CHAPTER 13A. Normal Distributions

Transcription

1 CHAPTER 13A Normal Distributions

2 SO FAR We always want to plot our data. We make a graph, usually a histogram or a stemplot. We want to look for an overall pattern (shape, center, spread) and for any striking deviations from that pattern. We choose either the five-number summary or the mean/standard deviation to describe the center and spread of a distribution numerically. Sometimes, the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. 2

3 SO FAR If we draw a curve through the tops of the bars in a histogram, we get what is called a density curve. Unlike histograms, density curves don't show counts but proportions. The total percentage under the curve is always 100%. 3

4 THE CENTER OF A DENSITY CURVE As with histograms or stemplots, we can use the median M, the quartiles Q 1, Q 3 and the mean x to describe a density curve. We describe the shape of density curves with the same vocabulary as for histograms (symmetric, skewed). If we have a symmetric density curve, the mean is roughly equal to the median. 4

5 NORMAL DISTRIBUTION Normal curves are symmetric, bell-shaped curves that have the following properties: A specific Normal curve is completely described by its mean and its standard deviation. The tails of the distribution fall off quickly, so we do not expect any outliers. 5

6 NORMAL DISTRIBUTION The mean determines the center of the distribution. The standard deviation determines the shape of the curve. It is the distance from the mean to the change-of-curvature points on either side. 6

7 THE RULE In any Normal distribution, approximately: 68% of the observations fall within one standard deviation of the mean. 95% of the observations fall within 2 standard deviations of the mean. 99.7% of the observations fall within 3 standard deviations of the mean. This is also known as the Empirical Rule. 7

8 THE RULE Here is a picture of the rule. 8

9 EXAMPLE 13.1 The distribution of heights of women aged 18 to 24 is approximately Normal with mean 65 inches and standard deviation 2.5 inches. To use the rule, always start by drawing a picture. The middle 68% of all women will be between and inches tall. The middle 95% of all women will be between and inches tall. The middle 99.7% of all women will be between and inches tall. 9

10 EXAMPLE 13.2 Researchers in Chicago collected the daily high temperature during the month of August from 1944 to After drawing a histogram, they noticed it was approximately Normal with mean 80 F and standard deviation 8 F. About what percent of the days in Chicago in August will have a high temperature between 72 F and 88 F? 68% 10

11 EXAMPLE 13.3 Heights of adults, ages Men mean: 70.0 inches standard deviation: 2.8 inches So 68% of men are between and inches 95% of men are between and inches 99.7% of men are between and inches 11

12 EXAMPLE 13.3 What proportion of men are less than 72.8 inches tall? What proportion of men are more than 75.6 inches tall? 12

13 EXAMPLE 13.4 The figure below is a stemplot of the IQ test scores of 74 seventh-grade students. This distribution is very close to Normal with mean 111 and standard deviation 11. It includes all the seventh-graders in a rural Midwest school. Take the Normal distribution with mean 111 and standard deviation 11 as a description of the IQ test scores of all rural Midwest seventh-grade students. Use this distribution and the rule to answer the following questions Key 6,9 8 6 represents 86 0,1,3,3,6,7,7,8 0,0,2,2,3,3,3,3,4,4,5,5,5,6,6,6,7,7,7,7,8,9 0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,8,8,9,9,9 0,0,3,3,4,4,6,7,7,8,8,8 0,2,6 13

14 EXAMPLE 13.4 Between what values do the IQ scores of 95% of all rural Midwest seventh-graders lie? What percentage of IQ scores for rural Midwest seventh-graders are less than 100? What percentage of all students have IQ scores 144 or higher? None of the 74 students in our sample school had a score this high, does this surprise you? 14

15 REMINDERS We will discuss Chapter 13b next. Chapter 13 homework is posted online and is due tomorrow. 15