Questions: A simple random sample of size n=36 is obtained from a population that is skewed right with μ=76 and σ=24. (a) Describe the sampling distribution of x̄. (b) What is P(x̄>81)? (c) What is P(x̄ ≤ 67.6)? (d) What is P(72<x̄<82.8)? Find the mean and standard deviation of the sampling distribution of x̄. μx̄=□ σx̄⁻=□ (Type integers or decimals. Do not round (b) P(x>81)= □ (Round to four decimal places as needed.) (c) P(x̄ ≤ 67.6)= □ (Round to four decimal places as needed.)

A simple random sample of size n=36 is obtained from a population that is skewed right with μ=76 and σ=24.
(a) Describe the sampling distribution of x̄.
(b) What is P(x̄>81)?
(c) What is P(x̄ ≤ 67.6)?
(d) What is P(72<x̄<82.8)?

Find the mean and standard deviation of the sampling distribution of x̄.

μx̄=□
σx̄⁻=□
(Type integers or decimals. Do not round
(b) P(x>81)= □ (Round to four decimal places as needed.)
(c) P(x̄ ≤ 67.6)= □ (Round to four decimal places as needed.)

Transcript text: A simple random sample of size $\mathrm{n}=36$ is obtained from a population that is skewed right with $\mu=76$ and $\sigma=24$. (a) Describe the sampling distribution of $\bar{x}$. (b) What is $P(\bar{x}>81)$ ? (c) What is $P(\bar{x} \leq 67.6)$ ? (d) What is $P(72<\bar{x}<82.8)$ ? Find the mean and standard deviation of the sampling distribution of $\bar{x}$. \[ \begin{array}{l} \mu_{x}=\square \\ \sigma_{x}^{-}=\square \end{array} \] (Type integers or decimals. Do not roun (b) $P(x>81)=$ $\square$ (Round to four decimal places as needed.) (c) $P(\bar{x} \leq 67.6)=$ $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Describe the Sampling Distribution

The sampling distribution of the sample mean $ \bar{x} $ for a sample size $ n = 36 $ drawn from a population with mean $ \mu = 76 $ and standard deviation $ \sigma = 24 $ is characterized by:

Mean: \[ \mu_{\bar{x}} = \mu = 76 \]
Standard deviation (standard error): \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{36}} = 4.0 \]

Step 2: Calculate $ P(\bar{x} > 81) $

To find $ P(\bar{x} > 81) $, we first calculate the Z-score for $ \bar{x} = 81 $: \[ Z = \frac{81 - \mu_{\bar{x}}}{\sigma_{\bar{x}}} = \frac{81 - 76}{4} = 1.25 \] Using the standard normal distribution, we find: \[ P(\bar{x} > 81) = 1 - P(Z \leq 1.25) = 1 - 0.8944 = 0.1056 \] Thus, \[ P(\bar{x} > 81) = 0.8944 \]

Step 3: Calculate $ P(\bar{x} \leq 67.6) $

Next, we calculate $ P(\bar{x} \leq 67.6) $ by finding the Z-score for $ \bar{x} = 67.6 $: \[ Z = \frac{67.6 - \mu_{\bar{x}}}{\sigma_{\bar{x}}} = \frac{67.6 - 76}{4} = -2.1 \] Using the standard normal distribution, we find: \[ P(\bar{x} \leq 67.6) = P(Z \leq -2.1) = 0.0179 \]