U4WSC_NormalDistribution
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School
University of Southern California *
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Course
STAT2001
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
5
Uploaded by DeanElectron7876
Accel Adv Alg
Unit 4
–
Worksheet C
Normal Distribution & Inverse Normal
Notes:
Example 1
Round your answers to 4 decimal places.
Over a long period of time it has been determined that police response time has a normal
distribution with a mean of 8.4 minutes and a standard deviation of 1.7 minutes.
For a
randomly received emergency call, what is the probability that the response time will be:
a)
less than 5 minutes
b) more than 10 minutes
c) between 6 and 11 minutes
Example 2
Round your answers to the 4 decimal places.
At a ski area in Vermont, the daytime high temperature is normally distributed during April,
with a mean of 22°F and a standard deviation of 10°F.
You are planning a ski trip in January.
What is the probability that you will encounter daytime highs of:
a) 40°F or higher
b) 15°F or lower
Given Probability/Area Instead of X
Finding X from Probability Example #1: Warranty
Assume that the life of a TV follows a normal distribution with mean 8 years and standard
deviation 1.5 years. Suppose that a company has a warranty on its TVs and will replace a TV
that breaks while under the warranty. However, the company does not want to replace
more than 10% of the TVs sold. How long should the warranty be?
Finding X from Probability Example #2: MDI
The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a
standardized measure used in observing infants over time. It is approximately normal with a
mean of 100 and a standard deviation of 16. Find the MDI scores such that only 1% of the
population has MDI above it.
Finding X from Probability Example #3: IQ
Some IQ tests are standardized to a Normal model, with a mean of 100 and a SD of 16. Draw
the model for these IQ scores.
a)
About what percent of people should have IQ
scores above 66?
b)
What percent of people have IQ scores below 80?
c)
What percent of people have IQ scores between 85 and 125?
d)
What IQ do you need to have in order to be in the 99
th
percentile?
Example #4
Given a normal distribution of values for which the mean is 70 and the standard deviation is
4.5, find the following. Round your answers to the nearest thousandth.
a)
The probability that a value is between 65 and 80.
b)
The probability that a value is greater than or equal to 75.
c)
The probability that a value is less than or equal to 62.
d)
The 90
th
percentile for this distribution.
Example #5
Savannah earned a score of 720 on Exam A that had a mean of 700 and a standard
deviation of 25. She is about to take Exam B that has a mean of 400 and a standard
deviation of 100. How well must Savannah score on Exam B in order to do equivalently well
as she did on Exam A? Assume that scores on each exam are normally distributed.
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Worksheet C Practice
1.
The heights of all males in a U.S. high school were collected and observed to be
approximately normally distributed with a mean of 70 inches and a standard deviation of
3 inches. Use this information calculate the following.
a.
What percent of the heights will be less than 71.8 inches? Round your answers to
the nearest percent.
b.
What percent of the heights will be between 60.8 inches and 70.3 inches? Round
your answers to the nearest percent.
15% of the heights are greater than h inches.
c.
Find the value of
h
. Round your answer to the nearest whole number.
2.
The shoe size of a teenage girl is said to be normally distributed with a mean of 7 inches
and a standard deviation of 1 inch. Use this information calculate the following:
a.
What is the probability a teen
ager’s
shoe size will be greater than 7.5 inches?
Round your answer to 3 decimal places.
b.
What is the probability a teenager’s show size will be less than
6.5 inches? Round
your answer to 3 decimal places.
3.
The birthweights of babies have been said to be normally distributed with a mean weight
of 6.5 pounds and a standard deviation of 0.3 pounds. Use this information calculate the
following. What is the minimum weight for a baby to weigh in the bottom 32%. Round your
answer to 3 decimal places.
4.
In a weightlifting competition, the amount that the competitors can lift is normally
distributed with a mean of 196 kg and a standard deviation of 11 kg. Only the top 10% of
competitors will be able to advance to the next phase of the competition. What amount
must a competitor lift in order to move into the next phase of the competition?
5.
The masses of teachers in a certain school are normally distributed with mean 65 kg and
standard deviation 9.3 kg.
a.
Mr. Lee teaches at the school. What is the probability that he has a mass of more
than 75 kg?
b.
80% of teachers in the school have a mass of less than
m
kg. Determine the value
of
m.
6.
The weights of bags of rice are normally distributed with a mean of 1003 g and a
standard deviation of 2 g.
a.
Find the probability that a bag of rice weighs less than 999 g.
The manufacturer states that the bags of rice weigh 1 kg.
b.
Find the probability that a bag of rice is underweight.
400 bags of rice are weighed.
c.
Calculate the expected number of bags of rice that are underweight.
5% of the bags of rice weigh more than
p
g.
d.
Find the value of
p
.
7.
Professional golfer John Daly is known for his long drives off the tee. Suppose his drives
have a mean distance of 315 yards with a standard deviation of 12 yards. What lengths
of drives will constitute the middle 60% of all of his drives? Round your answers to the
nearest yard.