# Probabilities based on normal distribution

In a study of how external clues influence performance, psychology professors at the University of Alberta and Pennsylvania State University gave two different forms of a midterm examination to a large group of introductory psychology students.

In one form of the exam the questions were printed on blue paper, and in the other form of the exam the questions were printed on red paper. Grading only the difficult questions, the researchers found the scores on the blue exam had a distribution with a mean of 53% and a standard deviation of 15%, while the scores on the red exam had a distribution with a mean of 39% and a standard deviation of 12%.

Assuming that both distributions are approximately normal, on which exam is a student more likely to score below 20% on the difficult questions, the blue one or the red one? Explain.

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In a study of how external clues influence performance, psychology professors at the University of Alberta and Pennsylvania State University gave two different forms of a midterm examination to a large group of introductory psychology students. In one form of the exam the questions were printed on blue paper, and in the other form of the exam the questions were printed on red paper. Grading only the difficult questions, the researchers found the scores on the blue exam had a distribution with a mean of 53% and a standard deviation of 15%, while the scores on the red exam had a distribution with a mean of 39% and a standard deviation of 12%. Assuming that both distributions are approximately normal, on which exam is a student more likely to score below 20% on the difficult questions, the blue one or the red one? Explain.

Answer

Let X be the score of blue exam. Given that X is normal with mean =53% and standard deviation =15%. Let Y be the score of red exam. Given that Y is normal with mean =39% and standard deviation =12%.

We need P(X<20) and P(Y<20). Standardizing both variables using Z score and from normal distribution we can compute the probability.

Normal Probabilities Normal Probabilities

Common Data:Blue Common Data :Red

Mean 53 Mean 39

Standard Deviation 15 Standard Deviation 12

Probability for X <20 Probability for Y <20

X Value 20 Y Value 20

Z Value -2.2 Z Value -1.583333

P(X<=20) 0.0139034 P(X<=20) 0.056673

Thus probability of scoring below 20% is high for red exams.

Â© BrainMass Inc. brainmass.com October 3, 2022, 11:10 pm ad1c9bdddf>https://brainmass.com/statistics/normal-distribution/probabilities-based-normal-distribution-348681