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# Quantitative Methods Problems

1. The data in the Excel spreadsheet linked below give the ages and salaries of the chief executive officers of 59 companies with sales between \$5 million and \$350 million.   The correlation between age and salary can be characterized as:
1.  Strong and positive.
2.  Strong and negative.
3.  Weak and positive.
4. Weak and negative.

Age Salary
(\$thousands)
53 145
43 621
33 262
45 208
46 362
55 424
41 339
55 736
36 291
45 58
55 498
50 643
49 390
47 332
69 750
51 368
48 659
62 234
45 396
37 300
50 343
50 536
50 543
58 217
53 298
57 1103
53 406
61 254
47 862
56 204
44 206
46 250
58 21
48 298
38 350
74 800
60 726
32 370
51 536
50 291
40 808
61 543
63 149
56 350
45 242
61 198
70 213
59 296
57 317
69 482
44 155
56 802
50 200
56 282
43 573
48 388
52 250
62 396
48 572

2. A political consultant conducts a survey to determine what position the mayoral candidate she works for should take on a proposed smoking ban in restaurants.   Which of the following survey questions will deliver an unbiased response?
1. Should the city ban smoking in restaurants to protect our children from second-hand smoke?
2.  Should tobacco smoke, a known cause of lung cancer, be banned from public spaces such as restaurants?
3. Does the city have the right to restrict recreational activities, such as moderate consumption of alcohol or tobacco, on the premises of privately-owned businesses?
4. None of the above.

3. A nutrition researcher wants to determine the mean fat content of hen's eggs. She collects a sample of 40 eggs. She calculates a mean fat content of 23 grams, with a sample standard deviation of 8 grams. From these statistics she calculates a 90% confidence interval of [20.9 grams, 25.1 grams].   What can the researcher do to decrease the width of the confidence interval?
1.  Increase the confidence level.
2. Decrease the confidence level.
3.  Decrease the sample size
4. None of the above.

4. In a random sample of 321 senior citizens, 61 were found to own a home computer.   Based on this sample, the 95% confidence interval for the proportion of computer-owners among senior citizens is:
1.  [2.6%; 7.4%].
2. [13.4%; 24.6%].
3.  [14.7%; 23.3%].
4. The answer cannot be determined from the information given.

5. Preliminary estimates suggest that about 58% of students at a state university favor implementing an honor code.   To obtain a 95% confidence interval for the proportion of all students at the university favoring the honor code, what is the minimum sample size needed if the total width of the confidence interval must be less than 5 percentage points (i.e., the confidence interval should extend at most 2.5 percentage points above and below the sample proportion)?
1.  375.
2.  264.
3. 1,498.
4. The answer cannot be determined from the information given.

6. In a survey of twelve Harbor Business School graduates, the mean starting salary was \$93,000, with a standard deviation of \$17,000.   The 95% confidence interval for the average starting salary among all Harbor graduates is:
1.  [\$83,382; \$102,618].
2. [\$82,727; \$103,327].
3.  [\$82,199; \$103,801].
4. [\$59,000; \$127,000].

7. In a survey of 53 randomly selected patrons of a shopping mall, the mean amount of currency carried is \$42, with a standard deviation of \$78.   What is the 95% confidence interval for the mean amount of currency carried by mall patrons?
[\$39.1; \$44.9].
[\$24.4; \$59.6].
[\$21.0; \$63.0].
[\$14.4; \$69.6].

8. A filling machine in a brewery is designed to fill bottles with 355 ml of hard cider. In practice, however, volumes vary slightly from bottle to bottle. In a sample of 49 bottles, the mean volume of cider is found to be 354 ml, with a standard deviation of 3.5 ml.   At a significance level of 0.01, which conclusion can the brewer draw?
1. The true mean volume of all bottles filled is 354 ml.
2. The machine is not filling bottles to an average volume of 355 ml.
3.  There is not enough evidence to indicate that the machine is not filling bottles to an average volume of 355 ml.
4. The machine is filling bottles to an average volume of 355 ml.

9. To conduct a one-sided hypothesis test of the claim that houses located on corner lots (corner-lot houses) have higher average selling prices than those located on non-corner lots, the following alternative hypothesis should be used:
1.  The average selling price of a corner-lot house is higher than it is commonly believed to be.
2. The average selling price of a corner-lot house is higher than the average selling price of all houses.
3.  The average selling price of a corner-lot house is the same as the average selling price of a house not located on a corner lot.
4. The average selling price of a corner-lot house is higher than the average selling price of a house not located on a corner lot.

