1. A large company introduced a range of new initiatives to improve employee        satisfaction. They selected 11 employees at random from the company and        measured their change in satisfaction ratings before and after the initiatives were introduced (change = after - before). The mean change was 9.36, with standard deviation 6.727. The values are displayed in the histogram.

−5       0        5        10       15      20      25

Change in rating (after − before)

(a)     i.  Briefly explain the commonly-used notation µ ,  ,  , σ ,  and s in the

context of this example.                                                          [3 marks]

ii.  Construct and interpret a 99% confidence interval.                 [7 marks]

iii.  List any assumptions that are made in the construction of the     confidence interval, and comment on how they could be assessed.

[6 marks]

(b)  Let U1 , U2 , ..., U40  be uniformly distributed random variables from a to b.

The probability density function is:

f(x)   =   , 

a < x < b

otherwise

40

E[Ui] =  (b + a) and Var(Ui ) =  (b - a)2 . Let T =z Ui .

i=1

i. Show E[T] = 20(a + b) and Var(T) =  (b - a)2 .                  [8 marks]

ii.  Let a = 5 and b = 15. Use the central limit theorem (CLT) to find       P (410 < T < 420), approximately. Explain in your own words (2-3      sentences) why the CLT is appropriate to use here.                 [9 marks]

(33 marks)

Oc Trinity College Dublin, The University of Dublin 2021

2. The functionality of an electrical component of a machine was tested under a  range of temperatures (temp = 0, 1, 2, ..., 14 一 C) in an experiment. A scatter plot of the data is shown, there were 30 data points in total.

0       2       4       6       8      10     12     14

Temperature (degrees Celsius)

(a) A simple linear regression model was fitted to this dataset using R. Here is

some of the output:

Estimate  Std .  Error  t  value  Pr(>|t|) (Intercept)    16 .8625          1 .3906    12 .126  1 .16e-12 temp                   0 .8625         0 .1691     5 .102  2 .10e-05

i. Write out the estimated equation of the line and interpret the

parameter estimates.

ii.  Interpret the hypothesis test for the temp parameter.

[7 marks]

[7 marks]

iii. The minimum acceptable functionality of the component is 20. Test    whether the average functionality for when temperature equals 0一 C is   lower than 20, using α = 0.05.                                                [7 marks]

iv. State the model assumptions and identify if the assumptions are           reasonably met using the following residual plots.                    [6 marks]

18     20     22     24     26     28

Predicted values

−2         −1          0           1           2

Theoretical Quantiles

(b) Some new information came to light during discussions between the

statistician and the person who carried out the experiment: there were two types of components used in the experiment and each was tested once        under each temperature 0, 1, 2,...,14 一 C. The scatter plot shows the data   with the points for component A shown in empty circles and for component B in filled circles.

0       2       4       6       8      10     12     14

Temperature (degrees Celsius)

The dataset was re-analysed by fitting this model:

yi  = β0 + β1 xi1 + β2 xi2 + β3 xi1xi2 + ∈i

where yi  is the functionality value for the ith experimental unit, xi1  is equal to 1 if the ith experimental unit was a type A component and 0 for a type  B component, and xi2  is the temperature for the ith experimental unit.

Write out this model in matrix notation, clearly showing the structure of     each matrix.                                                                                   [7 marks]

(34 marks)