Question

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. Do these data provide convincing evidence that the mean time of employment M for the population of postal employees has changed from the value of $7.5$ that was true $20$years ago? To determine this, we test the hypotheses $H_0:\mu =7.5$versus $H_a:\mu \ne 7.5$using a one-sample $t$test. What conclusion should we draw at the $5\%$significance level?

(a) There is convincing evidence that the mean time working with the postal service has changed.

(b) There is not convincing evidence that the mean time working with the postal service has changed.

(c) There is convincing evidence that the mean time working with the postal service is still $7.5$ years.

(d) There is convincing evidence that the mean time working with the postal service is now $7$years.

(e) We cannot draw a conclusion at the $5\%$ significance level. The sample size is too small.

Short answer

The answer is (a). There is convincing evidence that the mean time working with the postal service has changed.

Step-by-step solution

Given Information

$H_0:\mu =7.5$

$H_a:\mu \ne 7.5$

$\overline{x}=7$

$s=2$

$n=100$

Explanation

Determine the value of the test statistic:

$t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

$=\frac{7-7.5}{2/\sqrt{100}}$

$=-2.50$

The $P-$value is the chance of getting the test statistic's result, or a number that is more severe. The $P-$value is the number (or interval) in Table IV's column title that corresponds to the row's $t-$value.

localid="1650366176477" $n-1=100-1$

$=99>80$:

localid="1650366216288" $0.01=2\times 0.005<P<2\times 0.01$

$=0.02$

The null hypothesis is rejected if the $P-$value is less than the significance level.

$P<0.05=5\%\Rightarrow \text{Reject}{H}_0$

There is convincing evidence that the meantime working with the postal service has changed.