Question

A random sample of size n = 82 is taken from a population with mean μ = −15.6 and standard deviation σ = 5. [You may find it useful to reference the z table.]

a. Calculate the expected value and the standard error for the sampling distribution of the sample mean. (Negative values should be indicated by a minus sign. Round "expected value" to 1 decimal place and "standard error" to 4 decimal places.)

b. What is the probability that the sample mean is less than −16? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.)

c. What is the probability that the sample mean falls between −16 and −15? (Do not round intermediate calculations. Round "z" value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

a. The expected value for the sampling distribution of the sample mean is μ = -15.6, and the standard error is SE = σ / √n = 5 / √82 ≈ 0.5521 (rounded to 4 decimal places).

b. The probability that the sample mean is less than -16 corresponds to finding the z-score for this value. The z-score formula is z = (X - μ) / SE = (-16 - (-15.6)) / 0.5521 ≈ -0.7246. Referring to the z-table, the probability is approximately 0.2357 (rounded to 4 decimal places).

c. To find the probability that the sample mean falls between -16 and -15, calculate the z-scores for both values. For -16, z = (-16 - (-15.6)) / 0.5521 ≈ -0.7246, and for -15, z = (-15 - (-15.6)) / 0.5521 ≈ -1.0860. Using the z-table, the area between these z-scores is approximately 0.1566 (rounded to 4 decimal places).

Step-by-step explanation:

a. The expected value (mean) of the sampling distribution is the same as the population mean, which is μ = -15.6. The standard error (SE) for the sampling distribution of the sample mean is calculated using the formula SE = σ / √n, where σ is the population standard deviation and n is the sample size. Plugging in the values, SE ≈ 5 / √82 ≈ 0.5521, rounded to 4 decimal places.

b. To find the probability that the sample mean is less than -16, it involves calculating the z-score using the formula z = (X - μ) / SE, where X is the value of interest, μ is the population mean, and SE is the standard error. After computing the z-score for -16, which is approximately -0.7246, referencing the z-table provides the corresponding probability of around 0.2357, rounded to 4 decimal places.

c. Determining the probability that the sample mean falls between -16 and -15 requires calculating the z-scores for both values (-16 and -15) and then finding the difference in their areas under the standard normal curve using the z-table. After computing the z-scores and finding their respective probabilities, the area between these z-scores is approximately 0.1566, rounded to 4 decimal places, representing the probability that the sample mean falls within this range.