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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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Statistics - Multiple Choice: estimation, z value, confidence interval, t statistic, point estimate for population proportion, error of the estimation, point estimate of the population variance, chi-square values, sample size

Question 1: When a statistic calculated from sample data is used to estimate a population parameter, it is called :

an interval estimate
a point estimate
a statistical parameter
a good guess

Question 2: A large appliance company sends out technicians to unpack, assemble, and connect every gas dryer that is sold. In developing a pricing strategy, it is important to the company to have a "handle" on how long this process takes. The company hires a research assistant to follow technicians on 45 randomly selected jobs. The research assistant records how much time it takes the technicians to unpack, assemble, and connect each gas dryer. The resulting sample mean is 34.3 minutes. If research assistant concludes based on the sample mean that the average time for all such jobs is 34.3 minutes she is using a:

a range estimate
a statistical parameter
an interval estimate
a point estimate

Question 3: The Z value associated with a two sided 90% confidence interval is:

1.28
1.645
1.96
2.575

Question 4: Suppose a random sample of 36 is selected from a population with a standard deviation of 12. If the sample mean is 98, the 99% confidence interval to estimate the population mean is:

94.08 to 101.92
97.35 to 98.65
92.85 to 103.15
93.34 to 102.66

Question 5: What is the average length of a wireless phone call? Suppose researchers in the telecommunications industry want to estimate the average number of minutes of a wireless call. To do so, they randomly select one such call from 85 wireless phone bills around the country. The resulting sample mean is 2.54 minutes with a standard deviation of 1.20 minutes. From these data, the 86% confidence interval to estimate the average length of a wireless phone call for all users is:

2.285 to 2.795
2.326 to 2.754
2.347 to 2.733
2.519 to 2.561

Question 6: The t statistic was developed by:

Charles Student
William Gosset
Abraham deMoivre
Karl Gauss

Question 7: In order to find values in the t distribution table, you must convert the sample size or sizes to:

population sizes
z values
student values
degrees of freedom

Question 8: A researcher is taking a random sample of 18 items in an effort to estimate the population mean. He wants to be 95% confident of his results. The table t value that he should use is:

2.110
2.101
1.740
1.734

Question 9: A researcher is interested in estimating the mean value for a population. She takes a random sample of 17 items and computes a sample mean of 224 and a sample standard deviation of 32. She decides to construct a 98% confidence interval to estimate the mean. Assuming that the population is normally distributed, the resulting confidence interval is:

219.138 to 228.862
204.077 to 243.923
203.953 to 244.047
207.546 to 240.454

Question 10: The weights of aluminum castings produced by a process are normally distributed. A random sample of 5 castings is selected; the sample mean weight is 2.21 pounds; and the sample standard deviation is 0.12 pound. The 99% confidence interval for the population mean casting weight is:

2.009 to 2.411
2.100 to 2.320
1.825 to 2.595
1.963 to 2.457

Question 11: A researcher wants to estimate the proportion of the population which possess a given characteristic. A random sample of size 600 is taken resulting in 276 items which possess the characteristic. The point estimate for this population proportion is _______.

0.54
0.46
0.35
0.67

Question 12: A researcher wants to estimate the proportion of the population that possesses a given characteristic using a 90% confidence interval. A random sample of size 600 is taken resulting in 330 items that possess the characteristic. The error of the estimation of the confidence interval is:

0.0398
0.0068
0.0334
0.55

Question 13: To estimate the proportion of a population that possesses a given characteristic, a random sample of 1700 people are interviewed from the population. Seven hundred and fourteen of the people sampled possess the characteristic. Using this information, the researcher computes an 88% confidence interval to estimate the proportion of the population who possess the given characteristic. The resulting confidence interval is:

.401 to .439
.409 to .431
.392 to .448
.389 to .451

Question 14: From a sample of 42 items, a company wants to estimate the proportion of the population that is defective. Using the results of the sample given below, construct a 96% confidence interval to estimate that proportion. In the data below, a "y" denotes a defect.

.685 to .934
.049 to .332
.066 to .314
.072 to .309

Question 15: If a researcher is calculating a confidence interval and increases the confidence then the width of the confidence interval will do what, all other things being constant?

Remain the same
Increase
Decrease
None of the above

Question 16: The relationship of the sample variance to the population variance is captured by which distribution?

z distribution
normal distribution
t distribution
chi-square distribution

Question 17: A researcher wants to estimate the population variance. He is certain that the population is normally distributed. In an effort to construct a confidence interval, he randomly selects eight members of the population. The data are shown below. What is the point estimate of the population variance?

7.44
2.92
8.5
2.727

Question 18: A financial officer wants to estimate the population variance of daily deposits at the bank. The officer randomly records 14 deposits. The sample mean deposit is $235 with a sample standard deviation of $42. In estimating the population variance from these data, what is the point estimate?

$42
$235
$31.31
$1764

Question 19: A researcher wants to construct a 99% confidence interval to estimate a population variance using a random sample of 13 observations. The chi-square values for this confidence interval are:

4.40378, 23.3367
3.57055, 26.2170
3.56504, 29.8193
3.07379, 28.2997

Question 20: A fund manager manages a portfolio of 250 common stocks. The manager relies on various statistics, such as variance, to assess the overall risk of stocks in an economic sector. The manager's staff reported that for a sample 12 utility stocks the mean annualized return was 14% and that the variance was 3%. The 95% confidence interval for the population variance of annualized returns is

1.41 to 7.49
1.68 to 7.21
1.51 to 8.65
4.52 to 25.95

Question 21: In determining the sample size necessary to estimate a population mean, the error of estimation, E, is equal to

the distance between the sample mean and the population mean
the distance between the sample mean and the variance
the z score
the sample size

Question 22: In determining the sample size necessary to estimate p, if there is no good approximation for the value of p available, then what value should be used as an estimate of p in the formula?

0.10
0.40
0.50
1.96

Question 23: A researcher wants to estimate the average diameter of the population of a three foot long pipe. The researcher wants to be within .1" of the actual average and be 90% confident. The population variance of diameters for this type pipe is .25. How large of a sample size should be taken?

68
17
97
24

Question 24: A business researcher wants to estimate the proportion of all workers who feel stressed out with their job. The researcher is certain that the proportion is no more .22. She wants to be 99% confident of the results and be within .04 of the true population proportion. She needs to sample at least:

111
412
712
1036

Question 25: A researcher with a large national chain of fast food restaurants wants to estimate the proportion of customers who order French fries with their hamburger. The researcher is uncertain about what the proportion may actually be, wants to be 95% confident about the results, and wants to be within .03 of the actually figure. The researcher needs to sample at least:

33
1068
1842
2134

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