If we take a sample of size n = 100 from a population with $\mu$= 400 and $\sigma$= 20,find P(395.4

Question

If we take a sample of size n = 100 from a population with $\mu$= 400 and $\sigma$= 20,find P(395.4 < $\bar { x }$< 404.6).
A)0.9786
B)0.9878
C)0.1732
D)0.5398
E)0.8413

Quizplus · Accepted Answer

To find $P(395.4 < \bar{x} < 404.6)$, we use the Central Limit Theorem which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (n ≥ 30 is a common rule of thumb). Given $\mu = 400$, $\sigma = 20$, and $n = 100$, we first calculate the standard error of the mean (SEM) as $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = 2$. Next, we find the Z-scores for 395.4 and 404.6:$$Z_{395.4} = \frac{395.4 - 400}{2} = -2.3$$$$Z_{404.6} = \frac{404.6 - 400}{2} = 2.3$$Using the Z-table, we find the probabilities corresponding to these Z-scores and subtract the smaller from the larger to find the probability of the sample mean falling between 395.4 and 404.6. The Z-table gives us that the area to the left of $Z = 2.3$ is approximately 0.9893 and the area to the left of $Z = -2.3$ is approximately 0.0107. Subtracting these gives us $0.9893 - 0.0107 = 0.9786$, which is the probability of the sample mean falling within the specified range.

If We Take a Sample of Size N = 100 μ\muμ

If We Take a Sample of Size N = 100 $\mu$