Question

The Wall Street Journal (February \(15,\) 1972) reported that General Electric was sued in Texas for sex discrimination over a minimum height requirement of 5 feet, 7 inches. The suit claimed that this restriction eliminated more than \(94 \%\) of adult females from consideration. Let \(x\) represent the height of a randomly selected adult woman. Suppose that \(x\) is approximately normally distributed with mean 66 inches (5 ft. 6 in.) and standard deviation 2 inches. a. Is the claim that \(94 \%\) of all women are shorter than 5 feet, 7 inches correct? b. What proportion of adult women would be excluded from employment as a result of the height restriction?

Short answer

a. It's not correct to claim that 94% of all women are shorter than 5 feet 7 inches. The exact percentage will be obtained from the normal distribution table using the calculated z-score. b. The percentage of adult women excluded from employment because of the height restriction can be established by subtracting the obtained percentage from the table from 100%.

Step-by-step solution

Problem A Calculation

Using the information provided, consider that the mean height of women is 66 inches and the standard deviation is 2 inches. Subtract the height limit from the mean height to find the z-score. The z-score will therefore be: \( z = (67-66) / 2 = 0.5 \). After that, find the area to the left of the z-score 0.5, which represents the percentage of women who are shorter than 5 feet 7 inches. Finding this area allows a probability that corresponds to this z-score to be found in any normal distribution table. This will provide the percentage of women who are shorter than 5 feet 7 inches.

Problem A Analysis

After finding the exact percentage from the normal distribution table, it will show whether or not the claim that 94% of all women are shorter than 5 feet 7 inches is correct.

Problem B Calculation

Following the same steps as in Problem A, find the z-score using the height restriction as 5 feet 7 inches. Then find the corresponding percentage from the normal distribution table. Subtract the found percentage from 100% to find the percentage of women who do not meet the height restriction.

Problem B Analysis

The percentage obtained is the proportion of adult women who would be excluded from employment as a result of the height restriction.

Key concepts

Z-Score Calculation

Understanding the z-score is essential for analyzing how a single data point compares to a normal distribution. In simple terms, a z-score represents the number of standard deviations a data point is from the mean. To calculate a z-score, one must subtract the mean from the data point and then divide that by the standard deviation. The formula is:
\[ z = \frac{{x - \mu}}{{\sigma}} \]
where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. If the z-score is negative, the data point lies below the mean, and if it is positive, it lies above the mean. A z-score of 0 indicates that the data point is exactly at the mean. In the context of the height restriction problem, the z-score allows us to transform the height of 5 feet 7 inches into a standardized value that can be compared across a normal distribution of women's heights.

Standard Deviation

The concept of standard deviation provides us with a measure of the spread or dispersion within a set of values. It quantifies how much the numbers in a data set deviate from the mean, on average. Lower standard deviation means that the values tend to be close to the mean, while a higher standard deviation indicates that the values are spread out over a wider range. Mathematically, standard deviation is the square root of the variance.

Standard deviation is pivotal in z-score calculation because it serves as the scaling factor - determining how far a point is from the mean in terms of how variable the data set is. It's important to understand that in the given exercise, the standard deviation for adult women's height is provided as 2 inches, which implies that many women’s heights will fall within 2 inches of the mean height.

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed. It can take many forms, but one of the most common and important in statistics is the normal distribution, often referred to as the bell curve due to its shape. The normal distribution is characterized by a symmetrical curve where the mean, median, and mode of the distribution are equal.

An important property of the normal distribution is that approximately 68% of the data fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps to determine the likelihood of a particular event occurring and is key to making inferences about the population from a sample. In the height restriction problem, we use a normal distribution to calculate the probability of a woman being a certain height.

Normal Distribution Table

When dealing with normal distributions, a normal distribution table, sometimes called a z-table, is a very useful tool. This table lists z-scores and the corresponding percentile of the distribution to the left of the z-score. It's essentially a reference for determining what percentage of data falls below a certain z-score.

The table is organized in such a way that one can easily find the z-score obtained from the calculation and then read off the proportion of the population that lies below that z-score. This is exactly what we need for the exercise problem, as we need to assess what percentage of adult women fall below the height of 5 feet 7 inches. By looking up the z-score calculated earlier in the table, we effectively translate the standard score into a probability or percentage of the population that meets the criterion.