Question

The lengths of brook trout caught in a certain Colorado stream are normally distributed with a mean of 14 inches and a standard deviation of 3 inches. What proportion of brook trout caught will be between 12 and 18 inches in length?

Answer 1

Answer:

0.6563 or 65.63% of brook trout caught will be between 12 and 18 inches

Step-by-step explanation:

Mean trout length (μ) = 14 inches

Standard deviation (σ) = 3 inches

The z-score for any given trout length 'X' is defined as:  

$z=\frac{X-\mu}{\sigma}$  e interval

For a length of X =12 inches:

$z=\frac{12-14}{3}\\z=-0.6667$

According to a z-score table, a score of -0.6667 is equivalent to the 25.25th percentile of the distribution.

For a length of X =18 inches:

$z=\frac{18-14}{3}\\z=1.333$

According to a z-score table, a score of 1.333 is equivalent to the 90.88th percentile of the distribution.

The proportion of trout caught between 12 and 18 inches, assuming a normal distribution, is the interval between the equivalent percentile of each length:

$P(12\leq X\leq 18) = 90.88\% - 25.25\%\\P(12\leq X\leq 18) = 65.63\%$