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# Calculate probability, building confidence interval and performing hypothesis testing

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* * * For any hypothesis testing problems in this set, please do only the following: (1) set up the hypotheses in plain English, (2) set up the hypotheses in statistical terms, (3) specify type I and type II errors for the hypothesis testing context for the problem, and (4) provide a brief description of the cost implications of each of the two errors for the problem context * * *

1. In 1993, women took an average of 8.5 weeks unpaid leave from their jobs after the birth of a child (U.S. News & World Report, 12/27/1993). Assume that 8.5 weeks is the population mean and 2.2 weeks is the population standard deviation. What is the probability that a simple random sample of 50 women provides a sample mean unpaid leave of 7.5 to 9.5 weeks after the birth of a child?

2. A sample of 532 Business Week subscribers showed that the mean time a subscriber spends using the Internet and online services is 6.7 hours per week (Business Week 1996 Worldwide Subscriber Study). If the sample standard deviation is 5.8 hours, what is the 95% confidence interval for the mean time that the Business Week subscriber population spends on the Internet and online services?

3. Suppose that scores on an aptitude test used for determining admission to graduate study in business are distributed with a mean of 500 and a population standard deviation of 100. If a random sample of 64 applicants from Stephan College has a sample mean of 537, is there any evidence that their mean score is higher than the mean expected of all applicants? (Use  = .01)

4. To help your restaurant marketing campaign target the right age levels, you want to find out if there is a statistically significant difference, on the average, between the age of your customers and the age of the general population in town, 43.1 years. A random sample of 50 customers shows an average age of 33.6 years with a standard deviation of 16.2 years. What can you conclude? (Use  = .05)

5. External organizational development (OD) professionals provide consulting services in such areas as human resources, training, planning, skills education, industrial psychology, and organizational behavior. The Training and Development Journal (Feb. 1984) conducted a survey of OD professionals. A random sample of 440 external OD consultants yielded the following summary statistics on daily fees charged:

Sample mean = \$720 S = \$275

Suppose it is known that other management consultants charge, on average, \$800 per day. Do the data provide sufficient evidence to indicate that the mean daily fee charged by external OD consultants is less than \$800? Test using  = .10.

https://brainmass.com/statistics/hypothesis-testing/calculate-probability-building-confidence-interval-and-performing-hypothesis-testing-590754

#### Solution Preview

In 1993, women took an average of 8.5 weeks unpaid leave from their jobs after the birth of a child (U.S. News & World Report, 12/27/1993). Assume that 8.5 weeks is the population mean and 2.2 weeks is the population standard deviation. What is the probability that a simple random sample of 50 women provides a sample mean unpaid leave of 7.5 to 9.5 weeks after the birth of a child?

Since z=(xbar-mean)/[sd/sqrt(n)]=(xbar-8.5)/[2.2/sqrt(50)],
P(7.5<xbar<9.5)=P((7.5-8.5)/[2.2/sqrt(50]<Z<(9.5-8.5)/[2.2/sqrt(50])=P(-3.21<Z<3.21)=0.9987 from standard normal table.

A sample of 532 Business Week subscribers showed that the mean time a subscriber spends using the Internet and online services is 6.7 hours per week (Business Week 1996 Worldwide Subscriber Study). If the sample standard deviation is 5.8 hours, what is the 95% confidence interval for the mean time that the Business Week subscriber population spends on the Internet and online services?
From t table, at 95% level with df=532-1=531, the critical value is 1.96.
Hence, a 95% confidence interval for mean time is [6.7-1.96*5.8/sqrt(532), 6.7+1.96*5.8/sqrt(532)]=[6.21, 7.19]

Suppose that scores on an aptitude test used for determining admission to graduate study in business are distributed with a mean of 500 and a population standard deviation of 100. If a random sample of 64 ...

#### Solution Summary

The solution gives detailed steps on calculating probability, building confidence interval and performing hypothesis testing under either t or z distribution.

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## 2. The following table classifies 800 telephone calls by type (local or long distance) and by length: (6 points) a. What is the probability that a call sampled at random out of this population is local and 1 -5 minutes long? b. What is the probability that a call sampled at random is long distance? c. What is the probability that a call is not 5+ minutes given that is a long distance call?

Please answer the following questions in Microsoft Excel. There are six questions total. Work on how to do it should be shown.

2. The following table classifies 800 telephone calls by type (local or long distance) and by length: (6 points)
a. What is the probability that a call sampled at random out of this population is local and 1 -5 minutes long?
b. What is the probability that a call sampled at random is long distance?
c. What is the probability that a call is not 5+ minutes given that is a long distance call?

