Question

Direct Mailing Company sells computers and computer parts by mail. The company claims that at least 90% of all orders are mailed within 72 hours after they are received. The quality control department at the company often takes samples to check if this claim is valid. A recently taken sample of 150 orders showed that 115 of them were mailed within 72 hours. What are the decision and conclusion of test? Use α=2.5%.

Answer 1

We conclude that less than 90% of all orders are mailed within 72 hours after they are received.

Explanation:

We are given that the company claims that at least 90% of all orders are mailed within 72 hours after they are received.

A recently taken sample of 150 orders showed that 115 of them were mailed within 72 hours.

Let p = proportion of orders that are mailed within 72 hours after they are received.

So, Null Hypothesis,
`H_0` : p
`\geq` 90% {means that at least 90% of all orders are mailed within 72 hours after they are received}

Alternate Hypothesis,
`H_A` : p < 90% {means that less than 90% of all orders are mailed within 72 hours after they are received}

The test statistics that would be used here One-sample z proportion statistics;

T.S. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of orders that were mailed within 72 hours =
`(115)/(150)` = 0.767

n = sample of orders = 150

So, test statistics =
`\frac{0.767-0.90}{\sqrt{(0.767(1-0.767))/(150) } }`

= -3.853

The value of z test statistics is -3.853.

Now, at 2.5% significance level the z table gives critical value of -1.96 for left-tailed test. Since our test statistics is less than the critical values of z as -3.853 < -1.96, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that less than 90% of all orders are mailed within 72 hours after they are received.