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 > 38) which of the following normal curves corresponds to p(x > 38)?

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 50\), \(\sigma=7\), and \(x = 38\).

$$z=\frac{38 - 50}{7}=\frac{-12}{7}\approx - 1.71$$

Step2: Use the standard normal distribution property

We know that \(P(X>38)=1 - P(X\leq38)\). From the standard normal table (or using a calculator with a normal - distribution function), \(P(Z\leq - 1.71)\) can be found.
Using a standard normal table or a calculator, \(P(Z\leq - 1.71)=0.0436\)
Then \(P(X > 38)=1-0.0436 = 0.9564\)

For the normal - curve part:
The normal curve has a mean \(\mu = 50\). The value \(x = 38\) is to the left of the mean. The probability \(P(X>38)\) corresponds to the area under the curve to the right of \(x = 38\). So the correct curve is the one that has the area to the right of \(38\) shaded.

Answer:

The probability \(P(X>38)\approx0.9564\). For the multiple - choice part (assuming we are just asked about the probability calculation here, but if we consider the curve - selection based on the description of the area to the right of \(38\) shaded, the curve should have the region starting from \(38\) and going to the right shaded). If we assume the options:

• Option A: Shaded area from \(38\) to \(50\) (incorrect, because we need \(X>38\) which is from \(38\) to \(\infty\))
• Option B: Shaded a small left - tail (incorrect)
• Option C: Shaded area from \(38\) to the right (correct)

So the probability \(P(X>38)\approx0.9564\) and the correct curve is Option C.