a. Ms. Newell gave her class a 10-question multiple-choice quiz with 4 choices.

Question

a. Ms. Newell gave her class a 10-question multiple-choice quiz with 4 choices. Let X = the number of questions that a randomly selected student in the class answered correctly. The mean and standard deviation of X are 7.6 and 1.32 respectively. To determine each student’s grade on the quiz, Ms. Newell will multiply his or her number of correct answers by 10. Let RV ‘G’ be the grade of a randomly chosen student in the class.

i. Calculate the mean of RV ‘G’. Show your work.
ii. Calculate the standard deviation of RV ‘G’. Show your work.
ii. How are the mean and variance of G and X related?
iii. If Johnny was simply guessing, what is the probability that he would pass?

b. A report from Center for Health Statistics says that the height of a 20-year-old men when chosen at random, is a random variable (RV) X with mean height of 5.8 feet and standard deviation 0.24 feet. Height is "normally" distributed.

i. Find the mean and standard deviation of the height in inches of a randomly selected 20-year- old men. (Note: There are 12 inches in a foot)
ii. Calculate the 90th percentile for the height of 20 year old men.
iii. Calculate the probability that the man is taller than 5' 11" tall.

Explanation / Answer

a. (i) Here mean RV(G) = 10 * 7.6 = 76

(ii) Standard deviation G = 10 * 1.32 = 13.2

(iii) Here mea and standard deviation is 10 times of the mean and standard deviation of X.

(iv) Here passing marks are not given so we cant calculating the probability.

b.

Here mean height = 5.8 feet

standard deviation = 0.24 feet

(i) Mean height in inch = 5.8 * 12 = 69.6 inch

Standard devation of height = 0.24 * 12 = 2.88 inch

(ii) Here lets say the 90th percentile is x0

Pr(x < x0 ) = 0.90

NORM(x < x0 ; 69.6 inch ; 2.88 inch)

Here the z value if

Z = 1.2816

(x0 - 69.6)/2.88 = 1.2816

x0 = 69.6 + 2.88 * 1.2816 = 73.29 inch

(iii) if x is the height of 20 year old man.so the height of 5'11' means 71 inch

Pr(x > 71 inch ; 69.6 inch ; 2.88 inch)

Z = (71 - 69.6)/2.88 = 0.4861

Pr(x > 71 inch ; 69.6 inch ; 2.88 inch) = 1 - Pr(x < 71 inch ; 69.6 inch ; 2.88 inch)

= 1 - Pr(Z < 0.4861) = 1 - 0.6866 = 0.3134

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