Length (minutes)
Type of Phone Call 0-1 1-5 5+ Total
Local 150 250 100 500
Long distance 170 120 10 300
Total 320 370 110 800

3. Consider the following probability distributions a, b, and c. Calculate &#956; and &#963; for each. (10 points)
a. Distribution a:
x P(x)
0 0.2
1 0.8
b. Distribution b:
x P(x)
0 0.25
1 0.45
2 0.2
3 0.1
c. Distribution c:
x P(x)
-2 0.1
0 0.3
2 0.4
5 0.2

4. A company produces lightbulbs whose life follows a normal distribution with a mean of 1,200 hours and a standard deviation of 250 hours. If we choose a lightbulb at random, what is the probability that lifetime will last: (6 points)
a. Less than 1000 hours?
b. More than 750 hours?
c. Between 900 and 1,300 hours?

5. In an article in the Journal of Marketing, Morris and colleagues studied innovation by surveying firms to find (among other things) the number of new products introduced by the firms. Suppose a random sample of 100 California-based firms is selected and each firm is asked to report the number of new products introduced during the last year. The mean of the population is known to be 6.50, with a standard deviation of 8.70. The mean of the sample of 100 firms is 5.68. (10 points)
a. Construct a 95% confidence interval around the mean.
b. Construct a 99% confidence interval around the mean.
c. Discuss the trade-off between precision and confidence, using these confidence intervals as examples.

6. Suppose the federal government proposes to give a substantial tax break to automakers producing midsize cars that get a mean mileage exceeding 31 mpg. The standard deviation of the population is given at 0.8. A sample of 49 mileages has a mean of 31.55. Will the automaker get a tax break, using a 5% probability level? (12 points)
a. What type of test would you use to test the hypothesis this question?
b. Would you use a directional or non-directional hypothesis? Why or why not?
c. Following the steps of hypothesis testing:
i. State the hypotheses.
ii. State the decision rule.
iii. Calculate the test statistic.
iv. Make a decision and draw a conclusion.

7. In an article in the Journal of Management, Wright and Bonett studied the relationship between voluntary organizational tenure and such factors as work performance, work satisfaction, and company tenure. As part of the study, the authors compared work performance ratings for "stayers" (employees who stayed with the organization) and "leavers" (employees who voluntarily quit their jobs). Suppose that a random sample of 175 stayers had a mean performance rating of 12.8 (on a twenty point scale) and that a random sample of 140 leavers has a mean performance rating of 14.7. Assume these random samples are independent and that variance for the stayers was 3.7 and variance for the leavers was 4.5 (assume equal variances). Is there a difference in work performance, tested at the 5% level? (12 points)
a. What type of test would you use to test the hypothesis this question?
b. Would you use a directional or non-directional hypothesis? Why or why not?
c. Following the steps of hypothesis testing:
i. State the hypotheses.
ii. State the decision rule.
iii. Calculate the test statistic.
iv. Make a decision and draw a conclusion.

8. The table below shows the mean number of daily errors by air traffic controller trainees during the first two weeks on the job. Perform the appropriate statistical test at the 5% level to see if daily errors on average are decreasing. (12 points)
a. What type of test would you use to test the hypothesis this question?
b. Would you use a directional or non-directional hypothesis? Why or why not?
c. Following the steps of hypothesis testing:
i. State the hypotheses.
ii. State the decision rule.
iii. Calculate the test statistic.
iv. Make a decision and draw a conclusion

Trainee 1 Trainee 2 Trainee 3 Trainee 4 Trainee 5 Trainee 6 Trainee 7
Week 1 5.1 3.0 12.1 6.2 11.5 7.8 2.2
Week 2 3.2 2.2 8.7 7.7 9.4 7.8 3.1

9. A loan officer compares the interest rates for 48-month fixed-rate auto loans and 48-month variable-rate auto loans. Two independent, random samples of auto loans were selected. The table below shows the rates collected: 8 fixed-rate auto loans and five variable-rate auto loans. Assuming the variances are unequal, do the mean rates for 48-month fixed and variable-rate auto loans differ (use a 5% probability level)? (12 points)
a. What type of test would you use to test the hypothesis this question?
b. Would you use a directional or non-directional hypothesis? Why or why not?
c. Following the steps of hypothesis testing:
i. State the hypotheses.
ii. State the decision rule.
iii. Calculate the test statistic.
a. Make a decision and draw a conclusion

Fixed-rate auto loans Variable-rate auto loans
10.29% 9.59%
9.75% 8.75%
9.50% 8.99%
9.99% 8.50%
11.40% 9.00%
10.00%
9.75%
9.99%

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