Technology: Normal Model Computations

Many statistics textbooks explain how to perform Normal model computations using tables that can be found in the back of those books. We will NOT use table methods to find percentages associated with the Normal model, so make sure you know how to use your calculator to do these problems, as demonstrated below.

Standard Normal model
Suppose we want to find the percentage the Normal model with z-scores between 1 and 2. (That is, the percentage of persons/things we would expect to see with data values between 1 and 2 standard deviations above the mean data value.) We enter the following into the TI-84: normalcdf(1,2)

To do this, press 2ND then VARS to get to the DISTR (or distribution) menu. Move the cursor down to normalcdf(:

and press ENTER then type 1 followed by , (a comma) and 2 and then ) to match the opening parenthesis:

Finally press ENTER again and you should see:

which means that about 13.6% of the Normal model should be between 1 and 2 standard deviations above the mean.

You can approximate this answer with the 68-95-99.7 Rule; you should do this to check your understanding of that rule, and be sure to draw a picture:

In fact you should draw a picture for all of the problems of this type regardless of whether or not you use the TI-84 to solve the problem.

If instead we were interested in the percentage with a z-score between -1 and 2:

we would enter normalcdf(-1,2) ≈ 0.8186, or about 81.9%:

Be careful to use the negative sign (-) on the TI-84:

rather than the blue subtraction key when entering negative numbers.

Suppose now we wanted to compute the percentage with a z-score above 1, in other words between 1 and ∞ (infinity):

There is no ∞ button on the TI-84, so we need to use an extremely large number in place of ∞. For all practical purposes, the largest number the TI-84 can handle is 1×10⁹⁹. (This is written in scientific notation, otherwise it would be a 1 with 99 zeros after it!) The calculator notation for this number is 1E99 but to enter it into the calculator you actually have to type 1EE99. Notice that EE is found above the , (comma) key:

so you would press 1 2ND , 99 to enter 1E99. Getting back to the problem at hand, our answer is now given by normalcdf(1,1E99) ≈ 0.1587:

Check that you get approximately the same answer with the 68-95-99.7 Rule.

Similarly, to find the percentage with a z-score below 1.25:

we would get normalcdf(-1E99,1.25) ≈ 0.8944, or about 89%:

If you find yourself using normalcdf over and over again in succession, you can save some time by pressing 2ND ENTER after you get your first answer; the previous expression you typed into the calculator will reappear and you can use the arrow keys and the DEL and INS buttons to modify the expression slightly. Then just press ENTER to evaluate the modified expression.

You should now be in good shape if all of the questions are phrased in terms of z-scores, but they rarely are. Let's take a look at a more realistic example.

IQ scores
IQ tests are often designed to have a mean of µ = 100 and a standard deviation of σ = 15, and to follow the Normal model. Let's use the model N(100,15) to model IQ scores of U.S. adults. What percentage of all U.S. adults have an IQ score between 115 and 125?

First, we convert the IQ scores to z-scores: if IQ = 115, then:

`z = (115-100)/15 = 1`  

If IQ = 125, then:

`z = (125-100)/15 approx 1.67`

Note: We usually round z-scores to two decimal places, although if you can use a more precise number from a previous calculator computation, by all means do so.

So we've reduced the problem from asking, "What percentage of IQ scores are between 115 and 125?" to asking, "What percentage of z-scores are between 1.00 and 1.67?" The answer: normalcdf(1,1.67) ≈ 0.1112, or about 11%. We could also enter normalcdf(1,5/3) or normalcdf((115-100)/15,(125-100)/15) into the TI-84, both of which yield about 0.1109, essentially the same answer:

Notice what we did in this last example: we converted the given data values to z-scores, then used the calculator to find the desired percentage. You should practice this, as the technique will be necessary in other guises throughout the quarter. However, there is a shortcut available to us: if we can compute the z-scores, the calculator should be able to do so as well, given μ and σ. In fact, it can: if we enter normalcdf(115,125,100,15) we get about 0.1109, or the same answer as before:

Notice that we entered the original left and right data values, then the mean and standard deviation associated with our model.

Working backward
Sometimes we wish to solve this sort of problem in reverse. Suppose instead of asking something like, "What percentage of people/things have a z-score less than 1.36?" we ask a question like, "What z-score cuts off the lowest 10% of data values from the highest 90%?" Here we start out knowing the percentage (or area under the curve) and what we want to find is the corresponding z-score that separates these two regions under the curve:

We do this on the calculator by entering invNorm(0.10); the invNorm command is found right below normalcdf in the DISTR menu. You should get an answer of about -1.28:

which tells us that the data values that are more than 1.28 standard deviations below the mean account for the lowest 10% of all scores in the population for which a Normal model applies. The number we use with the invNorm command is always the area to the LEFT of the desired z-score.

Now let's return to our IQ example from above. Suppose we want to determine which people have IQ scores in the top 15% of the population; in other words we need to know the IQ score that cuts off those in the top 15% from those in the bottom 85%. We compute invNorm(0.85) ≈ 1.04:

so those people with a z-score above z = 1.04 will be in the top 15%. However, we still need to convert this z-score back to an IQ score. Using our z-score formula, we have:

`1.036 = (IQ - 100)/15 => 1.036 times 15 = IQ - 100 => 15.54 = IQ - 100 => 115.54 = IQ`

Thus anyone with an IQ of 116 or above should be in the top 15% of the population:

If we wanted, we could accomplish this more directly by entering invNorm(0.85,100,15) into the TI-84:

Check that you get roughly the same answer as above. Notice that we type the percentage to the left of the desired data value, then the mean and then the standard deviation.

Normal probability plots
If we have a data set and want to decide whether or not a Normal model applies, we should first check that it is unimodal and symmetric. We can also construct a Normal probability plot for the data set by follow the instructions for creating a histogram on the TI-84, but for the Type icon, select the last one (the one farthest to the right on the second row of icons):

To access it, move the cursor to the histogram icon, then keep pressing the right arrow key. Specify the list with the data you wish to plot and select any type of Mark you like.

Now use ZoomStat to get the Normal percentile plot. Using the 2007 assessed home values (from the house data set we examined previously) we get a plot like this:

What we want to see in such a plot is a scatterplot with a perfectly straight line: if the points in the Normal probability more or less line up, it should be OK to use a Normal model; otherwise, we probably shouldn't use a Normal model. (Don't worry right now about how this plot is constructed.)

The plot for the house data isn't perfectly straight, and it bends a bit on the far ends, but we might consider it nearly straight, so a Normal model might be appropriate here.