Solutions

Activity 1 Small flat-screen televisions

Discussion

For a batch size of 20, the median position is fraction 1 over 2 end open bracket 20 + 1 close bracket = 10 fraction 1 over 2 end. So, the median will be halfway between x subscript open bracket 10 close bracket end and x subscript open bracket 11 close bracket end. These are both 150, so the median is £150.

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Activity 2 The price of gas in UK cities

Part

Discussion

A stemplot of all 14 prices in the table is shown below.

Figure 6 Stemplot of 14 gas prices

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Part

Discussion

Stemplots for the prices for northern and southern cities are shown below.

Figure 7 Stemplots for northern and southern cities separately.

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Part

Discussion

For a batch size of 14, the median position is fraction 1 over 2 end open bracket 14 + 1 close bracket = 7 fraction 1 over 2 end. So, the all-cities median will be halfway between x subscript open bracket 7 close bracket end and x subscript open bracket 8 close bracket end. These are 3.784 and 3.795, so the median is 3.7895, which is 3.790 when rounded to three decimal places. (The rounded median should be written as 3.790 and not 3.79, to show it is accurate to three decimal places and not just two.)

For the northern and southern batches, both of size 7, the median for each is the value of x subscript open bracket 4 close bracket end (that is, fraction 1 over 2 end open bracket 7 + 1 close bracket = 4). This is 3.776 for the northern batch and 3.795 for the southern batch.

The range is the difference between the upper extreme, uppercase E subscript uppercase U end, and the lower extreme, uppercase E subscript uppercase L end (range = uppercase E subscript uppercase U end minus uppercase E subscript uppercase L end). So the all-cities range is

the range for the northern batch is

and the range for the southern batch is

The medians and ranges are summarised below.

Batch Median Range

All cities

3.790

0.078

Northern cities

3.776

0.064

Southern cities

3.795

0.075

Thus the general level of gas prices in the country as a whole was about 3.790p per kWh. The average price differed by only 0.078p per kWh across the 14 cities.

The difference between the median prices for the northern and southern cities is 0.019p per kWh open bracket 3.795 minus 3.776=0.019 close bracket, with the south having the higher median.

The analysis does not clearly reveal whether the general level of gas prices for typical consumers in 2010 was higher in the south or in the north, though there is an indication that prices were a little higher in the south. The range of prices was also rather greater in the south. It is worth noting that the differences in gas prices between the cities in Table 3 were generally small, when measured in pence per kWh – although, with a typical annual gas usage of 18 000 kWh, the price difference between the most expensive city and the cheapest would amount to an annual difference in bills of about £14 on a typical bill of somewhere around £700.

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Activity 3 Small televisions: the mean

Discussion

Using the data for the prices from Activity 1:

Or using the sum notation, sum x = 90 + 100 + ellipsis +270 = 3240 and n = 20, so

The prices were rounded to the nearest £10, so it is appropriate to keep one more significant figure for the mean, that is, to show it accurate to the nearest £1. So since the exact value is £162, it needs no further rounding.

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Activity 4 Changing the gas prices

Discussion

The completed table is:

Batch Mean Median

Seven southern cities

3.7859

3.795

Five southern cities (excluding Cardiff and Ipswich)

3.7996

3.796

Whereas deletion of Cardiff and Ipswich has the effect of increasing the mean price by 0.0137p per kWh, the median price increases by only 0.001p per kWh. This is what we would expect as, in general, the more resistant a measure is, the less it changes when a few extreme values are deleted.

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Activity 5 A misprint in the gas prices

Discussion

The completed table is:

Batch Mean Median

Five cities (correct data)

3.7996

3.796

Five cities (with misprint)

4.6996

3.796

Here the median is completely unaffected by the misprint, although the mean changes considerably.

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Exercise 1 Finding medians

Part

Discussion

For the arithmetic scores, the position of the median is fraction 1 over 2 end open bracket 33+1 close bracket = 17, so the median is 79%.

