Question

Accidents at construction sites. In a study published in the Business & Economics Research Journal (April 2015), occupational accidents at three construction sites in Turkey were monitored. The total numbers of accidents at the three randomly selected sites were 51, 104, and 37.

Summary statistics for these three sites are:\(\bar x = 64\)and s = 35.3. Suppose an occupational safety inspector claims that the average number of occupational accidents at all Turkish construction sites is less than 70

a. Set up the null and alternative hypotheses for the test.

b. Find the rejection region for the test using\(\alpha = .01\)

c. Compute the test statistic.

d. Give the appropriate conclusion for the test.

e. What conditions are required for the test results to be valid?

Short answer

a)

\(\begin{aligned}{H_0}:\mu = 70\\{H_a}:\mu < 70\end{aligned}\)

b) The one-tailed rejection region is\(t < - {t_{0.01}} = - 6.965\).

c) The test statistic is -0.2944.

d) We cannot reject the null hypothesis.

e) The sample are taken from an average population is satisfied for the test result to be valid.

Step-by-step solution

(a) Construct the hypothesis

Given that, the average number of occupational accidents is less than 70.

Here we use a one-tailed test.

Therefore, the null and alternative hypotheses are given by

\(\begin{aligned}{H_0}:\mu = 70\\{H_a}:\mu < 70\end{aligned}\)

(b) Rejection region

For,

\(\begin{aligned}\alpha & = 0.01\,\\and\,\\df &= n - 1\\ &= 2\end{aligned}\)

The one-tailed rejection region is \(t < - {t_{0.01}} = - 6.965\).

(c) Test statistic

Given that, \(\bar x = 64\,,\,s = 35.3\)

The test statistic is computed as

\(\begin{aligned}t &= \frac{{\bar x - \mu }}{{\frac{s}{{\sqrt n }}}}\\ &= \frac{{64 - 70}}{{\frac{{35.3}}{{\sqrt 3 }}}}\\ &= \frac{{ - 6}}{{20.38}}\\ &= - 0.2944\end{aligned}\)

Therefore, the test statistic is -0.2944.

(d) Conclusion

We observed that the calculated test statistic is greater than the tabulated test statistic. The calculated t falls outside of the rejection region; the inspector cannot reject the null hypothesis.

(e) Need conditions

There is insufficient evidence to conclude that \(\mu < 70\) . Here the sample size is three. The sample is taken from an average population is satisfied for the test result to be valid.