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


The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.

Verified step by step guidance

1. Step 1: Identify the given information. The population mean (μ) is 20.7, the population standard deviation (σ) is 5.9, and the sample size (n) is 36. Since the sample size is large (n ≥ 30), the sampling distribution of the sample mean can be approximated by a normal distribution with mean μ and standard error (SE) calculated as SE = σ / √n.
2. Step 2: Calculate the standard error (SE) of the sampling distribution. Use the formula SE = σ / √n, where σ = 5.9 and n = 36. This will give you the standard deviation of the sampling distribution of the sample mean.
3. Step 3: Standardize the values for each part of the problem using the z-score formula. The z-score formula is z = (X̄ - μ) / SE, where X̄ is the sample mean, μ is the population mean, and SE is the standard error. For part (a), calculate the z-score for X̄ = 22. For part (b), calculate the z-score for X̄ = 23. For part (c), calculate the z-scores for X̄ = 20 and X̄ = 21.5.
4. Step 4: Use the standard normal distribution table (or a statistical software) to find the probabilities corresponding to the z-scores calculated in Step 3. For part (a), find the probability that the z-score is less than the value calculated for X̄ = 22. For part (b), find the probability that the z-score is greater than the value calculated for X̄ = 23. For part (c), find the probability that the z-score is between the values calculated for X̄ = 20 and X̄ = 21.5.
5. Step 5: Interpret the results. For each part, explain the probability in the context of the problem. For example, for part (a), the probability represents the likelihood that the mean score of a random sample of 36 ACT composite scores is less than 22. Similarly, interpret the results for parts (b) and (c).

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, such as the sample of 36 ACT scores in this question.

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. In this case, with sigma = 5.9 and a sample size of 36, the SEM will help determine the probability of the sample mean falling within specified ranges.

Z-scores

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 normal distribution, allowing us to determine the likelihood of the sample mean being less than, greater than, or between certain values.