2-3 Normal Distribution

Normal • A normal distribution is modeled by a bell-

shaped curve called a normal curve that is symmetric about the mean.

• A normal distribution with mean and standard deviation has the following properties. The total area under the related normal curve is 1.

x

• The percent of the area covered by each standard deviation is shown in the graph.

1

68

95

99.7

𝑥 𝑥 +2𝜎

34% 13.5% 47.5%

13.5% + 2.35% + 0.15% = 16%

a. The scores of 518 and 748 represent 2 standard deviations to the right of the mean. So, the percent of the test-takers that have scores between 518 and 748 is 34% + 13.5% = 47.5%.

b. A score of 403 is 1 standard deviation to the left of the mean. So, the percent of the test-takers that have scores less than 403 is 13.5% + 2.35% + 0.15% = 16%.

Standard Normal Distribution

x xz

58 − 64.5

2.5 -2.6

-2.6 -2.6 0.0047

.47%

34% + 34% = 68%

𝑍 =70−64.5

2.5=2.2, so 0.9861 or 98.61% is the

probability.

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