Answer:
The probability that the mean of the sample would differ from the population mean by less than 20 points is approximately 0.8546.
Explanation:
To find the probability that the mean of the sample would differ from the population mean by less than 20 points, we can use the Central Limit Theorem. This theorem states that for a large enough sample size, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution.
Let's break down the steps:
1. Calculate the standard error of the mean (SEM), which is the standard deviation of the population divided by the square root of the sample size:
SEM = Standard deviation / √Sample size
= 334 / √314
≈ 18.859
2. Calculate the z-score, which measures how many standard errors the desired difference is from the population mean:
z-score = (Desired difference) / SEM
= 20 / 18.859
≈ 1.059
3. Look up the probability corresponding to the z-score in the standard normal distribution table. In this case, we want to find the probability that the difference is less than 20 points, so we need the area to the left of the z-score.
From the standard normal distribution table, the probability corresponding to a z-score of 1.059 is approximately 0.8546.
4. Round the probability to four decimal places:
Probability = 0.8546