Question: Which of the following events are independent?

> From your results for the birth rate model, predict the birth rate for a country with either (a). GNP equal to $3000, (b). a growth rate of 3% p.a., or (c). an income ratio of 7. How does your prediction compare with one using Todaro’s results? Comme

> a. For the data given in Problem 7.2, estimate the sample regression line and calculate the R2 statistic. Comment upon the results. Problem 7.2: The data below show alcohol expenditure and income (both in £s per week) for a sample of 17 fa

> a. For the data in Problem 7.1, find the estimated regression line and calculate the R2 statistic. Comment upon the result. How does it compare with Todaro’s findings? b. Calculate the standard error of the estimate and the standard err

> As Problem 7.3, for Data from Problem 7.3: What would you expect to be the correlation coefficient between the following variables? Should the variables be measured contemporaneously or might there be a lag in the effect of one upon the other? (a) real

> Using the data from Problem 1.5: Data from Problem 1.5: The distribution of marketable wealth in 1979 in the United Kingdom is shown in the table below (adapted from Inland Revenue Statistics, 1981, contains public sector information licensed under the

> What would you expect to be the correlation coefficient between the following variables? Should the variables be measured contemporaneously or might there be a lag in the effect of one upon the other? a. Nominal consumption and nominal income. b. GDP and

> The data below show alcohol expenditure and income (both in £s per week) for a sample of 17 families. a. Draw an XY plot of the data and comment. b. From the chart, would you expect the line of best fit to slope up or down? In theory, whic

> Update Todaro’s study using more recent data.

> The other data which Todaro might have used to analyse the birth rate were: For one of the three possible explanatory variables (in class, different groups could examine each of the variables): a. Draw an XY chart of the data above and comment upon the

> a. Do the accountants’ job properly for them (see the Oops! box in the text (page 244)). b. It might be justifiable to omit the ‘no responses’ entirely from the calculation. What happens if you do thi

> Four different holiday firms which all carried equal numbers of holidaymakers reported the following numbers who expressed satisfaction with their holiday: Is there any significant difference between the firms? If told that the four firms carried 600 h

> A survey of 64 families with five children found the following gender distribution: Test whether the distribution can be adequately modelled by the Binomial distribution. Number of boys 1 3 4 Number of families 8 28 19 4 4

> Groups of children from four different classes in a school were randomly selected and sat a test, with the following test scores: a. Test whether there is any difference between the classes, using the 95% confidence level for the test. b. How would you

> A sample of 40 observations has a standard deviation of 20. Estimate the 95% confidence interval for the standard deviation of the population.

> A firm receives components from a supplier, which it uses in its own production. The components are delivered in batches of 2000. The supplier claims that there are only 1% defective components on average from its production. However, production occasion

> Given a standard pack of cards, calculate the following probabilities: a. drawing an ace; b. drawing a court card (i.e. jack, queen or king); c. drawing a red card; d. drawing three aces without replacement; e. drawing three aces with replacement.

> A coin which is either fair or has two heads is to be tossed twice. You decide on the following decision rule: if two heads occur you will conclude it is a two-headed coin, otherwise you will presume it is fair. Write down the null and alternative hypoth

> Can your class tell the difference between tap water and bottled water? Set up an experiment as follows: fill r glasses with tap water and n - r glasses with bottled water. The subject has to guess which is which. If he or she gets more than p correct, y

> Discuss in general terms how you might ‘test’ the following: a. astrology b. extra-sensory perception c. the proposition that company takeovers increase profits.

> Another group of workers were tested at the same times as those in Problem 5.23, although their department also introduced rest breaks into the working day. Does the introduction of rest days alone appear to improve performance? Problem 5.23: The out

> The output of a group of 11 workers before and after an improvement in the lighting in their factory is as follows: Test whether there is a significant improvement in performance a. assuming these are independent samples, b. assuming they are dependent

> a. A pharmaceutical company testing a new type of pain reliever administered the drug to 30 volunteers experiencing pain. Sixteen of them said that it eased their pain. Does this evidence support the claim that the drug is effective in combating pain? b.

