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# Confidence intervals and margins of error

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6. Salaries: Student services consider college officials in admissions, registration, counseling, financial aid, campus ministry, food service, and so on. How much money do these people make each year? Suppose you read in your local newspaper that 45 officials in student services earned an average of = \$50,340 each year.

a. Assume that x= \$16,920 for salaries of college officials in student services. Find a 90'% confidence interval for the population mean salary of such personnel. What is the margin of error?

b. Assume that x= \$10,780 for salaries of college officials in student services. Find a 90'% confidence interval for the population mean salary of such personnel. What is the margin of error?

c. Assume that x= \$4830 for salaries of college officials in student services. Find a 90'% confidence interval for the population mean salary of such personnel. What is the margin of error?

d. Compare the margins of error for parts (a) through (c). As the standard deviation decreases,, does the margin of error decrease?

e. Compare the lengths of the confidence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a 90% confidence interval decrease?

https://brainmass.com/statistics/confidence-interval/confidence-intervals-margins-error-216071

#### Solution Preview

Please see the attached Word document. I have shown the formulas with detailed steps for finding each of the three confidence intervals, taking care to label the margin of error for each. Note that the standard deviation decreases from part (a) through (c), causing a decrease in both the margin of error and ...

#### Solution Summary

Detailed solution with calculations.

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## Statistics 1: 10 problems of confidence intervals, sample sizes, distribution

You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. As part of your study, you randomly select 45 repair costs and find the mean to be \$117.00. The standard deviation is \$18.90. Complete parts (a) and (b).
a. Construct a 90% confidence interval for the population mean repair costs.
b. Change the sample size to n = 90. Construct a 90% confidence interval for the population mean repair cost.
Which confidence interval is wider? Explain. Choose the correct answer below.
2. People were polled on how many books they read the previous year. How many subjects are needed to estimate the number of books read the previous year within one book with 95% confidence? Initial survey results indicate that &#963; = 10.7 books

3. Repair Costs: Microwaves In a random sample of five microwave ovens, the mean repair cost was \$80.00 and the standard deviation was \$13.00. Assume the variable is normally distributed and use a t-distribution to construct a 90% confidence interval for the population mean µ. What is the margin of error of µ?

4. A researcher wishes to estimate, with 99% confidence, the proportion of adults who have high-speed internet access. Her estimate must be accurate within 5% of the true proportion.

a) Find the minimum sample size needed, using a prior study that found that 38% of the respondents said they have high-speed internet access.

b) No preliminary estimate is available. Find the minimum sample size needed.

5. An election poll reported that a candidate had an approval rating of 31% with a margin of error E of 2%. Construct a confidence interval for the proportion of adults who approve of the candidate.

6. Find the margin of error for the given values of c, s and n.

c = 0.95, s = 3.1, n = 36

7. Construct the 98% confidence interval for the population mean µ.

c = 0.98, x-bar = 16.2, s = 8.0 and n = 70

8. Construct the indicated confidence interval for the population mean µ using (a) a t-distribution. (b) If you had incorrectly used a normal distribution, which interval would be wider?

c = 0.99, x-bar = 14.2, s = 2.0, n = 9

9. Construct the indicated confidence interval for the population mean µ using a t-distribution.

c = 0.80, x-bar = 108, s = 10, n = 13

10. A researcher wishes to estimate, with 90% confidence, the percentage of adults who support abolishing the penny. His estimate must be accurate within 5% of the true proportion.

a) Find the minimum sample size needed, using a prior study that found that 38% of the respondents said they support abolishing the penny.

See attached file.

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