \(s_2\).
      2. Let \(\mu _1\) denote the mean SAT score for all males and \(\mu _2\) the mean SAT score for all females. Use the results of part (a) to construct a \(95\%\) confidence interval for the difference \(\mu _1-\mu _2\).
      3. Test, at the \(10\%\) level of significance, the hypothesis that the mean SAT scores among males exceeds that of females.
    3. Large \(\text{Data Sets 7A and 7B}\) list the survival times for \(65\) male and \(75\) female laboratory mice with thymic leukemia. Denote the population of all such male mice as \(\text{Population 1}\) and the population of all such female mice as \(\text{Population 2}\).
      1. Restricting attention to just the males, find \(n_1\), \(\bar{x_1}\) and \(s_1\). Restricting attention to just the females, find \(n_2\), \(\bar{x_2}\) and \(s_2\).
      2. Let \(\mu _1\) denote the mean survival for all males and \(\mu _2\) the mean survival time for all females. Use the results of part (a) to construct a \(99\%\) confidence interval for the difference \(\mu _1-\mu _2\).
      3. Test, at the \(1\%\) level of significance, the hypothesis that the mean survival time for males exceeds that for females by more than \(182\) days (half a year).
      4. Compute the observed significance of the test in part (c).

    Answers

      1. \((4.20,5.80)\)
      2. \((-18.54,-9.46)\)
      1. \((-12.81,-10.39)\)
      2. \((-76.50,-68.10)\)
      1. \(Z = 8.753, \pm z_{0.025}=\pm 1.960\), reject \(H_0\), \(p\)-value=\(0.0000\)
      2. \(Z = -0.687, -z_{0.10}=-1.282\), do not reject \(H_0\), \(p\)-value=\(0.2451\)
      1. \(Z = 2.444, \pm z_{0.005}=\pm 2.576\), do not reject \(H_0\), \(p\)-value=\(0.0146\)
      2. \(Z = 1.702, z_{0.05}=-1.645\), reject \(H_0\), \(p\)-value=\(0.0446\)
      1. \(Z = -1.19\), \(p\)-value=\(0.1170\), do not reject \(H_0\)
      2. \(Z = -0.92\), \(p\)-value=\(0.3576\), do not reject \(H_0\)
      1. \(Z = 2.68\), \(p\)-value=\(0.0037\), reject \(H_0\)
      2. \(Z = -1.34\), \(p\)-value=\(0.1802\), do not reject \(H_0\)
      1. \(0.2\pm 0.4\)
      2. \(Z = -1.466, -z_{0.050}=-1.645\), do not reject \(H_0\) (exceeds by \(6.3\) or more)
      3. \(p\)-value=\(0.0869\)
      1. \(5.2\pm 1.9\)
      2. \(Z = -1.466, -z_{0.050}=-1.645\), do not reject \(H_0\) (exceeds by \(6.3\) or more)
      3. \(p\)-value=\(0.0708\)
      1. \(Z = 3.888, z_{0.01}=2.326\), reject \(H_0\) (upperclassmen study more)
      2. \(p\)-value=\(0.0001\)
      1. \(5\pm 1.8\)
      2. \(Z = 4.454, z_{0.01}=2.326\), reject \(H_0\) (Test A is easier)
      3. \(p\)-value=\(0.0000\)
    1. \(Z = 0.738, \pm z_{0.025}=\pm 1.960\), do not reject \(H_0\) (no difference)
    2. \(Z = -1.398, -z_{0.10}=-1.282\), reject \(H_0\) (more offense oriented)
      1. \(n_1=419,\; \bar{x_1}=1540.33,\; s_1=205.40, \; n_2=581,\; \bar{x_2}=1520.38,\; s_2=217.34\)
      2. \((-2.24,42.15)\)
      3. \(H_0:\mu _1-\mu _2=0\; vs\; H_a:\mu _1-\mu _2>0\). Test Statistic: \(Z = 1.48\). Rejection Region: \([1.645,\infty )\). Decision: Fail to reject \(H_0\).
      1. \(n_1=65,\; \bar{x_1}=665.97,\; s_1=41.60, \; n_2=75,\; \bar{x_2}=455.89,\; s_2=63.22\)
      2. \((187.06,233.09)\)
      3. \(H_0:\mu _1-\mu _2=182\; vs\; H_a:\mu _1-\mu _2>182\). Test Statistic: \(Z = 3.14\). Rejection Region: \([2.33,\infty )\). Decision: Reject \(H_0\).
      4. \(p\)-value=\(0.0008\)

    9.2: Comparison of Two Population Means: Small, Independent Samples

    Basic

    In all exercises for this section assume that the populations are normal and have equal standard deviations.

