Calculate confidence interval, Sample Size Estimation for Proportion

1) In a random sample of 810 women employees, it is found that 81 would prefer working for a female boss. What is the width of the 95% confidence interval for the proportion of women who prefer a female boss? Please show your calculations

2)Suppose the President wants an estimate of the proportion of the population who oppose a policy toward gun control. The President wants the estimate to be within .04 of the true proportion. Assume a 95 percent level of confidence. The President’s political advisors estimated the proportion supporting the current policy to be 0.60. How large a sample is required?

3) Explain whether the width of a confidence interval would increase, decrease, or remain the same as a result of each of the following changes:

a) The sample size is doubled, from 400 to 800

b) The population size is doubled, from 25 million to 50 million

c) The level of confidence is lowered from 95% to 90%

4) A financial institution wishes to estimate the mean balances owed by its credit card customers. The population standard deviation is estimated to be $300. If a 99 percent confidence interval is used and an interval of ±$75 is desired, how many cardholders should be sampled?

5) A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches? [compute the confidence of interval using t-distribution since σ is unknown]

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