It has been found that scores on the Writing portion of the SAT (Scholastic Apti

Question

It has been found that scores on the Writing portion of the SAT (Scholastic Aptitude Test) exam are normally distributed with mean 492 and standard deviation 111. Use the normal distribution to answer the following questions.

(a) What is the estimated percentile for a student who scores 460 on Writing? Round your answer to the nearest integer. The estimated percentile for 460 is . the absolute tolerance is +/-1 Link to Text

(b) What is the approximate score for a student who is at the 89 Superscript th percentile for Writing? Round your answer to the nearest integer. The 89 Superscript th percentile is the score of . the absolute tolerance is +/-1

Heights of ten year old boys (5th graders) follow an approximate normal distribution with mean mu equals 55.5 inches and standard deviation equals 2.7 inches.1 1Centers for Disease Control and Prevention growth chart at http://www.cdc.gov/growthcharts/html_charts/statage.htm

. (b) According to this normal distribution, what proportion of 10-year-old boys are between 4 ft 4.5 in and 5 ft 3.5 in tall (between 52.5 inches and 63.5 inches)? Round your answer to four decimal places. proportionequals the absolute tolerance is +/-0.001 Link to Text

(c) A parent says his 10-year-old son is in the 98 Superscript th percentile in height. How tall is this boy? Round your answer to two decimal places. heightequals inches the absolute tolerance is +/-0.01

Heights of adult males in the US are approximately normally distributed with mean 70 inches (5 ft 10 in) and standard deviation 3 inches.

(a) What proportion of US men are between 5 ft 7.4 in and 6 ft 0.6 in tall (67.4 and 72.6 inches, respectively)? Round your answer to three decimal places. proportionequals the absolute tolerance is +/-0.001 Link to Text

(b) If a man is at the 10 Superscript th percentile in height, how tall is he? Round your answer to one decimal place. heightequals inches the absolute tolerance is +/-0.1

Explanation / Answer

Q1) SAT Question
mean = 492
std. dev. = 111

(A) P(X < 460) = P(z < (460 - 492)/111) = P(z < -0.2883) = 0.3866
The estimated percentile for 460 is . the absolute tolerance is 38.66

(B) Corresponding z value of 89th percentile, z = 1.2265
Using central limit theorem,
z = (x - mean)/std. dev
x = 1.2265*111 + 492
x = 628.1415

The 89 Superscript th percentile is the score of . the absolute tolerance is 628.1415

Q2) Height of 10 year old boys
mean = 55.5
std. dev. = 2.7

(B)
P(52.5 < X < 63.5) = P((52.5 - 55.5)/2.7 < z < (63.5 - 55.5)/2.7) = P(-1.11 < z < 2.963) = P(z < 2.963) - P(z < -1.11) = 0.9985 - 0.1333 = 0.8652

(C)
Corresponding z value of 98th percentile, z = 2.0538
Using central limit theorem,
z = (x - mean)/std. dev
x = 2.0538*2.7 + 55.5
x = 61.0453

Q3) Height of Adult Males
mean = 70
std. dev. = 3

(A)
P(67.4 < X < 72.6) = P((67.4 - 70)/3 < z < (72.6 - 70)/3) = P(-0.8667 < z < 0.8667) = P(z < 0.8667) - P(z < 0.8667) = 0.8069 - 0.1931 = 0.6139

(B)
Corresponding z value of 10th percentile, z = -1.28
Using central limit theorem,
z = (x - mean)/std. dev
x = -1.28*3 + 70
x = 66.16

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