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    Quantitative Methods Problems

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    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

    © BrainMass Inc. brainmass.com April 1, 2020, 9:22 pm ad1c9bdddf
    https://brainmass.com/statistics/survey-methodology/quantitative-methods-problems-514164

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    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