If we imagine all the possible random samples of 150 students at this university,95% of the samples should have means between what two numbers?
A)$2206.06 and $2793.94
B)$150.00 and $2695.96
C)$2304.04 and $2695.96
D)$300 and $4900
E)$150.00 and $2597.98

Question

Quizplus · Accepted Answer

By the central limit theorem, we know that the sampling distribution of the sample means will be approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$, where $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Since we don't know the population standard deviation, we can use the sample standard deviation $s$ as an estimate. From the information given in the problem, we know that $\bar{x} = 2500$ and $s = 300$.

To find the range of sample means that includes 95% of all possible sample means, we can use the formula:

$\bar{x} \pm 1.96\left(\frac{s}{\sqrt{n}}\right)$

Plugging in the values, we get:

$2500 \pm 1.96\left(\frac{300}{\sqrt{150}}\right) \approx 2304.04$ and $2695.96$

Therefore, we can be confident that 95% of all possible random samples of 150 students will have means between $2304.04 and $2695.96.

If We Imagine All the Possible Random Samples of 150