Question

One buyer claims that 10% of 400 products are defective.

The manufacturer says that 2% of the products may be defective.

50 products will be checked.

Let the random variable be the number of broken out of the 50 items checked.

If X> c, accept what the buyer says.

if X < c, Accept what the manufacturer says.

Determine the number c in this decision rule so that when the buyer says it is true, the probability of rejection due to randomness does not exceed 5%.

Answer 1

This problem can be solved using knowledge of one sample proportion hypothesis test.

We have to find critical region for observed proportion so that

Our null hypothesis is and alternative hypothesis is

Corresponding test statistic is given by

Here,

Number of observations

Random variable X denotes number of broken products out of the 50 items checked.

Probability of type I error i.e. level of significance is

Thus we obtain critical region

For the number c = 2 in our decision rule, if the buyer says it is true the probability of rejection due to randomness does not exceed 5%.