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Part

Discussion

For the television prices, the position of the median is fraction 1 over 2 end open bracket 26+1 close bracket = 13 fraction 1 over 2 end, so the median is halfway between x subscript open bracket 13 close bracket end and x subscript open bracket 14 close bracket end. Thus, the median is

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Exercise 2 Finding means

Discussion

For the batch of arithmetic scores in part (a) of Exercise 1, the sum of the 33 values is 2326 and

Therefore, the mean is 70.5%. (The original data are given to the nearest whole number, so the mean is rounded to one decimal place.)

For the batch of television prices in part (b) of Exercise 1, the sum of the 26 values is 7856 and

Therefore, the mean is £302.2.

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Exercise 3 The effect of removing values on the median and mean

Discussion

For the median, there are now 17 prices left in the batch, so the median is at position fraction 1 over 2 end open bracket 17+1 close bracket = 9. It is therefore 150.

The sum of the remaining 17 values is 2480, so the mean is

In this case, removing the three highest prices has not changed the median at all, but it has reduced the mean considerably. This illustrates that the median is a more resistant measure than the mean.

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Activity 6 Using the rules for weighted means

Discussion

You should expect the weighted mean price to be nearer the London price, because of Rule 2 for weighted means (Subsection 2.1) and given that London has a much larger weight then Edinburgh.

The weighted mean price given by the formula in Example 11 is (after rounding) 3.814p per kWh, which is indeed much closer to the London price than to the Edinburgh price.

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Activity 7 Weighted means of Open University marks

Part

Discussion

OCAS = fraction open bracket 80 times 50 close bracket + open bracket 60 times 50 close bracket over 50 +50 end = fraction 4000 +3000 over 100 end = fraction 7000 over 100 end = 70.

This is the same as a simple (unweighted) mean of the two scores, because the two component scores have equal weight. It lies exactly halfway between the two scores (fraction 1 over 2 end open bracket 80+60 close bracket =70).

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Part

Discussion

OCAS = fraction open bracket 80 times 40 close bracket + open bracket 60 times 60 close bracket over 40 +60 end = fraction 3200 +3600 over 100 end = fraction 6800 over 100 end = 68.

This is slightly less than the simple mean in (a) because the component with the lower score (TMA) has the greater weight.

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Part

Discussion

OCAS = fraction open bracket 80 times 65 close bracket + open bracket 60 times 55 close bracket over 65 +55 end = fraction 5200 +3300 over 120 end = fraction 8500 over 120 end simeq 70.8.

This is slightly higher than the simple mean in (a) because the component with the higher score (iCMA) has the greater weight.

(Note that the weights need not necessarily sum to 100, even when dealing with percentages.)

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Part

Discussion

OCAS = fraction open bracket 80 times 25 close bracket + open bracket 60 times 75 close bracket over 25 +75 end = fraction 2000 +4500 over 100 end = fraction 6500 over 100 end = 65.

This is even lower than (b), so even nearer the lower score (TMA), because the TMA score has even greater weight.

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Part

Discussion

OCAS = fraction open bracket 80 times 30 close bracket + open bracket 60 times 90 close bracket over 30 +90 end = fraction 2400 +5400 over 120 end = fraction 7800 over 120 end = 65.

This is the same as (d) because the ratios of the weights are the same; they are both in the ratio 1 to 3. That is, 25:75=30:90 (=1:3).

(We say this as follows: ‘the ratio 25 to 75 equals the ratio 30 to 90’.)

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Activity 9 Weighted mean electricity price

Discussion

The table showing the required sums (and the values in the x w column, that you may not have had to write down), is as follows.

City Price (p/kWh): x Weight: w Price times weight: x w

Aberdeen

13.76

19

261.44

Belfast

15.03

58

871.74

Edinburgh

13.86

42

582.12

Leeds

12.70

150

1 905.00

Liverpool

13.89

82

1 138.98

Manchester

12.65

224

2 833.60

Newcastle-upon-Tyne

12.97

88

1 141.36

Nottingham

12.64

67

846.88

Birmingham

12.89

228

2 938.92

Canterbury

12.92

5

64.60

Cardiff

13.83

33

456.39

Ipswich

12.84

14

179.76

London

13.17

828

10 904.76

Plymouth

13.61

24

326.64

Southampton

13.41

30

402.30

Sum

1892

24 854.49

Thus sum x w = 24854.49, sum w = 1892 and

So the weighted mean of electricity prices is 13.14p per kWh.