Corner-lot
House Price
(in \$hundreds) Non-corner Lot
House Price
(in \$hundreds)
2150 2050
1999 2080
1800 2150
1375 1900
1250 1560
1110 1450
1139 1449
995 1270
900 1235
1695 1170
1553 1180
1300 1155
1020 995
1020 975
925 975
725 960
1299 860
1250 1250
1080 922
1050 899
835 850
805 876
750 890
773 870
1295 700
975 720
700 720
2100 749
600 731
1844 670
699 2150
1330 1599
1129 1350
1050 1239
1000 1200
1030 1125
940 1100
874 1049
766 955
739 934
* 875
* 889
* 855
* 810
* 799
* 759
* 755
* 750
* 730
* 729
* 710
* 690
* 670
* 619
* 939
* 820
* 780
* 770
* 620
* 540
* 1070
* 725
* 660
* 580
* 1580
* 1160
* 1109
* 1050
* 1045
* 1020
* 975
* 950
* 920
* 945
* 872
* 870
* 869

10. The data in the Excel spreadsheet linked below indicate the selling prices of houses located on corner lots ("corner-lot houses") and of houses not located on corner lots.   Conduct a one-sided hypothesis test of the claim that corner-lot houses have higher average selling prices than those located on non-corner lots. Using a 99% confidence level, which of the following statements do the data support?
1. Upscale, expensive neighborhoods have more street corners.
2. The average selling price of a corner-lot house is higher than that of the average house not located on a corner lot.
3. The average selling price of a corner-lot house is no more than that of the average house not located on a corner lot.
4. There is not enough evidence to support the claim that the average selling price of a corner-lot house is higher than that of the average house not located on a corner lot.

Corner-lot
House Price
(in \$hundreds) Non-corner Lot
House Price
(in \$hundreds)
2150 2050
1999 2080
1800 2150
1375 1900
1250 1560
1110 1450
1139 1449
995 1270
900 1235
1695 1170
1553 1180
1300 1155
1020 995
1020 975
925 975
725 960
1299 860
1250 1250
1080 922
1050 899
835 850
805 876
750 890
773 870
1295 700
975 720
700 720
2100 749
600 731
1844 670
699 2150
1330 1599
1129 1350
1050 1239
1000 1200
1030 1125
940 1100
874 1049
766 955
739 934
* 875
* 889
* 855
* 810
* 799
* 759
* 755
* 750
* 730
* 729
* 710
* 690
* 670
* 619
* 939
* 820
* 780
* 770
* 620
* 540
* 1070
* 725
* 660
* 580
* 1580
* 1160
* 1109
* 1050
* 1045
* 1020
* 975
* 950
* 920
* 945
* 872
* 870
* 869

11. Two semiconductor factories are being compared to see if there is a difference in the average defect rates of the chips they produce. In the first factory, 250 chips are sampled. In the second factory, 350 chips are sampled. The proportions of defective chips are 4.0% and 6.0%, respectively.   Using a confidence level of 95%, which of the following statements is supported by the data?
1. There is not sufficient evidence to show a significant difference in the average defect rates of the two factories.
2. There is a significant difference in the average defect rates of the two factories.
3. The first factory's average defect rate is lower than the second factory's on 95 out of 100 days of operation.
4. None of the above.

12. The regression analysis below relates average annual per capita beef consumption (in pounds) and the independent variable "average annual beef price" (in dollars per pound).   The coefficient on beef price tells us that:
Beef Consumption and Price
1.  For every price increase of \$1, average beef consumption decreases by 9.31 pounds.
2. For every price increase of \$1, average beef consumption increases by 9.31 pounds.
3. For every price increase \$9.31, average beef consumption decreases by 1 pound.
4. For price increase of \$9.31, average beef consumption increases by 1 pound.

13. The regression analysis below relates average annual per capita beef consumption (in pounds) and the independent variable "average annual beef price" (in dollars per pound).   In a year in which the average price of beef is at \$3.51 per pound, we can expect average annual per capita beef consumption to be approximately:
Beef Consumption and Price

1.  55.2 pounds
2.  52.6 pounds
3.  53.6 pounds
4. 117.9 pounds

14. The regression analysis below relates average annual per capita beef consumption (in pounds) and the independent variable "average annual per capita pork consumption" (in pounds).   At what level is the coefficient of the independent variable pork consumption significant?
Beef Consumption and Pork Consumption
Source

1.  0.10.
2.  0.05.
3.  0.01.
4.  None of the above.

