Question

When working properly, a machine that is used to makes chips for calculators does not produce more than 4% defective chips. Whenever the machine produces more than 4% defective chips, it needs an adjustment. To check if the machine is working properly, the quality control department at the company often takes samples of chips and inspects them to determine if they are good or defective. One such random sample of 200 chips taken recently from the production line contained 12 defective chips. Find the p-value to test the hypothesis whether or not the machine needs an adjustment. What would your conclusion be if the significance level is 2.5%

Answer 1

Answer:

The proportion of defective chips produced by the machine is more than 4% so the machine needs an adjustment.

Step-by-step explanation:

In this case we need to test whether the proportion of defective chips produced by the machine is more than 4%.

The hypothesis can be defined as follows:

H₀: The proportion of defective chips produced by the machine is not more than 4%, i.e. p ≤ 0.04.

Hₐ: The proportion of defective chips produced by the machine is more than 4%, i.e. p > 0.04.

The information provided is:

X = 12

n = 200

α = 0.025

The sample proportion of defective chips is:

$\hat p=\frac{X}{n}\\\\=\frac{12}{200}\\\\=0.06$

Compute the test statistic as follows:

$z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}\\\\=\frac{0.06-0.04}{\sqrt{\frac{0.04(1-0.04)}{200}}}\\\\=1.44$

The test statistic value is 1.44.

Decision rule:

We reject a hypothesis if the p-value of a statistic is lower than the level of significance α.

Compute the p-value of the test:

$p-value=P(Z>1.44)\\=1-P(Z<1.44)\\=1-0.92507\\=0.07493\\\approx 0.075$

The p-value of the test is 0.075.

p-value = 0.075 > α = 0.025

The null hypothesis was failed to be rejected at 2.5% level of significance.

Thus, it can be concluded that the proportion of defective chips produced by the machine is more than 4% so the machine needs an adjustment.