Question

Women's Height The mean height of American women between the ages of 30 and 39 is \(64.5\) inches, and the standard deviation is \(2.7\) inches. Find the probability that a 30 - to 39 -year-old woman chosen at random is (a) between 5 and 6 feet tall. (b) 5 feet 8 inches or taller. (c) 6 feet or taller

Short answer

The probabilities that a 30 - to 39-year-old woman chosen at random is: (a) between 5 feet and 6 feet tall can be calculated as \(P(\text{Z}_2) - P(\text{Z}_1)\), (b) is 5 feet 8 inches or taller is \(1 - P(\text{Z})\) where \(\text{Z} = 1.3\), and (c) is six feet or taller is \(1 - P(\text{Z}))\) where \(\text{Z} = 2.78\). The exact values can be found using a standard normal table.

Step-by-step solution

Convert Heights and Calculate Z-Scores

First, convert the heights into inches for ease of calculation. 5 feet converts to \(5 x 12 = 60\) inches and 6 feet converts to \(6 x 12 = 72\) inches. Also, 5 feet 8 inches converts to \(5 x 12 + 8 = 68\) inches. Then calculate the z-scores using the formula \[ \text{Z} = \frac{X - μ}{σ} \] where \(X\) is the value, \(μ\) is the mean, and \(σ\) is the standard deviation.

Calculate Z-Scores and Probabilities for Each Range

(a) For heights between 5 feet (60 inches) and 6 feet (72 inches), find the z-scores. \[ \text{Z}_1 = \frac{60 - 64.5}{2.7} \approx -1.67 \] \[ \text{Z}_2 = \frac{72 - 64.5}{2.7} \approx 2.78 \] The probability that height is between 5 and 6 feet can be found by subtracting the probabilities associated with these z-scores in a standard normal table. \[P = P(\text{Z}_2) - P(\text{Z}_1)\] (b) For a height of 5 feet 8 inches (68 inches) or taller, determine the z-score. \[ \text{Z} = \frac{68 - 64.5}{2.7} \approx 1.3 \] The probability that height is 5 feet 8 inches or taller is \(1 - P(\text{Z})\). (c) For a height of 6 feet or taller, find the z-score. \[ \text{Z} = \frac{72 - 64.5}{2.7} \approx 2.78 \] The probability that height is 6 feet or taller is \(1 - P(\text{Z})\).

Use Standard Normal Table to Determine Probabilities

Look up the probabilities associated with each z-score in the standard normal table. The probabilities associated with \(\text{Z} = -1.67\) and \(\text{Z} = 2.78\) for (a), and \(\text{Z} = 1.3\) for (b) and \(\text{Z} = 2.78\) for (c) are all you need. These values subtracted from 1 or subtracted from each other will give the required probabilities.

Key concepts

Z-Score

In statistics, the z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score can be positive or negative, with a positive value indicating the number of standard deviations above the mean and a negative value indicating the number below.

Calculating a z-score involves the formula: \[ \text{Z} = \frac{X - \mu}{\sigma} \] where \(X\) is the value being compared to the mean, \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation of the population. By converting raw scores to z-scores, we can describe exactly how many standard deviations our value is from the mean, and subsequently use this information to determine its position within a normal distribution. This allows for comparison between different data sets with different means and standard deviations.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. It is crucial in determining the reliability of statistical conclusions.

In the exercise, the standard deviation of the heights of American women aged 30 to 39 years is given as \(2.7\) inches, which helps us understand how much individual women's heights deviate from the average height. This deviation is essential when calculating the probability of a woman falling within certain height ranges, which becomes particularly useful when we apply it to the z-score formula to find out how 'typical' or 'atypical' a certain height is relative to the average.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Often referred to as the bell curve due to its shape, it is defined by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).

In real-life scenarios, such as the exercise on women's height, many variables are normally distributed. This implies that most women will have heights close to the average, with increasingly fewer women being significantly shorter or taller. When the heights are plotted on a graph, they approximate the bell curve, with the peak representing the mean height. Understanding the normal distribution allows for making inferences about populations using probability calculation.

Statistical Probability

Statistical probability refers to the likelihood of an event occurring. In terms of a normal distribution, it is the area under the curve that represents a range or single value. Using the z-score, we can map any given value to a point in the normal distribution and then calculate the probability of a score falling within or beyond that point.

In the given exercise, we calculate the likelihood that a randomly selected woman's height falls within certain ranges (e.g., between 5 and 6 feet tall). Using standard normal tables (or a calculator), we look up the probability associated with the computed z-scores. For example, to find the chance of a height between 5 feet and 6 feet, we determine the probabilities for the z-scores at 5 feet and at 6 feet, then subtract one from the other. This approach provides a concrete way to quantify the statistical probability.