σ. Step 3: Find the corresponding area using the Z-table.

Transcription

1 Section 6.2: Applications of the Normal Distribution Suppose that the scores for a standardized test are normally distributed, have a mean of 100, and have a standard deviation of 15. Then we can use the formula z = X µ value mean or z = to convert the values of the σ standard deviation variable into standard units or z-scores. Once the variable is transformed, and then we can use the Z-table to solve problems. Thus, the procedure to finding area under a normal curve can be summarized into 3 steps. Step 1: Draw a normal curve and shade the desired area. Step 2: Convert the values of X to z-values, using the formula z = X µ σ. Step 3: Find the corresponding area using the Z-table. Exercise 1. An adult has on average 5.2 liters of blood. Assume the variable is normally distributed and has a standard deviation of 0.3. Find the percentage of people who have less than 5.4 liters of blood in their system. 1

2 Exercise 2. Each month, an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the variable is approximately normally distributed and the standard deviation is 2 pounds. If a household is selected at random, find the probability of it generating a) Between 27 and 31 pounds per month b) More than 30.2 pounds per month Exercise 3. A desktop PC uses 120 watts of electricity per hour basked on 4 hours of use per day. Assume the variable is approximately normally distributed and the standard deviation is 6. If 500 PCs are selected, approximately how many will use less than 106 watts of power? 2

3 Exercise 4. Newborn elephant calves usually weigh between 200 and 250 pounds until October 2006, that is. An Asian elephant at the Houston (Texas) Zoo gave birth to a male calf weighing in at a whopping 384 pounds! Mack (like the truck) is believed to be the heaviest elephant calf ever born at a facility accredited by the Association of Zoos and Aquariums. If, indeed, the mean weight for newborn elephant calves is 225 pounds with a standard deviation of 45 pounds, what is the probability of a newborn weighing at least 384 pounds? Assume that the weights of newborn elephants are normally distributed. Moreover, a normal distribution can also be used to find specific data values for given percentages. In this case you are given a probability or percentage and need to find the corresponding data value X. You can use the formula z = X µ and solve for X. That is, X = z σ + µ. Thus, the procedure for σ finding data values for specific probabilities can be summarize into 3 steps. Step 1: Draw a normal curve and shade the desired area that represents the probability, proportion, or percentile. Step 2: Find the z-value from the table that corresponds to the desired area. Step 3: Calculate the X value by using the formula X = z σ + µ 3

4 Exercise 5. To qualify for a police academy, candidates must score in the top 10% on a general abilities test. Assume the test scores are normally distributed and the test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Moreover, there are several ways statisticians check for normality. The easiest way is to draw a histogram for the data and check its shape. If the histogram is not approximately bell-shaped, then the data are not normally distributed. Skewness can be checked by using the Person Coefficient (PC) of skewness also called Pearson s index of skewness. The formula is ( ) PC = 3 X median. s If the index is greater than or equal to +1 or less than or equal to -1, it can be concluded that the data are significantly skewed. In addition, the data should be checked for outliers by using the method shown in chapter 3. Even one or two outliers can have a big effect on normality. 4

5 Exercise 6. A survey of 18 high-tech firms showed that the number of days inventory they had on hand. Determine if the data are approximately normally distributed by a) Constructing a frequency distribution (7classes) and drawing a histogram. b) Use PC test 5

6 Exercise 7. The data shown consist of the number of games played each year in the career of Baseball Hall of Famer Bill Mazeroski. Determine if the data are approximately normally disturbed. 6