Statistics Independent Study Page 1 of 2 Study Sheet Using a Normal Curve Table

_____________ Copyright 2000 Apex Learning Inc. All rights reserved. This material is intended for the exclusive use of registered users only. No portion of these materials may be reproduced or redistributed in any form without the express written permission of Apex Learning, Inc.

In this activity you'll use a standard normal curve table to convert raw and standardized scores into areas under the normal curve, and vice versa. Before you start, review the formulas necessary to translate raw scores into standardized z-scores. A normal curve table is attached at the end of this activity. This may differ from other normal curve tables you encounter, but all tables provide you with the same information about the areas in the normal distribution. The important elements of the normal curve table include:

First column: first two digits of the standardized z-score Second column: last digit of the standardized z-score Cells of the table: area under the curve below that z-score The normal curve table is based on a theoretical distribution of a population with an infinite number of observations. This means that any single observation has 0 width. The practical consequence is this: The area at or below a given z-score is the same as the area below a z-score. The "at or" does not change the area. Remember, when one question asks you to find "the proportion or probability below" a value and another question asks you to find "the proportion or the probability at or below" a value, they're asking about the same area. Elements of the table can sometimes be confusing if you don't completely understand what areas of the curve you're looking for. Since you'll often be asked to compute areas below, above, or between some raw scores or z-scores, you may find it helpful to draw the normal curve and then shade the area of interest. For example, if you want to find the area between z=1.15 and z=2.05, draw a normal curve, place vertical lines at z=1.15 and z=2.05, and shade the area between these lines. It's often easier to compute an area when you have a visual representation of the proportion you're looking for. 1. Find the following areas under the normal curve:

A. 1.43 < z < 0.68 B. 0.58 < z < 1.74 C. 1.55 < z < -0.44 D. z > 1.34

2. Find the standardized score (z-score) closest to the following percentiles:

A. 35th percentile (the point below which 35% or the observations fall) B. 56th percentile (the point below which 56% or the observations fall) C. 88th percentile (the point below which 88% or the observations fall)

3. Given a normal distribution of heights of 5 year olds (in inches) with a mean of 36

and a standard deviation of 10, find the following areas under the curve: A. x < 31 inches B. x > 49 inches C. 40 < x < 50 inches

Statistics Independent Study Page 2 of 2 Study Sheet Using a Normal Curve Table

Acknowledgements Question 1: This is question 6.32 (a, b, c) from page 228 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. _____________ Copyright 2000 Apex Learning Inc. All rights reserved. This material is intended for the exclusive use of registered users only. No portion of these materials may be reproduced or redistributed in any form without the express written permission of Apex Learning, Inc.

4. Although faculty salaries at colleges and universities in the United States continue to rise, they do not always keep pace with the cost of living. During the 1996-97 academic year, female assistant professors earned an average of $39,643 per year. Suppose that these salaries are normally distributed, with a standard deviation of $4,000.

Data Source: Denise K. Magner, "Increases in Faculty Salaries Fail to Keep Pace with Inflation," The Chronicle of Higher Education, 3 July 1997, p. A8. A. What proportion of female assistant professors will have salaries less than

$30,000? (Remember, proportion is similar to saying "area under the curve.")

B. What proportion of female assistant professors will have salaries between $35,000 and $40,000?

C. If a female assistant professor is in the 95th percentile of salary ranges, how much does she make per year? Explain what the 95th percentile means in the context of this problem.

5. You have a normal distribution of hypothetical batting averages of 500 baseball

players with a mean batting average of 0.273 and a standard deviation of 0.028. Answer the following questions: A. What proportion of players hit .300 or above? B. What proportion of players hit between .220 and .250? C. All players in the bottom 10 % of batting averages will be demoted to the minor

leagues. A player will be demoted if he has an average below what value? D. What is the lowest batting average among the top ten hitters in the league?

(Hint: think about percentiles of the distribution)

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