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: Recall Z - score formula

For a normal distribution, the Z - score is given by $Z=\frac{X-\mu}{\sigma}$, where $X$ is the value from the distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $P(X > 30)=0.10$, so $P(X\leq30)=1 - 0.10 = 0.90$.

Step2: Find the Z - score corresponding to 0.90

Using the standard normal distribution table (or Z - table), the Z - score that corresponds to a cumulative probability of 0.90 is approximately $z = 1.28$ (we can also use a calculator with the inverse of the normal CDF function, and for $P(Z\leq z)=0.9$, $z\approx1.28$).

Step3: Substitute into Z - score formula and solve for $\mu$

We know that $z=\frac{X - \mu}{\sigma}$, where $X = 30$, $\sigma=3.5$, and $z = 1.28$. Rearranging the formula for $\mu$ gives $\mu=X - z\sigma$.
Substitute the values: $\mu=30-1.28\times3.5$.
First, calculate $1.28\times3.5 = 4.48$.
Then, $\mu=30 - 4.48=25.52\approx26$.

Answer:

A. 26 inches