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11.7 Review Questions

  1. An American roulette wheel contains 18 red numbers, 18 black numbers, and 2 green numbers. The following table shows the frequency with which the ball landed on each color in 300 trials.
    Color Red Black Green
    Frequency 140 120 40

    At the 5% significance level, do the data suggest that the wheel is out of balance?

  2. A gambler thinks a die may be loaded and that the six numbers are not equally likely. To test his suspicion, he rolled the die 150 times and obtained the data in the following table.
    Number 1 2 3 4 5 6
    Frequency 23 26 23 21 31 26

    Do the data provide sufficient evidence to conclude that the die is loaded? Perform the hypothesis test at the 5% significance level.

  3. The following table reported the survey results on how members would prefer to receive ballots in annual elections. At the 5% significance level, do the data provide sufficient evidence to conclude that gender (column) and preference (row) are associated?
    Male Female Total
    Mail 60 30 90
    Email 150 90 240
    Both 70 40 110
    N/A 80 50 130
    Total 360 210 570
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  1. If the wheel is balanced, the chance of landing on a red number is p1=1818+18+2, on a black number is p2=1818+18+2, and on a green number is p3=218+18+2. If the wheel is out of balance, at least one proportion differs from the specified value. We can use the chi-square goodness of fit test to test this. The assumptions of a goodness-of-fit test are:
    1. All expected frequencies are at least 1.
    2. At most 20% of the expected frequencies are less than 5.
    3. Simple random sample (if you need to generalize the conclusion to a larger population)

    The expected frequencies are E1=np1=300×1838=142.1053, E2=np2=300×1838=142.1053, E3=np3=300×238=15.78947. All the expected frequencies are greater than 5. We assume the trials were conducted randomly.

    Steps:

    1. Hypotheses. H0:p1=1838,p2=1838,p3=238 versus Ha: at least one proportion is different from the specified value.
    2. Significance level α=0.05
    3. Test statistic.
      Color Observed Expected Contribution
      red 140 142.10526 0.0311891
      black 120 142.10526 3.4385965
      green 40 15.78947 37.1228070

      χo2=i=13(OiEi)2Ei=(140142.10526)2142.10526+(120142.10526)2142.10526+(4015.78947)215.78947=0.03118908+3.43859649+37.12280702=40.59259.
      with df=k1=31=2.

    4. P-value. The chi-square test is always right-tailed. P-value is the area to the right of the observed value χo2 under the chi-square density curve with degrees of freedom df=k1=31=2. P-value=P(χ2χo2)=P(χ240.593)<P(χ210.597)=0.005.
      That is, P-value<0.005.
      If the critical value approach is used, we need to find the rejection region. The chi-square test is always right-tailed. Make sure that the area of the rejection region is α, so the critical value is χα2=χ0.052=5.991.
    5. Decision. We reject H0 since P-value<0.005<0.05 (α). If the critical approach is used, we reject H0 since the observed test statistic χo2=40.593>5.991 falls in the rejection region.
    6. Conclusion. At the 5% significance level, we have sufficient evidence that the wheel is out of balance.
  2. If the die is balanced, the proportion of landing on each of 1, 2, , 6 should be the same, i.e., pi=16,i=1,2,,6. All expected frequencies are Ei=npi=150×16=25>5, the assumptions for a chi-square goodness-of-fit test are satisfied.
    Steps:

    1. Hypotheses. H0:pi=16,i=1,2,6 versus Ha: at least one proportion is equal to 16.
    2. Significance level α=0.01
    3. Test statistic. The following table is useful.
      χo2=all cells(OE)2E=(2325)225+(2625)225+(2325)225+(2125)225+(3125)225+(2625)225=2.48
      with the degrees of freedom of the test df=k1=61=5.
    4. P-value. The chi-square test is always right-tailed. P-value is the area to the right of the observed value χo2 under the chi-square density curve with degrees of freedom df=k1=61=5. P-value=P(χ2χo2)=P(χ22.48)>P(χ29.236)=0.1, i.e., P-value>0.1.
      Or
      Rejection region. The chi-square test is always right-tailed. Make sure that the area of the rejection region is α, so the critical value is χα2=χ0.012=15.086.
    5. Decision: We cannot reject H0 since P-value>0.1>0.01 (α).
      Or
      We cannot reject H0 since χo2=2.48<15.086 falls in the non-rejection region.
    6. Conclusion: At the 5% significance level, we do not have sufficient evidence that the die is loaded.
  3. We should use the chi-square independence test. The expected frequencies are given in the following table. All the expected frequencies are greater than 5, the assumptions for a chi-square independence test are satisfied.
    Preference Male Female
    Mail 56.84211 33.15789
    Email 151.57895 88.42105
    Both 69.47368 40.52632
    N/A 82.10526 47.89474

    Steps:

    1. Hypotheses. H0: gender and preference are independent. versus Ha: gender and preference are associated.
    2. Significance level α=0.05.
    3. Test statistic.
      χo2=all cells(OE)2E=(6056.842)256.842+(3033.158)233.158+(150151.579)2151.579+(9088.421)288.421+(7069.474)269.474+(4040.526)240.526+(8082.105)282.105+(5047.895)247.895=0.678
      with the degrees of freedom of the test df=(r1)×(c1)=(41)×(21)=3.
    4. P-value. The chi-square test is always right-tailed. P-value is the area to the right of the observed value χo2 under the chi-square density curve with degrees of freedom df=3. P-value=P(χ2χo2)=P(χ20.678)>P(χ26.251)=0.1, i.e., P-value >0.10.
      Or
      Rejection region. The chi-square test is always right-tailed. Make sure that the area of the rejection region is α, so the critical value is χα2=χ0.052=7.815.
    5. Decision: Don’t reject H0, since P-value>0.1>0.05(α).
      Or
      We cannot reject H0 since χo2=0.678<7.815 falls in the non-rejection region.
    6. Conclusion: At the 5% significance level, we do not have sufficient evidence that gender and preference are associated.