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Exercise 4 A combined batch of camera prices

Discussion

Mean price of all the cameras is

which is £79.3 (rounded to the same accuracy as the original means).

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Exercise 5 The mean price of fabric

Discussion

Mean price of all the material is

which is £11.67 (rounded to the nearest penny).

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Activity 10 Finding more quartiles

Part

Discussion

Here, because n=15, an appropriate picture of the data would be Figure 12 (Subsection 3.2). To find the lower and upper quartiles, uppercase Q sub 1 and uppercase Q sub 3, of this batch, first find fraction 1 over 4 end open bracket n+1 close bracket = 4 and fraction 3 over 4 end open bracket n+1 close bracket = 12. Therefore uppercase Q sub 1 =268 p and uppercase Q sub 3 =299 p.

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Part

Discussion

For this batch, n=14 so fraction 1 over 4 end open bracket n+1 close bracket = 3 fraction 3 over 4 end and fraction 3 over 4 end open bracket n+1 close bracket = 11 fraction 1 over 4 end.

and

So the lower quartile is 3.756 p per kWh and the upper quartile is 3.802p per kWh.

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Activity 11 Coffee prices again

Discussion

The range is the distance between the extremes:

The interquartile range is the distance between the quartiles:

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Activity 12 Interquartile range of gas prices

Discussion

The quartiles, before rounding, are uppercase Q sub 1 =3.75575 and uppercase Q sub 3 =3.80175. So

and the interquartile range is 0.046p per kWh.

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Activity 13 Skew gas prices?

Part

Discussion

All the necessary figures have already been calculated. You found the median (3.790) in Activity 2 and the quartiles (uppercase Q sub 1 = 3.756, uppercase Q sub 3 = 3.802) in Activity 10. The extremes (uppercase E subscript uppercase L end =3.740, uppercase E subscript uppercase U end =3.818) and the batch size (n=14) are clearly shown in the stemplot.

So the five-figure summary is as follows:

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Part

Discussion

Looking at the stemplot, on the whole the lower values are more spread out, indicating that the data are not symmetric and are left-skew.

The central box of the boxplot again shows left skewness, with the left-hand part of the box being clearly longer than the right-hand part. However, this skewness does not show up in the lengths of the whiskers in this batch – they are both the same length.

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Exercise 6 Finding quartiles and the interquartile range

Part

Discussion

For the arithmetic scores, n=33 so fraction 1 over 4 end open bracket n+1 close bracket = 8 fraction 1 over 2 end and fraction 3 over 4 end open bracket n+1 close bracket = 25 fraction 1 over 2 end.

The lower quartile is therefore

The upper quartile is

The interquartile range is

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Part

Discussion

For the television prices, n=26 so fraction 1 over 4 end open bracket n+1 close bracket = 6 fraction 3 over 4 end and fraction 3 over 4 end open bracket n+1 close bracket = 20 fraction 1 over 4 end.

The lower quartile is therefore

The upper quartile is

The interquartile range is

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Exercise 7 Some five-figure summaries

Part

Discussion

Arithmetic scores:

From the stemplot, n=33, uppercase E subscript uppercase L end = 7 and uppercase E subscript uppercase U end = 100.

Figure 31 Five-figure summary of arithmetic scores

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Part

Discussion

Television prices:

From the data table, n=26, uppercase E subscript uppercase L end = 170 and uppercase E subscript uppercase U end = 699.

Figure 32 Five-figure summary of television prices

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Exercise 8 Boxplots and the shape of distributions

Discussion

For the boxplot of arithmetic scores, the left part of the box is longer than the right part, and the left whisker is also considerably longer than the right. This batch is left-skew.

