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```Section 5.4
The Normal Distribution
Normal Distribution
Data that, when graphed as a histogram or a frequency polygon, results in a
unimodal symmetric distribution about the mean.
Normal Curve A symmetrical curve that represents the normal distribution; also called a bell curve.
See page 241. Explore…. There are 4 different graphs pictured, which one represents a normal
distribution?
Properties of a Normal Distribution:

Every normal distribution has a mean and a standard deviation.

The graph is symmetrical. The mean, median and mode are very similar, if not equal, and fall on
the line of symmetry.

50% of data lies above line and 50 % data is below that

Almost all of the data lies within 3 standard deviations of the mean

The total area under the curve is 1 (100%)

Generally, measurements of living things (i.e. mass, height, length, blood pressure, test scores)
have a normal distribution.

The higher the standard deviation the flatter the graph. The lower the standard deviation the
higher the graph
Sketch the following normally distributed samples for each set of data – example 3 pg.247
Weight
Team
Men
Women
u (kg)
Weight
u (kg)
Please look through Example #1 on page 242!
The 68-95-99.7 Rule
This rule is used to make predictions about data that is normally distributed.
Jim raises Siberian husky dogs. The weights of adult dogs are normally distributed with a mean of 52.5
lbs and a standard deviation of 2.4 lbs. What % of dogs would you expect to have a weight between 47.7
lbs and 54.9 lbs?
Let’s do together:
1)
95% of students at school are between 1.1m and 1.7m tall.
Assuming this data is normally distributed can you calculate the mean and standard deviation?
Step #1: Calculate the mean:
Step #2: Calculate one standard deviation. 95% is 2 standard deviations either side of the mean (a total
of 4 standard deviations), so…
Step #3: Draw a normal curve.
2)
The sitting height (from seat to top of head) of drivers must be considered in the design of a new car
model. Men have sitting heights that are normally distributed with a mean of 36.0 in and a standard deviation of
1.4 in.
Draw a normal distribution curve for the data.
What is the probability that a man has a sitting height between 34.6 in and 38.8 in?
What is the probability that a man has a sitting height between 33.2 in and 37.4 in?
What is the probability that a man has a sitting height greater than 37.4 in?
What is the probability that a man has a sitting height between 31.8 in and 36 in?
3)
The lengths of pregnancies are normally distributed with a mean of 268 days and a
standard deviation of 15 days.
Draw a normal distribution curve for the data.
What is the probability of a pregnancy lasting...
a.) between 223 and 283 days?
b.) between 223 and 268 days?
c.) between 253 and 313 days?
d.) less than 253 days?
Ex (1) The mean score on a math exam was 135 and the standard deviation was 15.
a)
What percent were between 105 and 165?
b)
What was the lowest score on the upper 2.5% of the class?
c)
Suppose that 930 students wrote the test. How many scored less than 120?
Ex (2) A fast food restaurant manager discovers that the time spent by customers is normally distributed with a
mean of 25 minutes and a deviation of 7 minutes. If 346 people visit ther restaurant in one day how many stay
longer than 32 minutes?
REVIEW:
For the following statements, identify those that are properties of normal curves (P) and those that are not
properties (NP).
The curve is bell shaped with the highest point over the mean μ.
The curves has most of its data concentrated to the right (highest values).
A horizontal line divides the curve into two symmetric parts.
The curve approaches the horizontal line but never crosses it.
The transition points between cupping upward and downward occur above μ σ and μ + σ.
It is symmetric about a vertical line through μ.
Approximately 98% of the data lies between μ σ and μ + σ.
Sketch the normal curve with a mean of 10 and a standard deviation of 3.
For a normal curve with a mean of 20 and a standard deviation of 3.5, where are the transition points?
For a normal curve with a mean of 100 and a standard deviation of 8.2, where are the transition points?