PGCC Statistics Normal Distribution Worksheet

Normal Distribution (concepts)1. If the area to the left of xx in a normal distribution is 0.492, what is the area to the right
of xx?
Answer:
If the area to the right of xx in a normal distribution is 0.492, what is the area to the left
of xx?
Answer:
2. If XX represents a random variable coming from a normal distribution
and P(X5.4)=0.69.
True
False
3. If a distribution is normal, then it is not possible to randomly select a value that is more
than 4 standard deviations from the mean.
•
True
•
False
4. Since the area under the normal curve within two standard deviations of the mean is 0.95,
the area under the normal curve that corresponds to values greater than 2 standard
deviations above the mean is 0.05.
•
True
•
False
5. A variable xx is normally distributed with mean 22 and standard deviation 8.
Round your answers to the nearest hundredth as needed.
a) Determine the zz-score for x=27x=27.
z=z=
b) Determine the zz-score for x=15x=15.
z=z=
c) What value of xx has a zz-score of 0.750.75?
x=x=
d) What value of xx has a zz-score of 00?
x=x=
e) What value of xx has a zz-score of 00?
x=x=
6. In a normal distribution, a data value located 1.2 standard deviations below the mean has
Standard Score: z =
In a normal distribution, a data value located 2.2 standard deviations above the mean has
Standard Score: z =
In a normal distribution, the mean has Standard Score: z =
Normal Distribution
Help: Help: https://www.youtube.com/watch?v=gdO15dxWONI

1. Suppose that the distance of fly balls hit to the outfield (in baseball) is
normally distributed with a mean of 256 feet and a standard deviation of 37
feet. Let X be the distance in feet for a fly ball.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that a randomly hit fly ball travels less than 312 feet.
Round to 4 decimal places.
c. Find the 70th percentile for the distribution of distance of fly balls. Round to
2 decimal places.
feet
2. In the 1992 presidential election, Alaska’s 40 election districts averaged 2153
votes per district for President Clinton. The standard deviation was 597. (There
are only 40 election districts in Alaska.) The distribution of the votes per
district for President Clinton was bell-shaped. Let X = number of votes for
President Clinton for an election district. (Source: The World Almanac and Book
of Facts) Round all answers except part e. to 4 decimal places.
a. What is the distribution of X? X ~ N(
,
)
b. Is 2153 a population mean or a sample mean?
c. Find the probability that a randomly selected district had fewer than 2208
votes for President Clinton.
d. Find the probability that a randomly selected district had between 2168 and
2340 votes for President Clinton.
e. Find the third quartile for votes for President Clinton. Round your answer to
the nearest whole number.
3. The amount of calories consumed by customers at the Chinese buffet is
normally distributed with mean 2864 and standard deviation 630. One randomly
selected customer is observed to see how many calories X that customer
consumes. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that the customer consumes less than 2539
calories.
c. What proportion of the customers consume over 3079 calories?
d. The Piggy award will given out to the 1% of customers who consume the
most calories. What is the fewest number of calories a person must consume to
receive the Piggy award?
calories. (Round to the nearest calorie)
4. On the planet of Mercury, 4-year-olds average 3 hours a day unsupervised. Most
of the unsupervised children live in rural areas, considered safe. Suppose that
the standard deviation is 1.7 hours and the amount of time spent alone is
normally distributed. We randomly survey one Mercurian 4-year-old living in a
rural area. We are interested in the amount of time X the child spends alone
per day. (Source: San Jose Mercury News) Round all answers to 4 decimal
places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that the child spends less than 2 hours per day
unsupervised.
c. What percent of the children spend over 4.3 hours per day
unsupervised.
% (Round to 2 decimal places)
d. 89% of all children spend at least how many hours per day
unsupervised?
hours.
5. The average THC content of marijuana sold on the street is 9.4%. Suppose the
THC content is normally distributed with standard deviation of 1%. Let X be the
THC content for a randomly selected bag of marijuana that is sold on the
street. Round all answers to 4 decimal places where possible,
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that a randomly selected bag of marijuana sold on the
street will have a THC content greater than 11.
c. Find the 74th percentile for this distribution.
%
6. Suppose that the speed at which cars go on the freeway is normally distributed
with mean 70 mph and standard deviation 8 miles per hour. Let X be the speed
for a randomly selected car. Round all answers to 4 decimal places where
possible.
a. What is the distribution of X? X ~ N(
,
)
b. If one car is randomly chosen, find the probability that it is traveling more
than 67 mph.
c. If one of the cars is randomly chosen, find the probability that it is traveling
between 72 and 75 mph.
d. 89% of all cars travel at least how fast on the freeway?
mph.
7. Private nonprofit four-year colleges charge, on average, $27,709 per year in
tuition and fees. The standard deviation is $7,205. Assume the distribution is
normal. Let X be the cost for a randomly selected college. Round all answers to
4 decimal places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that a randomly selected Private nonprofit four-year
college will cost less than 27,227 per year.
c. Find the 73rd percentile for this distribution. $
dollar.)
(Round to the nearest
8. Los Angeles workers have an average commute of 29 minutes. Suppose the LA
commute time is normally distributed with a standard deviation of 13 minutes.
Let X represent the commute time for a randomly selected LA worker. Round
all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that a randomly selected LA worker has a commute that
is longer than 25 minutes.
c. Find the 85th percentile for the commute time of LA
workers.
minutes
9. The average number of words in a romance novel is 64,212 and the standard
deviation is 17,467. Assume the distribution is normal. Let X be the number of
words in a randomly selected romance novel. Round all answers to 4 decimal
places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the proportion of all novels that are between 72,946 and 81,680
words.
c. The 80th percentile for novels is
words. (Round to the nearest word)
d. The middle 50% of romance novels have from
to
words
words. (Round to the nearest word)
10. According to a study done by UCB students, the height for Martian adult males
is normally distributed with an average of 64 inches and a standard deviation of
2.3 inches. Suppose one Martian adult male is randomly chosen. Let X = height
of the individual. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. Find the probability that the person is between 64.7 and 66.6
inches.
c. The middle 20% of Martian heights lie between what two numbers?
Low:
inches
High:
inches