Question

Human heights are one of many biological random variables that can be modeled by the normal distribution. Assume the heights of men have a mean of 69 inches with a standard deviation of 3.5 inches. a. What proportion of all men will be taller than \(6^{\prime} 0^{\prime \prime}\) ? (HINT: Convert the measurements to inches.) b. What is the probability that a randomly selected man will be between \(5^{\prime} 8^{\prime \prime}\) and \(6^{\prime} 1^{\prime \prime}\) tall? c. President George \(\mathrm{W}\). Bush is \(5^{\prime} 11^{\prime \prime}\) tall. Is this an unusual height? d. Of the 42 presidents elected from 1789 through 2006,18 were \(6^{\prime} 0^{\prime \prime}\) or taller. \(^{1}\) Would you consider this to be unusual, given the proportion found in part a?

Short answer

2. What is the probability of a randomly selected man being between 5 feet 8 inches (68 inches) and 6 feet 1 inch (73 inches) tall? 3. Is George W. Bush's height of 5 feet 11 inches (71 inches) unusual? 4. Is the number of presidents 6 feet or taller unusual based on the proportion found in part 1?

Step-by-step solution

Calculate z-scores for the given heights

First, we need to calculate the z-scores for the heights mentioned in the questions (72 inches, 68 inches, and 73 inches). The formula for the z-score is: \(z = \frac{x - \mu}{\sigma}\), where x is the individual value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. 1. Z-score for 72 inches: \(z_{72} = \frac{72 - 69}{3.5}\) 2. Z-score for 68 inches: \(z_{68} = \frac{68 - 69}{3.5}\) 3. Z-score for 73 inches: \(z_{73} = \frac{73 - 69}{3.5}\) Now we can calculate these values: 1. \(z_{72} = \frac{3}{3.5} = 0.857\) 2. \(z_{68} = \frac{-1}{3.5} = -0.286\) 3. \(z_{73} = \frac{4}{3.5} = 1.143\)

Calculate the proportions using the z-scores

Next, we'll use z-scores to calculate the proportions asked in the exercise. a. First, we need to find the proportion of men taller than 72 inches (z-score \(0.857\)). We will look up the value for \(0.857\) in the z-table (standard normal table) and subtract this value from 1. The resulting proportion is the desired answer. \(Proportion = 1 - P(Z \leq 0.857) \approx 1 - 0.8049 = 0.1951\) b. To find the probability of a randomly selected man being between 68 inches and 73 inches tall, we'll use the z-scores \(-0.286\) and \(1.143\). We'll subtract the value of \(P(Z \leq -0.286)\) from \(P(Z \leq 1.143)\). \(Probability = P( -0.286 \leq Z \leq 1.143) \approx 0.8672 - 0.3873 = 0.4799\)

Determine if George W. Bush's height is unusual

George W. Bush's height is 71 inches, so we can calculate the z-score for this height using the formula mentioned earlier: \(z_{71} = \frac{71 - 69}{3.5} = \frac{2}{3.5} = 0.571\) Now, we can check the value of \(P(Z \leq 0.571)\) in the z-table. \(P(Z \leq 0.571) \approx 0.7163\) Since this value is not in the tails of the distribution (let's say, outside of the 5% on either side), it can be considered not unusual.

Analyze whether the number of presidents 6 feet or taller is unusual

We found in Step 2a that the proportion of men taller than 6 feet is 0.1951. Now let's find the expected number of presidents, out of 42, who would be 6 feet or taller based on this proportion. \(Expected\_Number = 0.1951 * 42 \approx 8.2\) There were actually 18 presidents 6 feet or taller, while the expected number is 8.2. This difference seems substantial, so it could be considered unusual.

Key concepts

Z-Score

Understanding the z-score is essential for interpreting data on a standard normal distribution. It is a statistical measure that describes a value's relationship to the mean of a group of values. Put simply, a z-score tells you how many standard deviations an element is from the mean.

When you calculate the z-score of an individual measurement, you're comprehending its position relative to the average. In the context of human heights, if the average height of men is 69 inches, a z-score will indicate how tall or short someone is compared to the average man.

For instance, a positive z-score implies that the measurement is above the mean, while a negative z-score means it's below the mean. A z-score of 0 signifies that the measurement is exactly at the mean. Understanding this concept allows for better interpretation of where a particular measurement stands in a normally distributed population.

Probability

In statistics, probability is a measure of the likelihood of an event occurring. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty. In the problem, figuring out the probability that a randomly selected man falls within a certain height range requires knowledge of the normal distribution and the ability to use z-scores.

Probabilities in a normal distribution can be found with z-tables, which showcase the percentage of data within certain z-score values. For example, if we wanted to determine how likely it is to find a man between 68 and 73 inches tall, we would look at the cumulative probability associated with the z-scores of those heights and calculate the difference. This application of probability in normal distribution is crucial for making predictions and understanding data variability.

Statistical Significance

The concept of statistical significance is a cornerstone for determining whether a result from a data set is likely due to a specific factor or merely by chance. In statistical terms, a result is considered statistically significant if the likelihood of the result occurring by chance is low, often below an arbitrary threshold of 5% (the p-value).

For example, assessing whether the number of tall presidents is statistically significant involves comparing the expected proportion based on general height distributions to the observed proportion in presidents. If a significant difference exists, like the unexpectedly high number of tall presidents, we might infer that height plays a role in presidential elections or that stature was regarded differently in certain historical periods. However, statistical significance doesn't always equate to practical significance, something to keep in mind when interpreting these results.

Standard Deviation

The standard deviation is a measure of variation or dispersion within a set of values. In a normal distribution, it quantifies how spread out numbers are from the mean value. A high standard deviation indicates a wide range of values, while a low standard deviation signifies that the values tend to be closer to the mean.

In the context of human heights, the standard deviation dictates the typical amount of height variation we can expect from the average. The problem presented involves a standard deviation of 3.5 inches for men's heights, which guides us in understanding the range within which most men's heights fall. Through the lens of a normal distribution, approximately 68% of data falls within one standard deviation of the mean, providing a clear and useful picture of variability within the population.