1. Fitting to a Normal Distribution
8-7
Fitting to a Normal Distribution
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt Algebra 2

2. Find the mean and standard deviation of each data set.
Warm Up
Find the mean and standard deviation
of each data set.
1. {2, 10, 5, 3}
mean: 5; std. dev. ≈ 3.08
2. {30, 30, 60}
mean: 40; std. dev. ≈ 14.1
3. {2, 2, 2, 2,2}
mean: 2; std. dev. = 0

3. Objectives
Use tables to estimate areas under normal curves. Recognize data sets that are not normal.

4. Vocabulary
standard normal value

6. Example 2: Using Standard Normal Values
Scores on a test are normally distributed with a mean of 160 and a standard deviation of 12.
A. Estimate the probability that a randomly selected student scored less than 148.
First, find the standard normal value of 148, using μ = 160 and σ = 12. = Z X σ
148
160 12 1

7. Example 2: Continued
Use the table to find the area under the curve for all values less than 1, which is The probability of scoring less than 148 is about 0.16.
B. Estimate the probability that a randomly selected student scored between 154 and 184.
Find the standard normal values of 154 and 184. Use the table to find the areas under the curve for all values less than z.

8. Example 2: Using Standard Normal Values continued= Z X σ
154
160 12
0.5
Area=0.31 = Z X σ
184
160 12 2
Area=0.98
Subtract the areas to eliminate where the regions overlap. The probability of scoring between 154 and 184 is about 0.98 – 0.31 = 0.67.

9. Check It Out! Example 2
Scores on a test are normally distributed with a mean of 142 and a standard deviation of 18. Estimate the probability of scoring above 106.
First, find the standard normal value of 106, using μ = 142 and σ = 18. Z = X σ
106
142 18 2
Use the table to find the area under the curve for all values less than –2, which is The probability of scoring above 106 is 0.98.

10. Example 3: Determining Whether Data May Be Normally Distributed
The lengths of the 20 snakes at a zoo, in inches, are shown in the table. The mean is 34.1 inches and the standard deviation is 10.5 inches. Does the data appear to be normally distributed?

11. Example 3: Continued
No, the data does not appear to be normally distributed. There are only 5 values below the mean.

12. Check It Out! Example 3
A random sample of salaries at a company is shown. If the mean is $37,000 and the standard deviation is $16,000, does the data appear to be normally distributed?
No, the data does not appear to be normally distributed. 14 out of 18 values fall below the mean.

13. Lesson Quiz: Part I
Scores on a test are normally distributed with a mean of 200 and a standard deviation of 12. Find each probability.
1. A randomly selected student scored less than 218.
0.93
2. A randomly selected student scored between
182 and 200.
0.43
3. A randomly selected student scored between 182 and 188.
0.09

14. Lesson Quiz: Part II
4. A randomly selected student scored above 224.
0.02
5. The weights, in grams, of 30 randomly chosen apples from a large bin are shown below. The mean weight is 110 grams and the standard deviation is 5.5 grams. Does the data appear to be normally distributed?

15. Lesson Quiz: Part III
Yes; the projected number of values for each value of z is close to the actual number of data values.