Preparatory Work for Mathematics and Statistics

Preparatory work for Mathematics and Statistics

Many modules assume a good knowledge of mathematics and some statistics. Mathematical tools will be used in Macroeconomics 2 and Microeconomics 2, and Econometrics 1 assumes additionally some background in statistics. As part of your preparation for the course you should spend some time before starting the programme revising your mathematics.

Prerequisites

You should be familiar with the following: basic algebra: variables, formulae, rules for manipulation of algebraic expressions, solving linear equations in one variable and simultaneous linear equations, quadratic equations; functions of one variable; logarithms; exponential functions; differential calculus: the derivative of a function of one variable, rules of differentiation (xn, log(x), ln(x), ex, ax, the chain rule, the product rule, the quotient rule); integration: indefinite integral, definite integral, areas under graphs of functions; curve sketching, turning points and inflexion points of functions of one variable, asymptotes.

Suggested Reading

“Essential Mathematics for Economic Analysis” by Knut Sydsæter and Peter Hammond, Third Edition, Prentice Hall.

Also, you might want to look at some statistical appendices you find in some econometrics text books. For example, “Introductory Econometrics: A Modern Approach”by Wooldridge, 4th edition, has a nice chapter. However, note, the 5th edition does not have this.

Statistics Exercises

PLEASE BRING YOUR SOLUTIONS TO YOUR PRE-SESSIONAL CLASSES AT THE START OF THE COURSE.

Exercise 1: Descriptive Statistics

A random sample of incomes in a community yields the following values (in Thousands of £ per year): 42, 37, 78, 38, 35, 52.

1. Judging from this sample, the income random variable is:

(a) symmetric.

(b)negatively skewed.

(d) normally distributed.

(e) chi-squared distributed.

2. The sample variance of the income random variable is:

(a) 47

(b) 222.667

(d)14.922

(e) 267.2

3.The sample standard deviation of the income random variable is:

(a) 47

(b) 222.667

(d)14.922

(e) 267.2

4. What is an appropriate measure of central tendency for the income random variable?

(a) 40

(b) 47

(d) 42

(e) 56.4

5.A colleague has added a 7th observation to the above sample. The sample mean when calculated on all 7 observations is 49. What is the income value of the observation that was added?

(a) 47

(b) 49

(d) 61

(e) 343

6. You obtained a set of 40 observations and calculated a sample standard deviation of 3.564. What is the population variance for the same 40 observations?

(a) 12.702

(b) 495.382

(d) 13.028

(e) 12.385

The table below reports the A-level score in the best 3 A-levels for all leavers from a UK ‘old’ University in 1993, broken down by gender (where 10 points are awarded for a grade A, 8 points for a grade B, 6=C, 4=D and 2=E):

A-level score / No. females
14 / 1591
16 / 2413
18 / 3262
20 / 3846
22 / 4289
24 / 4241
26 / 3819
28 / 3063
30 / 3057

(a)Calculate the mean.

(b)Calculate the variance and standard deviation.

(c)Calculate the median.

(d)Calculate the mode.

(e)Calculate the interquartile range.

(f)What do you think are the interesting points about the distribution of A-level scores for males and females.

For a sample of 35 companies, the table below shows the percentage change in output over the last 3 months, grouped into classes.

% change / -2-0 / 0-2 / 2-4 / 4-6 / 6-8
Number of companies / 13 / 4 / 8 / 7 / 3

(a)Calculate the mean percentage change in output.

(b)Calculate the median.

(c)Find the modal class.

(d)Calculate the variance.

(e)Calculate the standard deviation.

(f)Calculate the interquartile range.

In question 2 suppose the 1st interval width were -4-0% instead of -2-0% and the last interval were 6-10% instead of 6-8% what would be the effect on the mean and variance of the distribution. In question 2, suppose an innovation in technology means all output grows by 2 percentage points more than previously, what would be the effect on the mean and variance of the distribution. If alternatively as a result of the new technology output increases by 20%, what would be the effect on the mean and variance of the distribution.

Exercise 2: Probability

This information relates to question 1-3. Thefrequencies of the number of GCSE passes at grades A to C, derivedfrom Youth Cohort Survey data for 2004, for boys and girls:

No. of GCSE passes / Boys / Girls
0 / 883 / 717
1-4 / 1056 / 1281
5-6 / 528 / 625
7-10 / 2120 / 2952
11-17 / 707 / 1136

Note: the numbers of GCSE passes shown are inclusive, so, for example, 1-4 means 1, 2, 3 and4 passes.

1. What proportion of pupils obtained 5 or 6 GCSE passes at grades A to C?

(a) 0.1.

(b) 0.093.

(d) 0.096.

(e) 0.167.

2. What proportion of boys got at least 7 GCSE passes at grades A to C?

(a) 0.866.

(b) 0.534.

(d) 0.830.

(e) 0.847.

3. What is the probability that the pupil is female?

(a) 0.169.

(b) 0.134.

