Suppose that a random sample of size 64 is to be selected from a population with mean 30 and standard
deviation 7.
(a) What are the mean and standard deviation of the sampling distribution of x
(b) What is the approximate probability that will be within 0.5 of the population mean 𝜇? (Round your
answer to four decimal places.)
(c) What is the approximate probability that will differ from 𝜇 by more than 0.9? (Round your answer to
four decimal places.)
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(a) The mean of the sampling distribution of x is equal to the population mean, which is 30. The standard deviation of the sampling distribution of x, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size. So, the standard deviation of the sampling distribution of x is 7 / sqrt(64) = 7 / 8 = 0.875.
(b) To find the approximate probability that x will be within 0.5 of the population mean 𝜇, we can use the standard deviation of the sampling distribution of x to determine the standard error. Then, we can use the standard normal distribution to find the probability. In this case, we want to find the probability that x is between 𝜇 - 0.5 and 𝜇 + 0.5.
Using the formula z = (x - 𝜇) / standard error, we can calculate the z-values for 𝜇 - 0.5 and 𝜇 + 0.5:
z1 = (𝜇 - 0.5 - 𝜇) / standard error = -0.5 / 0.875 = -0.5714
z2 = (𝜇 + 0.5 - 𝜇) / standard error = 0.5 / 0.875 = 0.5714
Now, we can use a standard normal distribution table or calculator to find the probability of being between these two z-values. The probability is the area under the curve between -0.5714 and 0.5714.
P(𝜇 - 0.5 < x < 𝜇 + 0.5) ≈ 2 * P(z < 0.5714) - 1 ≈ 2 * 0.7157 - 1 ≈ 0.4314
So, the approximate probability that x will be within 0.5 of the population mean 𝜇 is 0.4314.
(c) To find the approximate probability that x will differ from 𝜇 by more than 0.9, we can use the same method as in part (b). In this case, we want to find the probability that x is either less than 𝜇 - 0.9 or greater than 𝜇 + 0.9.
Using the same formula z = (x - 𝜇) / standard error, we can calculate the z-values for 𝜇 - 0.9 and 𝜇 + 0.9:
z1 = (𝜇 - 0.9 - 𝜇) / standard error = -0.9 / 0.875 = -1.0286
z2 = (𝜇 + 0.9 - 𝜇) / standard error = 0.9 / 0.875 = 1.0286
Now, we can find the probability of being less than -1.0286 or greater than 1.0286. This is the sum of the areas to the left of -1.0286 and to the right of 1.0286 on the standard normal distribution.
P(|x - 𝜇| > 0.9) ≈ P(z < -1.0286) + P(z > 1.0286) ≈ 2 * (1 - P(z < 1.0286)) ≈ 2 * (1 - 0.8438) ≈ 0.3124
So, the approximate probability that x will differ from 𝜇 by more than 0.9 is 0.3124.
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