Question

assume that the random variable x is normally distributed, with mean μ = 52 and standard deviation σ = 7. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded. p(x ≤ 44) which of the following shaded regions corresponds to p(x ≤ 44)?

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Here, \(x = 44\), \(\mu=52\), and \(\sigma = 7\).

$$z=\frac{44 - 52}{7}=\frac{-8}{7}\approx - 1.14$$

Step2: Find the probability using the standard normal distribution table

We want to find \(P(X\leq44)\), which is equivalent to \(P(Z\leq - 1.14)\) (since \(Z=\frac{X-\mu}{\sigma}\) and we found \(z\approx - 1.14\)).
Looking up \(z=-1.14\) in the standard - normal distribution table (the cumulative - distribution function for the standard normal distribution \(\varPhi(z)\)), we find that \(P(Z\leq - 1.14)=0.1271\)

For the normal - curve part, the normal curve is symmetric about \(x = 52\). The value \(x = 44\) is to the left of the mean \(x = 52\). The shaded region should be the area to the left of \(x = 44\) on the normal curve.

Answer:

The probability \(P(X\leq44)\approx0.1271\). For the multiple - choice part (assuming the options are about the position of the shaded region on the normal curve), the correct shaded region is the one to the left of \(x = 44\) (the option where the shaded area is on the left - hand side of the normal curve, typically corresponding to the left - tail probability).