AP Statistics: Section 2.2 C

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

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 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.

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

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

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

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

NOTE: In some text books, an outlier is defined as any value more than two standard deviations away from the mean. Thus, the 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.

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

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

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.

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

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.

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

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

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.