Question

The heights of men in a certain population follow a normal distribution with mean 69.7 inches and standard deviation 2.8 inches \({ }^{15}\). (a) If a man is chosen at random from the population, find the probability that he will be more than 76 inches tall. (b) If two men are chosen at random from the population, find the probability that (i) both of them will be more than 76 inches tall; (ii) their mean height will be more than 76 inches.

Short answer

The probability that a randomly chosen man is over 76 inches tall is the area to the right of the calculated Z-score from Part (a). The probability that both men are over 76 inches tall, for Part (b)(i), is the square of the probability from Part (a). And for Part (b)(ii), first calculate the Z-score using the standard error for two samples and then find the associated probability.

Step-by-step solution

- Calculate the Z-score for 76 inches

First, standardize the height of 76 inches using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X = 76 \) inches is the height of interest, \( \mu = 69.7 \) inches is the mean, and \( \sigma = 2.8 \) inches is the standard deviation. Calculate the Z-score to determine how many standard deviations 76 inches is above the mean.

- Find the probability for Part (a)

Use the Z-score obtained in Step 1 to find the probability that a man is more than 76 inches tall. This can be done using the standard normal distribution table, a calculator with normal distribution functions, or relevant software. The probability is the area to the right of the Z-score value.

- Calculate the probability that both men are taller than 76 inches

For part (b)(i), assume that the heights of the two men are independent. The probability of both men being taller than 76 inches is the product of the individual probabilities found in Step 2.

- Calculate the Z-score for the mean height of two men

For part (b)(ii), find the Z-score for the mean height of two men to be more than 76 inches, using the formula \( Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \), where \( \bar{X} = 76 \) inches, \( \mu = 69.7 \) inches, \( \sigma = 2.8 \) inches, and \( n = 2 \) since two men are chosen.

- Find the probability for Part (b)(ii)

With the Z-score calculated in Step 4, find the probability that the mean height of two randomly chosen men is more than 76 inches. This probability also comes from the standard normal distribution table or a calculator with normal distribution functions.

Key concepts

Z-Score Calculation

Understanding the concept of a Z-score is essential for analyzing data in relation to a mean, also known as the average. Imagine a bell curve where most values cluster around the center, and this center is the mean. A Z-score tells us how many standard deviations a particular value is from this mean.

For example, if we wanted to determine how unusual a man's height of 76 inches is in a population with a mean height of 69.7 inches and a standard deviation of 2.8 inches, we'd calculate the Z-score using the formula: \[ Z = \frac{X - \mu}{\sigma} \].

After substituting the given values into the formula: \[ Z = \frac{76 - 69.7}{2.8} \],we get a numerical value that quantifies the number of standard deviations from the mean. This Z-score can then be used to compare this man's height to the overall population's height distribution.

Standard Deviation

Standard deviation is a statistical measurement that shows how much variation or dispersion there is from the average (mean). In simpler terms, it indicates whether the data points are close to the mean or spread out over a wide range.

For the heights of men in a certain population, a standard deviation of 2.8 inches suggests that most men's heights fall within 2.8 inches of the mean, which is 69.7 inches in this case. If the standard deviation were larger, heights would be more spread out. Conversely, a smaller standard deviation would indicate that heights are more tightly clustered around the mean.

This concept is crucial because it allows us to calculate how unusual or typical a certain height is within that population. High standard deviation equals more variety, while low standard deviation signifies less diversity in heights.

Probability

Probability is the measure of how likely an event is to occur. Think about flipping a coin; there's a 50% probability of getting heads because two possible outcomes are equally likely.

In the context of the height example, probability can show us how common or rare it is to encounter a man over 76 inches tall. After getting the Z-score, we use this value to find the probability of a random man being this tall by looking up the score in the standard normal distribution table or by using a calculator with statistical functions.

The calculation of how likely it is for two random men to both be over 76 inches tall involves multiplying the individual probabilities because these events are independent of each other. Understanding these probabilities helps us grasp the rareness or commonness of an event based on the characteristics of the population in question.

Mean Height

The mean height is simply the average height of a group, calculated by adding up all the heights and dividing by the number of people in the group. For normally distributed data like our population of men's heights, the mean is also the peak of the bell curve where most data points cluster.

When evaluating the probability that two random men have a mean height greater than 76 inches, we're actually considering a different probability scenario from looking at a single individual's height. To calculate this probability, we adjust our Z-score formula to incorporate the sample size; in this case, the sample size is two men. The formula becomes \[ Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \]

The resulting Z-score represents how many standard deviations the average or mean height of these two men is from the population’s mean height. Essentially, it's a measure of whether this pair of heights is typical or an outlier when compared to the average height of all men in the population.