In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for Level 1 actuaries in the United States is about \$72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (a) less than \(75,000? Assume sigma = \)11,000.

Verified step by step guidance

1. Step 1: Identify the key information provided in the problem. The population mean (μ) is \$72,000, the population standard deviation (σ) is \$11,000, the sample size (n) is 45, and we are tasked with finding the probability that the sample mean is less than \$75,000.
2. Step 2: Recognize that the sampling distribution of the sample mean follows a normal distribution because the sample size is sufficiently large (n > 30). The mean of the sampling distribution is equal to the population mean (μ = \$72,000), and the standard error of the mean (SE) is calculated as SE = σ / √n.
3. Step 3: Calculate the standard error of the mean using the formula SE = σ / √n. Substitute the values: σ = \$11,000 and n = 45. This will give you the standard error.
4. Step 4: Convert the sample mean of \(75,000 into a z-score using the formula z = (X̄ - μ) / SE, where X̄ is the sample mean, μ is the population mean, and SE is the standard error. Substitute the values X̄ = \)75,000, μ = \$72,000, and the SE calculated in Step 3.
5. Step 5: Use the z-score obtained in Step 4 to find the corresponding probability from the standard normal distribution table (or use statistical software). This probability represents the likelihood that the sample mean is less than \$75,000. Interpret the result in the context of the problem.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n > 30). This theorem is crucial for calculating probabilities related to sample means, especially when dealing with large samples.

Standard Error of the Mean

The Standard Error of the Mean (SEM) quantifies how much the sample mean is expected to vary from the true population mean. It is calculated by dividing the population standard deviation (sigma) by the square root of the sample size (n). In this case, SEM helps determine the probability of the sample mean being less than a specific value, such as $75,000.

Z-Score

A Z-score measures how many standard deviations an element is from the mean. In the context of this problem, the Z-score is used to standardize the sample mean to find the probability that it falls below a certain threshold. By calculating the Z-score for the sample mean of $75,000, we can use the standard normal distribution to find the corresponding probability.