For the boxplot of television prices, the right part of the box is rather longer than the left part. The right whisker is also rather longer than the left, and if one also takes into account the fact that two potential outliers have been marked, the top 25% of the data are clearly much more spread out than the bottom 25%. This batch is right-skew.

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Activity 14 Gradgrind’s gas price increase

Discussion

The increase (in £/MWh) is 29 minus 24=5. This is fraction 5 over 24 end simeq 0.208 as a proportion of the 2007 price. That is, fraction 5 over 24 end times 100 % simeq 20.8 % of the 2007 price. Or you might have worked this out by finding that the 2008 price is fraction 29 over 24 end times 100 % simeq 120.8 % of the 2007 price, so that again the increase is 20.8% of the 2007 price.

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Activity 15 Gradgrind’s electricity price index

Discussion

The 2008 electricity price is 1.145 times 100 % = 114.5 % of the 2007 price, so that the increase is 14.5% of the 2007 price.

The 2008 value of the electricity price index is

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Activity 16 How much fuel did Gradgrind use?

Discussion

The expenditure on a particular fuel in a particular year can be calculated as expenditure = quantity used times price. Therefore, if the expenditure and price are known, the quantity used can be calculated as

In 2007, Gradgrind’s gas cost £24 per MWh, and they spent £9298 on gas, so the amount of gas they used in MWh was

The other amounts, in MWh, are found in a similar way, and all are shown in the following table.

Energy type 2007 2008

Gas

387.4

280.9

Electricity

42.2

34.4

The reason that the expenditures went down is simply that Gradgrind used less of each fuel in 2008 than in 2007.

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Activity 17 Gradgrind’s energy price ratio for 2009 relative to 2008

Part

Discussion

The gas price ratio for 2009 relative to 2008 is

The electricity price ratio for 2009 relative to 2008 is

(Over this year, electricity prices rose a lot more than gas prices.)

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Part

Discussion

The overall energy price ratio for 2009 relative to 2008 is

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Part

Discussion

Using the 2009 expenditures for weights instead of the 2008 expenditures, the overall energy price ratio for 2009 relative to 2008 is

This price ratio is considerably less than the one found in part (b).

(Note that if full calculator accuracy is retained throughout the calculations, the price ratio is 1.043 to three decimal places.)

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Activity 18 Gradgrind’s energy price index for 2010

Discussion

The gas price ratio for 2010 relative to 2009 is

The electricity price ratio for 2010 relative to 2009 is

(Both price ratios are less than 1 because, over this year, Gradgrind’s gas and electricity prices both fell.)

The overall energy price ratio for 2010 relative to 2009 is

Then the value of the index for 2010 is found by multiplying the 2009 value of the index by this overall price ratio, giving

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Exercise 9 Gradgrind’s energy price index for 2011

Discussion

The gas price ratio for 2011 relative to 2010 is

The electricity price ratio for 2011 relative to 2010 is

The overall energy price ratio for 2011 relative to 2010 is

Then the value of the index for 2011 is found by multiplying the 2010 value of the index by this overall price ratio, giving

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Activity 19 The expenditure of a typical household

Part

Discussion

What you need to remember here is that the size of an area represents the proportion of expenditure on that class of goods or services. (Also, it is admittedly not very easy to estimate these areas ‘by eye’! Your estimates might quite reasonably differ from those given here.)

  • The sector for ‘Personal expenditure’ looks as if it is approximately a tenth of the whole inner circle – so approximately a tenth of total expenditure is personal expenditure.

  • ‘Housing and household expenditure’ looks as if it is somewhere between a third and a half of the inner circle – perhaps approximately two fifths – so approximately two fifths of expenditure is on housing and household expenditure.

  • The area for ‘Housing’ takes up about a quarter of the outer ring, so about a quarter of expenditure is on housing.

