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AP Statistics Review: The Normal Distribution Page 3 of 13

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Objective 4 Use your graphing calculator to find the areas between two values under a normal curve.

Objective 5 Use your graphing calculator to find the raw score value, or z-score, when given an area to the left of a value under a normal curve.

Examples A. Use your graphing calculator to find the area under a standard normal curve between z =

-2.34 and z = -.5. B. Given a normal distribution whose mean is 72 and whose standard deviation is 8, use a

graphing calculator to find the area between scores of 60 and 75. C. For the distribution in B, use a graphing calculator to find the percent of scores less than

65. D. For a distribution that has N(72,8), as above, find the raw score that has a percentile rank

of 80.

Tips The functions to do these problems are all in the DISTR menu on the graphing calculator. On the TI-83/84, if you only put two parameters in normalcdf, the calculator will assume

you've given it two z-scores for a standard normal curve and return the area between them. Similarly, if you only put one parameter in invNorm, the calculator will assume an area under a standard normal curve and return the appropriate z-score. On the TI-89, you may choose to work with either z-scores or actual values. Ti use z-scores simply enter 0 for the mean and 1 for the standard deviation.

The syntax for normalcdf is normalcdf(lower bound, upper bound, [ , ] , and for invNorm is invNorm(Area, [ , ] . On the TI-83/84, you enter the mean and standard deviation only if it isn't a standard normal curve.

To find an area less than a given value (or greater than a given value), remember the 68-95-99.7 rule. Enter an extreme lower bound, at least four or more standard deviations from the mean. That way, there's very little area left beyond that value in the tail of the curve.

Normalpdf gives the height of the normal curve above the specified value. There's almost no reason to ever use this function in this course.

Now that you know how to use your calculator to figure areas, you may not want to use the normal probability table any more. Still, the table is a good way for you to learn about areas, probabilities, and frequencies, and you may want to continue to use it if you're comfortable with it.

Answers A. Normalcdf(-2.34,-.5) = .2989. Since there are only two parameters entered in the

function, the calculator assumes they're z-scores. (Note that the calculator will give you 12-place accuracy, unlike the tables, which are usually good for 4.)

B. Normalcdf(60,75,72,8) = .5794. C. Normalcdf(0,65,72,8) = .1908. The choice of 0 as a lower bound is motivated by the fact

it's many standard deviations from the mean. A choice of 25 for the lower bound would have given the same answer, correct to 4 decimal places. However, a choice of 51, which is three standard deviations below the mean, would have given a value of .1865.

D. A percentile rank of 80 means that the area to the left of the desired value is .8. Therefore, invNorm(.8,72,80) = 78.73.



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Well, the percent within one standard deviation is a little low, but the discrepancy is probably not great enough to claim that MINITAB isn't doing what it claims to be doing (that is, generating data from a normal distribution). Objective 2 Describe a normal probability plot.

Example Describe a normal probability plot and how you'd construct one.

Answer A normal probability plot, or NPP (also called a normal quantile plot), plots data sorted from least to greatest value (ordered data) against corresponding z-scores on an x- z coordinate axis. For example, suppose there are 100 values in your sample, all different. Order the data from smallest to largest. Consider the 20th term as you move from smallest to largest. This term is at the 20th percentile and the z-score for a value at the 20th percentile (lower tail area = .2) is -.84. You'd plot the term at the 20th position vs. -.84. The important thing about a normal probability plot is that, if the data are a sample from a normal distribution, the NPP will be a more or less straight diagonal line. Therefore an NPP provides a method for testing to see if a given data set is likely to be normal.

If you had to construct an NPP, you'd do so with your calculator. In the TI-83/84, it's the last graph in a STAT PLOT under "Type." Put your data in a list, (say L1), choose the icon for NPP, choose the proper Data List, leave the Data Axis set on X, and choose your favorite mark. Then do a ZOOM STAT. (Be sure that no Y= graphs are on.) On the TI-89, it's option 2 under the F2: Plots menu in List Editor. You will choose the plot number, the list containing the values, set the Data Axis to X, choose your mark type, and leave the Store Zscores setting to its default setting, which should be statvars\z.

Objective 3 Construct a normal probability plot on a graphing calculator and use it to check for normalcy.

Example Use randNorm on your calculator to generate 100 random numbers from N(50,5), (a normal distribution with a mean of 50 and a standard deviation of 5) store the data in L1 (TI-83/84) or list1 (TI-89), and the check to see how well the calculator does using a normal probability plot. Answer On the TI-83/84: Press MATH [PRB" [randNorm"(50,3,100) STO 2nd L1 ENTER. (Each person doing this will get a different set of values in L1.) Then, do an NPP as follows: 2nd STAT PLOT (go into plot 1 and turn it on). Select the NPP icon from "Type" (the last icon in the list; it looks like a scatterplot). Set Data List to L1, and Data Axis to X. Make sure there are no active equations in Y=. Press ZOOM 9. The resulting NPP will, in most cases, look pretty straight, although it'll probably tail off some at the ends. Marked departures from straightness indicate non-normality.

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