Question 3: Four factories producing soft-drinks are randomly chosen …

1. Four factories producing soft-drinks are randomly chosen to check if the amount of caffeine in 1L bottles varies throughout the factories. For this the amount of caffeine (in a hundred mg) in three 1L bottles of soft-drinks of each of the selected factories is recorded in the following table:

Based on the given information, answer the following questions:

(a) Suppose we assume the linear model:

$y_{ij} = \mu + \alpha_i + \epsilon_{ij} \;\; ; i = 1, 2, 3, 4 \;\text{ and }\; j = 1, 2, 3$

where, $\alpha_i \sim N(0, \sigma^2_\mu)$ and $\epsilon \sim N(0, \sigma^2)$

Identify whether the given model is the fixed effect model or the random effect model. Give reason. [2 Marks]

(b) Compute the treatment means for each of the given treatment, i.e. $\overline{Y}_{i.}$ and the overall mean of the given observations, i.e.$\overline{Y}_{..}$. [2 Marks]

(c) Find the sum of squares due to treatment, i.e., $SS_{treatment}$ and sum of squares due to error, i.e. $SS_{error}$. [4 Marks]

(d) Compute mean sum of squares due to treatment and error, i.e. $MS_{treatment}$ and $MS_{error}$. [2 Marks]

(e) Find the estimate value of $\sigma^2_\mu$. [2 Marks]

(f) Draw the ANOVA table for the above given model. [3 Marks]

(g) An analyst wish to test if there is a variability in amount of caffeine across all the 4 factories. Write down the null and alternative hypothesis to be tested for it. [2 Marks]

(h) Perform hypothesis testing for the above defined hypothesis and conclude your results. [4 Marks]

(i) The analyst also wishes to test if the grand mean caffeine content in 1L soft-drinks bottles is 5 (in a hundred mg) or not. Perform hypothesis test for the same. [5 Marks]