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
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 2
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 3
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.00.01.02 0.7.7580.7611.7642 0.8.7881.7910.7939 0.9.8159.8186.8212 The Practice of Statistics, 5 th Edition 4
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 5
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 = 1.53. 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 0.9370. This is the area to the left of z = 1.53. To find the area above z = 1.53, subtract 0.9370 from 1 to get 0.0630. The Practice of Statistics, 5 th Edition 6
Finding areas under the standard Normal curve Problem: Find the proportion of observations from the standard Normal distribution that are between 0.58 and 1.79. 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 = 0.58. The difference in these proportions is the proportion of observations that are between z = 0.58 and z = 1.79. The Practice of Statistics, 5 th Edition 7
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 0.8000 because the table always lists area to the left of a boundary. The closest area to 0.8000 is 0.7995, which corresponds to a z-score of z = 0.84. Thus, approximately 20% of the observations in a standard Normal distribution are above z = 0.84. The Practice of Statistics, 5 th Edition 8
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 9
Serving speed The Practice of Statistics, 5 th Edition 10
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 11
Heights of three-year-old females The Practice of Statistics, 5 th Edition 12
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 13
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? 12.9 13.7 14.1 14.2 14.5 14.5 14.6 14.7 15.1 15.2 15.3 15.3 15.3 15.3 15.5 15.6 15.6 15.8 16.0 16.0 16.2 16.2 16.3 16.4 16.5 16.6 16.6 16.6 16.8 17.0 17.0 17.2 17.4 17.4 17.9 18.4 These percents are quite close to what we would expect based on the 68 95 99.7 rule. Combined with the graph, this gives good evidence that this distribution is close to Normal. The Practice of Statistics, 5 th Edition 14
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 15
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 16
PAGE 129 46, 48, 50, 52, 54 Homework The Practice of Statistics, 5 th Edition 17