Questions: A simple random sample of size n=81 is obtained from a population that is skewed right with μ=74 and σ=27. (a) Describe the sampling distribution of x̄. (b) What is P(x̄>78.2) ? (c) What is P(x ≤ 66.95) ? (d) What is P(69.5<x̄<79.25) ?

A simple random sample of size n=81 is obtained from a population that is skewed right with μ=74 and σ=27.
(a) Describe the sampling distribution of x̄.
(b) What is P(x̄>78.2) ?
(c) What is P(x ≤ 66.95) ?
(d) What is P(69.5<x̄<79.25) ?

Transcript text: A simple random sample of size $n=81$ is obtained from a population that is skewed right with $\mu=74$ and $\sigma=27$. (a) Describe the sampling distribution of $\bar{x}$. (b) What is $\mathrm{P}(\bar{x}>78.2)$ ? (c) What is $\mathrm{P}(\mathrm{x} \leq 66.95)$ ? (d) What is $\mathrm{P}(69.5<\overline{\mathrm{x}}<79.25)$ ?

Solution

Solution Steps

Step 1: Describe the Sampling Distribution

The sampling distribution of the sample mean $ \bar{x} $ is approximately normal due to the Central Limit Theorem. It has the following parameters:

Mean: $ \mu = 74 $
Standard Deviation: $ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{27}{\sqrt{81}} = 3.0 $

Thus, the sampling distribution can be expressed as: \[ \bar{x} \sim N(74, 3.0) \]

Step 2: Calculate $ P(\bar{x} > 78.2) $

To find $ P(\bar{x} > 78.2) $, we first convert $ 78.2 $ to a Z-score: \[ Z = \frac{78.2 - \mu}{\sigma_{\bar{x}}} = \frac{78.2 - 74}{3.0} = 1.4 \] Using the standard normal distribution, we find: \[ P(\bar{x} > 78.2) = 1 - P(Z \leq 1.4) = 1 - 0.9192 = 0.0808 \]

Step 3: Calculate $ P(\bar{x} \leq 66.95) $

Next, we calculate $ P(\bar{x} \leq 66.95) $ by converting $ 66.95 $ to a Z-score: \[ Z = \frac{66.95 - \mu}{\sigma_{\bar{x}}} = \frac{66.95 - 74}{3.0} = -2.35 \] Thus, we have: \[ P(\bar{x} \leq 66.95) = P(Z \leq -2.35) = 0.0094 \]