Use of Normal Approximation to Binomial in Test of Hypothesis

Question

Let $Y \sim \mathsf{Binom}(192, p).$ We reject $H_0$ : $p = 0.75$ and accept $H_1$: $p > 0.75$ if and only if $Y \geq 152$. Use the normal approximation to determine

(a) $\alpha = P(Y \geq 152;\, p = 0.75).$

(b) $\beta = P(Y < 152;\, p = 0.80).$

This is a question from probability and statistical inference, 9th edition. I know how to solve this using the poisson distribution but not normal approximation.

Answer 1

You have $X \sim \mathsf{Binom}(n = 192; p),$ with unspecified Success probability $p.$ Then

$$\alpha = P(Y \ge 152;\, p = 0.75) = P\left(\frac{X - np}{\sqrt{np(1-p)}} \ge \frac{151.5 - 144}{6} = 1.25\right) \approx P(Z \ge 1.25).$$

For the continuity correction, I use 151.5 instead of 152. Also, $np = 144,\,np(1-p)=36.$ Now it remains to use printed tables of the normal CDF to find $P(Z \ge 1.25),$ where $Z$ is standard normal. The other part, for $\beta,$ is done similarly.

An exact computation of the binomial probability in R statistical software goes as follows:

1 - pbinom(151, 192, .75)
##  0.1040372

So it appears that your test of hypothesis is to be tested at about the 10% level of significance.

You will not get exactly the value 0.10404 from the normal approximation, but it will be close.