Question

A textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes to test the hypothesis H0: µ = 12 against H1: µ < 12 using a random sample of n = 4 specimens. Calculate the P-value if the observed statistic is Xbar (average) = 11.25. Suppose that the distribution of the sample mean is approximately normal.

Answer 1

Answer:

The p-value of the test is 0.0013.

Step-by-step explanation:

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.

12 is tested at the null hypothesis:

This means that $\mu = 12$

Standard deviation of 0.5 kilograms.

This means that $\sigma = 0.5$

Sample of n = 4 specimens. Observed statistic is Xbar (average) = 11.25.  

This means that $n = 4, X = 11.25$

Value of the test statistic:

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

$z = \frac{11.25 - 12}{\frac{0.5}{\sqrt{4}}}$

$z = -3$

P-value:

Probability of finding a sample mean belo 11.25, which is the p-value of z = -3.

Looking at the z-table, z = -3 has a p-value of 0.0013, thus the this is the p-value of the test.