Question

3. (a) Scores on a statistics evaluation follow a normal distribution What is the probability, rounded to four digits, that a randomly selected student will achieve a score that exceeds the mean score by more than 1.5 standard deviations? (Remember that 1.5 standard deviations means a z-score of 1.5.) (b) A statistics instructor gives an evaluation to a large group of students. The scores are normally distributed with μ = 70 and σ = 10. (i) Compute the probability that one randomly sampled student scores more than 80? (Round to four digits.) (ii) Four students are chosen at random, what is the probability, rounded to four digits, that at least one of them scores more than 80 on the exam? For this final calculation, use your rounded answer to b(i). (Note: this is a simple binomial problem! Use the probability you computed in b(i) and find the binomial P(X≥1) for n = 4. All of this is related, eh? You could, of course, also find the complement of P(X≥1), i.e., 1-P((X = 0), to solve this problem.)

Answer 1

(a) Scores on a statistics evaluation follow a normal distribution What is the probability, rounded to four digits, that a randomly selected student will achieve a score that exceeds the mean score by more than 1.5 standard deviations? (Remember that 1.5 standard deviations means a z-score of 1.5.)

For normal distribution mean = 0 and std.deviation = 1

(b) A statistics instructor gives an evaluation to a large group of students. The scores are normally distributed with μ = 70 and σ = 10.
(i) Compute the probability that one randomly sampled student scores more than 80? (Round to four digits.)

(ii) Four students are chosen at random, what is the probability, rounded to four digits, that at least one of them scores more than 80 on the exam? For this final calculation, use your rounded answer to b(i). (Note: this is a simple binomial problem! Use the probability you computed in b(i) and find the binomial P(X≥1) for n = 4. All of this is related, eh? You could, of course, also find the complement of P(X≥1), i.e., 1-P((X = 0), to solve this problem.)