Reference no: EM13710229

Question 1: 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?

b) What is the probability that someone who has bought a meal is a postgraduate?

c) Give two different examples from the question of mutually exclusive events.

Question 2: 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?

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?

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 e^-k What is n?

Question 3:

If P(A) = 0.2 and P(A^- ∩ B) = 0.4

a) find P(B|A^-)

b) Can P(B) = 0? Explain your answer.

c) What is the smallest value that P(A^-|B) can take? Explain your answer.

d) What would P(A^-|B) have to be, for the complement of A, and B to be independent of each other?

Question 4: Large random samples of size n are taken from a population which follows a uniform distribution with mean 25 and variance of 22.

a) What is the expected value of the sample mean?

b) Can the Central Limit Theorem be applied in this case?

c) The probability that a sample mean is greater than 27 is 0.1587. Find n.

d) If the population followed a Chi-Squared distribution, how would you approach part c) of this question?

Question 5: A random sample is taken from a population which is Normally distributed. The sample size is 9, the sample mean is 5 and s is 9.

a) Construct an 80% confidence interval for the population mean.
b) If a different sample was taken and n was 1000, what would the 80% confidence interval for the population mean be?
c) Comment upon the change in the confidence interval, if any.

Question 6: A regression analysis is performed. The depende nt variable is ‘exam grade', x1 is ‘time spent studying in hours' and x2 is ‘time spent in the library in hours'.

a) Explain what is meant by homoscedasticity in relation to the error terms.

b) The regression output is as follows:

y^{^} = 0.2 + 0.8x₁ +0.2x₂

        (2.68) (2.3)    (2.85)  (standard errors in brackets)
N 20

R squared 0.5

Adjusted R squared 0.44

Standard error 47.5

Test whether or not the coefficient for x1 is significant at the 5% level.

c) Comment on the result for part b.

d) A researcher wants to use a new variable x3 (number of textbooks bought) instead of x2. She is also concerned about a possible gender effect and so uses a variable m for ‘male' and another variable f for ‘female'. The R squared is now 0.52.

Discuss whether the researcher's ideas are sensible and any potential problems with the researcher's approach.

e) How would you go about formulating the model, using only the variables mentioned here. Explain your choices.

Question 7: A regression analysis relates the sales of cake s (S) to the price per cake in £ (P) and to the time in hours that the cake stall is open (T).

The output is as follows:

R squared                 0.521
Adjusted R squared   0.44
Standard error          47.46
Observations            43
F                             7.81

                   Coefficients       t statistics
Intercept             24                 2.10
P                        2.3                2.52
T                        0.5                2.81

a) Comment upon the coefficients of the regression.

b) Test the overall significance of the model at the 5% level.

c) The stall-holder believes that the day of the week might affect sales. How would you include this idea in the model?

d) When you include the days of the week into your model, would you expect your answer to b to change?

e) Ignore the day of the week and also price. The new model now looks like this:

y^{^} = 16 + 1.3T + 1.9T²
        (111) (2.9)   (3.1) (Standard errors in brackets)

R squared               0.53
Adjusted R squared 0.45
F                            8.1

If you plotted T on the x axis and residuals on the y axis of a graph, what might indicate that the line is a good fit?

f) Test for the significance (at the 5% level) of the quadratic effect in your new model.

Question 8: Light bulbs are tested to see if they work or n ot. The probability of a bulb working properly is 0.8. 100 bulbs are checked.

a) What is the probability that exactly 92 bulbs work?

b) What is the probability that fewer than 80 bulbs work properly?

c) You now discover that there are two completely different machines which make the bulbs, Machine A and Machine B. How would this affect the assumptions which you made in calculating part a and b?

d) You are then told that (working) bulbs made by Machine A are used to light the main warehouse. Usually, one bulb burns out every 2 days. You are asked to calculate the probability that three bulbs will burn out in the space of one day.

Question 9: The heights of all people who are members of a university sports club are measured. There are 1000 people in total. They make up a quarter of the student population.

a) On average, science students make up 40% of a sports club's membership. What is the probability that a club of 20 people will have exactly 12 scientists?

b) What is the probability that the number of students joining sporting activities is less than or equal to 1050?

Question 10: Trains arrive at a station at the rate of 25 per hour.

a) What is the probability that the time between trains arriving is 6 minutes or less?

b) The train company wants to state that waiting times (in minutes) between trains have been targeted for improvement. It fails to meet its target 40% of the time. What target has it set itself?

c) A train enthusiast notes that a goods train arrives, on average, once an hour. What is the probability that two goods trains will arrive in one hour?

d) If the enthusiast wishes to stay at the station for an hour, what is the probability that he will see at least one goods train in that time?

Question 11: A probability distribution can be described as follows:

a) What is the probability that X = 9?

b) P(X=1) = P(X=11) = 0.04. Draw out the distribution and find m.

c) What is F(X=2) where F represents the cumulative probability function?

d) Random samples of n = 1000 are taken from the population. The sample means are calculated. Describe what the sampling distribution will
look like for the sample means. Explain your answer fully.

e) What is F(X=12)? Can it be found?