12.4. The Normal Distribution: A Problem-Solving Tool

Transcription

1 12.4. The Normal Distribution: A Problem-Solving Tool 1

2 Objectives A. Find the mean and standard deviation from a normal curve. B. Find the z-score of a measurement from a normally distributed set of data. C. Know the distribution of z-scores and their relationship to the probability of an event under a normal curve. 2

3 Introduction (Normal Curve) S.A.T. Deviations Do you remember your S.A.T. writing and mathematics scores? In a recent year, the mean score for the writing portion was (read mu ) = 489 with a standard deviation of (read sigma ) = 113, whereas the mathematics scores had a mean of 514 and a standard deviation of 117. Suppose you scored 603 on the verbal portion and your friend scored 631 on the mathematics portion. 3

4 Introduction (Normal Curve) Which is the better score? It might not be the 631. Statistically, you may still beat your friend but to see how, you have to learn about the normal curve shown in Figure Area under a normal curve. Figure

5 Introduction (Normal Curve) A normal curve describes data that have a very large (or infinite) number of values distributed among the population in a bell shape. A large number of the values are near the middle with a few values trailing off in either direction. 5

6 Introduction (Normal Curve) Statisticians call a distribution with a bell-shaped curve a normal distribution. 6

7 Introduction (Normal Curve) Now, for the rest of the story. If we assume that scores on the writing and mathematics portions of the S.A.T. are normally distributed and have the means and standard deviations mentioned, we can label the curves in Figures and with their respective means, = 489 and 514. Writing S.A.T. scores. Math S.A.T. scores. Figures Figures

8 Introduction (Normal Curve) Remember that is the mean of a sample population and s is its standard deviation, but is the mean of the entire population and is its standard deviation. In Figure 12.11, 1 standard deviation to the right of the mean will be +, or = 602, whereas in Figure 12.12, 1 standard deviation to the right of the 514 mean will be 631 ( ). 8

9 Introduction (Normal Curve) Now, a score of 603 on the writing will be slightly to the right of 1 standard deviation (602), whereas a score of 631 on the mathematics will be exactly 1 standard deviation from the mean. Believe it or not, a 603 writing score is comparatively better than a 631 mathematics score! 9

10 Normal Distribution Look at the curves in Figure Figure They are not normal distributions! The curve labeled (a) is not symmetric, (b) is not bell shaped, (c) crosses the x axis, and (d) has tails turning up away from the x axis. 10

11 Normal Distribution We show some normal curves in Figure Figure

12 Example Consider the normal curves in Figure (a) What is the mean for A? (b) What is the mean for B? (c) What is the standard deviation for A? (d) What is the standard deviation for B? (e) What percent of the values would you expect to lie between 3 and 1 in B? (f) What percent of the values would you expect to lie between 0 and 1 in A? 12

13 Solution (a) The mean for A is 0 (under the highest point). (b) The mean for B is also 0. (c) The interval from 0 to 1 must have 3 standard deviations, so each of them must be unit. (d) The standard deviation for B is 1 (there are 3 to the right of 0). 13

14 Solution (e) Since there are 2 standard deviations between 3 and 1, 2.5% % = 16% of the values would be in that region. (Refer to Figure 12.10, for the values.) Area under a normal curve. Figure (f) Half (50%) of the values should be between 0 and 1. 14

15 Remark Now suppose Rudie earned a score of 80 on her U.S. history test and a score of 80 on her geometry test. Which of these is the better score? Without additional information, we cannot answer this question. However, if we are told that the mean score in the U.S. history test was 60, with a standard deviation of 25.5, and the mean score in the geometry test was 70, with a standard deviation of 14.5, then we can use a technique similar to the one used in Getting Started to compare Rudie s two scores. 15

16 Standardized Score( z-score ) In order to make a valid comparison, we have to restate the scores on a common scale. A score on this scale is known as a standardized score or a z-score. 16

17 Standardized Score( z-score ) Since the numerator of z is the difference between x and the mean, the z-score gives the number of standard deviations that x is from the mean. 17

18 Example Compare Rudie s scores in U.S. history and geometry, given all the preceding information. 18

19 Rudie s z-scores are Solution Thus, Rudie did better in U.S. history than in geometry. 19

20 Distribution of z-scores For a normal distribution of scores, if we subtract from each score, the resulting numbers will have a mean of 0. If we then divide each number by the standard deviation, the resulting numbers will have a standard deviation of 1. Thus, the z-scores are distributed as shown in Figure Distribution of z-scores. Figure

21 Distribution of z-scores For instance, 34% of the z-scores lie between 0 and 1, 13.5% lie between 1 and 2, and 2.5% are greater than 2. For such a distribution of scores, the probabilities of randomly selecting z-scores between 0 and a given point to the right of 0 have been calculated and appear in tables. To read the probability that a score falls between 0 and 0.25 standard deviation above the mean, we go down the column under z to 0.2 and then across to the column under 5; the number there is 0.099, the desired probability. This probability is actually the area under the curve between 0 and

22 Example Time for a break: We are headed to the ice cream parlor! The combined weight of the two scoops on a double-dip ice cream cone satisfies a normal distribution with a mean of 8 oz and a standard deviation of oz. But one double-dip looks a little bit larger. As a matter of fact, it weighs 8.5 oz, and you picked it! What is the probability that a randomly selected cone is smaller than yours? 22

23 Solution To find the answer, we find the probability that the amount of ice cream in a randomly selected cone is less than yours, that is, that the z-score for the other cones is less than the z-score of yours. The z-score for your 8.5-oz ice cream cone is 23

24 Solution The value for (red area), and the total value for all dips with z-scores under 2 is the yellow area plus the red area, that is, z-scores This means that the probability that the other dips are smaller than yours is 97.7%. You have a good eye for ice cream! 24

25 Solution You also can do this problem by looking at Figure and observing that the area under the curve to the left of is approximately Distribution of z-scores. Figure