Problem 35 - Basic Statistics for Business and Economics with Student

Question

The mean age at which men in the United States marry for the first time follows the normal distribution with a mean of 24.8 years. The standard deviation of the distribution is 2.5 years. For a random sample of 60 men, what is the likelihood that the age at which they were married for the first time is less than 25.1 years?

Step-by-Step Solution

Answer

The likelihood is approximately 82.38%.

Step 1: Understand the Problem

We need to find the probability that the mean age of a sample of 60 men is less than 25.1 years. The population mean is 24.8 years and the standard deviation is 2.5 years. We'll use the normal distribution model to determine this probability.

Step 2: Calculate the Standard Error

The standard error (SE) of the mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} \]where \( \sigma = 2.5 \) is the standard deviation and \( n = 60 \) is the sample size. So,\[ SE = \frac{2.5}{\sqrt{60}} \approx 0.322 \]

Step 3: Find the Z-Score

The Z-score tells us how many standard deviations an element is from the mean. It is calculated using the formula:\[ Z = \frac{X - \mu}{SE} \]where \( X = 25.1 \) is the sample mean we're comparing to, \( \mu = 24.8 \) is the population mean, and \( SE = 0.322 \). So,\[ Z = \frac{25.1 - 24.8}{0.322} \approx 0.93 \]

Step 4: Use the Z-Table

Look up the Z-score of 0.93 in the standard normal distribution (Z) table to find the probability. For \( Z = 0.93 \), the cumulative probability is approximately 0.8238. This means there is about an 82.38% chance that the sample mean will be less than 25.1 years.

Key Concepts

Mean Age

The mean age is an essential concept in statistics, especially when examining how data tends to cluster. In the context of our exercise, the mean age represents the average age at which men in the United States enter into their first marriage.

The mean age acts as a central point of a normal distribution, which is a bell-shaped curve that shows how data values are spread. In this exercise, the mean age is given as 24.8 years. This value tells us that most men marry around this age.

Understanding the mean allows us to compare individual data points and calculate additional statistics, such as variance and standard deviation. It is a fundamental measure of central tendency, crucial for drawing insights from datasets.

Standard Deviation

Standard deviation is a measure that shows how much variation or dispersion there is from the mean. In simpler terms, it lets us know how spread out the values are in a dataset.

In our example with the mean age of first marriages being 24.8 years, the standard deviation is 2.5 years. This means that the ages at which men marry are generally within 2.5 years above or below the mean.

A low standard deviation indicates that most of the ages are clustered around the mean. A high standard deviation would suggest that there is more variation in the ages at which men marry. For statistical analyses such as calculating probabilities, knowing the standard deviation helps us understand the extent to which data deviates from the average.

Z-Score

The Z-score is a statistical measure that describes a data point's relation to the mean of a group of data. It represents the number of standard deviations a data point is from the mean.

In this exercise, we calculated the Z-score to determine where 25.1 years lies concerning the mean age of 24.8 years, given the standard deviation. Using the Z-score formula: \[ Z = \frac{X - \mu}{SE} \] We find that 25.1 years is about 0.93 standard deviations above the mean.

Z-scores are especially helpful in identifying how unusual or typical a data point is within a distribution. A Z-score close to 0 suggests the value is near the mean, while a high positive or negative Z-score indicates the value is far from the mean.

Probability

Probability measures the likelihood of a specific outcome or event occurring. It's often expressed as a number between 0 (impossible) to 1 (certain). In this exercise, we want to know the probability that the average age of marriage for a sample of men is less than 25.1 years.

With the calculated Z-score of 0.93, we use the standard normal distribution table (also known as the Z-table) to find this probability. For a Z-score of 0.93, the cumulative probability is approximately 0.8238, or 82.38%.

This means there is an 82.38% chance that the sample mean of ages will be less than 25.1 years, which reflects how often this outcome is expected under the specified conditions. Probability helps in understanding and predicting the behavior of a dataset in the context of real-world scenarios.