Using a printed table to find what fraction of a Gaussian population lies within various intervals I'm having trouble understanding the question. The solution looks simple, but I do not understand the logic behind the problem. For example, (a). How is it that z=1 when z=(x - mu)/(sigma). See problem image
Normal table
 A: Let $X$ have a normal distribution with mean $\mu$ and standard deviation $\sigma.$
It is not possible to make printed distribution tables for all possible choices
of $\mu$ and $\sigma.$ So we transform $X$ to $Z$ with the 'standardization'
formula $Z = (X - \mu)/\sigma.$ If $X \sim \mathcal{Norm}(\mu = 100,\, \sigma=15)$ and you want $P(X \le 115),$ you write
$$P(X \le 115) = P\left(\frac{X - \mu}{\sigma} \le \frac{115-100}{15} = 1\right) = P(Z \le 1) =  0.8413.$$ 
You can get $P(Z \le 1) = 0.8413$ from the printed standard normal table. In your
table, find the entry for 1.0 in the left margin, and read 0.3413--to which
you need to add 0.5000.
In the sketch below (following @SeanRobertson's suggestion), you want the area under the curve to the left of the
dotted red line. The total area under the curve is $1,$ half on either side
of the vertical green line. (Your table gives the area between the red and green lines.)

The answers to parts (a) and (b) are about 0.68 and about 0.95, respectively.
Parts (c), (d) and (e) are (let me see how to put this delicately) total garbage as written. That's one of the reasons
I'm trying to give you some help. (I guess that in (c) "$\mu$ to $\sigma$" was intended to mean
"between $\mu$ and $\mu + \sigma."$)
