Question

the distribution of lengths of salmon from a certain river is approximately normal with standard deviation 3.5 inches. if 10 percent of salmon are longer than 30 inches, which of the following is closest to the mean of the distribution?
a 26 inches
b 28 inches
c 30 inches
d 33 inches
e 34 inches

Explanation:

Step1: Use z - score table

If 10% of salmon are longer than 30 inches, then the area to the right of 30 under the normal - distribution curve is 0.1. So the area to the left of 30 is \(1 - 0.1=0.9\). Looking up in the standard normal distribution (z - score) table, the z - score corresponding to an area of 0.9 is approximately \(z = 1.28\).

Step2: Apply z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x = 30\) (the value from the original distribution), \(\mu\) is the mean of the original distribution, \(\sigma = 3.5\) (the standard deviation of the original distribution), and \(z = 1.28\).
Substitute the values into the formula: \(1.28=\frac{30 - \mu}{3.5}\).

Step3: Solve for the mean \(\mu\)

First, multiply both sides of the equation \(1.28=\frac{30 - \mu}{3.5}\) by 3.5: \(1.28\times3.5=30 - \mu\).
\(1.28\times3.5 = 4.48\), so \(4.48=30 - \mu\).
Then, solve for \(\mu\): \(\mu=30 - 4.48 = 25.52\approx26\).

Answer:

A. 26 inches