# 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

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