Question

question 14
a simple random sample of size ( n = 20 ) is obtained from a population with ( mu = 154 ) and ( sigma = 47 ). enter your answer as an area under the curve with 4 decimal places.
( p(overline{x}>170)= )______

Explanation:

Step1: Calculate the standard error

The standard error formula is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 47$ and $n = 20$, then $\sigma_{\bar{x}}=\frac{47}{\sqrt{20}}\approx10.51$.

Step2: Calculate the z - score

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$.
Here, $\bar{x} = 170$, $\mu=154$, so $z=\frac{170 - 154}{10.51}\approx1.52$.

Step3: Find the probability

We want $P(\bar{X}>170)$, which is equivalent to $P(Z > 1.52)$.
Since $P(Z>z)=1 - P(Z\leq z)$, and from the standard normal table $P(Z\leq1.52)=0.9357$.
So $P(Z > 1.52)=1 - 0.9357=0.0643$.

Answer:

$0.0643$