The following data shows the annual salaries of 12 randomly selected employees of a

Question

company (in £ 1000’s) on the X-axis and the average number of days they have taken days
off work in the last 10 years on the Y-axis.
Y X Y X
11.1 29.6 9.6 39.2
8.4 58.6 10.9 46.8
8.7 71.2 6.8 70.0
6.3 73.9 10.4 33.5
7.8 70.7 11.0 35.3
8.2 46.3 12.5 30.8
[Note: Use hand calculation to answer this Question. You may use a calculator to help
you. However, a statistical package or worksheet output will not be accepted for this
question.]
(i) Calculate the least-squares estimates of slope and intercept for the simple linear
regression of Days off (Y) on Annual Salaries (X).
(ii) Construct the analysis of variance table and perform an F-test to find out the
overall significance of the regression model. What conclusion can you make?
(iii) Perform a T-test for the estimate of the slope and comment on the result. Do you
see a relationship between the test statistic for the T-test and the F-test?
Investigate why an F-distribution is used to model the quantity MSR/MSE while
testing the overall significance of the regression model.
(iv) Using the fitted regression line, find the number of days that an employee with an
annual salary of £20,000 is expected to take in an year and obtain a 95%
prediction interval for it. Give the meaning of this prediction interval.
(v) The manager of the study suggested that another variable, money spent on health
benefits of an employee, might be important. To analyse the same data as earlier
with the additional variable, a multiple linear regression model was proposed with
the form , 1 1 2 2 Y = β + β X + β X + ε o where ε ∼ N(0, σ2
), X1 represents the annual
salary, and X2 represents the money spent on health benefits of an employee. It
was found that the increase of the R2
-value of this new model was 2.40%. Using
this result, apply an appropriate test to decide whether it was necessary to include
the money spent on health benefits of an employee (X2) in the final model

Answer 1

You want to set up a games stall at the Winter Wonderland. You have a single pack of 52
cards and decide to play card games with your customers. The game is designed as follows. A
customer draws 4 cards at random. For every “Ace” that the customer draws, he/she wins £3
and for every face card (“Jack”, “Queen” or “King”), he/she wins £1.
Let the two discrete random variables U and V, be the number of aces and face cards
obtained, respectively.
(i) Derive the joint probability mass function p(U, V).
(ii) Find the marginal probability mass functions of U and V.
(iii) Are U and V independent? Specify the reason.
(iv) Find E(U), E(V), Var(U), Var(V) and Cov(U, V) and interpret the obtained values.
(v) Find the minimum price for the game, so that you don’t bear a loss if a very large
number of games are played.
(vi) If you set the price for each game by rounding up the value found in (