Question

a simple random sample of size n = 20 is obtained from a population with μ = 154 and σ = 47. enter your answer as an area under the curve with 4 decimal places. p(\\(\overline{x}\\) ≤ 134) = ________

Explanation:

Step1: Calculate the standard error

The standard error \(SE=\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 47\) and \(n = 20\).
\(SE=\frac{47}{\sqrt{20}}\approx\frac{47}{4.4721}\approx10.51\)

Step2: Calculate the z - score

The z - score formula is \(z=\frac{\bar{x}-\mu}{SE}\), where \(\bar{x} = 134\), \(\mu=154\) and \(SE\approx10.51\)
\(z=\frac{134 - 154}{10.51}=\frac{- 20}{10.51}\approx - 1.90\)

Step3: Find the probability

Using the standard normal distribution table (or a calculator with a normal - distribution function), we find \(P(Z\leq - 1.90)\)
From the standard normal table, \(P(Z\leq - 1.90)=0.0287\)

Answer:

\(0.0287\)