Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12. What is the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman’s final exam?Multiple ChoiceCannot be determined.0.02620.97380.6915

To solve this problem, we need to use the Central Limit Theorem which states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.

Step 1: Identify the population mean (μ) and the population standard deviation (σ). From the problem, we know that μ = 76 and σ = 12.

Step 2: Identify the sample size (n) and the sample mean (X̄). From the problem, we know that n = 15 and X̄ = 70.

Step 3: Calculate the standard error (SE) of the sample mean. The standard error is calculated as σ/√n. Substituting the given values, we get SE = 12/√15 ≈ 3.098.

Step 4: Calculate the z-score. The z-score is calculated as (X̄ - μ)/SE. Substituting the given values, we get z = (70 - 76)/3.098 ≈ -1.94.

Step 5: Look up the z-score in a standard normal distribution table or use a calculator to find the probability associated with that z-score. The probability that a z-score is less than -1.94 is approximately 0.0262.

However, we want the probability that the class average is greater than 70, so we need to subtract the probability we found from 1 (since the total probability under the normal curve is 1).

So, 1 - 0.0262 = 0.9738.

Therefore, the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman’s final exam is 0.9738 or 97.38%. So, the correct answer is 0.9738.