# Normal Probabilities and Confidence Interval

Use the Student_Data.xls file which consists of 200 MBA students at Whatsamattu U. The file includes variables regarding students' age, gender, major, GPA, Bachelors GPA, course load, English speaking status, family, weekly hours spent studying. Each of the three assigned problems should be formatted as a one page memo. Answers to all three of the problems should be submitted via the Drop Box as a single Word document with each of the problems clearly labeled. Be sure to include your name on the document itself!

COMPLETE PROBLEM 1: Let's assume (for this problem only) that the Student_Data.xls file was the entire population. We know the mean and standard deviation of student ages to be 42.3 and 8.9, respectively. Using the Normal_ Probability.xls file:

a) compute the percentage of students that are older than 50,

b) compute the percentage of students that are younger than 40,

c) compute the percentage of students that are between 41 and 46, inclusive.

d) compute the age range of the oldest 10% of the students.

Then compare each of the answers you computed to the truth as found in the actual file. Briefly discuss your results (100+ words, 3 or more sentences).

COMPLETE PROBLEM 2: You wish to know the average GPA of MBA students at Whatsamatta U. Compute the 95% confidence interval of the mean using the sample of 200 students. Report on your findings (50+ words, 2 or more sentences) along with a normal probability chart that illustrates the points raised in your report.

COMPLETE PROBLEM 3: You wish to know the proportion of MBA students that are majoring in Finance. Compute the 95% confidence interval of the proportion using the sample of 200 students. Report on your findings (50+ words, 2 or more sentences) along with a normal probability chart that illustrates the points raised in your report.

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#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score and confidence interval. Formula for the calculation and Interpretations of the results are also included.

Statistics: Sample variability, and Introduction to statistical inferences

1. Consider a normal population with µ = 25 and σ = 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(α/2) = 1.96 and σ = 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(α/2) and σ, 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 , 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