Question

Suppose that scores on a statistics exam are normally distributed with a mean of 77 and a standard deviation of 4. Find the probability of a student scoring less than 80 on the exam using the following steps.

(a) What region of the normal distribution are you looking to find the area of? (to the left of a zscore, to the right of a z-score, between two z-scores, or to the left of one z-score and to the right of another z-score)

(b)Calculate the z-score(s) needed to find the probability

(c) Find probability P(X < 80)

Answer 1

Facts : 1) If X and Z are the normal and standard normal distibuted random variables respectively. X-interval can be converted to Z-interval by using Z = Where u = mean(X) o = standard deviation(X) Let X = score of a student in a statistics exam. Given that X follows normal distribution with mean(u) = 77 and standard deviation(o) = 4 Now we have to find the probability of a student scoring less than 80 on the exam.

a student scoring less than 80. X< 80. X < 80 we are looking for region to the left of a Z-score.

b) Now we have to find the Z-score for a student score of 80. X-u = 0.75 .: Z = 0.75 Now we have to find P(X < 80). P(X < 80) = P(Z < 0.75) = 0.7734 .: P(X < 80) = 0.7734

Note: Use standard normal table to get P(Z < 0.75). (Or) Use Excel formula: ENORM.S.DIST(0.75, TRUE) to get P(Z < 0.75).

thank you.