Statistics Helps

WEEK TWO

z scores percentages and z scores percentages above or below some raw score percentages between two raw scores

probabilities

z scores

Use the z score formulas in your Horror book. In general,

If your problem asks you to find a raw score, you need to use the transformation of the z formula that starts out with X =.

Percentages and z scores

What we are doing here, is trying to segment off portions of the normal curve and then figuring out what percent of the whole, the segment is. We do that by first determining exactly where is the area of interest and then we determine how big the area is by using the table in the back of the book, the z table on page 546. First start by considering the area between the mean and some z score, say, 1.27.

Note on the normal curve, where that 1.27 is located. Then go to page 546 in your book. Move down the page to 1.2, and then across the page to .07. You should find 39.80.

This means that 39.80 percent of the entire normal curve is found between the mean and the z score of 1.27.

Now, if we wanted to know how much of the normal curve was located below the z score of 1.27, we would have to add the 39.8 percent to the 50 percent located below the mean.

The percentile for a z score of 1.27 is 89.8.

Percentages above or below some raw score

Suppose you wanted to throw rotten eggs at your stats professor. You know that eggs rot at an average rate of 14 hours at 75 degrees F. If the standard deviation is 3 hours, what percent of your eggs would be rotten and ready for tossing at 9 hours? (assuming, of course that you have a normally distributed population of eggs)

For this problem, you want to know what is the percent of the curve falling below the raw score of 9 hours.

First, determine the z score at 9 hours. The mean is 14 and the standard deviation is three. The z would be 9 minus 14, divided by 3, or -1.67. So, now you are getting closer to knowing what percent of eggs could be thrown at the end of just 9 hours. That would be the amount of time you'd have to spend doing just one of these problems, right?

Now go to page 523 to find the area between the mean and -1.67. That percent is 45.25. So the green area, estimated by the curve to the right, is 45.25%. The yellow area, then, must be 50% minus 45.25%, or 4.74%. Therefore, however many eggs you had to begin with, you'd be able to throw 4.74% of them at your stats professor.

Percentages between two raw scores

Let's first look at the logic of these kinds of problems. Here, we have two z scores on opposite sides of the mean. What would you do to find the percent between them?

Yes, find the area between each z score and the mean and ADD them together. Here's an example: Suppose you wanted to find the percentage between the two raw scores of 45 and 57 for a distribution with a mean of 50 and a standard deviation of 3.

The z score for 45 is (45 - 50)/3, or -1.67. The z score for the 57 is (57 - 50)/3, or 2.33. The first z is on the left side of the mean and the second is on the right side of the mean. The percentage for the -1.67 is 45.25. The percentage for the 2.33 is 49.01. Together, the percentage is 94.26. Therefore,94.26 percent of the population resides between those two raw scores, given the mean of 50 and the SD of 3.