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# Normal Probability, Confidence Interval and Sample Size

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1. Consider a normal population with µ = 25 and &#963; = 8.0.

(A) Calculate the standard score for a value x of 27.
(B) Calculate the standard score for a randomly selected sample of 30 with = 27.
(C) Explain why the standard scores of 27 are different between A and B above

2. Assume that the mean score on a certain aptitude test across the nation is 100, and that the standard deviation is 20 points. Find the probability that the mean aptitude test score for a randomly selected group of 150 8th graders is between 99.5 and 100.5.

3. Assume that a sample is drawn and z(&#945;/2) = 1.96 and &#963; = 20. Answer the following questions:

(A) If the Maximum Error of Estimate is 0.02 for this sample, what would be the sample size?
(B) Given that the sample Size is 400 with this same z(&#945;/2) and &#963;, what would be the Maximum Error of Estimate?
(C) What happens to the Maximum Error of Estimate as the sample size gets smaller?
(D) What effect does the answer to C above have to the size of the confidence interval?

4. By measuring the amount of time it takes a component of a product to move from one workstation to the next, an engineer has estimated that the standard deviation is 4.17 seconds.
Answer each of the following (show all work):

(A) How many measurements should be made in order to be 95% certain that the maximum error of estimation will not exceed 0.5 seconds?
(B) What sample size is required for a maximum error of 2.0 seconds?

5. A 98% confidence interval estimate for a population mean was computed to be (36.5, 52.9). Determine the mean of the sample, which was used to determine the interval estimate (show all work).

6. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 160 was taken, and the mean amount spent was \$223.24. Assuming a standard deviation equal to \$49.78, find the 95% confidence interval for &#61549;, the mean for all such families (show all work).

7. A confidence interval estimate for the population mean is given to be (39.86, 47.87). If the standard deviation is 16.219 and the sample size is 63, answer each of the following (show all work):

(A) Determine the maximum error of the estimate, E.
(B) Determine the confidence level used for the given confidence interval