Download Distribution Review Problems

Distribution Review Problems
ANSWERS
1.
The heights of young American women, X, follows the normal
distribution with mean =65.5 inches and standard deviation =2.5
inches.
a.
What is the chance that a randomly selected female is at least
67 inches tall?
b.
P(X<62)
c.
P(X65.5)
2.
Among applicants to one law school in 1976 the average LSAT score
was 600 with a standard deviation of 100. The LSAT scores followed
a normal distribution.
a.
What is the chance a selected applicant scored above a 750?
b.
If an applicant’s score was in the 90th percentile, what was her
score?
c.
Do we know what the median score was? What is the
difference between mean and median?
3.
A flashlight battery is guaranteed to last for 40 hours. Tests indicate
that the length of life of these batteries is normally distributed with
mean 50 and variance 64.
a.
What is the chance a battery will fail the guarantee?
b.
What is the chance a battery will not fail the guarantee?
4.
Suppose the population mean subscription price for daily newspapers
in the US is $6.50/week (all 7 days); the population standard deviation
is $2.00. (Assume the prices are for local subscriptions and the
distribution of prices is normal).
a.
Identify what the population is in this problem.
b.
If a newspaper is randomly selected from the population, what
is the probability its local subscription rate will be at least
$8.00/week?
c.
What is the chance a randomly selected newspaper will have a
local subscription rate of less than $4.25 per week?
5.
Suppose the population mean family income of Cal State LA students
this quarter is $45,500. The population standard deviation is
$2738.61.
a.
Identify what the population is in this problem.
b.
Calculate the probability that a randomly selected individual
from the population has a family income within $500 of the
population mean (assume the population of family incomes
follows a normal distribution).
c.
If a random sample of 30 students were taken from the
population, calculate the probability that the sample mean
income will be within $500 of the actual population mean.
d.
Suppose instead a sample of 60 students were taken. What is
the probability the sample mean will be within $500 of the
actual population mean?
e.
Explain why taking a larger sample is better than taking a
smaller one.
f.
Explain why the normality assumption for the population may
not be appropriate for this problem.
Distribution Review Problems
1.
The heights of young American women, X, follows the normal distribution with
mean =65.5 inches and standard deviation =2.5 inches.
a. What is the chance that a randomly selected female is at least 67 inches tall?
.2743
b. P(X<62) .0808
c. P(X65.5) .5
2.
Among applicants to one law school in 1976 the average LSAT score was 600
with a standard deviation of 100. The LSAT scores followed a normal
distribution.
a. What is the chance a selected applicant scored above a 750? .0668
b. If an applicant’s score was in the 90th percentile, what was her score? 728
c. Do we know what the median score was? What is the difference between
mean and median?
The median is the middle observation in a data set ordered from lowest to highest.
For a symmetric distribution the mean would equal the median.
3.
A flashlight battery is guaranteed to last for 40 hours. Tests indicate that the
length of life of these batteries is normally distributed with mean 50 and variance
64.
a. What is the chance a battery will fail the guarantee? .1056
b. What is the chance a battery will not fail the guarantee? .8944
4.
Suppose the population mean subscription price for daily newspapers in the US is
$6.50/week (all 7 days); the population standard deviation is $2.00. (Assume the
prices are for local subscriptions and the distribution of prices is normal).
a. Identify what the population is in this problem. US daily newspapers
b. If a newspaper is randomly selected from the population, what is the
probability its local subscription rate will be at least $8.00/week? .2266
c. What is the chance a randomly selected newspaper will have a local
subscription rate of less than $4.25 per week? .1292
5.
Suppose the population mean family income of Cal State LA students this quarter
is $45,500. The population standard deviation is $2738.61.
a. Identify what the population is in this problem. Cal State LA students this
quarter
b. Calculate the probability that a randomly selected individual from the
population has a family income within $500 of the population mean (assume
the population of family incomes follows the normal distribution). .1428
c. If a random sample of 30 students were taken from the population, calculate
the probability that the sample mean income will be within $500 of the actual
population mean. .6826
d. Suppose instead a sample of 60 students were taken. What is the probability
the sample mean will be within $500 of the actual population mean? .84
e. Explain why taking a larger sample is better than taking a smaller one.
A larger sample decreases the standard deviation of the sample mean.
f. Explain why the normality assumption for the population may not be
appropriate for this problem. Large incomes in the population normally skew
the distribution to the right. The normal distribution is symmetric. If the
sample size is at least 30 observations, the distribution of x is approximately
normal, regardless of the distribution of the population.