Question

the dodge reports are used by many companies in the construction field to estimate the time required to complete various jobs. the company managers want to know if the time required to install 130 square feet of bathroom tile is different from the eight hours reported in the current manual. a researcher for dodge randomly selects 56 construction workers and determines the time required to install 130 square feet of bath tile. the average time required to install the tile for the sample is 6.8 hours with a standard deviation of 4.5 hours. use a hypothesis test to determine whether the managers assumptions are substantiated by the data. use a significance level of $alpha = 0.01$. assume the population of tile installation times is approximately normally distributed. step 3 of 3: draw a conclusion and interpret the decision. answer we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours. we reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours. we reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours.

Explanation:

Step1: Calculate the test - statistic

The null hypothesis $H_0:\mu = 8$ and the alternative hypothesis $H_1:\mu
eq8$. The formula for the one - sample z - test statistic is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x} = 6.8$, $\mu = 8$, $\sigma = 4.5$, and $n = 56$.
$z=\frac{6.8 - 8}{\frac{4.5}{\sqrt{56}}}=\frac{- 1.2}{\frac{4.5}{7.4833}}\approx\frac{-1.2}{0.6013}\approx - 1.996$.

Step2: Determine the critical values

For a two - tailed test with $\alpha=0.01$, the critical values are $z_{\alpha/2}=\pm 2.576$.

Step3: Make a decision

Since $-2.576<-1.996 < 2.576$, we fail to reject the null hypothesis. This means there is insufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours.

Answer:

We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance to support the claim that the average time required to install the tile differs from 8 hours.