HOMEWORK 4:
NORMAL CURVE AND Z SCORES (70 PTS POSSIBLE)
This homework requires both your text and your calculator. The objective of your fifth homework assignment involves answering questions related to the normal and standard normal curves. Excel will not be a part of the assignment this week. Instead, you will submit this assignment as a Word document. You will need to use the “Normal Curve Table” in Table A-1 of the Appendix of your text that is gone over in detail (with examples) in Chapter 3. First, be sure you view the PointCast presentation for this week, found in the Course Content under Module 5. This presentation provides information and goes through the steps you will need to be familiar with in order to complete this assignment. The standard normal curve is provided here to the right for your reference. Be sure to show your work in each problem.
1. Reading readiness of preschoolers from an impoverished neighborhood (n = 20) was measured using a standardized test. Nationally, the mean on this test for preschoolers is 30.9, with SD = 2.08.
a. Children below the 30th percentile (in the bottom 30%) are in need of special assistance prior to attending school. What raw score marks the cut-off score for these children? (8 pts)
b. What percentage of children score between 25 and 28.5? (8 pts)
c. How many children would we expect to find with scores between 28 and 31.5?
(8 pts)
d. Children in the top 25% are considered accelerated readers and qualify for different placement in school. What raw score would mark the cutoff for such placement? (11 pts)
2. Age at onset of dementia was determined for a sample of adults between the ages of 60 and 75. For 15 subjects, the results were ΣX = 1008, and Σ(X-M)2 = 140.4. Use this information to answer the following:
a. What is the mean and SD for this data? (8 pts)
b. Based on the data you have and the Normal Curve Tables, what percentage of people might start to show signs of dementia at or before age 62? (8 pts)
c. If we consider the normal range of onset in this population to be +/-1 z-score from the mean, what two ages correspond to this? (8 pts)
d. A neuropsychologist is interested only in studying the most deviant portion of this population, that is, those individuals who fall within the top 10% and the bottom 10% of the distribution. She must determine the ages that mark these boundaries. What are these ages? (11 pts)
HOMEWORK 4 REVIEW AND STUDY GUIDE
This week’s homework focuses on using your knowledge of the normal curve, the normal curve tables, and Z scores to calculate percentages and find raw scores for different scenarios. This study guide is meant to help you think about how to set up and work the homework problems. The problems are based on information from your textbook and the presentations.
General Review:
Converting scores: Recall that any raw score can be converted to a Z score using the formula
Z = (X-M) where X represents the raw score, M is the mean of the raw scores, and SD is
SD the standard deviation of the raw scores
and any Z score can be converted to a raw score using the formula
X = M + SD(Z) where M and SD are the mean and standard deviation of the raw scores
Z scores: Recall that Z scores help you to standardize data so it is easy to make comparisons between data sets, or to answer questions about one data set without having to do a lot of complex figuring. A Z score simply represents where a score lies in terms of the number of standard deviations from the mean. A Z score of +1 = 1 standard deviation above the mean; a Z score of -2 = 2 standard deviations below the mean; a Z score of +1.7 = 1.7 SD above the mean, and so on. (See the presentation, especially the example concerning IQ/SAT scores and the standard normal distribution, for a review.)
Standard normal distribution/curve: Recall that the standard normal distribution is a distribution based on Z scores, with a mean of 0 and a standard deviation of 1. Most scores within a data set will fall between -3 and +3 on this distribution (Z=-3 / Z=+3). This means that most scores fall within -3 and +3 standard deviations from the mean (see below).
68/95/99 Rule: Recall that roughly 68% of scores fall between -1 and +1 SD, 95% of scores fall between -2 and +2 SD, and 99% of scores fall between -3 and +3. This means that, if you calculate an answer and the Z score lies far outside of the -3/+3 boundaries, you should probably check your math! A Z score of 8.7, for example, is highly improbable.
Guide for Working Problems:
Finding percentages: Recall that the normal curve tables will give you the percentage of scores that lie below or above a certain Z score. The tables in your text give you two values: the % between the mean (center) and the score, and the percent from the score out into the tail. These two percentages will always add up to be 50% (because they represent all of the area on one side of the mean, or 50% of the normal curve). These tables take advantage of the fact that the normal distribution is symmetrical. Therefore, the percent given in the tables for the right side of the distribution will be exactly the same on the left side of the distribution. For example, the % mean to Z of .60 = 22.57%. Therefore, the % mean to Z of -.60 also equals 22.57%. The tables only give the values for positive Z scores, because these are going to be exactly the same as the corresponding negative Z scores. If you need to find % in tail of Z = -1.33, look for the % in tail for Z = 1.33, which is 40.82%. Again, this percentage applies to both the negative and positive Z scores because the distribution is symmetrical.
You can also use the table to find the percentage of scores between two Z scores. This is a more complex operation, but not impossible with a little figuring! We go over this in more detail later.
Make sure you know the Z score: Any question dealing with percentages will require that you know the Z score(s). “The Z is the key.”
If the question provides only the raw score(s), be sure to convert to Z score(s) first using the standard formula, then use the normal curve tables to answer the question.
If the question provides only percentages, use the normal curve table to find the corresponding Z score(s), then answer the question, converting to the raw score(s) if necessary.
The following is a general guide to using the tables in your appendix to find percentages (or percentiles) for different cut-off scores, and for finding Z or raw scores for certain percentages (or percentiles). Your text covers some of these in more detail. Most questions on your homework involve this kind of figuring.
1. Finding the total percent (or area) below a negative Z score:
Find % in tail for that Z score
2. Finding the total percent (or area) below a positive Z score:
Find % to mean for that Z score, then add 50%
3. Finding the total percent (or area) above a negative Z score:
Find % to mean for that Z score, then add 50%
4. Finding the total percent (or area) above a positive Z score:
Find % in tail for that Z score
5. Finding the total percent (or area) between 2 Z scores on the same side of the distribution (both positive or both negative): Steps a-d
a. Find % to mean for the Z score closest to the mean
b. Find % in tail for the Z score closest to the tail
c. Add a. and b. together
d. Subtract your total in c. (above) from 50% (this gives you the remaining area between the 2 scores)
6. Finding the total percent (or area) between 2 Z scores on different sides of the distribution (one negative, one positive): Steps a-b
a. Find % to mean for each Z score
b. Add these together (this gives you the total area between the 2 scores)
7. Finding Z score(s) from percentage(s):
Use the table, look up the percentage, and find the corresponding Z
8. Finding raw score(s) from percentage(s): Steps a-b
a. Use the table, look up the percentage, and find the corresponding Z
b. Convert the Z score(s) to raw score(s) using the standard formula
Most of the homework problems require that you use at least one of the methods above to find an answer. As recommended in your text, if you go through each step above, using a drawing of a normal curve and shading in the areas represented, you will get a better understanding of the mechanics behind the math.
