Chapter 11.8, Problem 11.41E

Bidding on Construction Jobs A building contractor employs three construction engineers. A, B, and C, to estimate and bid on jobs. To determine whether one tends to be a more conservative (or liberal) estimator than the others, the contractor selects four projected construction jobs and has each estimator independently estimate the cost (in dollars per square foot) of each job. The data are shown in the table:

Analyze the experiment using the appropriate methods. Identity the blocks and treatments, and investigate any possible differences in treatment means. If any differences exist, use an appropriate method to specifically identity where the differences lie. Has blocking been effective in this experiment? What are the practical implications of the experiment’? Present your results in the form of a report.

To determine

To find: the blocks and treatments, whether the blocking iseffective, the practical implications.

Answer to Problem 11.41E

There is not sufficient evidence to support the claim that there are differences in the mean cost for the different construction jobs and there is not sufficient evidence to support the claim that there are differences in the mean cost for the different estimators.

Blocking was not effective.

The practical implications are that it does not matter which estimator is used, as we will always obtain the same mean cost.

Explanation of Solution

Given:

  

Calculation:

${\displaystyle \sum x}=$ Sum of given data $=406.75$

The necessary sum is,

${\displaystyle \sum x^2=}$ Sum of squares of the given data $=3769.1125$

The treatments are the estimators, while the blocks are the construction jobs, because we suspect that there could be a difference in the cost for each job.

The value of $\text{total }SS$

  $\text{total }SS={\displaystyle \sum x^2-\frac{{\left({\displaystyle \sum x}\right)}^2}{bk}=3769.1125-\frac{{\left(406.75\right)}^2}{4\left(3\right)}}=459.5$

The value of $SST$ is,

  $SST={\displaystyle \sum \frac{T_i^2}{b}-\frac{{\left({\displaystyle \sum x}\right)}^2}{bk}}$

  $SST=\frac{{\left(130.45\right)}^2}{4}+\frac{{\left(139.55\right)}^2}{4}+\frac{{\left(136.75\right)}^2}{4}-\frac{{\left(406.75\right)}^2}{4\left(3\right)}=65.86$

The value of $SSB$

  $SSB={\displaystyle \sum \frac{B_i^2}{k}-\frac{{\left({\displaystyle \sum x}\right)}^2}{bk}}$

  $SSB=\frac{{\left(108.85\right)}^2}{3}+\frac{{\left(104.20\right)}^2}{3}+\frac{{\left(94.80\right)}^2}{3}+\frac{{\left(78.90\right)}^2}{3}-\frac{{\left(406.75\right)}^2}{4\left(3\right)}=174.77$

The value of $SSE$

  $SSE=\text{total }SS-SST-SSB=459.5-65.86-174.77=218.87$

  $d{f}_T$ is the number of groups k decreased by 1

  $d{f}_T=k-1=3-1=2$

  $d{f}_B$ is the number of groups b decreased by 1

  $d{f}_B=b-1=4-1=3$

$d{f}_E$ is the product of $d_T$ and $d_B$

  $d{f}_E=d_T{d}_B=3\cdot 2=6$

Total $df$ is the sum of the separate degrees of freedom $d{f}_T$ , $d{f}_B$ and $d{f}_E$

  $\text{total }df=d{f}_T+d{f}_B+d{f}_E=2+3+6=11$

$MSB$ is $SSB$ divided by $d{f}_B$

  $MSB=\frac{SSB}{d{f}_B}=\frac{174.77}{3}=58.26$

$MST$ is $SST$ divided by $d{f}_T$

$MST=\frac{SST}{d{f}_T}=\frac{65.86}{2}=32.93$

$MSE$ is $SSE$ divided by $d{f}_E$

  $MSE=\frac{SSE}{d{f}_E}=\frac{218.87}{6}=36.48$

The value of the test statistic is calculated as

$F_T=\frac{MST}{MSE}=\frac{32.93}{36.48}=0.9027$

$F_B=\frac{MSB}{MSE}=\frac{58.26}{36.48}=1.5970$

Source	$df$	SS	MS	F
Construction job	2	65.86	32.93	0.9027
Estimator	3	174.77	58.26	1.5970
Error	6	218.87	36.48
total	11	459.57

Construction job:

The P-value is the number in the row title of the F-distribution table in the appendix containing the F-value $F_T=0.9027$ in the row $d{f}_2=d{f}_E=6$ and in the column $d{f}_1=d{f}_T=2$

  $P>0.10$

If the P-value is less than the significance level, then reject the null hypothesis.

$P>0.05\Rightarrow$ Fail to reject ${\text{H}}_0$

There is not sufficient evidence to support the claim that there are differences in the mean cost for the different construction jobs.

Blocking was not effective, because there was no significant difference in the mean cost for the different construction jobs.

Estimator:

The P-value is the number in the row title of the F-distribution table in the appendix containing the F-value $F_B=1.5970$ in the row $d{f}_2=d{f}_E=6$ and in the column $d{f}_1=d{f}_T=2$

  $P>0.10$

If the P-value is less than the significance level, then reject the null hypothesis.

$P>0.05\Rightarrow$ Fail to reject ${\text{H}}_0$

There is not sufficient evidence to support the claim that there are differences in the mean cost for the different estimators.

The practical implications are that it does not matter which estimator is used, as we will always obtain the same mean cost.

Conclusions:

Therefore,treatments: estimator A, B, C

Block: Construction job 1, 2, 3, 4

There is not sufficient evidence to support the claim that there are differences in the mean cost for the different construction jobs and there is not sufficient evidence to support the claim that there are differences in the mean cost for the different estimators.

Blocking was not effective.

The practical implications are that it does not matter which estimator is used, as we will always obtain the same mean cost.