7-7 Statistics The Normal Curve

WHAT YOU WILL LEARN • To transform data into z-scores and understand the uses for z-scores • To classify histograms into standard curves

Types of Distributions Rectangular Distribution J-shaped distribution

Types of Distributions (continued) Bimodal Skewed to right

Types of Distributions (continued) Skewed to left Normal

Properties of a Normal Distribution The graph of a normal distribution is called the normal curve. The normal curve is bell shaped and symmetric about the mean. In a normal distribution, the mean, median, and mode all have the same value and all occur at the center of the distribution.

Empirical Rule Approximately 68% of all the data lie within one standard deviation of the mean (in both directions). Approximately 95% of all the data lie within two standard deviations of the mean (in both directions). Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions).

z-Scores z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

Example: z-scores A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values. a) 55 b) 60 c) 43 a) A score of 55 is one standard deviation above the mean.

Example: z-scores (continued) b) A score of 60 is 2 standard deviations above the mean. c) A score of 43 is 1.4 standard deviations below the mean.

To Find the Percent of Data Between any Two Values 1. Draw a diagram of the normal curve, indicating the area or percent to be determined. 2. Use the formula to convert the given values to z-scores. Indicate these z- scores on the diagram. 3. Look up the percent that corresponds to each z-score in Table A.

To Find the Percent of Data Between any Two Values (continued) 4. a) When finding the percent of data to the left of a negative z-score, use Table A. b) When finding the percent of data to the left of a positive z-score, use Table A. c) When finding the percent of data to the right of a z-score, subtract the percent of data to the left of that z-score from 100%. d) When finding the percent of data between two z-scores, subtract the smaller percent from the larger percent.

Example Assume that the waiting times for customers at a popular restaurant before being seated for lunch are normally distributed with a mean of 12 minutes and a standard deviation of 3 min. a) Find the percent of customers who wait for at least 12 minutes before being seated. b) Find the percent of customers who wait between 9 and 18 minutes before being seated. c) Find the percent of customers who wait at least 17 minutes before being seated. d) Find the percent of customers who wait less than 8 minutes before being seated.

Solution a. wait for at least 12 minutes Since 12 minutes is the mean, half, or 50% of customers wait at least 12 min before being seated. b. between 9 and 18 minutes Use table 13.7 on pages 889-89 in the 8th edition. 97.7% - 15.9% = 81.8%

Solution (continued) c. at least 17 min Use Table A 100% - 95.3% = 4.7% Thus, 4.7% of customers wait at least 17 minutes. d. less than 8 min Use Table A. Thus, 9.2% of customers wait less than 8 minutes.

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are between 20 and 24? a. 79.7% b. 61.6% c. 29.7%% d. 18.1%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are between 20 and 24? a. 79.7% b. 61.6% c. 29.7%% d. 18.1%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are older than 23? a. 34.8% b. 39.1% c. 60.9% d. 65.2%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are older than 23? a. 34.8% b. 39.1% c. 60.9% d. 65.2%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are older than 20.5? a. 24.2% b. 69.6% c. 75.8% d. 30.4%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are older than 20.5? a. 24.2% b. 69.6% c. 75.8% d. 30.4%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are younger than 25.5? a. 6.9% b. 43.1% c. 75.8% d. 93.1%

The average age of students at Tri-County Community College is normally distributed with a mean of 22.1 and a standard deviation of 2.3. What percent of students are younger than 25.5? a. 6.9% b. 43.1% c. 75.8% d. 93.1%