Forty-four of the 61 members of the Yahoo Optimist Club are 50 years of age or older.
Seventeen are under 50 years old. Of those aged 50 and over, 32 are married. Of those under age 50, 15 are married. Suppose you selected a member at random.
A. What is the probability that the member would be 1. < 50 years old: _____________________________
2. married: _________________________________
3. married and ≥ 50 years old: ___________________
B. What proportion of the club is
1. unmarried: _______________________________
2. married and < 50 years old: __________________
THE NORMAL DISTRIBUTION
The normal distribution was briefly encountered in an earlier chapter. Here, we will examine it in greater detail. It is a bell-shaped curve of probabilities and proportions whose units of measurement are in z-scores. Other important characteristics of the normal distribution are:
• The tails are asymptotic (i.e., they never touch the baseline); this is because the normal distribution is based on an infinite number of cases.
• The area under the curve is equal to 1.
• The proportion of the area that lies between the mean and a z-score of +1.00 is .3413.
• .1359 of the area lies between z-scores of +1.00 and +2.00.
• .0215 of the area lies between z-scores of +2.00 and +3.00.
• .0013 of the area lies beyond a z-score of +3.00.
• Because the distribution is symmetrical, the same proportions that apply to the right side of the distribution (positive z-values) also apply to the left (negative z-values).
The theoretical normal distribution is important in the social sciences because the dis- tributions of many human characteristics, when measured, approximate normally shaped empirical distributions. A few people will score very high and a few people will score very low. But most people are average and will score in the middle. Thus, we can apply our knowl- edge of the mathematical properties and known probabilities of the theoretical distribution to answer questions about probability for our empirically derived distributions.
RELATIONSHIP BETWEEN THE NORMAL DISTRIBUTION AND PROBABILITY/
PROPORTIONS/PERCENTAGES
Because of the precise relationship between the normal distribution and z-score values, we can use it to determine probabilities, proportions, and percentages of specified z-scores. Keep the following points in mind:
• Proportions and probability are interchangeable terms.
• Proportions can be translated into percentages by multiplying by 100 and adding a percent sign (%):
For example: .1359 100 13 59 . %
• Likewise, percentages can be translated into proportions by dropping the percent sign and dividing by 100:
For example :13 59 100. .1359
Other Examples
• The probability of obtaining a z-score between −1.00 and +1.00 is .6826.
• The proportion of the population that will have a z-score below −3.00 is .0013.
• The percentage of the population that will have a z-score between +2.00 and +3.00 is 2.15%.
THE Z-TABLE
A z-score usually will not be a whole number, but tables have been constructed that identify precise proportions under the curve associated with specific z-score values. Table 1 near the end of this book, the Table of the Normal Distribution Curve, is one such listing. We can also refer to this table as the z-table. Notice that we will refer to four columns in the table, as follows:
• Column A – indicates a particular z-score value.
• Column B – indicates the proportion in the body of the curve (i.e., the larger portion).
• Column C – indicates the proportion in the tail (i.e., the smaller portion).
• Column D – indicates the proportion between the mean and the z-score value in Col- umn A. (Remember that the mean of a z-scale is 0 and it is located in the center of the distribution.)
Let us examine the proportions in the chart for a particular z-score, say +1.25, which will be found in Column A. Column B tells us that .8944 proportion of the curve lies below that z-score value. Column C tells us that .1056 proportion of the curve lies above that z-score value. And Column D tells us that .3944 proportion of the curve lies between the mean and that z-score value.
The z-table can be used to answer a number of questions about probabilities, proportions, and percentages.
Probabilities/Proportions/Percentages for Specified z-Scores
We may be interested in knowing the proportion of the normal distribution that is associ- ated with particular z-scores.
For Example
What proportion of the normal distribution is associated with z-scores greater than +2.50?
The proper notation for this question is:
p(z > + 2.50) = ?
Tip!
to find p values in the z-table that are:
• above a positive number, look in Column C (tail).
• below a positive number, look in Column b (body).
• number, look in Column b (body).
• below a negative number, look in Column C (tail).
Notice that the proportions for columns B and C, when added together, equal 1 because these two columns together encompass the entire distribution.
Tip!
it is helpful to sketch a picture of the normal distribution curve when answering questions about z-scores, proportions, and probabilities.
To find this proportion, look in Column C (in the tail) of the z-table. You will see that .0062 of the distribution is associated with z-scores greater than +2.50. Remember that this value is not only a proportion but also a probability. Hence, the probability of obtaining a z-score value greater than +2.50 is also .0062. Keep in mind that this value can be changed
can say that .62% of the scores in a normal distribution are above a z-score value of +2.50, which, as you can tell, is a very low probability of occurrence.
p(z > + 2.50) = .0062
Other Examples Are as Follows
p(z < +2.00) = .9772 (look in the body, Column B) p(z < −1.50) = .0668 (look in the tail, Column C) p(z > −1.23) = .8907 (look in the body, Column B)
z-Scores for Specific Probabilities/Proportions/Percentages
In the previous section, you were given a z-score and asked to find the associated proportion, probability, or percentage. However, you may also be given a proportion or percentage and asked to find the associated z-score. In this case, follow the steps below:
• If you are given a percentage, translate it into a proportion by dividing by 100 because the language of the z-table is in proportions.
• Next, look in Column B or Column C for the closest proportion.
• Then look in Column A for the corresponding z-score.
For Example
What z-score value is associated with the lowest 10% of the distribution?
• First, translate 10% into a proportion (10 ÷ 100 = .1000).
• Next, look in Column C, the tail, for the proportion closest to .1000, which is .1003.
• Finally, look in Column A for the z-score that corresponds to the lowest .1003 of the distribution. This z-score is 1.28. We know that the z-score will be negative because we were interested in the lowest 10% of the distribution.
−1.28 is the z-score associated with the lowest 10% of the scores.
Another Example
What z-score separates the lowest 75% from the highest 25% of the distribution?
• In this case, we can locate the z-score associated with either the lower .7500 (75 ÷ 100) or the upper .2500 (25 ÷ 100).
• The closest proportion to .7500 is .7486 (Column B). The closest proportion to .2500 is .2514 (Column C).
Thus, +.67 is the z -score that separates the lowest 75% of the distribution from the upper 25%.
Your Turn!