Question

An inspector inspects large truckloads of potatoes to determine the proportion p in the shipment with major defects prior to using the potatoes to make potato chips. Unless there is clear evidence that p is less than 0.10, he will reject the shipment. He selects an SRS of 200 potatoes from the truck. Suppose that 12 of the potatoes are found to have major defects. Do the hypotheses test:
H0:p=0.10, H0:p<0.10 and help him to make a decision

a. Calculate the sample proportion p.
b. Calculate the test statistic.
c. Find the p-value.
d. Given α=0.05 what is your conclusion? Should he reject the shipment?

Answer 1

Answer:

a) The sample proportion is 0.06.

b) The test statistic is z = -1.89.

c) 0.0294

d) 0.294 < 0.05, which means that there is enough evidence that p is less than 0.1, which means that we reject the null hypothesis, and accept the shipment.

Step-by-step explanation:

The null hypothesis is:

$H_{0} = 0.1$

The alternate hypotesis is:

$H_{1} < 0.1$

The test statistic is:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

In which X is the sample mean, $\mu$ is the value tested at the null hypothesis, $\sigma$ is the standard deviation and n is the size of the sample.

a. Calculate the sample proportion p.

200 potatoes, 12 defective.

This means that $p = \frac{12}{200} = 0.06$

The sample proportion is 0.06.

b. Calculate the test statistic.

For a proportion p, we have that:

$\sigma = \sqrt{p(1-p)}$

In this question, since $\mu = 0.1$

$\sigma = \sqrt{0.1}{0.9}$

SRS of 200 means that $n = 200$

$H{0} = 0.1$ means that $\mu = 0.1$. So

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

$z = \frac{0.06 - 0.1}{\frac{\sqrt{0.1*0.9}}{\sqrt{200}}}$

$z = -1.89$

The test statistic is z = -1.89.

c. Find the p-value.

Looking at the z-table, $z = -1.89$ has a pvalue of 0.0294, which is the pvalue of this hypothesis test.

d. Given α=0.05 what is your conclusion? Should he reject the shipment?

0.294 < 0.05, which means that there is enough evidence that p is less than 0.1, which means that we reject the null hypothesis, and accept the shipment.