15. The regression analysis below relates average annual per capita beef consumption (in pounds) and the independent variable "average annual per capita pork consumption" (in pounds).   Which of the following statements is true?
Beef Consumption and Pork Consumption
Source

1.  Beef consumption can never be less than 65.09 pounds.
2. Beef consumption can never be greater than 65.09 pounds.
3.  The y-intercept of the regression line is 65.09 pounds.
4.  The x-intercept of the regression line is 65.09 pounds.

16. The regression analysis at the bottom relates average annual per capita beef consumption (in pounds) and the independent variables "average annual per capita pork consumption" (in pounds) and "average annual beef price" (in dollars per pound).   Which of the independent variables is significant at the 0.01 level?
Beef Consumption, Pork Consumption, and Beef Price
Source

1. Beef price only.
2. Pork consumption only.
3. Both independent variables.
4.  Neither independent variable

#### Solution Preview

Hi there,

Thanks for letting me work on your post. I've included my explanation in both Word and Excel documents.

1. The data in the Excel spreadsheet linked below give the ages and salaries of the chief executive officers of 59 companies with sales between \$5 million and \$350 million.   The correlation between age and salary can be characterized as:
1. Strong and positive.
2.  Strong and negative.
3.  Weak and positive.
4. Weak and negative.
By running "correlation" under "data analysis" in excel, we could obtain the following output:
Age Salary
Age 1
Salary 0.127555 1
Since r value is 0.127555, positive but far away from 1, the correlation between these two variables is weak and positive. In this case, choice "3" is the right one.
Age Salary
(\$thousands)
53 145
43 621
33 262
45 208
46 362
55 424
41 339
55 736
36 291
45 58
55 498
50 643
49 390
47 332
69 750
51 368
48 659
62 234
45 396
37 300
50 343
50 536
50 543
58 217
53 298
57 1103
53 406
61 254
47 862
56 204
44 206
46 250
58 21
48 298
38 350
74 800
60 726
32 370
51 536
50 291
40 808
61 543
63 149
56 350
45 242
61 198
70 213
59 296
57 317
69 482
44 155
56 802
50 200
56 282
43 573
48 388
52 250
62 396
48 572

2. A political consultant conducts a survey to determine what position the mayoral candidate she works for should take on a proposed smoking ban in restaurants.   Which of the following survey questions will deliver an unbiased response?
1. Should the city ban smoking in restaurants to protect our children from second-hand smoke?
2.  Should tobacco smoke, a known cause of lung cancer, be banned from public spaces such as restaurants?
3. Does the city have the right to restrict recreational activities, such as moderate consumption of alcohol or tobacco, on the premises of privately-owned businesses?
4. None of the above.
For 1, "to protect our children from second-hand smoke" is some kind of bias; for 2, "a known cause of lung cancer" is some kind of bias; for 3, it is not biased. Therefore, 3 is the right choice.

3. A nutrition researcher wants to determine the mean fat content of hen's eggs. She collects a sample of 40 eggs. She calculates a mean fat content of 23 grams, with a sample standard deviation of 8 grams. From these statistics she calculates a 90% confidence interval of [20.9 grams, 25.1 grams].   What can the researcher do to decrease the width of the confidence interval?
1.  Increase the confidence level.
2. Decrease the confidence level.
3.  Decrease the sample size
4. None of the above.
When the confidence level increases, the width increases. So 1 is the right choice. Meanwhile, when the sample size decreases, the margin of error increases (critical value*standard deviation/sqrt(n), n is the sample size) and thus the width increases. So choice 3 is not right. In this case, only choice 2 is the right one.

4. In a random sample of 321 senior citizens, 61 were found to own a home computer.   Based on this sample, the 95% confidence interval for the proportion of computer-owners among senior citizens is:
1.  [2.6%; 7.4%].
2. [13.4%; 24.6%].
3.  [14.7%; 23.3%].
4. The answer cannot be determined from the information given.

The critical value for 95% confidence interval is 1.96
Proportion p=61/321=0.19
Margin of error=1.96*sqrt(0.19*(1-0.19)/321)=0.043
Upper limit: 0.19+0.043=0.233
Lower limit: 0.19-0.043=0.147
Therefore, 3 is the right choice.
5. Preliminary estimates suggest that about 58% of students at a state university favor implementing an honor ...

#### Solution Summary

Quantitative methods problems for independent variables are examined. The average annual beef price are determined.

\$2.19