CHAPTER 2 Modeling Distributions of Data

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1 CHAPTER 2 Modeling Distributions of Data 2.2 Density Curves and Normal Distributions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers

2 HW 34. Sketch a density curve that might describe a distribution that ha a single peak and is skewed to the left. 40. The figure below displays two density curves, each with three points marked. At which of these points on the curve do the mean and median fall? a) Mean is A, median is A (the distribution is symmetric, so the mean equals the median). b) Mean is A, median is B (the left skew pulls the mean to the left of the median). The Practice of Statistics, 5 th Edition 2

3 HW 42. The distribution of heights of adult American men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Draw an accurate sketch of the distribution of men s heights. Be sure to label the mean as well as the points 1, 2, and 3 standard deviations away from the mean on the horizontal axis. 44. Refer to Exercise 42. Use the rule to answer the following questions. a) Between what weights do the middle 68% of bags fall? ounces b) What percent of bags weigh less than 9.02 ounces? 2.5% c) What percent of 9-ounce bags of this brand of potato chips weigh between 8.97 and 9.17 ounces? = 83.85% d) A bag that weighs 9.07 ounces is at what percentile in this distribution? 16 th percentile The Practice of Statistics, 5 th Edition 3

4 Try This IQ Scores Some IQ tests are standardized to a Normal model, with a mean of 100 and a standard deviation of 16. Draw an accurate sketch of the distribution of IQ tests. Be sure to label the mean as well as the points 1, 2, and 3 standard deviations away from the mean on the horizontal axis. a) In what interval would you expect the central 95% of IQ scores to be found? 68 to 132 points b) About what percent of people should have IQ scores above 116? 16% c) About what percent of people should have IQ scores between 68 and 84? (95-68)/2 = 13.5% d) What is the range of scores for the top 2.5% 132 or greater The Practice of Statistics, 5 th Edition 4

5 Density Curves and Normal Distributions Learning Objectives After this section, you should be able to: FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence. The Practice of Statistics, 5 th Edition 5

6 The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ from the mean µ as center. The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable z = x - m s has the standard Normal distribution, N(0,1). The Practice of Statistics, 5 th Edition 6

7 The Standard Normal Table The standard Normal Table (Table A) is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81 standard deviations above the mean. We can use Table A: P(z < 0.81) =.7910 Z The Practice of Statistics, 5 th Edition 7

8 Normal Distribution Calculations We can answer a question about areas in any Normal distribution by standardizing and using Table A or by using technology. How To Find Areas In Any Normal Distribution Step 1: State the distribution and the values of interest. Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and boundary value(s) clearly identified. Step 2: Perform calculations show your work! Do one of the following: (i) Compute a z-score for each boundary value and use Table A or technology to find the desired area under the standard Normal curve; or (ii) use the normalcdf command and label each of the inputs. Step 3: Answer the question. The Practice of Statistics, 5 th Edition 8

9 Finding area to the right Suppose we wanted to find the proportion of observations in a Normal distribution that were more than 1.53 standard deviations above the mean. That is, we want to know what proportion of observations in the standard Normal distribution are greater than z = To find this proportion, locate the value 1.5 in the left-hand column of Table A, then locate the remaining digit 3 as.03 in the top row. The corresponding entry is This is the area to the left of z = To find the area above z = 1.53, subtract from 1 to get The Practice of Statistics, 5 th Edition 9

10 Finding areas under the standard Normal curve Problem: Find the proportion of observations from the standard Normal distribution that are between 0.58 and Solution: To find this proportion, we must find the proportion of values that are less than z = 1.79 and then subtract the proportion of values that are less than z = The difference in these proportions is the proportion of observations that are between z = 0.58 and z = The Practice of Statistics, 5 th Edition 10

11 Working Backwards In a standard Normal distribution, 20% of the observations are above what value? Using Table A, we should look up an area of because the table always lists area to the left of a boundary. The closest area to is , which corresponds to a z-score of z = Thus, approximately 20% of the observations in a standard Normal distribution are above z = The Practice of Statistics, 5 th Edition 11

12 Working Backwards: Normal Distribution Calculations Sometimes, we may want to find the observed value that corresponds to a given percentile. There are again three steps. How To Find Values From Areas In Any Normal Distribution Step 1: State the distribution and the values of interest. Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and unknown boundary value clearly identified. Step 2: Perform calculations show your work! Do one of the following: (i) Use Table A or technology to find the value of z with the indicated area under the standard Normal curve, then unstandardize to transform back to the original distribution; or (ii) Use the invnorm command and label each of the inputs. Step 3: Answer the question. The Practice of Statistics, 5 th Edition 12

13 Serving speed The Practice of Statistics, 5 th Edition 13

14 Serving speed (continued) Problem: What percent of Rafael Nadal s serves are between 100 and 110 mph? Step 1: State the distribution and the values of interest. Rafael Nadal s serve speed follows a Normal distribution with mean 115 and standard deviation 6. We want to find the percent of his serves that are between 100 and 110 mph. The Practice of Statistics, 5 th Edition 14

15 Heights of three-year-old females The Practice of Statistics, 5 th Edition 15

16 Assessing Normality The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. A Normal probability plot provides a good assessment of whether a data set follows a Normal distribution. Interpreting Normal Probability Plots If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-normal distribution. Outliers appear as points that are far away from the overall pattern of the plot. The Practice of Statistics, 5 th Edition 16

17 No space in the fridge? The measurements listed below describe the usable capacity (in cubic feet) of a sample of 36 side-by-side refrigerators (Consumer Reports, May 2010).Are the data close to Normal? These percents are quite close to what we would expect based on the rule. Combined with the graph, this gives good evidence that this distribution is close to Normal. The Practice of Statistics, 5 th Edition 17

18 No space in the fridge? (continued) Here is a Normal probability plot (also called a Normal quantile plot) of the refrigerator data from the previous page. It is quite linear, supporting our earlier decision that the distribution is close to Normal. Go to stat plot, and choose the sixth graph. Enter you data list and x for the axis. The Practice of Statistics, 5 th Edition 18

19 Density Curves and Normal Distributions Section Summary In this section, we learned how to ESTIMATE the relative locations of the median and mean on a density curve. ESTIMATE areas (proportions of values) in a Normal distribution. FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence. The Practice of Statistics, 5 th Edition 19

20 PAGE , 48, 50, 52, 54 Homework The Practice of Statistics, 5 th Edition 20

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