Normal Distributions

1
Normal Distributions
2
Normal Distribution

• Many common statistics (human height, weight,
  blood pressure) gathered from samples in the
  natural world tend to have a normal distribution
  about the mean.

3
Normal Distribution Curve

• A normal distribution curve is symmetrical,
  bell-shaped curve defined by the mean and
  standard deviation of a data set.
• Maximum at the center
• Graph is symmetric about the mean
• The mean, mode and median are very close to equal

4
42
23
23
100 of the population is graphed.
23 42 23 88
Note in this situation we do not know the
standard deviation
5
Normal Distribution

• In a normal distribution
• 68 of data fall within one standard deviation of
  the mean
• 95 of data fall within two standard deviations
  of the mean
• 99.7 of data fall within three standard
  deviations of the mean

If we know that standard deviation and the mean,
we can use them to find probabilities for the
population
6
2.8
2.8
2.8
2.8
2.8
2.8
15.7
18.5
21.3
24.1
12.9
10.1
7.3
In a sample of male European eels, the mean body
length was 15.7 inches. The standard deviation
was 2.8 inches. Sketch the normal curve.
7
34
34
13.5
13.5
2.35
2.35
13.5 3434 81.5
2.8
2.8
2.8
2.8
2.8
2.8
15.7
18.5
21.3
24.1
12.9
10.1
7.3
What percent of the male population of European
eels is between 10.1 inches and 18.5 inches long?

81.5
8

• The heights of adult American males are
  approximately normally distributed with a mean of
  69.5 in and a standard deviation of 2.5 in.
• a. Sketch the curve
• b. What percent of adult American males are
  between 67 in and 74.5 in tall?

343413.581.5
9
Normal Distribution Probability
TI 83/84
A Calculus exam is given to 500 students. The
scores have a normal distribution with a mean of
78 and a standard deviation of 5. What percent of
the students have scores between 82 and 90?
Example

• TI 83/84 directions
• Press 2ndVARS(DISTR) 2 (normalcdf)

b. Press 82 , 90 , 78 , 5
)Enter
There is a 20.37 probability that a student
scored between 82 and 90 on the Calculus exam.
normalcdf(82,90,78,5) .2036578048
10
Normal Distribution Probability
A Calculus exam is given to 500 students. The
scores have a normal distribution with a mean of
78 and a standard deviation of 5. How many
students have scores between 82 and 90?
Extension
Using the probability previously found
500 .2037 101.85
There are about 102 students who scored between
82 and 90 on the Calculus exam.
11
Normal Distribution Probability
Practice
An Algebra 2 exam is given to 1,000 students. The
scores have a normal distribution with a mean of
80 and a standard deviation of 4.5. What percent
of the students have scores above 70?
12
Normal Distribution Probability
An Algebra 2 exam is given to 1,000 students. The
scores have a normal distribution with a mean of
80 and a standard deviation of 4.5. How many
students have scores above 70?
Practice
1000.9452 986.86 About 987 students have a
score above 70 on the Algebra 2 exam.
13
Normal Distribution Probability
Find the probability of scoring below a 1400 on
the SAT if the scores are normal distributed with
a mean of 1500 and a standard deviation of 200.
Practice
Hint Use -1E99 for lower limit when you dont
know it Use 1E99 for upper limit when you dont
know it E is 2nd, on T I84
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Normal Distribution Probability
Find the probability of scoring below a 1400 on
the SAT if the scores are normal distributed with
a mean of 1500 and a standard deviation of 200.
Practice
TI 84
There is a 30.85 probability that a student
will score below a 1400 on the SAT.
Normalcdf(-1E99,1400,1500,200) .3085375322