Question

a simple random sample of size ( n = 14 ) is obtained from a population with ( mu = 66 ) and ( sigma = 15 ).
(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 ( \bar{x} ).
(b) assuming the normal model can be used, determine ( p(\bar{x}<69.6) ).
(c) assuming the normal model can be used, determine ( p(\bar{x} geq 68.3) ).
(a) what must be true regarding the distribution of the population?
a. the population must be normally distributed and the sample size must be large.
b. the population must be normally distributed.
c. since the sample size is large enough, the population distribution does not need to be normal.
d. the sampling distribution must be assumed to be normal.
assuming the normal model can be used, describe the sampling distribution ( \bar{x} ). choose the correct answer below.
a. normal, with ( mu_{\bar{x}} = 66 ) and ( sigma_{\bar{x}}=\frac{14}{sqrt{15}} ).
b. normal, with ( mu_{\bar{x}} = 66 ) and ( sigma_{\bar{x}} = 15 ).
c. normal, with ( mu_{\bar{x}} = 66 ) and ( sigma_{\bar{x}}=\frac{15}{sqrt{14}} ).
(b) ( p(\bar{x}<69.6)=square ) (round to four decimal places as needed.)
(c) ( p(\bar{x} geq 68.3)=square ) (round to four decimal places as needed.)

Explanation:

Step1: Check population distribution condition

For the normal model to be used for the sampling distribution of the sample mean when \(n = 14\) (a small sample size, \(n<30\)), the population must be normally distributed.

Step2: Determine sampling distribution parameters

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}=\mu = 66\), and the standard deviation of the sampling distribution of the sample mean \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{15}{\sqrt{14}}\)

Step3: Calculate \(P(\bar{x}<69.6)\)

First, calculate the z - score \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}=\frac{69.6 - 66}{\frac{15}{\sqrt{14}}}\approx\frac{3.6}{4.0056}\approx0.90\)
Then, using the standard normal table \(P(Z < 0.90)=0.8159\)

Step4: Calculate \(P(\bar{x}\geq68.3)\)

Calculate the z - score \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}=\frac{68.3 - 66}{\frac{15}{\sqrt{14}}}\approx\frac{2.3}{4.0056}\approx0.57\)
\(P(\bar{x}\geq68.3)=1 - P(\bar{x}<68.3)=1 - P(Z < 0.57)\)
From the standard normal table \(P(Z < 0.57)=0.7157\), so \(P(\bar{x}\geq68.3)=1 - 0.7157 = 0.2843\)

Answer:

(a) B. The population must be normally distributed
Sampling distribution: C. Normal, with \(\mu_{\bar{x}} = 66\) and \(\sigma_{\bar{x}}=\frac{15}{\sqrt{14}}\)
(b) \(0.8159\)
(c) \(0.2843\)