Final answer:
Using the Central Limit Theorem, we can determine the probability of the average of a random sample of 15 people being at least 46. The probability is approximately 0.9177 or 91.77%.
Step-by-step explanation:
To solve this problem, we need to use the Central Limit Theorem, which states that the sampling distribution of the sample mean from any population with a finite mean and variance will be approximately normally distributed, regardless of the shape of the population distribution. In this case, the mean of the sample mean is the same as the population mean, which is 40.5. The standard deviation of the sample mean, also known as the standard error, is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard error is equal to the square root of 75 divided by 15, which is approximately 3.87.
Next, we want to find the probability that the average of a random sample of 15 people is at least 46. To do this, we can standardize the sample mean using the z-score formula: z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the mean of the distribution, and σ is the standard deviation of the distribution. In this case, we have x = 46, μ = 40.5, and σ = 3.87. Plugging these values into the formula, we find that the z-score is approximately 1.40. We can then look up the probability associated with a z-score of 1.40 in the standard normal distribution table, which is approximately 0.9177.
Therefore, the probability that the average of a random sample of 15 people is at least 46 is approximately 0.9177 or 91.77%.