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    30 Multiple Choice Problems in Statistics

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    Problem A
    The manager of a grocery store claims that the average time that customers spend in checkout lines is 20 minutes or less. A sample of 36 customers is taken. The average time spent on checkout lines for the sample is 24.6 minutes; and the sample standard deviation is 12 minutes. Conduct a hypothesis test (at 0.05 level of significance) to determine if the mean waiting time for the customer population is significantly more than 20 minutes.

    The observed value of the test statistic is:
    a. 2.3 b. 0.38 c. -2.3 d. -0.38

    The p-value is:
    a. 0.5107 b. 0.0214 c. 0.0137 d. 0.4893

    Problem E:
    A company wants to measure the relationship between its employee productivity (measured in output/employee) and the number of employees. Sample data for the last four months are shown below. Use simple linear regression to estimate this relationship.

    Independent Variable Dependent Variable
    Number of Employees Employee Productivity
    15 5
    12 7
    10 9
    7 11

    ANSWER QUESTIONS 16 THROUGH 19 BELOW.

    16. The least squares estimate of the slope b1 is:
    a. -0.7647 b. -0.13 c. 21.4 d. 16.41

    17. The least squares estimate of the intercept b0 is:
    a. -7.647 b. -0.13 c. 21.4 d. 16.41

    18. The estimated employee productivity when the number of employees is 5 is:
    a. 78 b. 12.59 c. 5.8 d. 32.6

    19. If the sample covariance is -8.67; estimate the coefficient of correlation between the number of employees and employee productivity:
    a. -0.997 b. 0.997 c. 1.23 d. 1.02

    Problem F:
    Consumer Research is an independent agency that is collecting data on annual income (INCOME) and household size (SIZE), to predict annual credit card charges. It runs a regression analysis on the data and an incomplete MS Excel output is shown below.
    ANSWER QUESTIONS 20 THROUGH 30 BELOW.

    Regression Statistics
    Multiple R 0.88038239
    R Square
    Adjusted R Square
    Standard Error 510.495493
    Observations

    ANOVA
    df SS MS F Significance F
    Regression 2 17960368.3 3.31446E-07
    Residual 20 260605.648
    Total 22

    Coefficients Standard Error t Stat P-value Lower 95%
    Intercept 352.694714 4.15578994 0.00048872 730.0172039
    INCOME 25.062956 8.47147285 2.95851223 0.00776734 7.391781505
    SIZE 408.400776 71.808401 1.447E-05 258.6111461

    20. The sample size is:
    a. 23 b. 22 c. 20 d. 21

    21. The coefficient of determination is:
    a. 0.88 b. 0.775 c. 0.92 d. -0.38

    22. The Sum of Squares for Error (i.e., Residual) is:
    a. 17960368.3 b. 5212112.97 c. 23172481.3 d. 260605.648

    23. The Sum of Squares for Total (SST) is:
    a. 17960368.3 b. 5212112.97 c. 23172481.3 d. 260605.648

    24. The Mean Square for Regression is
    a. 17960368.3 b. 5212112.97 c. 260605.648 d. 8980184.17

    25. The observed or computed F-value is:
    a. 34.459 b. 0.029 c. 3.445 d. 0.29

    26. The hypothesis to be tested is:
    H0: B1 = B2 = 0
    Ha: At least one of the B is not equal to 0.
    The hypothesis is to be tested at the 5% level of significance. The null hypothesis is:
    a. not rejected
    b. rejected
    c. the test is inconclusive
    d. none of the above answers are correct

    27. The hypothesis to be tested is:
    H0: B1 = 0
    Ha: B1 ≠ 0

    The hypothesis is to be tested at the 1% level of significance. The null hypothesis is:
    a. not rejected
    b. rejected
    c. the test is inconclusive
    d. none of the above answers are correct

    28. The estimate of the intercept b0 is:
    a. 10010.2 b. 2810.3 c. 1465.5 d. 2641.5

    29. The observed or computed t-stat (i.e., t-value) for the independent variable SIZE is:
    a. 2.96 b. 3.445 c. 4.16 d. 5.687

    30. What is the estimated annual credit charges if INCOME = 20, and SIZE = 3?
    a. 9700 b. 12600 c. 3189 d. 5300

    Problem G:
    Last year, the student body of a local university consisted of 30% freshmen, 24% sophomores, 26% juniors, and 20% seniors. A sample of 300 students taken from this year's student body showed the number of students in each classification.
    Freshmen 83
    Sophomores 68
    Juniors 85
    Seniors 64

    We want to know if there has been a significant change in the proportions of student classifications between the two years.
    ANSWER QUESTIONS 31 THROUGH 34 BELOW.