    Q9.2.1

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(95\%\) confidence, \[n_1=10,\; \bar{x_1}=120,\; s_1=2\\ n_2=15,\; \bar{x_2}=101,\; s_1=4\]
    2. \(99\%\) confidence, \[n_1=6,\; \bar{x_1}=25,\; s_1=1\\ n_2=12,\; \bar{x_2}=17,\; s_1=3\]

    Q9.2.2

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(90\%\) confidence, \[n_1=28,\; \bar{x_1}=212,\; s_1=6\\ n_2=23,\; \bar{x_2}=198,\; s_1=5\]
    2. \(99\%\) confidence, \[n_1=14,\; \bar{x_1}=68,\; s_1=8\\ n_2=20,\; \bar{x_2}=43,\; s_1=3\]

    Q9.2.3

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(99.9\%\) confidence, \[n_1=35,\; \bar{x_1}=6.5,\; s_1=0.2\\ n_2=20,\; \bar{x_2}=6.2,\; s_1=0.1\]
    2. \(99\%\) confidence, \[n_1=18,\; \bar{x_1}=77.3,\; s_1=1.2\\ n_2=32,\; \bar{x_2}=75.0,\; s_1=1.6\]

    Q9.2.4

    Construct the confidence interval for \(\mu _1-\mu _2\) for the level of confidence and the data from independent samples given.

    1. \(99.5\%\) confidence, \[n_1=40,\; \bar{x_1}=85.6,\; s_1=2.8\\ n_2=20,\; \bar{x_2}=73.1,\; s_1=2.1\]
    2. \(99.9\%\) confidence, \[n_1=25,\; \bar{x_1}=215,\; s_1=7\\ n_2=35,\; \bar{x_2}=185,\; s_1=12\]

    Q9.2.5

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

    1. Test \(H_0:\mu _1-\mu _2=11\; vs\; H_a:\mu _1-\mu _2>11\; @\; \alpha =0.025\) \[n_1=6,\; \bar{x_1}=32,\; s_1=2\\ n_2=11,\; \bar{x_2}=19,\; s_1=1\]
    2. Test \(H_0:\mu _1-\mu _2=26\; vs\; H_a:\mu _1-\mu _2\neq 26\; @\; \alpha =0.05\) \[n_1=17,\; \bar{x_1}=166,\; s_1=4\\ n_2=24,\; \bar{x_2}=138,\; s_1=3\]

    Q9.2.6

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

    1. Test \(H_0:\mu _1-\mu _2=40\; vs\; H_a:\mu _1-\mu _2<40\; @\; \alpha =0.10\) \[n_1=14,\; \bar{x_1}=289,\; s_1=11\\ n_2=12,\; \bar{x_2}=254,\; s_1=9\]
    2. Test \(H_0:\mu _1-\mu _2=21\; vs\; H_a:\mu _1-\mu _2\neq 21\; @\; \alpha =0.05\) \[n_1=23,\; \bar{x_1}=130,\; s_1=6\\ n_2=27,\; \bar{x_2}=113,\; s_1=8\]

    Q9.2.7

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

    1. Test \(H_0:\mu _1-\mu _2=-15\; vs\; H_a:\mu _1-\mu _2<-15\; @\; \alpha =0.10\) \[n_1=30,\; \bar{x_1}=42,\; s_1=7\\ n_2=12,\; \bar{x_2}=60,\; s_1=5\]
    2. Test \(H_0:\mu _1-\mu _2=103\; vs\; H_a:\mu _1-\mu _2\neq 103\; @\; \alpha =0.10\) \[n_1=17,\; \bar{x_1}=711,\; s_1=28\\ n_2=32,\; \bar{x_2}=598,\; s_1=21\]

    Q9.2.8

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

    1. Test \(H_0:\mu _1-\mu _2=75\; vs\; H_a:\mu _1-\mu _2>75\; @\; \alpha =0.025\) \[n_1=45,\; \bar{x_1}=674,\; s_1=18\\ n_2=29,\; \bar{x_2}=591,\; s_1=13\]
    2. Test \(H_0:\mu _1-\mu _2=-20\; vs\; H_a:\mu _1-\mu _2\neq -20\; @\; \alpha =0.005\) \[n_1=30,\; \bar{x_1}=137,\; s_1=8\\ n_2=19,\; \bar{x_2}=166,\; s_1=11\]

    Q9.2.9

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach. (The \(p\)-value can be only approximated.)

