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PSY 216 Assignment 5 Answers 1. Problem 1 from the text What information is provided by the sign (+ / -) of a z-score? What information is provided by the numerical value of the z-score? The sign tells you whether the score (X) is above (+) or below (-) the mean. The numerical value of the z-score tells you how far (in standard deviation units) the score is from the mean. 2. Problem 2 from the text A distribution has a standard deviation of σ = 12. Find the z-score for each of the following locations in the distribution. (Problem 2 from the text) a. Above the mean by 3 points z = (X – M) / σ z = (M + 3 – M) / 12 = 0.25 b. Above the mean by 12 points z = (X – M) / σ z = (M + 12 – M) / 12 = 1.00 c. Below the mean by 24 points z = (X – M) / σ z = (M - 24 – M) / 12 = -2.00 d. Below the mean by 18 points z = (X – M) / σ z = (M - 18 – M) / 12 = -1.50 3. Problem 6 from the text For a population with a mean of μ = 100 and a standard deviation σ = 12, a. Find the z-score for each of the following X values. X = 106 X = 115 X = 130 X = 91 X - μ 106 100 0.5 σ 12 X - μ 115 100 Z 1.25 σ 12 Z X - μ 130 100 2.5 σ 12 X - μ 91 100 Z 0.75 σ 12 Z X = 88 Z X - μ 88 100 1.0 σ 12 X = 64 Z X - μ 64 100 3.0 σ 12 b. Find the score (X value) that corresponds to each of the following z-scores. z = -1.00 z = -0.50 z = 2.00 z = 0.75 z = 1.50 z = -1.25 X μ z σ 100 - 1.00 12 88 X μ z σ 100 - 0.50 12 94 X μ z σ 100 2.00 12 124 X μ z σ 100 0.75 12 109 X μ z σ 100 1.50 12 118 X μ z σ 100 - 1.25 12 85 4. Problem 12 from the text A score that is 6 points below the mean corresponds to a z-score of z = -0.50. What is the population standard deviation? z = (X – M) / σ -0.50 = (M – 6 – M) / σ σ = -6 / -0.50 σ = 12 5. Which of the following exam scores should lead to the better grade? a. A score of X = 55 on an exam with μ = 60 and σ = 5. b. A score of X = 40 on an exam with μ = 50 and σ = 20. Explain your answer Because the distributions have different means and/or standard deviations, they are not directly comparable. You should first convert each score to a z score so the scales of the distributions are comparable: X - μ 55 60 1 σ 5 X - μ 40 50 zb 0.5 σ 20 za Because zb is larger than za, the score on exam a should lead to be a better grade.