Question

A supplier of 3.5" disks claims that no more than 1% of the disks are defective. In a randomsample of 600 disks, it is found that 3% are defective, but the supplier claims that this isonly a sample fluctuation. At the 0.01 level of significance, do the data provide sufficientevidence that the percentage of defects exceeds 1%

Answer 1

At a significance level of 0.01, there is enough evidence to support the claim that the percentage of defective disks exceeds 1%.

Explanation:

This is a hypothesis test for a proportion.

The claim is that the percentage of defective disks exceeds 1%.

Then, the null and alternative hypothesis are:

`H_0: \pi=0.01\\\\H_a:\pi>0.01`

The significance level is 0.01.

The sample has a size n=600.

The sample proportion is p=0.03.

The standard error of the proportion is:

`\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.01*0.99)/(600)}\\\\\\ \sigma_p=√(0.000017)=0.004`

Then, we can calculate the z-statistic as:

`z=(p-\pi-0.5/n)/(\sigma_p)=(0.03-0.01-0.5/600)/(0.004)=(0.019)/(0.004)=4.719`

This test is a right-tailed test, so the P-value for this test is calculated as:

`P-value=P(z>4.719)=0.000001`

As the P-value (0.000001) is smaller than the significance level (0.01), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the percentage of defective disks exceeds 1%.