Question

a simple random sample of size n = 42 is obtained from a population with μ = 64 and σ = 17
(a) what must be true regarding the distribution of the population in order to use the normal model to compute
probabilities involving the sample mean? assuming that this condition is true, describe the sampling distribution of x
(b) assuming the normal model can be used, determine p(\\(\bar{x}<68.2\\))
(c) assuming the normal model can be used, determine p(\\(\bar{x}\geq65.7\\))
assuming the normal model can be used, describe the sampling distribution \\(\bar{x}\\). choose the correct answer below.
a. approximately normal, with \\(\mu_{\bar{x}} = 64\\) and \\(\sigma_{\bar{x}}=\frac{17}{\sqrt{42}}\\)
b. approximately normal, with \\(\mu_{\bar{x}} = 64\\) and \\(\sigma_{\bar{x}} = 17\\)
c. approximately normal, with \\(\mu_{\bar{x}} = 64\\) and \\(\sigma_{\bar{x}}=\frac{42}{\sqrt{17}}\\)
(b) p(\\(\bar{x}<68.2\\)) = 0.9452 (round to four decimal places as needed.)
(c) p(\\(\bar{x}\geq65.7\\)) = (round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\). We know that \(\mu_{\bar{x}} = 64\), \(\sigma_{\bar{x}}=\frac{17}{\sqrt{42}}\approx2.62\), and \(\bar{x} = 65.7\).

$$z=\frac{65.7 - 64}{\frac{17}{\sqrt{42}}}=\frac{1.7}{2.62}\approx0.65$$

Step2: Find the probability

We want to find \(P(\bar{X}\geq65.7)\), which is equivalent to \(1 - P(\bar{X}<65.7)\). Using the standard normal distribution table, \(P(Z < 0.65)=0.7422\).

$$P(\bar{X}\geq65.7)=1 - 0.7422 = 0.2578$$

Answer:

\(0.2578\)