1. AP Statistics: Section 2.2 C

2. Example 1: Determine if each of the following is likely to have a Normal distribution (N) or a non-normal distribution (nn). _____ gas mileage of 2006 Corvettes _____ prices of homes in Westlake _____ gross sales of business firms _____ weights of 9-oz bags of potato chips

3. While experience can suggest whether or not a Normal distribution is plausible in a certain case, it is very risky (especially on the AP test) to assume that a distribution is Normal without actually inspecting the data. Chapters 10 -15 of our text deal with inference, and an important condition that will need to be met is that the data comes from a population that is approximately Normal.

4. Our goal in this section is to explore methods for determining if a distribution is Normal or, at least, approximately Normal.

5. Method 1: Construct a histogram, stemplot or dotplot. What are we looking for?

6. Non-Normal features to look for include ________, pronounced _________, or ______ and ________. outliers skewnessgaps clusters

7. To be truly Normal, such a bell- shaped plot should conform to the 68-95-99.7 rule. Mark the points on the horizontal axis and determine the percentage of counts in each interval.

9. NOTE: In some text books, an outlier is defined as any value more than two standard deviations away from the mean. Thus, the 68-95-99.7 rule would suggest that a Normal distribution might have outliers. BUT, if we are examining the distribution of a sample and trying to determine if the corresponding population is Normally distributed, any outliers in the sample would indicate that the population is probably not Normally distributed. Why? If only 5% of the original population could be outliers, it is highly unlikely that our sample would actually contain any outliers.

10. Method 2: Examine the 5-number summary. What are we looking for.

11. Note: What should be true about the mean and the median in a Normal distribution?

12. Method 3: Construct a Normal probability plot. Statistical packages, as well as TI calculators, can construct a Normal probability plot. Before we look at constructing a Normal probability plot, let’s look at some basic ideas behind constructing a Normal probability plot.

13. 1. Arrange the observed data values from _________ to _________. Record what percentile of the data each value represents. For example, the smallest observation in a set of 20 would be at the _____ percentile, the second smallest would be at the _____ percentile, etc. smallestlargest 5 th 10 th

14. 2. Determine the z-scores for each of the observations. For example, z = _______ is the 5% point of the standard Normal distribution and z = _______ is the 10% point.

15. 3. Plot each data point x against the corresponding z. If the data distribution is close to Normal, the plotted points will be __________________. Systematic deviations from a straight line indicate a non-Normal distribution. approximately linear

16. In a right-skewed distribution, the largest observations fall distinctly _______ a line drawn through the main body of points. In a left-skewed distribution, the smallest observations fall distinctly ________ a line drawn through the main body of points. Outliers appear as points that are far away from the overall pattern. above below

17. Construct a Normal probability plot on your calculator as follows: 1. Enter the data in L1. (STATS, EDIT). 2. Go to STAT PLOT (2 nd Y=). ENTER. Highlight ON. First draw a histogram to see that the data appears to be Normally distributed. Key ZOOM 9 to get an appropriate window. 3. Go back to STAT PLOT. Change “Type” to the 6 th graph choice. For “Data List”, put in L1 and for “Data Axis”, highlight Y. 4. Push GRAPH and key ZOOM 9 to get an appropriate window.