4.2 Relative position measurements (Value: 38 points)

Instructions:

In this task, you will work with measures of relative position, the empirical rule, and Chebyshev's theorem. You will calculate measures of relative position and use the empirical rule in various situations. You should present the processes necessary to support the answers to the exercises and round the final results not the intermediate values that appear during the calculations to two decimal places, when necessary.

Question 1

The weight of a group of Business Administration students has a normal distribution, with a mean of 148 pounds and a standard deviation of 22 pounds. Using the empirical rule, what is the approximate percentage of students between: a) 104 and 192 pounds, b) 82 and 214 pounds? (3 points)

Question # 2

The mean height of men is 67 inches and their standard deviation is two point five (2.5) inches. Jos is 69 inches tall (Triola, 2004). Answers:

What is the difference between Jos's height and the average? (1 point)

Approximately how many standard deviations from the mean is Jos's height? Use the empirical rule to answer this question. (2 points)

Convert Jose's height into a Z score. (2 marks)

If we consider that the "common" heights are those that correspond to Z scores between -2 and 2, is Jos's height common or uncommon? (2 points)

Question # 3

With a mean of 98.20 and a standard deviation of 0.62, convert the following temperatures to Z scores.

a) 97.38 (2 points)

b) 98.70 (2 points)

c) 98.30 (2 points)

Question # 4

Which is relatively better: a score of 88 on a psychology test or a score of 45 on an economics test? Scores on the Psychology test have a mean of 90 and a standard deviation of 10. Scores on the economics test have a mean of 51 and a standard deviation of 5.5. Use the measures of relative position to answer this exercise (Z Score) (Triola, 2004). (3 points)

Question # 5

What is the highest relative score of the three scores below? Use the Z value to answer. (3 points)

a score of 145 with a mean of 138 and a standard deviation of 35

a score of 92 on a test with a mean of 88 and a standard deviation of 16

a score of 19 on a test with a mean of 15 and a standard deviation of 6

Question # 6

In a time study conducted in a manufacturing plant, the time to complete a specified operation was measured for each of the 40 workers. The mean and standard deviation were found to be 12.9 and 1.8, respectively. Describe the sample data using the empirical rule (Mendenhall et al., 2015). (3 points)

Question # 7

The mean and variance of a sample of 25 data are 85 and 100, respectively. Use Chebyshev's theorem to describe the distribution of the data. (3 points)

Question # 8

The data below are the weights (in pounds) of 15 packages of ground beef in Supermax supermarket refrigerators: 1.28, .90, 1.29, 1.15, .93, 1.18, .89, 1.12, 2.33, 1.94, 2.14, 2.6, 1.11, 1.9, 1.2 (Mendenhall et al., 2015).

a) Construct a stem-and-leaf plot to show the distribution of these weights. (3 points)

b) Find the mean and standard deviation of the data set. (4 points)

c) Find the percentage of measurements in the intervals of the empirical rule. (3 points)