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Date: _____________________
Period: _______
Chapter 14: Probability & Statistics
Topic #5: Normal Distribution
Normal Distribution is a very important statistical data distribution pattern occurring in many natural
phenomena, such as height, weight, blood pressure, IQ scores, SAT scores, and so on. Certain data when
graphed as a histogram creates a bell-shaped curve known as a normal curve, or a normal distribution.
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Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the
mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
The spread of a normal distribution is controlled by the standard deviation, 𝜎.
The smaller the standard deviation the more concentrated the data.
The mean and the median are the same in a normal distribution.
Testing Tips:
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If you see the words “normally distributed” in a Regents problem, you should consider using the
normal curve graph to solve the problem.
Data is more “consistent” when the standard deviation is smaller.
Utilize the curve on the reference sheet. Mark it up as much as you need to (in pencil, in case you
need it for another question!)
Name: _______________________________________________________________
Examples:
Date: _____________________
Period: _______
1. If the scores of the 40 students on the math test are normally distributed with a mean of 85 and a standard
deviation of 2:
a) What is the probability that a student will receive between an 84 and an 87 on the test?
b) How many students got above an 88 on the test?
2. The NuBolt Company manufactures nuts and bolts. The measures of the diameters of the bolts
manufactured produce a normal distribution. The mean size of a certain bolt is 3 centimeters, with a
standard deviation of 0.1 centimeters. Bolts that vary from the mean by more than 0.3 centimeters cannot be
sold. If the company manufactures 150,000 of the 3-centimeter bolts, approximately how many of them
cannot be sold?
3. In a study of 200 students at St. Francis Prep, the researchers found that the hours that these students
studied a week were normally distributed, with a mean of 4.5 hours and a standard deviation of 1 hour.
Determine if there were 6 students who studied less than 2.5 hours.
Name: _______________________________________________________________
Date: _____________________
Period: _______
4. In a national sit-up test with a normal distribution, the mean was 42 sit-ups per minute and the standard
deviation was 2.6. Which score could be expected to occur less than 5% of the time?
(1) 50
(2) 45
(3) 39
(4) 37
5. Each year, the College Board publishes the mean SAT score and the standard deviation for the students
taking the test. SAT scores are normally distributed. Assume for a group of students that the mean SAT score
is 500 with a standard deviation of approximately 100 points. In 2007, approximately 1,500,000 students
took the SAT. Approximately how many of them would be expected to score:
a. between 400 and 450
b. between 500 and 550
c. between 650 and 750
d. between 300 and 400
6. The mean score in a national jump rope competition is 82.75 jumps per minute and the standard deviation
is 2.25. If the scores were normally distributed, which of the following scores would be most likely to occur?
(1) 90
(2) 87.25
(3) 80.5
(4) 77
7. The number of hours students watch TV a week has a normal distribution with a mean of 20.5 hours and a
standard deviation of 3 hours. If Dylan watches 25 hours of TV a week, what percentile will he lie in?
Name: _______________________________________________________________
Date: _____________________
Regents Questions
Period: _______
These are to be done IN YOUR NOTEBOOK
2009 – Fall Sampler:
1. The lengths of 100 pipes have a normal distribution with a mean of 102.4 inches and a standard
deviation of 0.2 inch. If one of the pipes measures exactly 102.1 inches, its length lies
(1) below the 16th percentile
(3) between the 16th and 50th percentiles
(2) between the 50th and 84th percentiles
(4) above the 84th percentile
August 2010
2. An amateur bowler calculated his bowling average for the season. If the data are normally distributed,
about how many of his 50 games were within one standard deviation of the mean?
(1) 14
(2) 17
(3) 34
(4) 48
January 2011
3. Assume that the ages of first-year college students are normally distributed with a mean of 19 years and
a standard deviation of 1 year.
 To the nearest integer, find the percentage of first year college students who are between the ages of
18 years and 20 years, inclusive.
 To the nearest integer, find the percentage of first-year college students who are 20 years or older.
June 2011
4. In a study of 82 game players, the researchers found that the ages of these players were normally
distributed, with a mean age of 17 years and a standard deviation of 3 years. Determine if there were 15
video game players in this study over the age of 20. Justify your answer.
January 2012
5. If the amount of time students work in any given week is normally distributed with a mean of 10 hours
per week and a standard deviation of 2 hours, what is the probability a student works between 8 and 11
hours per week?
(1) 34.1%
(2) 38.2%
(3) 53.2%
(4) 68.2%