Question

assume the random variable x is normally distributed with mean μ = 50 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 > 40)
which of the following normal curves corresponds to p(x > 40)?

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 50\), \(\sigma=7\), and \(x = 40\), we have \(z=\frac{40 - 50}{7}=\frac{-10}{7}\approx - 1.43\)

Step2: Use the standard normal distribution property

We know that \(P(X>40)=1 - P(X\leqslant40)\). From the standard normal table (or using a calculator with a normal - distribution function), \(P(Z\leqslant-1.43)\) can be found. Using a standard normal table (or a calculator like \(P(Z\leqslant z)\) where \(z=-1.43\)), \(P(Z\leqslant - 1.43)=0.0764\)

Step3: Compute the probability

\(P(X > 40)=1-0.0764 = 0.9236\)

For the normal - curve selection:
The normal curve for \(P(X>40)\) should have the area to the right of \(x = 40\) shaded. Since the mean \(\mu=50\), the curve is symmetric about \(x = 50\). The value \(x = 40\) is to the left of the mean. So the curve with the area to the right of \(x = 40\) (including the area from \(x = 40\) to \(x=50\) and from \(x = 50\) to \(\infty\)) shaded is the correct one.

If we assume that in option A the area from \(x = 40\) to \(x=\infty\) is shaded (the left tail is cut at \(x = 40\) and the rest of the curve is shaded), option B has a wrong shading (maybe only the left - of - mean part of \(x\in[40,50]\) is shaded in a wrong proportion), and option C has only a small left - tail shaded.

Answer:

The probability \(P(X>40)\approx0.9236\). The correct normal curve is the one where the area to the right of \(x = 40\) (including the part from \(x = 40\) to \(x = 50\) and \(x=50\) to \(\infty\)) is shaded (assuming option A is such a curve).