Chapter 9, Problem 9PT

To determine

To verify:the conclusion drawn at the $5\text{\%}$ significance level hypotheses $H_0:\mu =7.5$ versus $H_a:\mu \ne 7.5$ that the mean time working with the postal service has changed using one sample $t$ test

Answer to Problem 9PT

The answer is (a)

There is convincing evidence that he mean time working in the postal service has changed

Explanation of Solution

Given:

An SRS of 100 postal employees found that the average time these employees had worked at the postal service was 7 years with standard deviation 2 years.

Let $\mu$ denote the average time of employment of a postal employee

Now to test whether mean time of employment for the population of postal employees has changed from the value 7.5

That is $H_0:\mu =7.5$ versus $H_a:\mu \ne 7.5$

Given that

Sample mean, $\overline{x}=7$

Sample standard deviation, $s_x=2$

Test statistic,

  $t=\frac{\overline{x}-\mu }{s_x/\sqrt{n}}$

  $=\frac{7-7.5}{2/\sqrt{100}}$

  $=\frac{0.5}{0.2}$

  $=2.5$

Degrees of freedom

  $\begin{array}{cc}df& =n-1\\ & =100-1\\ & =99\end{array}$

To find $P$ -value, first find the area in two tail.

Using table showing $t$ distribution probabilities,

P-value

  $=2P(t\ge 2.5)$

Using technology,

P-value

  $=2tcdf(2.5,100,99)$

  $=0.014$

Since the $P$ -values is less than $0.05,$ the hypothesis is rejected. So it is concluded that there is convincing evidence that the mean time working in postal service has changed.

So the answer is (a)

There is convincing evidence that he mean time working in the postal service has changed