Final answer:
Using the Central Limit Theorem, one would calculate the z-scores for each given condition using the formula z = (X - μ) / SE and then find the corresponding probabilities from a standard normal distribution table.
Step-by-step explanation:
To calculate the probability of a sample mean being greater than or less than a certain value, we use the Central Limit Theorem which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough, regardless of the distribution of the population. Given a population with a mean (μ) of 50 and a standard deviation (σ) of 19, for a sample size (n) of 64, we can find the standard error (SE) of the mean by dividing the population standard deviation by the square root of the sample size:
SE = σ / √n = 19 / √64 = 19 / 8 = 2.375
Next, we calculate the z-score for each scenario. The z-score indicates how many standard errors the sample mean is away from the population mean. The formula to calculate the z-score is:
z = (X - μ) / SE
where X is the sample mean. Using this formula, we calculate the z-scores for each part of the question and then consult the standard normal distribution table to find the probabilities:
- For a sample mean greater than 53, z = (53 - 50) / 2.375 = 1.263. Use this z-score to find the corresponding probability.
- For a sample mean less than 53, z = (53 - 50) / 2.375 = 1.263, but we need to look at the lower tail of the normal distribution.
- For a sample mean less than 47, z = (47 - 50) / 2.375 = -1.263.
- For a sample mean between 47.5 and 51.5, calculate the z-scores for both values and find the probability of the sample mean lying within this range.
- For a sample mean between 50.9 and 51.5, calculate the z-scores for both values and find the probability of the sample mean lying within this range.
Finally, round the z-scores to two decimal places and the probabilities to four decimal places as instructed in the question.