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Part

Discussion

The amount spent each week on ‘Personal expenditure’ is approximately

The amount spent each week on ‘Housing and household expenditure’ is approximately

The amount spent each week on ‘Housing’ is approximately

Recall, however, that the weights represent average proportions of expenditure, and the spending patterns of the selected household may differ from those of the ‘typical’ household.

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Activity 20 Your own household’s expenditure

Discussion

Every household will be different, but think about the reasons for any large differences between your weights and those for the RPI.

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Activity 21 Calculating the RPI for July 2011

Discussion

Group Price ratio for July 2011 relative to January 2011: r 2011 weights: w Price ratio times weight: r w

Food and catering

1.024

165

168.960

Alcohol and tobacco

1.042

88

91.696

Housing and household expenditure

1.012

408

412.896

Personal expenditure

1.053

82

86.346

Travel and leisure

1.030

257

264.710

Sum

1000

1024.608

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Activity 22 The effects of particular price changes on the RPI

Part

Discussion

The RPI is calculated using the price ratio and weight of each item. Since the weights of items change very little from one year to the next, the price ratio alone will normally tell you whether a change in price is likely to lead to an increase or a decrease in the value of the RPI. If a price rises, then the price ratio is greater than one, so the RPI is likely to increase as a result. If a price falls, then the price ratio is less than one, so the RPI is likely to decrease. Therefore, since the price of leisure goods fell, this is likely to lead to a decrease in the value of the RPI. For a similar reason, the increase in the price of canteen meals is likely to lead to an increase in the value of the RPI.

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Part

Discussion

Both changes are likely to be small for two reasons. First, the price changes are themselves fairly small. Second, leisure goods and canteen meals form only part of a household’s expenditure: no single group, subgroup or section will have a large effect on the RPI on its own, unless there is a very large change in its price.

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Part

Discussion

The weight of ‘Leisure goods’ was 33 in 2012 (see Table 12). Since ‘Canteen meals’ is only one section in the subgroup ‘Catering’, which had weight 47 in 2012, the weight of ‘Canteen meals’ will be much smaller than 47. (In fact it was 3.) So the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’.

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Part

Discussion

Since the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’, and the percentage change in the prices are not too different in size, the change in the price of leisure goods is likely to have a much larger effect on the value of the RPI as a whole.

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Activity 23 The annual inflation rate in February 2012

Discussion

The ratio of the two RPI values is

or 103.7%. Therefore the annual inflation rate, based on the RPI was 3.7%. (Note that this is slightly higher than the annual inflation rate measured using the CPI.)

Back to - Activity 23 The annual inflation rate in February 2012



Activity 24 Index-linking a pension using the CPI

Discussion

The weekly amount in November 2011 should be

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Activity 25 Annual inflation and the purchasing power of the pound

Part

Discussion

For May 2010, the ratio of the value of the RPI to its value one year earlier is

so the annual inflation rate is 5.1%.

The purchasing power of the pound compared to one year previously is

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Part

Discussion

For October 2011, the ratio of the value of the RPI to its value one year earlier is

so the annual inflation rate is 5.4%.

The purchasing power of the pound compared to one year previously is

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Part

Discussion

For March 2011, the ratio of the value of the RPI to its value one year earlier is

so the annual inflation rate is 5.3%.

The purchasing power of the pound compared to one year previously is

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Exercise 10 Calculating the RPI for February 2012

Discussion

(The published index was 239.9. Again, the difference between this and your calculated value is because the ONS statisticians used more accuracy in their intermediate calculations.)

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Exercise 11 Annual inflation rates and the purchasing power of the pound

Part

Discussion

For October 2010, the ratio of the value of the RPI to its value one year earlier is

so the annual inflation rate is 4.5%.

The purchasing power of the pound compared to one year previously is

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Part

Discussion

For January 2011, the ratio of the value of the RPI to its value one year earlier is

so the annual inflation rate is 5.1%.

The purchasing power of the pound compared to one year previously is

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Exercise 12 Index-linking another pension

Discussion

The RPI for April 2011 was 234.4 and the RPI for April 2010 was 222.8. So in April 2011, the pension should be

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