Questions: Suppose a simple random sample of size n=36 is obtained from a population that is skewed right with μ=79 and σ=6. (a) Describe the sampling distribution of x̄. (b) What is P(x̄>80.5) ? (c) What is P(x̄ ≤ 77) ? (d) What is P(77.55<x̄<81.4) ? (a) Choose the correct description of the shape of the sampling distribution of x̄. A. The distribution is skewed right. B. The distribution is approximately normal. C. The distribution is skewed left. D. The distribution is uniform. E. The shape of the distribution is unknown. Find the mean and standard deviation of the sampling distribution of x̄. μx̄=□ σx̄=□ (Type integers or decimals. Do not round.) (b) P(x̄>80.5)= □ (Round to four decimal places as needed.) (c) P(x̄ ≤ 77)= □ (Round to four decimal places as needed.) (d) P(77.55<x̄<81.4)= □ (Round to four decimal places as needed.)

Suppose a simple random sample of size n=36 is obtained from a population that is skewed right with μ=79 and σ=6. (a) Describe the sampling distribution of x̄. (b) What is P(x̄>80.5) ? (c) What is P(x̄ ≤ 77) ? (d) What is P(77.55<x̄<81.4) ? (a) Choose the correct description of the shape of the sampling distribution of x̄. A. The distribution is skewed right. B. The distribution is approximately normal. C. The distribution is skewed left. D. The distribution is uniform. E. The shape of the distribution is unknown. Find the mean and standard deviation of the sampling distribution of x̄. μx̄=□ σx̄=□ (Type integers or decimals. Do not round.) (b) P(x̄>80.5)= □ (Round to four decimal places as needed.) (c) P(x̄ ≤ 77)= □ (Round to four decimal places as needed.) (d) P(77.55<x̄<81.4)= □ (Round to four decimal places as needed.)
Transcript text: Suppose a simple random sample of size $n=36$ is obtained from a population that is skewed right with $\mu=79$ and $\sigma=6$. (a) Describe the sampling distribution of $\bar{x}$. (b) What is $P(\bar{x}>80.5)$ ? (c) What is $P(\bar{x} \leq 77)$ ? (d) What is $P(77.55<\bar{x}<81.4)$ ? (a) Choose the correct description of the shape of the sampling distribution of $\overline{\mathrm{x}}$. A. The distribution is skewed right. B. The distribution is approximately normal. C. The distribution is skewed left. D. The distribution is uniform. E. The shape of the distribution is unknown. Find the mean and standard deviation of the sampling distribution of $\bar{x}$. \[ \begin{array}{l} \mu_{\bar{x}}=\square \\ \sigma_{\bar{x}}=\square \end{array} \] (Type integers or decimals. Do not round.) (b) $P(\bar{x}>80.5)=$ $\square$ (Round to four decimal places as needed.) (c) $P(\bar{x} \leq 77)=$ $\square$ (Round to four decimal places as needed.) (d) $P(77.55<\bar{x}<81.4)=$ $\square$ (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Determine the Sampling Distribution of \(\bar{x}\)

The mean of the sampling distribution (\(\mu_{\bar{x}}\)) is equal to the population mean (\(\mu\)) = 79. The standard deviation of the sampling distribution (\(\sigma_{\bar{x}}\)) is equal to the population standard deviation (\(\sigma\)) divided by the square root of the sample size (\(n\)), i.e., \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = 1\).

Step 2: Use the Standard Normal Distribution to Find Probabilities

Convert the given value for comparison (\(x\)) to its corresponding Z-score using the formula \(Z = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}} = 1.5\). Then, use the standard normal distribution to find the probability associated with these Z-scores, which is P(\bar{x} > 80.5) = 0.0668.

Final Answer:

The probability that P(\bar{x} > 80.5) is approximately 0.0668.

Step 1: Determine the Sampling Distribution of \(\bar{x}\)

The mean of the sampling distribution (\(\mu_{\bar{x}}\)) is equal to the population mean (\(\mu\)) = 79. The standard deviation of the sampling distribution (\(\sigma_{\bar{x}}\)) is equal to the population standard deviation (\(\sigma\)) divided by the square root of the sample size (\(n\)), i.e., \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = 1\).

Step 2: Use the Standard Normal Distribution to Find Probabilities

Convert the given value for comparison (\(x\)) to its corresponding Z-score using the formula \(Z = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}} = -2\). Then, use the standard normal distribution to find the probability associated with these Z-scores, which is P(\bar{x} \leq 77) = 0.0228.

Final Answer:

The probability that P(\bar{x} \leq 77) is approximately 0.0228.

Step 1: Determine the Sampling Distribution of \(\bar{x}\)

The mean of the sampling distribution (\(\mu_{\bar{x}}\)) is equal to the population mean (\(\mu\)) = 79. The standard deviation of the sampling distribution (\(\sigma_{\bar{x}}\)) is equal to the population standard deviation (\(\sigma\)) divided by the square root of the sample size (\(n\)), i.e., \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = 1\).

Step 2: Use the Standard Normal Distribution to Find Probabilities

Convert the given values for comparison (\(x_1\) and \(x_2\)) to their corresponding Z-scores using the formula \(Z = \frac{x - \mu_{\bar{x}}}{\sigma_{\bar{x}}}\), resulting in \(Z_1 = -1.45\) and \(Z_2 = 2.4\). Then, use the standard normal distribution to find the probability associated with these Z-scores, which is P(77.55 < \bar{x} < 81.4) = 0.918.

Final Answer:

The probability that P(77.55 < \bar{x} < 81.4) is approximately 0.918.

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