Final answer:
To use the normal model for sample means, the distribution of the population must be approximately normal. The sampling distribution of the sample mean will be approximately normal and its mean will equal the population mean. For P(x<70.1), calculate the z-score and find the associated probability. For P(x≥67.1), standardize the value and find the probability.
Step-by-step explanation:
To use the normal model to compute probabilities regarding the sample mean, the distribution of the population must be approximately normal. With a simple random sample of size n=11, the Central Limit Theorem states that if the sample size is sufficiently large, the sampling distribution of the sample mean will be approximately normal, regardless of the distribution of the population. The mean of the sampling distribution will be equal to the population mean, and the standard deviation of the sampling distribution (also known as the standard error of the mean) will be equal to the population standard deviation divided by the square root of the sample size.
(b) Assuming the normal model can be used, we can calculate P(x<70.1) by standardizing the value using the sampling distribution. The z-score formula is z = (x - μ) / (σ / √n), where x is the value we're interested in, μ is the population mean, σ is the population standard deviation, and n is the sample size. In this case, x = 70.1, μ = 66, σ = 17, and n = 11. We calculate the z-score as z = (70.1 - 66) / (17 / √11), which is approximately 0.732. We can then use a z-table or a calculator to find the probability associated with a z-score of 0.732, which is approximately 0.767.
(c) Similarly, we can calculate P(x≥67.1) by standardizing the value and finding the probability associated with the z-score. In this case, x = 67.1, μ = 66, σ = 17, and n = 11. The z-score is calculated as z = (67.1 - 66) / (17 / √11), which is approximately 0.227. The probability associated with a z-score of 0.227 is approximately 0.411.