ECON1007 W1
Q1) Both undergraduates and postgraduates can use the university
cafeteria. Each diner can choose between buying a meal or bringing a
packed lunch. (Everyone has exactly one meal each, no more and no less).
The cafeteria offers a daily choice between a hot meal or a cold meal. A
survey of undergraduate diners finds that 40% of them bring their own food.
Overall, only 25% of the diners bring their own food. Postgraduates make
up one fifth of the diners in the cafeteria.
a) What is the probability that a diner is an undergraduate and has
bought a meal? [8]
b) What is the probability that someone who has bought a meal is a
postgraduate? [8]
c) Give two different examples from the question of mutually exclusive
events. [4]
Q2) Two types of vehicle use the university central bus station: Uni-link
buses and National Express coaches.
a) On average, a National Express coach arrives every two hours. What
is the probability that only one bus will arrive in a four hour period? [5]
b) On the U1 Uni-link bus route, buses either go to the airport or to the
docks. Other Uni-link routes do not go to the airport. At the bus station, half
of the U1 buses which arrive are going to the airport. At midday, 10% of the
Uni-link buses are full. Of these particular buses, 30% are airport buses on
the U1 route. U1 buses make up 40% of the Uni-link arrivals at the bus
station. What is the probability that a bus is full at midday, given that it is an
Airport U1 bus? [6]
c) Are the events ‘bus is full’ and ‘bus is going to the Airport’
independent? Show your reasoning. [4]
Question 2 continues on the next page
ECON1007 W1
d) On average, a certain number, k, of National Express coaches go to
Bournemouth every day. It is found that the probability of n coaches going
to Bournemouth in a day is .
k e- What is n? [5]
Q3) The standard error of the mean is given by the formula:
n
x
s
s =
a) Explain what the standard error of the mean represents. [5]
b) Is it possible for the standard error of the mean to be zero? Explain
your answer. [5]
c) Does the sampling distribution of x look like a Normal distribution if n
is 10? [5]
d) As n increases, what happens to the population variance 2 s ? [5]
Q4) A random sample is taken from a population. The data points are:
5 4 9 2 1
a) What is the mode? [2]
b) Calculate the unbiased estimate for the population variance. [4]
c) Another random sample is taken from the population with an unbiased
estimate of the population variance of 9.2. Does this result imply anything
about whether the estimator is truly unbiased or not? Explain your answer.
[4]
d) A random sample of 92 observations is taken from another population.
The sample mean is 5.5. The unbiased estimate for the population variance
is 6.4. Test whether the population mean is equal to 3 or not at a
significance level of 5%. [10]
/PLEASE TURN OVER
ECON1007 W1
Q5) Consider the following regression model:
i i i i Y = a + ß x + ß x + e 1 1 2 2
a) Suppose that ß 2 was equal to zero. You are given a sample of data for
the variables 1 2 x , x and y. You are then asked to fit a regression line to the
data. In simple Ordinary Least Squares regression, the fitted line must
always go through a particular point, which is found from the x and y data.
What is the point in question? [4]
b) Suppose that ß 2 was now not equal to zero. A researcher using the
regression model finds that the values of the term i e are correlated with one
another. The researcher argues that this shows that the model is properly
specified as the terms follow a coherent pattern. Is this a reasonable
approach? Explain your answer. [3]
c) The researcher then estimates the model again, but uses a different
set of data with 12 observations. The intercept term, seems to be zero. Is
this cause for concern? Explain your answer. [3]
d) You are given the following statistics about the regression:
The t statistic for the first independent variable is -2.3 and the estimate is
4.5. The t statistic for the second independent variable is -0.5 and the
estimate is 5.1. The t statistic for the intercept term is 1.5.
The R squared is 0.7 and the F statistic is 2.4. Test the overall significance
of the model at the 5% level. [6]
e) The researcher would like to create another regression model and use
data from people who are currently students. Then he would use
extrapolation to predict what the dependent variable would be for older
people. Is this approach valid? Explain your answer. [4]
5 ECON1007 W1
Q6) You are given a large sample of data to analyse using regression
analysis.
a) Firstly, you wish to discover whether using simple linear regression
analysis is appropriate. Which two graphs could you create with your data
to help you? How would you interpret them? [4]
b) You also decide to consider using a quadratic regression model. Write
out the regression model for this case. The independent variable is x and
the dependent variable is y. [2]
c) How would you test whether the coefficient of the quadratic term was
statistically significant at the 5% level? [4]
d) How would you compare the two models in order to choose the most
appropriate one? [4]
e) Suppose that you wished to study differences in income in the UK.
One of the factors which you believe is important is gender. You are given
statistical data from a national survey. Respondents can state that they are
male or female or refuse to disclose their gender. How would you
incorporate gender into your regression equation or would you choose to
discard it from your analysis? [5]
f) What pitfall should you avoid with part e) above? [1]
END OF PAPER