(d) 0.384.

(e) 0.616.

This information relates to question 4-6. Consider the experiment of rolling a single dice. Events A and B are defined as

A = {an odd numbered face isshowing}

B = {faces 1 or 2 or 3 or 4 areshowing}

4. Which one of the following statements is true?

(a) A and B are mutually exclusive events.

(b) A and B are disjoint events.

(d) A is the complementary event to B.

(e) A and B are independent events.

5. What is the probability of the event ?

(a) 3/4.

(b) 2/3.

(d) 1/3.

(e) 1/6.

6.What is the probability of the event?

(a) 3/4.

(b) 2/3.

(d) 1/3.

(e) 1/6.

This information relates to questions 7-11. We have a sample of 150 students where we have information on the kind of school they go to: Type of school: S1=State non-selective, S2=State Selective or S3=Private Selective and how they travel to school. Type of transport: T1=Bike, T2=Foot, T3=Public Transport or T4=Car:

School – S
Transport – T / S1 / S2 / S3 / Total
T1 / 12 / 9 / 3 / 24
T2 / 12 / 9 / 15 / 36
T3 / 33 / 24 / 6 / 63
T4 / 6 / 6 / 15 / 27
Total / 63 / 48 / 39 / 150

7.What is the value for P(S2)?

(a) 48

(b) 0.68

(d) 0.06

(e)None of the above

8. What is the value for?

(a) 0.22

(b) 0.42

(d) 26.46

(e)None of the above

9. What is the value for?

(a) 0.02

(b) 0.125

(d) 0.16

(e)None of the above

10. How many students would you expect to go to a private School and use public transport if the two variables T and S were independent?

(a) 6

(d) 4.26

(d) 16.38

(e)None of the above

The Government Economic Service is concerned about the Economics and Econometric skills of its employees. 40% of these employees signed up to Economic classes and 50% to Econometric classes. Of those signing up for Economic classes, 30% signed up for Econometric classes.

(a)What is the probability that a randomly selected employee signed up for both classes?

(b)What is the probability that a randomly selected worked who signed up for the econometrics classes also signed up for the Economics classes?

(c)What is the probability that a randomly chosen worker signed up for at least one of these two classes?

(d)Are the two events statistically independent?

The alumni association at a leading UK university solicits donations by telephone. It is estimated that for any individual the probability of an instant donation was 0.05, 0.20 give no immediate donation, but a request for further information through mail and 0.75 have no interest. Mailed information is sent to all persons requesting it and 0.15 of these eventually give a donation. An operator makes a sequence of calls, which can be assumed to be independent.

(a)What is the probability of a donation?

(b)What is the probability that an instant credit card donation is immediately preceded by at least 4 no interest calls?

[Hint for this question you need to remember that:

]

Exercise 3: Discrete and joint distributions

This information relates to questions 1-2. The joint probability distributionof the random variablesX and Y, as given in the table

X
0 / 1
Y / 0.37 / 0.17
0.39 / 0.07

1.The expected value and variance of X is (to 3 d.p.)

(a) 0.46, 0.46

(b) 0.24, 0.24

(d) 0.46, 0.248

(e) 0.24, 0.182

2. The expected value and variance of Y is (to 3 d.p.)

(a) 0.46, 0.46.

(b) 0.24, 0.24.

(d) 0.46, 0.248.

(e) 0.24, 0.182.

This information relates to questions 3-5. The discrete random variable X with probability distributiongiven by

Values of X / -3 / -1 / 1 / 3
Probabilities / 1/3 / 1/6 / 1/6 / 1/3

3.The variance var(X) of X is:

(a) 8.

(b) 0.

(d) 6.333.

(e) 1.

4. If Zis defined as , the expected value of Z is

(a)-1.

(b) 14.222.

(d) 6.333.

(e) 5.333.

5. If Zis defined as , the variance of Z is

(a)-1.

(b) 14.222.

(d) 6.333.

(e) 5.333.

This information relates to questions 3-5. A random sample of size 2 is drawn from the probability distribution of the randomvariable X with probability distribution

Values of X / -2 / 0 / 2
Probabilities / 0.4 / 0.2 / 0.4

Drawing a random sample of size 2 from the population represented by this randomvariable.

6. What are the probabilities of the samples (-2, 0) and(-2, 2)?

(a)2/25and4/25.

(b)4/25and2/25.

(d) 4/25and4/25.

(e)2/25and1/25.

7. What is the value of the sample mean in each of the samples(-2, 0) and(-2, 2)?

(a)-2 and 0.

(b)-2 and -1.

(c)-1 and 2.

(d) 0 and -1.

(e)-1 and 0.

8. What is the relationship between the population mean and the expected value ofthe sample mean ? Choose just one of the alternatives that best reflects these

properties.

(a) The sample mean is derived from the population mean.

(b) There is no relationship.

(d) The population mean is smaller than the expected value of the sample mean.

(e) They have equal values.

