Question

A bottle maker believes that 14% of his bottles are defective. If the bottle maker is accurate, what is the probability that the proportion of defective bottles in a sample of 622 bottles would be less than 11%

Answer 1

`z = (0.11-0.14)/(0.0139) = -2.156`

And we can use the normal standard distribution table and we got:

`P(Z<-2.156) =0.0155`

Explanation:

For this case we know the following info given:

`p =0.14` represent the population proportion

`n = 622` represent the sample size selected

We want to find the following proportion:

`P(\hat p <0.11)`

For this case we can use the normal approximation since we have the following conditions:

i) np = 622*0.14 = 87.08>10

ii) n(1-p) = 622*(1-0.14) =534.92>10

The distribution for the sample proportion would be given by:

`\hat p \sim N (p ,\sqrt{(p(1-p))/(n)})`

The mean is given by:

`\mu_(\hat p)= 0.14`

And the deviation:

`\sigma_(\hat p)= \sqrt{(0.14*(1-0.14))/(622)}= 0.0139`

We can use the z score formula given by:

`z=(\hat p -\mu_(\hat p))/(\sigma_(\hat p))`

And replacing we got:

`z = (0.11-0.14)/(0.0139) = -2.156`

And we can use the normal standard distribution table and we got:

`P(Z<-2.156) =0.0155`