Download 5.2 - normal distribution practice ws

The Normal Distribution
Name:____________________________________ Date:____________________
-Draw and label a sketch of the distribution, indicating what it is that you are looking for
-Use Table 4 (in back of textbook) to find the answer to each question
-Clearly show all steps to getting your final answer.
1. Methacton Physics students have tested several cannon balls for its newly designed catapult,
and obtained a normal distribution for the weights of the cannon balls. Their mean weight is 110
oz. and has a standard deviation of 15 oz.
a. What percentage of the balls between 100 oz. and 125 oz.?
b. What percentage of the balls weigh more than 160 oz. (that’s 10 lbs) ?
c. What cannon ball weight corresponds to the 65th percentile?
2. Scores on the SAT-I test are normally distributed with a mean of 1017 and a standard
deviation of 209.
a. What proportion of students who take the test score above 1310?
b. The director of admissions for Temple University wants to admit as many students as
possible, but knows that his University is not right for all students. Therefore, he decides that
he will not admit students who score in the lower 4% or the upper 4% on the SAT. What are the
cutoff scores on the SAT for admission to Temple?
3. According to Runner’s World, the times of the finishers in the New York City 10k run are
normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes.
a. What proportion of runners take more than 52 minutes to finish the run?
b. What proportion of runners finish in less than 43 minutes?
c. What finish time corresponds to the 84th percentile?
4. Men’s heights in the United States are normally distributed with a mean of 69.0 in. and a
standard deviation of 2.5 in. In order to join the U.S. Marine Corps, a man must be between 64
and 78 in. tall.
a. What percentage of men are rejected by the Marines based on the height requirement?
SOLUTIONS – Statistics worksheet – the Normal Distribution
1. Birth weights of infants at a local hospital have a normal distribution with a mean of 110 oz. and a
standard deviation of 15 oz.
a. What percentage of infants weigh between 100 oz. and 125 oz. at birth?
58.99% of infants weigh between 100oz and 125 oz at birth.
(this comes from .8413 - .2514, which we get from the table for z = 1.00 and z = -0.67.)
b. What percentage of infants weigh more than 160 oz. at birth? (that’s 10 lbs)
0.04% of infants weight more than 160 oz at birth. (z = 3.33, giving .9996 from the table. 1- .9996)
c. What birth weight corresponds to the 65th percentile?
A birth weight of 115.85 oz corresponds to the 65th percentile, meaning 65% of newborns weigh
115.85 oz or less at birth.
(look for .6500 in the center of the table. Closest value is .6517, giving z = .39. convert to oz)
2. Scores on the SAT-I test are normally distributed with a mean of 1017 and a standard deviation of 209.
a. What proportion of students who take the test score above 1310?
0.8008 or 8.08% of students score above 1310 on the SAT-I. (z = 1.40 gives .9192. 1- .9192)
b. The director of admissions for a junior college in Mississippi wants to admit as many students as
possible, but knows that his college is not right for all students. Therefore, he decides that he will not
admit students who score in the lower 4% or the upper 4% on the SAT. What are the cutoff scores on the
SAT for admission to this college?
A student must score between 651 (651.25) and 1382 (1382.75) to be admitted to this college.
(two part solution: look for .0400 in the center of the table – closest is .0401, corresponding to z = 1.75, which gives the lower cutoff score. Part 2 – look for .9600 in the center of the table – closest
is .9604,corresponding to z = 1.75, giving upper cutoff score)
3. According to Runner’s World, the times of the finishers in the New York City 10k run are normally
distributed with a mean of 61 minutes and a standard deviation of 9 minutes.
a. What proportion of runners take more than 52 minutes to finish the run?
.8413 or 84.13% of runners take longer than 52 minutes to finish the run.
(z = -1.00, giving .1587. 1-.1587)
b. What proportion of runners finish in less than 43 minutes?
.0228 is the proportion of runners who finish in less than 43 minutes. (or 2.28% of runners)
Z = -2.00, giving .0228)
c. What finish time corresponds to the 84th percentile?
A finish time of 69.91 minutes corresponds to the 84th percentile, meaning 84% of runners finish
in 69.91 minutes or less.
(look for .8400 in body of the table – closest is .8389, giving z = .99. convert)
4. Men’s heights in the United States are normally distributed with a mean of 69.0 in. and a standard
deviation of 2.5 in. In order to join the U.S. Marine Corps, a man must be between 64 and 78 in. tall.
a. What percentage of men are rejected by the Marines based on the height requirement?
2.29% of men are rejected from the Marine Corps based on the height requirement.
(the proportion who are shorter than 64 inches is .0228, and the proportion who are taller than 78
inches is .0001. Add them together to get the total.)