Section 13.6 The Normal Curve

What You Will Learn Rectangular Distribution J-shaped Distribution Bimodal Distribution Skewed Distribution Normal Distribution z-Scores

Rectangular Distribution All the observed values occur with the same frequency.

J-shaped Distribution The frequency is either constantly increasing or constantly decreasing.

Bimodal Distribution Two nonadjacent values occur more frequently than any other values in a set of data.

Skewed Distribution Has more of a “tail” on one side than the other.

Skewed Distribution Smoothing the histograms of the skewed distributions to form curves.

Skewed Distribution The relationship between the mean, median, and mode for curves that are skewed to the right and left.

Normal Distribution The most important distribution is the normal distribution.

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 (or standard scores) determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

z-Scores The formula for finding z-scores (or standard scores) is

Example 2: Finding z-scores A normal distribution has a mean of 80 and a standard deviation of 10. Find z-scores for the following values. a) 90 b) 95 c) 80 d) 64

Example 2: Finding z-scores Solution a) 90 A value of 90 is 1 standard deviation above the mean.

Example 2: Finding z-scores Solution b) 95 A value of 90 is 1.5 standard deviations above the mean.

Example 2: Finding z-scores Solution c) 80 The mean always has a z-score of 0.

Example 2: Finding z-scores Solution d) 64 A value of 64 is 1.6 standard deviations below the mean.

To Determine 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 13.7.

To Determine the Percent of Data Between any Two Values a) When finding the percent of data to the left of a negative z-score, use Table 13.7(a).

To Determine the Percent of Data Between any Two Values b) When finding the percent of data to the left of a positive z-score, use Table 13.7(b).

To Determine the Percent of Data Between any Two Values 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%.

To Determine the Percent of Data Between any Two Values c) Or use the symmetry of a normal distribution.

To Determine the Percent of Data Between any Two Values d) When finding the percent of data between two z-scores, subtract the smaller percent from the larger percent.

To Determine the Percent of Data Between any Two Values 4. Change the areas you found in Step 3 to percents as explained earlier.

Example 5: Horseback Rides Assume that the length of time for a horseback ride on the trail at Triple R Ranch is normally distributed with a mean of 3.2 hours and a standard deviation of 0.4 hour. a) What percent of horseback rides last at least 3.2 hours? Solution In a normal distribution, half the data are above the mean. Since 3.2 hours is the mean, 50%, of the horseback rides last at least 3.2 hours.

Example 5: Horseback Rides b) What percent of horseback rides last less than 2.8 hours? Solution Convert 2.8 to a z-score. The area to the left of –1.00 is 0.1587. The percent of horseback rides that last less than 2.8 hours is 15.87%.

Example 5: Horseback Rides c) What percent of horseback rides are at least 3.7 hours? Solution Convert 3.7 to a z-score. Area to left of 1.25 is .8944 = 89.44%. % above 1.25: 1 – 89.44% = 10.56%. Thus, 10.56% of horseback rides last at least 3.7 hours.

Example 5: Horseback Rides d) What percent of horseback rides are between 2.8 hours and 4.0 hours? Solution Convert 4.0 to a z-score. Area to left of 2.00 is .9722 = 97.22%. Percent below 2.8 is 15.87%. The percent of data between –1.00 and 2.00 is 97.22% – 15.87% = 81.58%.

Example 5: Horseback Rides Solution Thus, the percent of horseback rides that last between 2.8 hours and 4.0 hours is 81.85%.

Example 5: Horseback Rides e) In a random sample of 500 horseback rides at Triple R Ranch, how many are at least 3.7 hours? Solution In part (c), we determined that 10.56% of all horseback rides last at least 3.7 hours. Thus, 0.1056 × 500 = 52.8, or approximately 53, horseback rides last at least 3.7 hours.