Question

a simple random sample of size n = 36 is obtained from a population with μ = 104 and σ = 23.
enter your answer as an area under the curve with 4 decimal places.
p(\\(\bar{x}\\) ≥ 107) = ______

Explanation:

Step1: Calculate the standard error

The standard error $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 23$ and $n = 36$, then $\sigma_{\bar{x}}=\frac{23}{\sqrt{36}}=\frac{23}{6}\approx3.8333$.

Step2: Calculate the z - score

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x} = 107$, $\mu = 104$, and $\sigma_{\bar{x}}\approx3.8333$. So $z=\frac{107 - 104}{3.8333}=\frac{3}{3.8333}\approx0.78$.

Step3: Find the probability

We want $P(\bar{X}\geq107)$, which is equivalent to $P(Z\geq0.78)$. Using the standard normal table $P(Z\geq z)=1 - P(Z < z)$. From the standard normal table, $P(Z < 0.78)=0.7823$. So $P(Z\geq0.78)=1 - 0.7823 = 0.2177$.

Answer:

$0.2177$