the life spans of a computer manufacturers hard drives are normally distributed, with a mean of 3 years 6…

Explanation:

Step1: Convert time to months

The mean $\mu = 3\times12 + 6=42$ months, the standard - deviation $\sigma = 9$ months. The lower value $x_1=2\times12 + 3 = 27$ months and the upper value $x_2=3\times12+3 = 39$ months.

Step2: Calculate the z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$. For $x = x_1 = 27$ months, $z_1=\frac{27 - 42}{9}=\frac{- 15}{9}\approx - 1.67$. For $x = x_2 = 39$ months, $z_2=\frac{39 - 42}{9}=\frac{-3}{9}\approx - 0.33$. We know that the standard normal distribution is symmetric about $z = 0$. So, $P(-1.67<Z<-0.33)=P(0.33 < Z<1.67)$. $P(0.33 < Z<1.67)=P(Z < 1.67)-P(Z < 0.33)$. From the standard - normal table, $P(Z < 1.67)=0.9525$ and $P(Z < 0.33)=0.6293$.

Step3: Calculate the probability

$P(0.33 < Z<1.67)=0.9525 - 0.6293=0.3232\approx32%$.

Answer:

32%