Question

It is thought that the proportion of defective items produced by a particular machine is 10%. A random sample of 100 items is inspected and found to contain 15 defective items.

(a) State the sample proportion in this test.

(b) Compute its statistical value

Answer 1

a) the sample proportion

$\hat p=\frac x n$

where x is the number of defective items and n is the sample size.

$\hat p=\frac{15}{100}$

=0.15

b) statistical value

we wish to test the hypotheses;

H₀:p=0.1

H₁:p≠ 0.1

$np>5; nq>5$ implies that the proportions follow a normal distribution.

we create a 95% confidence interval to test the claim.

$C.I=\hat p ±z*\sqrt{\frac{(\hat p)(1-\hat p)}{n}}$

=$0.15±1.96*\sqrt{\frac{0.15*0.85}{100}}$

=(0.08,0.22)

The population proportion(0.1) is within the 95% confidence interval. we are 95% confident that the proportion of defective items produced by a particular machine is 10%.