    31. The expected number of freshmen in this year is:
    a. 83 b. 90 c. 30 d. 10

    32. The number of degrees of freedom is:
    a. 4 b. 2 c. 3 d. 1

    33. The hypothesis is to be tested at the 5% level of significance. The critical chi-square value from the table equals:
    a. 1.645 b. 1.96 c. 2.75 d. 7.815

    34. If the chi-square value that is calculated equals 1.6615, then the null hypothesis is:
    a. not rejected
    b. rejected
    c. the test is inconclusive
    d. none of the above answers are correct

    Problem H: Use the following Excel Output to answer questions 35-39:

    Source Sum of Squares d.f.
    Between Groups 213.88125 3
    Within Groups 11.208333 20
    Total 225.0895 23

    35. Consider the above one-way ANOVA table. What is the treatment mean square?

    A) 71.297 B) 0.5604 C) 1.297 D) 213.881 E) 9.7

    36. Consider the above one-way ANOVA table. What is the mean square error?

    A) 71.297 B) 0.5604 C) 1.297 D) 213.8810 E) 9.7

    37. Consider the above one-way ANOVA table. How many groups (treatment levels) are included in the study?
    A) 3 B) 4 C) 6 D) 20 E) 24

    38. Consider the above one-way ANOVA table. If there are equal number of observations in each group, then each group (treatment level) consists of ______ observations.

    A) 3 B) 4 C) 6 D) 20 E) 24

    39. What is the critical F-value at an alpha of 0.05?

    A) 3.1 B) 3.86 C) 14.17 D) 4.94 E) 8.66

    Problem I:
    Use the following to answer questions 40-42:

    The following results were obtained from a simple regression analysis:

    = 37.2895 - (1.2024) * X
    r = - 0.6774

    40. For each unit change in X (independent variable), the estimated change in Y (dependent variable) is equal to:
    A) -1.2024 B) 0.6774 C) 37.2895 D) 0.2934

    41. When X (independent variable) is equal to zero, the estimated value of Y (dependent variable) is equal to:
    A) -1.2024 B) 0.6774 C) 37.2895 D) 0.2934

    42. __________ is the proportion of the variation explained by the simple linear regression model:
    A) 0.8230 B) 0.6774 C) 0.4589 D) 0.2934 E) 37.2895

    43. Given the following information about a hypothesis test of the difference between two means based on independent random samples, which one of the following is the correct rejection region at a significance level of .05?

    HA: μA >μB , μ1 = 12, μ2 = 9, s1 = 4, s2 = 2, n1 = 13, n2 = 10.

    A) Reject H0 if Z > 1.96
    B) Reject H0 if Z > 1.645
    C) Reject H0 if t > 1.721
    D) Reject H0 if t > 2.08
    E) Reject H0 if t > 1.734

    Problem K: Business travelers were asked to rate Miami Airport (on a scale of 1-10). Similarly business travelers were asked to rate Los Angeles airport. A hypothesis test (at alpha = 0.05) is conducted for any difference in the population means in the ratings. The Excel output is shown below. Use the following to answer questions 47- 48:

    t-Test: Two-Sample Assuming Unequal Variances
    Miami Los Angeles
    Mean 6.34 6.72
    Variance 4.677959184 5.63428571
    Observations 50 50
    Hypothesized Mean Difference 0
    df 97
    t Stat -0.836742811
    P(T<=t) one-tail 0.202396923
    t Critical one-tail 1.660714588
    P(T<=t) two-tail 0.404793846
    t Critical two-tail 1.984722076

    48. A 95% confidence interval of the difference between the mean ratings is:
    a. - 0.52 to 1.25
    b. 1.67 to 2.43
    c. -0.51 to 1.27
    d. -1.28 to 0.52
    e. -2.43 to 1.67

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    This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

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