    1. Test \(H_0:\mu _1-\mu _2=12\; vs\; H_a:\mu _1-\mu _2>12\; @\; \alpha =0.01\) \[n_1=20,\; \bar{x_1}=133,\; s_1=7\\ n_2=10,\; \bar{x_2}=115,\; s_1=5\]
    2. Test \(H_0:\mu _1-\mu _2=46\; vs\; H_a:\mu _1-\mu _2\neq 46\; @\; \alpha =0.10\) \[n_1=24,\; \bar{x_1}=586,\; s_1=11\\ n_2=27,\; \bar{x_2}=535,\; s_1=13\]

    Q9.2.10

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach. (The \(p\)-value can be only approximated.)

    1. Test \(H_0:\mu _1-\mu _2=38\; vs\; H_a:\mu _1-\mu _2<38\; @\; \alpha =0.01\) \[n_1=12,\; \bar{x_1}=464,\; s_1=5\\ n_2=10,\; \bar{x_2}=432,\; s_1=6\]
    2. Test \(H_0:\mu _1-\mu _2=4\; vs\; H_a:\mu _1-\mu _2\neq 4\; @\; \alpha =0.005\) \[n_1=14,\; \bar{x_1}=68,\; s_1=2\\ n_2=17,\; \bar{x_2}=67,\; s_1=3\]

    Q9.2.11

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach. (The \(p\)-value can be only approximated.)

    1. Test \(H_0:\mu _1-\mu _2=50\; vs\; H_a:\mu _1-\mu _2>50\; @\; \alpha =0.01\) \[n_1=30,\; \bar{x_1}=681,\; s_1=8\\ n_2=27,\; \bar{x_2}=625,\; s_1=8\]
    2. Test \(H_0:\mu _1-\mu _2=35\; vs\; H_a:\mu _1-\mu _2\neq 35\; @\; \alpha =0.10\) \[n_1=36,\; \bar{x_1}=325,\; s_1=11\\ n_2=29,\; \bar{x_2}=286,\; s_1=7\]

    Q9.2.12

    Perform the test of hypotheses indicated, using the data from independent samples given. Use the \(p\)-value approach. (The \(p\)-value can be only approximated.)

    1. Test \(H_0:\mu _1-\mu _2=-4\; vs\; H_a:\mu _1-\mu _2<-4\; @\; \alpha =0.05\) \[n_1=40,\; \bar{x_1}=80,\; s_1=5\\ n_2=25,\; \bar{x_2}=87,\; s_1=5\]
    2. Test \(H_0:\mu _1-\mu _2=21\; vs\; H_a:\mu _1-\mu _2\neq 21\; @\; \alpha =0.01\) \[n_1=15,\; \bar{x_1}=192,\; s_1=12\\ n_2=34,\; \bar{x_2}=180,\; s_1=8\]

    Applications

    Q9.2.13

    A county environmental agency suspects that the fish in a particular polluted lake have elevated mercury level. To confirm that suspicion, five striped bass in that lake were caught and their tissues were tested for mercury. For the purpose of comparison, four striped bass in an unpolluted lake were also caught and tested. The fish tissue mercury levels in mg/kg are given below.

    Sample 1 (from polluted lake) Sample 2(from unpolluted lake)
    0.580 0.382
    0.711 0.276
    0.571 0.570
    0.666 0.366
    0.598
    1. Construct the \(95\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(5\%\) level of significance, whether the data provide sufficient evidence to conclude that fish in the polluted lake have elevated levels of mercury in their tissue.

    Q9.2.14

    A genetic engineering company claims that it has developed a genetically modified tomato plant that yields on average more tomatoes than other varieties. A farmer wants to test the claim on a small scale before committing to a full-scale planting. Ten genetically modified tomato plants are grown from seeds along with ten other tomato plants. At the season’s end, the resulting yields in pound are recorded as below.

    Sample 1(genetically modified) Sample 2(regular)
    20 21
    23 21
    27 22
    25 18
    25 20
    25 20
    27 18
    23 25
    24 23
    22 20
    1. Construct the \(99\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(1\%\) level of significance, whether the data provide sufficient evidence to conclude that the mean yield of the genetically modified variety is greater than that for the standard variety.

