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Statistics Exam Questions and Answers

Before voting on a proposal to offer day care service for employeesâ€™ children, the board of directors of a corporation is seeking more information. The human resources (HR) manager polls 300 employees and 90 say they would seriously consider utilizing the service.

- What is the point estimate of the proportion of employees who would seriously consider utilizing the service?
- Is the sample large? Explain.
- Build a 90% confidence interval for the proportion of employees who would seriously consider utilizing the service.
- How large a sample should be taken in order to obtain a margin of error no more than 3.5% with 90% confidence?

The board of directors has agreed to allow the human resources manager to move to the next step in planning day care service for employeesâ€™ children if the HR manager can prove that at least 25% of the employees have interest in using the service.

- State your hypotheses.
- Interpret your hypotheses.
- What would a type I error consist in?
- What would a type II error consist in?
- At the ?= .10 level of significance, is there enough interest in the service to move to the next planning step?
- Interpret your conclusion in context. What action is taken?
- If you have committed an error, which one is it?

A marketing professor at Givens College is interested in the relationship between hours spent studying and total points earned in a course. Data collected on 10 students who took the course last quarter follow

Hours |
Points |

45 |
40 |

30 |
35 |

90 |
75 |

60 |
65 |

105 |
90 |

65 |
50 |

90 |
90 |

80 |
80 |

55 |
45 |

75 |
65 |

Â

- Draw a scatter plot. Does the relationship look linear?
- Develop the least square regression line.
- Interpret the value of b
_{0}. - Interpret the value of b
_{1}. - Compute the coefficient of determination.
- Interpret the value of the coefficient of determination. Is there a good fit between the two variables? Why or why not?
- Compute and interpret the correlation coefficient.

**You are about to conduct a F test.**

- State your hypotheses.
- Interpret your hypotheses.
- Compute the test statistic and the p-value.
- What do you conclude? Interpret in context.
- Can you conclude the relationship between hours spent studying and total points earned is linear? Why or why not?
- Can you conclude that more hours spent studying leads to higher total points earned? Why or why not?

**Confidence interval**

- Build a 95% confidence interval for the slope of the regression line.
- Interpret the confidence interval.

**Using the regression line**

- Mark Sweeney has spent 95 hours studying. What do you expect his total points earned to be?
- Build a 95% prediction interval for Mark Sweneyâ€™s total points earned.
- What is the 95% confidence interval for all students who have spent 95 hours studying?

**Model validation**

- Compute the residuals.
- Draw the residual plot.
- Is there any evidence of a violation of OLS assumptions? Explain your answer.

**Question #3 (8%)**

A random sample of 36 magazine subscribers is selected to estimate the mean age of all subscribers. The data follow. Construct a 90% confidence interval estimate of the mean age of all of this magazine's subscribers.

Subscriber |
Age |
Subscriber |
Age |
Subscriber |
Age |

1 |
39 |
13 |
40 |
25 |
38 |

2 |
27 |
14 |
35 |
26 |
51 |

3 |
38 |
15 |
35 |
27 |
26 |

4 |
33 |
16 |
41 |
28 |
39 |

5 |
40 |
17 |
34 |
29 |
35 |

6 |
35 |
18 |
46 |
30 |
37 |

7 |
51 |
19 |
44 |
31 |
33 |

8 |
36 |
20 |
44 |
32 |
41 |

9 |
47 |
21 |
43 |
33 |
36 |

10 |
28 |
22 |
32 |
34 |
33 |

11 |
33 |
23 |
29 |
35 |
46 |

12 |
35 |
24 |
33 |
36 |
37 |

Â

- What is the point estimate of the mean age of all subscribers?
- What is the point estimate of the standard deviation of the age of all subscribers?
- How large is the margin of error?
- Provide the 90% confidence interval.

**Question #4 (14%)**

At a certain manufacturing plant, a machine produces ball bearings that should have a diameter of 0.500 mm. If the machine produces ball bearings that are either too small or too large, the ball bearings must be scrapped. Every hour, a quality control manager takes a random sample of 36 ball bearings to test to see if the process is "out of control" (i.e. to test to see if the average diameter differs from 0.500 mm). Assume that the process is maintaining the desired standard deviation of 0.06 mm. The results from the latest sample follow:

0.468 |
0.521 |
0.421 |
0.476 |
0.448 |
0.346 |

0.452 |
0.513 |
0.465 |
0.395 |
0.558 |
0.526 |

0.354 |
0.474 |
0.447 |
0.405 |
0.411 |
0.453 |

0.456 |
0.477 |
0.529 |
0.440 |
0.570 |
0.319 |

0.471 |
0.480 |
0.499 |
0.446 |
0.405 |
0.557 |

0.468 |
0.521 |
0.421 |
0.476 |
0.448 |
0.346 |

- Define the population parameter in words. Define the variable in words.
- State your hypotheses.
- Interpret your hypotheses.
- At a .01 level of significance, test whether the process is out of control.
- Interpret your conclusion in context.
- What action, if any, should the quality control manager take?
- What is the probability the quality control manager has made a type I error?

The proportion of Americans who support the death penalty is .53. A sample of 1000 randomly selected Americans is surveyed by telephone interview.

- What is the probability that the sample proportion of those supporting the death penalty will be less than .50?
- What is the probability that the sample proportion of those supporting the death penalty will be at least .55?
- What is the probability that the sample proportion of those supporting the death penalty will be between .50 and .55?
- For samples of size 1000, 15% of all sample proportions are at most what value?

Is the following statement true, false or uncertain? Justify your answer.

â€œIf you toss a coin once, you either win or lose: if you toss a coin thirty times, you will have ten or more wins 98 per cent of the time.â€

John Kay, Other Peopleâ€™s Money: The Real Business of Finance, PublicAffairs, New York, 2015, p. 91.