A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows:

Table 1: Distribution of days of completion

Time (Days) / 1 / 2 / 3 / 4 / 5
Probability / 0.05 / 0.20 / 0.35 / 0.30 / 0.10

(a)What is the probability a randomly chosen project will take less than 3 days to complete?

(b)Find the expected time to complete.

(c)Find the variance of time required to complete a project.

(d)The contractor’s cost is made up of two parts – a fixed cost of £10,000 plus £1,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost.

(e)If 3 projects are undertaken, what is the probability that at least 2 of them will take at least 4 days to complete, assuming independence of individual project completion times.

(f)Consider the new distribution, which is the old distribution plus 10 days:

Time (Days) / 11 / 12 / 13 / 14 / 15
Probability / 0.05 / 0.20 / 0.35 / 0.30 / 0.10

What is the expected time and variance for time to complete this project and how do these relate to the answers to (a) and (b), respectively.

(g)Consider the new distribution, which is the old distribution times 5 days:

Time (Days) / 5 / 10 / 15 / 20 / 25
Probability / 0.05 / 0.20 / 0.35 / 0.30 / 0.10

What is the expected time and variance for time to complete this project and how do these relate to the answers to (a) and (b), respectively.

(h)Define as the expected value of the random variable X andas the variance ofthat distribution. Show the in general:

(i), (ii), (iii), (iv)

Consider the following bivariate distribution between Y and X

x
-1 / 0 / 1
1 / 0.2 / 0.1 / 0.1
y / 2 / 0.1 / 0.0 / 0.2
3 / 0.0 / 0.2 / 0.1

(a)Calculate

(b)Calculate

(c)Write out the univariate distribution of X+Y.Using this univariate distribution calculate. How do these relate to your answers to parts (a) and (b)?

(d)Write out the univariate distribution of X-Y. Using this univariatedistribution calculate. How do these relate to your answers to parts (a) and (b)?

(e)Define the joint probability density function for the random variables X, Y as. Define the marginal density of X [Y] as p(x) [p(y)], as p(x) [p(y)], as[]. Now, , ,and.Show that:

(i), (ii)

Exercise 4: Special and Probability distributions

The following information relates to questions 1-2. The probability mass function of the random variable X, thenumber of successes in 6 Bernoulli trials, where the success probability is = 0.4. Theprobability mass function of X is

1. What is

(a) 0.047.

(b) 0.311.

(d) 0.544.

(e) 0.746.

2. What is (to 3 d.p.)

(a) 0.047.

(b) 0.311.

(d) 0.544.

(e) 0.746.

The following information relates to questions 3-4. The random variable V with probability density function givenby

3. What is ?

(a) 0.5.

(b) 2.

(d)-1.

(e) 0.

4. What is the variance, var(V), of V?

(a) 1/8.

(b) 1/2.

(d) 1/10.

(e) 1/12.

The following information relates to questions 5-6. The random variable X with probability

distribution N(-2, 4).

5. What is (to 3 d.p.)?

(a) 0.841.

(b) 0.309.

(d) 0.955.

(e) 0.691.

6. What is the value of xwhich satisfies (to 3 d.p.)?

(a) 2.326.

(b) 1.96.

(c)-5.29.

(d) 1.29.

(e) 1.645.

7.What is the value of x which satisfies (to 3 d.p.)?

(a) 2.326.

(b) 1.96.

(c)-5.29.

(d) 1.29.

(e) 1.645.

8. Suppose the probability is 0.5 that the value of the $US will rise against the Japanese yen over any given week, and that the outcomes week on week are independent.

(a)What is the probability that the value of the $US will rise against the Japanese yen in a majority of the weeks over a period of 7 weeks?

(b)What is the probability that the value of the $US will rise against the Japanese yen in each of the 7 weeks?

(c)What is the probability that the value of the $US will rise against the Japanese yen more than once in the 7 weeks?

A university health centre receives walk-in patients at an average rate of 5 per minute, during mid-day hours.

(a)Find the probability that there will be fewer than 2 walk-in patients in a particular mid-day hour.

(b)Find the probability that there will be more than 8 walk-in patients in a particular mid-day hour.

(c)Find the probability that there will be between 2 and 5 walk-in patients in a particular mid-day hour.

Anticipated consumer demand for a product next month can be represented by a normal distribution with mean 1,200 units and standard deviation, 100 units.

(a)What is the probability that sales will exceed 1000 units?

(b)What is the probability that sales will be between 1100 and 1300 units?

(c)The probability is 0.10 that sales will be more than how many units?

A lecturer found that time spent by students on an exercise sheet follow a normal distribution with mean 60 minutes and standard deviation 18 minutes.

(a)The probability is 0.9 that a randomly chosen student spends more than how many minutes on this exercise sheet.

(b)The probability is 0.8 that a randomly chosen student spends less than how many minutes on this exercise sheet.

(c)Two students are chosen at random. What is the probability that at least one of them spends at least 75 minutes on the exercise sheet?