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


The mean MCAT total score in a recent year is 500.9. A random sample of 32 MCAT total scores is selected. What is the probability that the mean score for the sample is (b) more than 502? Assume sigma=10.6.

Verified step by step guidance

1. Step 1: Identify the key information provided in the problem. The population mean (μ) is 500.9, the population standard deviation (σ) is 10.6, the sample size (n) is 32, and we are tasked with finding the probability that the sample mean (x̄) is greater than 502.
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 the same as the population mean (μ = 500.9), 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 = 10.6 and n = 32. This will give you the standard error.
4. Step 4: Convert the sample mean (502) to a z-score using the formula z = (x̄ - μ) / SE. Substitute the values: x̄ = 502, μ = 500.9, and the SE calculated in the previous step.
5. Step 5: Use the z-score obtained in Step 4 to find the corresponding probability from the standard normal distribution table or a statistical software. Since the problem asks for the probability that the sample mean is greater than 502, calculate the area to the right of the z-score (1 - cumulative probability).

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 original population distribution. This theorem is crucial for calculating probabilities related to sample means, especially when the sample size is sufficiently large, typically n > 30.

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 exceeding a certain value.

Z-Score

A Z-score measures how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the sample mean and then dividing by the standard error. Z-scores are essential for finding probabilities in a standard normal distribution, allowing us to interpret the likelihood of observing a sample mean greater than a specified value.