rson.com/Student/PlayerTest.aspx?testld=257861046 202324 TERM3 Deanna Gordon 11/21/23 12:58 PM (?) $8.1 \& 8.2$ Question 2 of 7 This test: 7 point(s) possible This question: 1 point(s) possible Submit test A simple random sample of size $n=12$ is obtained from a population with $\mu=62$ and $\sigma=19$. (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 $\mathrm{P}(\bar{x}<65.9)$. (c) Assuming the normal model can be used, determine $\mathrm{P}(\bar{x} \geq 64.2)$. A. Normal, with $\mu_{\bar{x}}=62$ and $\sigma_{\bar{x}}=\frac{12}{\sqrt{19}}$ B. Normal, with $\mu_{\bar{x}}=62$ and $\sigma_{\bar{x}}=\frac{19}{\sqrt{12}}$ C. Normal, with $\mu_{\bar{x}}=62$ and $\sigma_{\bar{x}}=19$ (b) $\mathrm{P}(\overline{\mathrm{x}}<65.9)=\square$ (Round to four decimal places as needed.) (c) $P(\bar{x} \geq 64.2)=\square$ (Round to four decimal places as needed.) Next
Solution
Step 1 ：The problem is asking for the conditions that must be true for us to use the normal model to compute probabilities involving the sample mean. It also asks us to describe the sampling distribution of the sample mean, assuming that these conditions are true.
Step 2 ：We can use the normal model to compute probabilities involving the sample mean if the population from which the sample is drawn is normally distributed or if the sample size is large enough (usually n > 30). In this case, the sample size is 12, which is not large enough. However, the question does not provide information about the distribution of the population. Therefore, we have to assume that the population is normally distributed.
Step 3 ：The sampling distribution of the sample mean, denoted as \(\bar{x}\), is also normally distributed. Its mean is equal to the mean of the population, and its standard deviation, often called the standard error, is equal to the standard deviation of the population divided by the square root of the sample size.
Step 4 ：In this case, the mean of the sampling distribution, \(\mu_{\bar{x}}\), is 62, and the standard deviation of the sampling distribution, \(\sigma_{\bar{x}}\), is \(\frac{19}{\sqrt{12}}\).
Step 5 ：Final Answer: The population must be normally distributed in order to use the normal model to compute probabilities involving the sample mean. The sampling distribution of the sample mean is normal, with \(\boxed{\mu_{\bar{x}}=62}\) and \(\boxed{\sigma_{\bar{x}}=\frac{19}{\sqrt{12}}}\).
