One-Way Analysis of Variance

A study was conducted to assess the role of pure motivation (knowledge of the goal of the work) in performing boring monotonous operations when turning a metal workpiece. Eighteen workers of equal qualification were randomly divided into three groups. Group $A$ had no information about the required labor productivity, group $B$ received only general information, and group $C$ had exact information about the task. Table 8.7 gives the number of processed workpieces. Can it be considered, at significance level 0,05, that labor productivity increases with awareness?

Table 8.7

Prove that the statistic (8.15) $\hat{\gamma }$ is an unbiased estimator of the parameter $\gamma$ from Definition 8.1.

Find the distribution of the statistic (8.16) and prove that the statistic (8.16) is a pivotal statistic for the parameter contrast $\gamma$ from Definition 8.1.

The following are data on the subsistence minimum for the working-age population (rubles per capita per month) established in the constituent entities of the Russian Federation for the 4th quarter of 2008 (Table 8.8).

Table 8.8

Can it be considered that the subsistence minimum is, on average, the same in all the presented federal districts? Choose the significance level equal to 0,05.

Let the subsistence-minimum values for the working-age population $X_{i j}$, where $j = 1,2,3$ is the number of the federal district and $i = 1, \ldots , n_{j}$ is the number of the region in the corresponding federal district, given in Problem 8.3 , be described by model (8.12), in which the $\varepsilon_{i j}$ have the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$. Construct point estimates and confidence intervals with confidence level 0,95 for the contrasts $\gamma_{1} = \theta_{1}-\theta_{2}, \gamma_{2} = \theta_{1}-\theta_{3}$ and $\gamma_{3} = \theta_{2}-\theta_{3}$ in model (8.12).

Three groups of randomly selected people were taught speed-reading skills by three different methods. At the end of the training, a test was held at which the reading speed was assessed. The trainees showed the following results (pages per ten minutes). First group: $20,23,24,24,25,26$, $28,30,31,32$. Second group: $38,42,42,44,47,48,49,50,51,52$. Third group: $29,32,33,35,35,37,38,39,40,42$. Can it be considered, at significance level 0,05 , that the proposed teaching methods differ in effectiveness?

The time (in sec.) of a chemical reaction for different catalyst contents was distributed as follows (Table 8.9)

Table 8.9

Can one assume, at the significance level 0,01, that increasing the catalyst content decreases the chemical reaction time on average?

Prove the equivalence of formulas (8.3) and (8.4).