Problem 21 - A First Course in Probability

Question

Suppose that the height, in inches, of a 25 -year-old man is a normal random variable with parameters \(\mu=71\) and \(\sigma^{2}=6.25 .\) What percentage of 25 year-old men are over 6 feet, 2 inches tall? What percentage of men in the 6 -footer club are over 6 feet, 5 inches?

Step-by-Step Solution

Answer

Approximately 11.51% of 25-year-old men are over 6 feet 2 inches tall, and approximately 2.38% of men in the 6-footer club are over 6 feet 5 inches tall.

Step 1: Convert heights from feet to inches

To answer the questions, we need to first convert the given heights in feet and inches into inches for consistency. To convert 6 feet 2 inches into inches: 6 feet = 6 * 12 = 72 inches Total height = 72 + 2 = 74 inches To convert 6 feet 5 inches into inches: 6 feet = 6 * 12 = 72 inches Total height = 72 + 5 = 77 inches

Step 2: Calculate the z-scores

For each height, we will calculate the z-score, which represents how many standard deviations away a value is from the mean. The formula for calculating the z-score is: \(Z = \frac{(x - \mu)}{\sigma}\) where \(x\) is the value we want to find the z-score for, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Since the variance is given as \(\sigma^2 = 6.25\), it means that \(\sigma = \sqrt{6.25}= 2.5\). For 74 inches, the z-score is: \(Z_{74} = \frac{(74 - 71)}{2.5} = \frac{3}{2.5} = 1.2\) For 77 inches, the z-score is: \(Z_{77} = \frac{(77 - 71)}{2.5} = \frac{6}{2.5} = 2.4\)

Step 3: Find the probabilities using the z-scores

Using the z-scores, we can find the probability that a man's height will be above the heights in question by using a standard normal distribution table or a calculator. For men over 74 inches (6 feet 2 inches): P(height > 74) = P(Z > 1.2) ≈ 0.1151 (or 11.51%) For the 6-footer club (height >= 72 inches), we need to find the probability of men over 77 inches (6 feet 5 inches): P(height > 77) = P(Z > 2.4) ≈ 0.0082 (or 0.82%) However, to get the percentage of men in the 6-footer club who are over 77 inches tall, we need to first find the probability of being in the 6-footer club: P(height >= 72) = P(Z >= 0.4) = 1 - P(Z < 0.4) ≈ 1 - 0.6554 = 0.3446 (or 34.46%) Then, we will divide the probability of being over 77 inches tall by the probability of being in the 6-footer club: P(height > 77 | height >= 72) = \(\frac{0.0082}{0.3446}\) ≈ 0.0238 (or 2.38%) In conclusion, approximately 11.51% of 25-year-old men are over 6 feet 2 inches tall, and approximately 2.38% of men in the 6-footer club are over 6 feet 5 inches tall.

Key Concepts

Z-Score Calculation

The z-score is a numerical measurement that tells us how many standard deviations an element is from the mean. Z-scores are used in statistics to determine how unusual or common a data point is within a normal distribution.

The formula for the z-score is given by:

• \( Z = \frac{(x - \mu)}{\sigma} \)

where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Let's see how it's calculated:

• For 74 inches: \( Z_{74} = \frac{(74 - 71)}{2.5} = 1.2 \)


• For 77 inches: \( Z_{77} = \frac{(77 - 71)}{2.5} = 2.4 \)

These z-scores tell us how these heights compare to the average height in terms of standard deviations. A positive z-score means the value is above the mean.

Standard Deviation

Understanding standard deviation is crucial as it provides insight into the spread of a set of data points. It tells us how much these data points deviate from the mean. In our exercise, the standard deviation \(\sigma\) is calculated from the given variance, which is \(\sigma^2 = 6.25\).

So, \(\sigma = \sqrt{6.25} = 2.5\).

This means that most height values fall within 2.5 inches above or below the mean height of 71 inches for the group we are studying. A smaller standard deviation would indicate that the data points are clustered closely around the mean, while a larger standard deviation indicates a broader spread.

Probability Calculation

Probability calculation using the normal distribution is a way to measure the likelihood of a certain outcome. We use z-scores to find these probabilities. A higher z-score means the data point is far from the mean, thus affecting the probability.

Once we've determined our z-score, a standard normal distribution table (or calculator) helps us find the probability. Here’s how it works in our example:

• For men over 74 inches: \(P(Z > 1.2) \approx 0.1151\), meaning 11.51% of men are taller than 74 inches.


• For men over 77 inches, \(P(Z > 2.4) \approx 0.0082\), or 0.82% of men are taller than 77 inches.

Moreover, to find a conditional probability, such as men in the "6-footer club" also being over 77 inches, you have to divide the smaller probability by the probability of being in the larger group.

Variance

Variance measures how far a set of numbers is spread out from their average value. It is the expectation of the squared deviation of a random variable from its mean, and it gives us insight into the degree of spread in the data.

The formula for variance is:\

• \( \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \)

In our case, the exercise provides a variance of 6.25 for the height of 25-year-old men. This is calculated under the assumption that heights follow a normal distribution pattern. The square root of this variance gives us the standard deviation, \( \sigma = 2.5 \), which demonstrates the average deviation in height from the mean of 71 inches. Understanding variance is vital as it explains the dispersion, which influences the reliability of the mean as a representation of the dataset.