Question

assume that the random variable x is normally distributed, with mean μ = 48 and standard deviation σ = 9. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded. p(x ≤ 42) which of the following shaded regions corresponds to p(x ≤ 42)? p(x ≤ 42) = (round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
Given \(\mu = 48\), \(\sigma=9\), and \(x = 42\).
Substitute the values into the formula: \(z=\frac{42 - 48}{9}=\frac{-6}{9}\approx - 0.67\).

Step2: Find the probability using the standard normal distribution table

We want to find \(P(X\leq42)\), which is equivalent to \(P(Z\leq - 0.67)\) (since \(X\) is normally distributed and we have transformed it to the standard normal variable \(Z\)).
Looking up the value of \(P(Z\leq - 0.67)\) in the standard - normal distribution table (or using a calculator with a normal - distribution function, e.g., in Excel: NORM.S.DIST(-0.67,TRUE)).
The value of \(P(Z\leq - 0.67)=0.2514\)

Answer:

\(0.2514\)