> a. A random sample of 180 men who took the driving test found that 103 passed. A similar sample of 225 women found that 105 passed. Test whether pass rates are the same for men and women. b. If you test whether the group of people who passed the driving

> Given the two hypotheses and σ2 = 1000 (for both hypotheses): a. Draw the distribution of x under both hypotheses. b. If the decision rule is chosen to be: reject H0 if x > 410 from a sample of size 40, find the probability of a Type II

> Answer true or false, with reasons if necessary. a. There is no way of reducing the probability of a Type I error without simultaneously increasing the probability of a Type II error. b. The probability of a Type I error is associated with an area under

> a. A sample of 954 adults in early 1987 found that 23% of them held shares. Given a UK adult population of 41 million and assuming a proper random sample was taken, find the 95% confidence interval estimate for the number of shareholders in the United Ki

> The distribution of marketable wealth in 1979 in the United Kingdom is shown in the table below (adapted from Inland Revenue Statistics, 1981, contains public sector information licensed under the Open Government Licence (OGL) v3.0, http://www.nationalar

> a. A sample of 200 women from the labour force found an average wage of £26 000 p.a. with standard deviation £3500. A sample of 100 men found an average wage of £28 000 with standard deviation £2500. Es

> A firm receives components from a supplier in large batches, for use in its production process. Production is uneconomic if a batch containing 10% or more defective components is used. The firm checks the quality of each incoming batch by taking a sample

> If the probability of a boy in a single birth is 1 2 and is independent of the sex of previous babies, then the number of boys in a family of 10 children follows a Binomial distribution with mean 5 and variance 2.5. In each of the following instances,

> A report in The Guardian newspaper (20 June 2010, http://www.guardian.co.uk/education/ 2010/jun/20/internet-plagiarism-rising-in-schools) reports ‘Half of university students also prepared to submit essays bought off the internet, according to research.’

> Using a weekend’s football results from the Premier (or other) League, see if the number of goals per game can be adequately modelled by a Poisson process. First calculate the average number of goals per game for the whole league, and then derive the dis

> An extremely numerate newsagent (with a spreadsheet program, as you will need) is trying to work out how many copies of a newspaper he should order. The cost to him per copy is 40 pence, which he then sells at £1.20. Sales are distributed No

> An experienced invoice clerk makes an error once in every 100 invoices, on average. (a) What is the probability of finding a batch of 100 invoices without error? (b) What is the probability of finding such a batch with more than two errors? Calculate the

> A machine producing electronic circuits has an average failure rate of 15% (they are difficult to make). The cost of making a batch of 500 circuits is £8400 and the good ones sell for £20 each. What is the probability of the firm making a loss on any one

> Two dice are thrown and the absolute difference of the two scores is recorded. Graph the resulting probability distribution and calculate its mean and variance. What is the probability that the absolute difference is 4 or more?

> Ten adults are selected at random from the population and their IQ measured. (Assume a population mean of 100 and standard deviation of 16 as in Problem 3.16.) (a) What is the probability distribution of the sample average IQ? (b) What is the probability

> Using the data from Problem 1.2: Data from Problem 1.2: The data below show the average hourly earnings (in £s) of those in full-time employment, by category of education (NVQ levels. NVQ 4 corresponds to a university degree). a. What is

> Two dice are thrown and the sum of the two scores is recorded. Draw a graph of the resulting probability distribution of the sum and calculate its mean and variance. What is the probability that the sum is 9 or greater?

> A news item revealed that a London ‘gender’ clinic (which reportedly enables you to choose the sex of your child) had just set up in business. Of its first six births, two were of the ‘wrong’ sex. Assess this from a probability point of view.

> Judy is 33, unmarried and assertive. She is a graduate in political science, and involved in union activities and anti-discrimination movements. Which of the following statements do you think is more probable? a. Judy is a bank clerk. b. Judy is a bank c

> How might you estimate the probability of Peru defaulting on its debt repayments next year? What type of probability estimate is this?

> a. Translate the following odds to ‘probabilities’: 13/8, 2/1 on, 100/30. b. In the 2.45 race at Plumpton the odds for the five runners were: Philips Woody…………………………………1/1 Gallant Effort………………………………….5/2 Satin Noir………………………………………11/2 Victory Anthem……………

> This problem is tricky, but amusing. Three gunmen, A, B and C, are shooting at each other. The probabilities that each will hit what they aim at are 1, 0.75 and 0.5, respectively. They take it in turns to shoot (in alphabetical order) and continue until

> A man is mugged and claims that the mugger had red hair. In police investigations of such cases, the victim was able correctly to identify the assailant’s hair colour 80% of the time. Assuming that 10% of the population have red hair, what is the probabi

> a. Your initial belief is that a defendant in a court case is guilty with probability 0.5. A witness comes forward claiming he saw the defendant commit the crime. You know the witness is not totally reliable and tells the truth with probability p. Use Ba

> The UK national lottery originally worked as follows. You choose six (different) numbers in the range 1 to 49. If all six come up in the draw (in any order), you win the first prize, generally valued at around £2m (which could be shared if s

