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Multiple Choice Statistics Problems

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3. A chemical engineer is investigating the effect of process operating temperature on product yield. The study results in the following data; use your knowledge of least squares regression to construct a linear model for predicting yield from temperature. These data apply throughout question 3.

Temp(Celsius) Yield (grams)
100 43.6215
110 46.6231
120 58.2752
130 58.8906
140 65.4354
150 74.5960
160 72.5202
170 79.0639
180 83.6280
190 84.4351

Calculate the slope and intercept of the least squares regression model by hand, which requires only the means and standard deviations of X and Y, and the correlation coefficient (here r = 0.9805).

3 a) If the yield were measured in ounces instead of grams (note that 1 gram is 0.35274 ounces), the slope would increase by a factor of:

Answer
a. 0.35274
b. 1/0.35274
c. would not change

3 b) If the yield were measured in ounces instead of grams (note that 1 gram is 0.35274 ounces), the correlation coefficient would increase by a factor of:

Answer
a. 0.35274
b. 1/0.35274
c. would not change

4. (Look at the file attachment for the table)
Results of a two year study of the effects of calcium supplementation on bone loss are summarized below. The rate of bone loss, computed for each subject, was expressed as a percentage of their initial bone mass. Subjects were randomly allocated to three treatment groups. Group 1 received estrogen creme and a calcium placebo - Estrogen Group. Group 2 received placebo estrogen creme and 200 mg/day calcium - Calcium Group. Group 3 received placebo estrogen creme and a calcium placebo - Placebo Group.

Use one-way ANOVA to compare mean bone mass change per year for the three treatment groups and perform an F test to see if the treatment group means differ. Proceed in a step-by-step fashion doing the computations by hand (with a calculator), and answer throughout question 4.

4 a) What is the value of the grand mean computed from the above data?

Answer
a. 1.50 - 2.00
b. 2.01 - 2.25
c. 2.26 - 2.50
d. 2.51 - 2.75
e. 2.76 - 3.00

4 b) What is the value of the error sum of squares (SSE) for the above data?
(Pick the interval containing the best answer.)

Answer
a. 150 - 160
b. 161 - 170
c. 171 - 180
d. 181 - 190
e. 191 - 200

4 c) Assuming a significance level of 0.05, what is the critical value of F (F crit) for this test?

Answer
a. 3.0 - 3.5
b. 4.0 - 4.5
c. 5.0 - 5.5
d. 6.0 - 6.5
e. 7.0 - 7.5

4 d) What is the value of the F statistic for this sample of data (Fdata)?

Answer
a. 3.50 - 4.00
b. 4.01 - 4.50
c. 4.51 - 5.00
d. 5.01 - 5.50
e. 5.51 - 6.00

4 e) Is the strength of evidence against the H0 provided by the data strong enough to reject it in favor of the alternative hypothesis?

Answer
a. yes, reject H0
b. no, do not reject H0
c. can't tell.

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3.
A chemical engineer is investigating the effect of process operating temperature on product yield. The study results in the following data; use your knowledge of least squares regression to construct a linear model for predicting yield from temperature. These data apply throughout question 3.

Temp(Celsius) Yield (grams)
100 43.6215
110 46.6231
120 58.2752
130 58.8906
140 65.4354
150 74.5960
160 72.5202
170 79.0639
180 83.6280
190 84.4351

Calculate the slope and intercept of the least squares regression model by hand, which requires only the means and standard deviations of X and Y, and the correlation coefficient (here r = 0.9805).
Mean of x(temp)=(100+110+...+190)/10=145
Sxx=(100-145)^2+(110-145)^2+...+(190-145)^2=8250
Mean of yield ...

Solution Summary

The solution assists with answering the multiple choice statistics problems with a brief explanation.

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30 Multiple Choice Problems in Statistics

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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