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Probability Confidence Intervals

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I am currently studying probability and am having a bit of trouble with the following questions. Can you help me? If you can, thank you very much. The questions are:

1.) An office manager wishes to estimate the mean time required to handle customer complaints. A sample of 38 complaints shows a mean time of 28.7 minutes with a standard deviation of 12 minutes.

a) Construct a 90% confidence interval for the true mean time required to handle customer complaints.
b) How large a sample should be taken, if the manager wishes to estimate the mean time to handle a complaint to within +/- one minute? Assume that the confidence interval is to be 90%.

2.) A hospital claims that the average length of stay is 5 days. A study of the length of stay for a random sample of 25 patients found the mean stay to be 6.2 days with a standard deviation of 1.2 days. Do these data present sufficient evidence to support the hospital's claim? Use a 0.05 significance level.

3.) A survey was conducted to determine the proportion of registered nurses in a particular state that are actively employed. A random sample of 400 nurses selected from the state registry showed 274 actively employed. Find a 95% confidence interval estimate for the true proportion of registered nurses actively employed in the state.

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Solution Summary

This solution provides a step-by-step answer for each of the three probability questions.

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1. In part b of the first question, the 90% confidence limits are given by:

28.7 - xm
----------- = +/- 1.645

where xm = population mean. Thus:

xm = 28.7 +/- 12/sqrt(38)*1.645
= 28.7 +/- 3.20225

And so:

25.498 < xm < 31.902

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