Question

assume that the random variable x is normally distributed, with mean μ = 50 and standard deviation σ = 10. compute the probability. be sure to draw a normal curve with the area corresponding to the probability shaded. p(x ≤ 45) which of the following shaded regions corresponds to p(x ≤ 45)?

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 50\), \(\sigma=10\), and \(x = 45\).

$$z=\frac{45 - 50}{10}=\frac{-5}{10}=- 0.5$$

Step2: Find the probability using the standard normal distribution table

We want to find \(P(X\leq45)\), which is equivalent to \(P(Z\leq - 0.5)\) using the standard normal distribution \(Z\sim N(0,1)\).
Looking up the value of \(P(Z\leq - 0.5)\) in the standard normal table (or using a calculator with a normal - distribution function), we know that \(P(Z\leq - 0.5)=0.3085\)

Answer:

The probability \(P(X\leq45)\) is \(0.3085\). For the graph, the region to the left of \(x = 45\) (since we are calculating \(P(X\leq45)\)) should be shaded. If we assume that in the options:

• Option A: If it is shaded to the left of \(45\) (a small region on the left - hand side of the normal curve near \(45\))
• Option B: If it is shaded from the left - tail up to \(45\) (the left - hand half of the curve up to \(45\) when \(\mu = 50\))
• Option C: If it is shaded from \(45\) to the right - tail (incorrect as we want \(X\leq45\))

The correct graph is the one where the area to the left of \(x = 45\) is shaded.