Objectives Students will learn how to use tables to estimate areas under normal curves and recognize data sets that are not normal.

Mathematical Symbols A population is a whole, it’s every member of a group. A sample mean is a part of a population; which is a fraction of a whole group. Note: The sample mean is useful because it allows you to estimate what the whole population is doing, without surveying everyone The measurement of how spread out numbers are from mean.  The sum of a sequence of things that are all to be added together.

When a survey is used to gather data, it is important to consider how the sample is selected for the survey. If the sampling method is biased, the survey will not accurately reflect the population. Most national polls that are reported in the news are conducted using careful sampling methods in order to minimize bias. Other polls, such as those where people phone in to express their opinion, are not usually reliable as a reflection of the general population Remember that a random sample is one that involves chance. Six different types of samples are shown below.

432 450 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. 432 450

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.

Example 1, You try it! 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 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 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.

Example 2: Using Standard Normal Values Continued… 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. = 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.

Example 2, You try it! 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.

Z Area Below z X Values Below z Proj. Act. -2 0.02 13.1 1 -1 0.16 23.6 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? 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.

Example 3, You Try it! 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.