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

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

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 \\eq 0.12`

The test statistic is:

`z = (X - \mu)/((\sigma)/(√(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 = √(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 = (127)/(1000) = 0.127`

What is the value of the z-test statistic?

`z = (X - \mu)/((\sigma)/(√(n)))`

`z = (0.127 - 0.12)/((√(0.12*0.88))/(√(1000)))`

`z = 0.68`

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