Question

An article claims that 12% of trees are infested by a bark beetle. A random sample of 1,000 trees were tested for traces of the infestation and found that 127 trees were affected. what is the value of the z-test statistic?

Answer 1

Answer:

The value of the z-test statistic is $z = 0.68$

Step-by-step explanation:

An article claims that 12% of trees are infested by a bark beetle.

At the null hypothesis, we test if the proportion is of 12%, that is:

$H_0: p = 0.12$

At the alternative hypothesis, we test if the proportion is different of 12%, that is:

$H_1: p \neq 0.12$

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.

0.12 is tested at the null hypothesis:

This means that $\mu = 0.12, \sigma = \sqrt{0.12*0.88}$

A random sample of 1,000 trees were tested for traces of the infestation and found that 127 trees were affected.

This means that $n = 1000, X = \frac{127}{1000} = 0.127$

What is the value of the z-test statistic?

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

$z = \frac{0.127 - 0.12}{\frac{\sqrt{0.12*0.88}}{\sqrt{1000}}}$

$z = 0.68$

The value of the z-test statistic is $z = 0.68$