Question

let x be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of 75. find the value of x so that the area under the normal curve to the right of x is approximately.0275.

Explanation:

Step1: Find the z - score

The area to the right of \(x\) is \(0.0275\). So the area to the left of \(x\) is \(1 - 0.0275=0.9725\).
Looking up in the standard normal distribution table (z - table), the \(z\) - score corresponding to an area of \(0.9725\) is \(z = 1.92\).

Step2: Use the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 550\), \(\sigma=75\) and \(z = 1.92\).
Substitute the values into the formula: \(1.92=\frac{x - 550}{75}\).
Multiply both sides by \(75\): \(1.92\times75=x - 550\).
\(1.92\times75 = 144\). Then \(x=144 + 550\).

Answer:

\(x = 694\)