Question

The one-sample z-test for proportions determines the p-value using the “normal approximation method”. This method approximates the exact p-value that would have been found if the binomial formula were used. But, the “normal approximation method” only works well when the sample size is “large enough”.

a. If the hypothesis test is done by hand, why do you think we use the “normal approximation method” for large sample sizes instead of using the binomial formula to find p-values?

b Why do you think it’s not as necessary today to use the normal approximation method compared to, say, 50 years ago?

Answer 1

a)

When we use the exact test using binomial distribution, the hypothesis and the test statistics looks lies as follow,

The distrbution of random variable X is,

and the P-value is obtained using the formula,

If the x value is large, the calculation goes more and more complex which is difficult to calculate by hand.

Since the binomial distribution for large sample follows the normal distribution curve, it is better to approximate the binomial curve to normal by taking transformation as follow,

Now, the calculation can be easily computed by hand and the P-value can be obtained using the standard normal distibution table.

b)

We know that microsoft excel and the other statistical software were not in existance before 50 years ago from today. That is why, it was better to take approximation at that time to reduce time and complexity of the calculation.