> An important numerical calculation on a spacecraft is carried out independently by three computers. If all arrive at the same answer, it is deemed correct. If one disagrees, it is overruled. If there is no agreement, then a fourth computer does the calcu

> Odds’ in horserace betting are defined as follows: 3/1 (three-to-one against) means a horse is expected to win once for every three times it loses; 3/2 means two wins out of five races; 4/5 (five to four on) means five wins for every four defeats, etc. a

> Roll six sixes to win a Mercedes!’ is the announcement at a fair. You have to roll six dice. If you get six sixes you win the car, valued at £40 000. The entry ticket costs £1. What is your expected gain or loss on this game? If there are 400 people who

> The BMAT test (see http://www.ucl.ac.uk/lapt/bmat/) is an on-line test for prospective medical students. It uses ‘certainty-based marking’. After choosing your answer from the alternatives available, you then have to give your level of confidence that yo

> Criticise the following statistical reasoning. Amongst arts graduates, 10% fail to find employment. Amongst science graduates only 8% remain out of work. Therefore, science graduates are better than arts graduates. Hint: imagine there are two types of jo

> There are 25 people at a party. What is the probability that there are at least two with a birthday in common? They do not need to have been born in the same year, just the same day and month of the year. Also, ignore leap year dates. (Hint: the compleme

> A firm can build a small, medium or large factory, with anticipated profits from each dependent upon the state of demand, as in the table below. a. Which project should be chosen on the expected value criterion? b. Which project should be chosen on the

> A firm has a choice of three projects, with profits as indicated below, dependent upon the state of demand. a. Which project should be chosen on the expected value criterion? b. Which project should be chosen on the maximin and maximax criteria? c. Whi

> A bond is issued which promises to pay £400 p.a. in perpetuity. How much is the bond worth now, if the interest rate is 5%? (Hint: the sum of an infinite series of the form is 1/r, as long as r > 0.) 1 1 + 1 +r (1 + r)? (1 + r)}

> Find the anti-log of the following values: -0.09 691, 2.3, 3.3, 6.3.

> Find the anti-log of the following values: -0.823 909, 1.1, 2.1, 3.1, 12.

> Find the ln of the following values: 0.3, e, 3, 33, -1.

> Find the natural logarithms of: 0.15, 1.5, 15, 225, -4.

> Find the log of the following values: 0.8, 8, 80, 4, 16, -37.

> Find the common logarithms of: 0.15, 1.5, 15, 150, 1500, 83.7225, 9.15, -12.

> Demonstrate that Ef(x – µ)? Efx? ,2 μ Ef Ef

> Demonstrate that Ef(x – k)_ Efx Ef Ef k where k is a constant.

> Depreciation of BMW and Mercedes cars is given in the following table of new and used car prices: a. Calculate the average rate of depreciation of each type of car. b. Use the calculated depreciation rates to estimate the value of the car after 1, 2, e

> Given the following frequencies, fi associated with the x values in Problem 1A.2: {10, 6, 6, 16, 10}, evaluate: Problem 1A.2: Given the following data on xi: {8, 12, 6, 4, 10} evaluate: Efx, £fx², £f(x – 3), Efx – 3 4 Σχ, Σχ?, ( Σx;) ?, Σ (x; - 3

> Given the following frequencies, fi associated with the x values in Problem 1A.1: {5, 3, 3, 8, 5}, evaluate: Problem 1A.1: Given the following data on xi : {4, 6, 3, 2, 5} evaluate: Efx, Efx², £f(x – 3), Efx – 3 4 Ex; Ex}, ( Ex;)², > (x; – 3), Ex

> Given the following data on xi : {8, 12, 6, 4, 10} evaluate: 4 Σχ, Σχ?, ( Σx;) ?, Σ (x; - 3), Σx3. ΣΧ

> Given the following data on xi : {4, 6, 3, 2, 5} evaluate: 4 Ex; Ex}, ( Ex;)², > (x; – 3), Ex; – 3. Ex; Xi

> Evaluate: V30, V17,81/4, 15°, 12°, 3-1/3

> Evaluate: V10, V3.7,4/4,12-3, 25-3/2.

> Find the anti-ln of the following values: 3.496 508, 14, 15, -1.

> Find the anti-ln of the following values: 2.708 05, 3.708 05, 1, 10.

> A firm purchases for £30,000 a machine which is expected to last for 10 years, after which it will be sold for its scrap value of £3000. Calculate the average rate of depreciation p.a., and calculate the written-down value of the machine after one, two a

> The data below show the average hourly earnings (in £s) of those in full-time employment, by category of education (NVQ levels. NVQ 4 corresponds to a university degree). a. In what fundamental way do the data in this table differ from tho

> Using the data from Problem 1.1: Data From Problem 1.1: The following data show the education and employment status of women aged 20–29: a. Which education category has the highest proportion of women in work? What is the proportion?

