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# Confidence Interval and Margin of Error

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In an effort to estimate the mean amount spent per customer for dinner at a major Atlanta restaurant, data were collected for a sample of 49 customers. Assume a population standard deviation of \$5.

a. At 95% confidence, what is the margin of error?
b. If the sample mean is \$24.80, what is the 95% confidence interval for the population mean?

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#### Solution Summary

This solution is comprised of detailed step-by-step calculation and analysis of the given problem and provides students with a clear perspective of the underlying concept.

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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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