Calculate continuity correction for approximate binomial test Let p be the proportion of plants of a certain kind that can be attacked by late blight. In an experiment with 160 plants
50 of them were attacked. Test the following hypotheses with a significance level 5%
$H_0 : p = 0.4$ against $H_1 : p < 0.4.$
$N = 160, n = 50, \mu = 64.$ 
So I notice that I have to use normal approximation since 
$Np$ and $N(1-p) \ge 5.$
$\sigma = \sqrt{ Np(1-p) } = \sqrt{ 160*0.4*0.6 } = 6.1968.$
$Z = (x - \mu)/\sigma$
According to the answer x is 50.5, what rule in the continuity for correction table have they used? I am having trouble interpreting it. 
My guess would be that they have used $P(X\le n)$ and $P(X < n + 0.5).$ 
Is it because $p \le 0.4?$ What variables in the question do I need to use when navigating the table?

According to your null hypothesis the number $X$ attacked by blight
is distributed as $\mathsf{Binom}(160, .4).$ The exact P-value of your test
is $P(X \le 50) = 0.0138,$ which is smaller than 0.05. Accordingly, you
would reject $H_0: p = 0.4$ at the 5% level, concluding that the proportion
attacked by blight must be smaller than 0.4.$ (The exact binomial computation
from R statistical software is shown below.) 
pbinom(50, 160, .4)
## 0.01375971

You are correct that you could get a reasonable approximation to the P-value
using the normal approximation:
$$P(X \le 50) = P(X < 50.5) = 
P\left(\frac{X-\mu}{\sigma} < \frac{50.5 - 64}{ 6.1968}\right)
\approx P(Z < -2.18) = 0.0146,$$
which is smaller than 0.05. Here $Z$ is standard normal and the value 0.0146 can be obtained from
printed normal tables. The normal approximation is accurate to about
two decimal places, which is as good as you can expect. It is close
enough to the correct value to let you know that you should reject $H_0.$
The normal approximation uses 50.5 (instead of 50) in order to more nearly match the discrete
binomial distribution with the continuous normal distribution.
In the figure below, you are interested in the probability to the left
of the vertical dotted line.