> Is the 95% confidence interval (a) twice as wide, (b) more than twice as wide and (c) less than twice as wide, as the 47.5% interval? Explain your reasoning.

> a. Why is an interval estimate better than a point estimate? b. What factors determine the width of a confidence interval?

> a. Using the data in Problem 10.3, calculate the expenditure shares on each fuel in 1999 and the individual price index number series for each fuel, with 1999 = 100. b. Use these data to construct the Laspeyres price index using the expenditures shares a

> The prices of different house types in south-east England are given in the table below: a. If the numbers of each type of house in 1991 were 1898, 1600, 1601, 499 and 1702, respectively, calculate the Laspeyres price index for 1991–94

> The following tables show energy prices and consumption in 1999–2003 (analogous to the data in the chapter for the years 2006–10). a. Construct a Laspeyres price index using 1999 as the base year. b. Construct a Paa

> The World Development Report contains data on the income distributions of many countries around the world (by quintile). Use these data to compare income distributions across countries, focusing particularly on the differences between poor countries, mid

> Compare the degrees of concentration in the following two industries. Can you say which is likely to be more competitive? Firm A В E F G H Sales 337 384 696 321 769 265 358 521 880 334 Sales 556 899 104 565 782 463 477 846 911 227

> Calculate the three-firm concentration ratio for employment in the following industry: Firm A D E F G H Employees 3350 290 440 1345 821 112 244 352 B

> For the Kravis, Heston and Summers data (Table 10.26), combine the deciles into quintiles and calculate the Gini coefficient from the quintile data. How does your answer compare with the answer given in the text, based on deciles? What do you conclude ab

> a. A government bond is issued, promising to pay the bearer £1000 in five years’ time. The prevailing market rate of interest is 7%. What price would you expect to pay now for the bond? What would its price be after two years? If, after two years, the ma

> The following data show the gross operating surplus of companies, 2005–10, in the United Kingdom, in £m. a. Turn the data into an index number series with 2005 as the reference year. b. Transform the series so that 2008 i

> a. Draw a Lorenz curve and calculate the Gini coefficient for the 1979 wealth data contained in Problem 1.5 (Chapter 1). Draw the Lorenz curve on the same diagram as you used in Problem 10.17. b. How does the answer compare to 2005 wealth data? Data fro

> Calculate the internal rates of return for the projects in Problem 10.14. Data from Problem 10.14: A firm uses a discount rate of 12% for all its investment projects. Faced with the following choice of projects, which yields the higher NPV? Projec

> A firm uses a discount rate of 12% for all its investment projects. Faced with the following choice of projects, which yields the higher NPV? Project Outlay Income stream 1 2 3 4 6 A 5600 1000 1400 1500 2100 1450 700 В 6000 800 1400 1750 2500 1925 1

> A firm is investing in a project and wishes to receive a rate of return of at least 15% on it. The stream of net income is: a. What is the present value of this income stream? b. If the investment costs £1600, should the firm invest? What

> a. If w represents the wage rate and p the price level, what is w/p? b. If ∆w represents the annual growth in wages and i is the inflation rate, what is ∆w - i? c. What does ln (w) - ln (p) represent? (ln= natural logarithm.)

> Using the data in Problem 10.6, calculate how much the average consumer would need to be compensated for the rise in prices between 1990 and 1994. Problem 10.6: The following table shows the weights in the retail price index and the values of the index

> The data below show exports and imports for the United Kingdom, 2005–10, in £bn at current prices. a. Construct index number series for exports and imports, setting the index equal to 100 in 2005 in each case. b. Is it po

> A firm has £10 000 to spend on a survey. It wishes to know the average expenditure on gas by businesses to within £30 with 99% confidence. The variance of expenditure is believed to be about 40 000. The survey costs £7000 to set up and then £15 to survey

> A firm wishes to know the average weekly expenditure on food by households to within £2, with 95% confidence. If the variance of food expenditure is thought to be about 400, what sample size does the firm need to achieve its aim?

> How would you expect the following time-series variables to look when graphed? a. The price level. b. The inflation rate. c. The £/$ exchange rate.

> Find figures for the monetary aggregate M0 for the years 1995 to 2003 in the United Kingdom, in nominal terms.

> What issues of definition arise in trying to measure ‘unemployment’?

> What issues of definition arise in trying to measure ‘output’?