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

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 4. Determine which data set has the greater standard deviation without calculating it. Explain.

Warm Up : Continued Set A: {73, 120, 54, 81, 66} Set B: {83, 95, 106, 99, 82}. Set A; the values are further apart.

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

Vocabulary standard normal value

Example 1: Finding Joint and Marginal Relative Frequencies Jamie can drive her car an average of 432 gallons per tank of gas, with a standard deviation of 36 miles. Use the graph to estimate the probability that Jamie will be able to drive more than 450 miles on her next tank of gas.

Example 1 : Continued

Example 1 : Continued The area under the normal curve is always equal to 1. Each square on the grid has an area of 10(0.001) = 0.01. Count the number of grid squares under the curve for values of x greater than 450. There are about 31 squares under the graph, so the probability is about 31(0.01) = 0.31 that she will be able to drive more than 450 miles on her next tank of gas.

Check It Out! Example 1 estimate the probability that Jamie will be able to drive less than 400 miles on her next tank of gas?

Check It Out! Example 1 continued There are about 19 squares under curve less than 400, so the probability is about 19(0.01) = 0.19 that she will be able to drive less than 400 miles on the next tank of gas.

Example 2: Using Standard Normal Values First, find the standard normal value of 148, using μ = 160 and σ = 12. 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. = µ Z X σ 148 160 12 1

Example 2: Continued Use the table to find the area under the curve for all values less than 1, which is 0.16. 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.

Example 2: Using Standard Normal Values continued = µ Z X σ 154 160 12 0.5 Area=0.31 = µ 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.

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. µ 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 0.02. The probability of scoring above 106 is 0.98.

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?

Example 3: Continued Z Area Below z X Values Below z Proj. Act. -2 0.02 13.1 1 -1 0.16 23.6 3 5 0.5 34.1 10 0.84 44.6 17 19 2 0.98 55.1 20 No, the data does not appear to be normally distributed. There are only 5 values below the mean.

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.

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

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?

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