    Q9.2.15

    The coaching staff of a professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, \(20\) randomly selected games from one year’s schedule were compared to \(11\) randomly selected games from the schedule five years later. The sample information on rushing yards per game (rypg) is summarized below.

    n \(\bar{x}\) s
    rypg previously 20 112 24
    rypg recently 11 114 21
    1. Construct the \(95\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(5\%\) level of significance, whether the data on rushing yards per game provide sufficient evidence to conclude that the rushing offense has become more potent in recent years.

    Q9.2.16

    The coaching staff of professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, \(20\) randomly selected games from one year’s schedule were compared to \(11\) randomly selected games from the schedule five years later. The sample information on passing yards per game (pypg) is summarized below.

    n \(\bar{x}\) s
    pypg previously 20 203 38
    pypg recently 11 232 33
    1. Construct the \(95\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(5\%\) level of significance, whether the data on passing yards per game provide sufficient evidence to conclude that the passing offense has become more potent in recent years.

    Q9.2.17

    A university administrator wishes to know if there is a difference in average starting salary for graduates with master’s degrees in engineering and those with master’s degrees in business. Fifteen recent graduates with master’s degree in engineering and \(11\) with master’s degrees in business are surveyed and the results are summarized below.

    n \(\bar{x}\) s
    Engineering 15 68,535 1627
    Business 11 63,230 2033
    1. Construct the \(90\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(10\%\) level of significance, whether the data provide sufficient evidence to conclude that the average starting salaries are different.

    Q9.2.18

    A gardener sets up a flower stand in a busy business district and sells bouquets of assorted fresh flowers on weekdays. To find a more profitable pricing, she sells bouquets for \(15\) dollars each for ten days, then for \(10\) dollars each for five days. Her average daily profit for the two different prices are given below.

    n \(\bar{x}\) s
    $15 10 171 26
    $10 5 198 29
    1. Construct the \(90\%\) confidence interval for the difference in the population means based on these data.
    2. Test, at the \(10\%\) level of significance, whether the data provide sufficient evidence to conclude the gardener’s average daily profit will be higher if the bouquets are sold at \(\$10\) each.

    Answers

      1. \((16.16,21.84)\)
      2. \((4.28,11.72)\)
      1. \((0.13,0.47)\)
      2. (1.14,3.46)\((1.14,3.46)\)
      1. \(T = 2.787,\; t_{0.025}=2.131\), reject \(H_0\)
      2. \(T = 1.831,\; \pm t_{0.025}=\pm 2.023\), do not reject \(H_0\)
      1. \(T = -1.349,\; -t_{0.10}=-1.303\), reject \(H_0\)
      2. \(T = 1.411,\; \pm t_{0.05}=\pm 1.678\), do not reject \(H_0\)
      1. \(T = 2.411,\; df=28,\; \text{p-value}>0.01\), do not reject \(H_0\)
      2. \(T = 1.473,\; df=49,\; \text{p-value}<0.10\), reject \(H_0\)
      1. \(T = 2.827,\; df=55,\; \text{p-value}<0.01\), reject \(H_0\)
      2. \(T = 1.699,\; df=63,\; \text{p-value}<0.10\), reject \(H_0\)
      1. \(0.2267\pm 0.2182\)
      2. \(T = 1.699,\; df=63,\; t_{0.05}=1.895\), reject \(H_0\) (elevated levels)
      1. \(-2\pm 17.7\)
      2. \(T = -0.232,\; df=29,\; -t_{0.05}=-1.699\), do not reject \(H_0\) (not more potent)
      1. \(5305\pm 1227\)
      2. \(T = 7.395,\; df=24,\; \pm t_{0.05}=\pm 1.711\), reject \(H_0\) (different)

    9.3 Comparison of Two Population Means: Paired Samples

    Basic

    In all exercises for this section assume that the population of differences is normal.

    1. Use the following paired sample data for this exercise. \[\begin{matrix} Population\: 1 & 35 & 32 & 35 & 35 & 36 & 35 & 35\\ Population\: 2 & 28 & 26 & 27 & 26 & 29 & 27 & 29 \end{matrix}\]
      1. Compute \(\bar{d}\) and \(s_d\).
      2. Give a point estimate for