GCSE Statistics · Higher · Chapter 8: Index numbers, rates and quality assurance

Index numbers, rates and control charts

About 85 minutesCalculator allowed86 marks

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Date

Do Now

The first two are what this chapter needs. The rest bring back earlier chapters so nothing is forgotten.

A1
Needed today
After a $5%$ increase, the price of a bike is £ $252$ . What was the price before the increase?
[2 marks]
Answer

A2
Needed today
Write $\frac{3}{5} : \frac{2}{3}$ as a ratio of whole numbers in its simplest form.
[2 marks]
Answer
A3
Maths skill
Given that $\frac{6}{36} = \frac{5}{x}$ , find $x$ .
[2 marks]
Answer
A4
Maths skill
The point $(4, 0)$ is reflected in the line $x = 2$ . Find the coordinates of its image.
[2 marks]
Answer
A5
Maths skill
$\frac{2}{3}$ of a number is $10$ . What is the number?
[2 marks]
Answer

Examples, then your turn

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

The table shows the index numbers and weights for three ingredients a school canteen buys. Work out the weighted index number.

Hint Multiply each index number by its weight, add, then divide by the total weight.

B1
The table shows the index numbers and weights for a family's costs. Work out the weighted index number.
[2 marks]
Answer

Example 2

The chain base index numbers for a bus fare in three years were $103$ , $106$ and $104$ . Work out the average yearly rate of change. Give your answer to 1 decimal place and interpret it.

Hint Multiply the three as decimals, then take the cube root.

B2
A phone's price had chain base index numbers of $98$ , $101$ , $95$ and $99$ over four months. Work out the average monthly rate of change. Give your answer to 1 decimal place and interpret it.
[3 marks]

Example 3

A town's death rates are $3$ , $9$ and $60$ per 1000 for people aged under 40, 40 to 69, and 70 and over. The standard population is $25%$ , $55%$ and $20%$ in these groups. Work out the standardised death rate.

Hint Weight each rate by the standard population's percentage.

B3
A district's death rates are $1.4$ , $6$ and $52$ per 1000 for people aged under 40, 40 to 69, and 70 and over. The standard population is $30%$ , $50%$ and $20%$ in these groups. Work out the standardised death rate. Give your answer to 1 decimal place.
[3 marks]
Answer

Example 4

A town has $64000$ people, of whom $13500$ are women aged 15 to 44. There were $945$ births last year. Work out the crude birth rate and the general fertility rate.

Hint The crude rate divides by everyone; the general fertility rate divides by the women aged 15 to 44 only.

B4
A city has $82000$ people, of whom $17600$ are women aged 15 to 44. There were $1144$ births last year. Work out the crude birth rate and the general fertility rate.
[3 marks]
Answer

Example 5

Jars of jam have a target mass of 750 g, and the sample means have a standard deviation of 3 g. A sample of four jars has masses 754, 761, 757 and 760 g. What should be done?

Hint Work out the limits first, then the sample mean.

B5
Bottles of water have a target of 2000 ml, and the sample means have a standard deviation of 8 ml. A sample of four bottles has volumes 2023, 2031, 2026 and 2028 ml. What should be done?
[3 marks]

Example 6

One sample mean on a control chart is between the upper warning limit and the upper action limit. Explain why the process is not stopped straight away.

Hint How often is a mean outside the warning limits when nothing is wrong?

B6
A sample mean on a control chart is above the upper action limit. Explain why the process should be stopped and adjusted.
[2 marks]

Practice

Each question asks for something different. Show your working.

C1
A loaf of bread cost £3.60 in 2022 and £4.14 in 2025. Work out the index number for 2025 with 2022 as the base year, and say what it means.
[2 marks]

C2
A bike cost £350 in 2020. Its index number for 2025, with 2020 as the base year, is $92$ . Work out the price of the bike in 2025.
[2 marks]
Answer
C3
The chain base index numbers for a cinema ticket in three years were $104$ , $105$ and $110$ . Work out the percentage increase in the price over the three years.
[2 marks]
Answer
C4
There were $1248$ deaths last year in a city with a population of $96000$ . What was its crude death rate?
[2 marks]
Answer
C5
A county's general fertility rate is $58$ per 1000, and it has $21500$ women aged 15 to 44. How many births were there?
[2 marks]
Answer
C6
A control chart for the lengths of nails has warning limits of $37.2$ mm and $42.8$ mm. Work out the target length and the standard deviation of the sample means.
[2 marks]
Answer
C7
The range chart for bags of rice has an upper warning limit of $10$ g and an upper action limit of $14$ g. A sample of five bags has masses 48, 57, 53, 61 and 50 g (above 1 kg). What should be done? Give a reason.
[2 marks]
C8
A process is working correctly and 400 sample means are plotted. About how many would you expect to be outside the action limits?
[2 marks]
Answer

Exam-style questions

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The index numbers for a train season ticket, with 2020 as the base year, are $110$ in 2023 and $121$ in 2024. Amy says, "The price rose by $11%$ from 2023 to 2024."
Is Amy correct? You must show your working.
[2 marks]

D2
The table shows the index numbers and weights for the costs of running a car.
(a)
Work out the weighted index number for the cost of running the car.
[2 marks]
Answer
(b)
Over the same time the Consumer Price Index rose from $100$ to $106$ . Compare the cost of running the car with general prices.
[2 marks]
(c)
Give one reason why this index may not show the change in costs for one particular driver.
[1 mark]
Total for D2: 5 marks
D3
The table shows the chain base index numbers for the mean price of a house in a village. In 2020 the mean price was £180 000.
(a)
Work out the mean price in 2024. Give your answer to the nearest pound.
[2 marks]
Answer
(b)
Work out the average yearly rate of change in the price. Interpret your answer.
[2 marks]
Total for D3: 4 marks
D4
The table shows the people and the deaths in one year in two towns, P and Q, and a standard population.
(a)
Work out the crude death rate for town P. Give your answer to 1 decimal place.
[1 mark]
Answer
(b)
Use the standard population to find each town's standardised death rate.
[3 marks]
Answer
(c)
The crude death rate for town Q is $7.9$ per 1000. Use your answers to compare the two towns.
[2 marks]
Total for D4: 6 marks
D5
The table gives facts about two regions, R and S.
(a)
Work out the general fertility rate for each region.
[2 marks]
Answer
(b)
The crude birth rate is $11.9$ for R and $15.5$ for S. Which region has the higher fertility? Explain why the crude birth rates give a different answer.
[2 marks]
Total for D5: 4 marks
D6
A machine fills bags of nuts with a target mean of 60 g. The control chart shows the sample means for the first 9 samples.
(a)
What should have been done after sample 3?
[1 mark]
(b)
What should be done after sample 9? Give a reason.
[1 mark]
(c)
Describe the pattern in samples 5 to 9, and say what it shows about the machine.
[2 marks]
Total for D6: 4 marks
D7
Cans of drink have a target volume of 330 ml. The standard deviation of the sample means is $1.5$ ml.
(a)
Find the two warning limits and the two action limits.
[2 marks]
Answer
(b)
A sample of five cans has volumes 333.1, 334.0, 332.6, 333.8 and 335.0 ml. What should be done? You must give a reason.
[2 marks]
Total for D7: 4 marks
D8
The Consumer Price Index rose from $128.4$ to $133.5$ over one year. Leon's pension rose by $3%$ over the same year. Is Leon's pension keeping up with prices? You must show your working.
[3 marks]

Extension

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A gym membership has an index number of $108$ in 2024, using 2022 as the base year. The chain base index number for 2024 is $104$ . Work out the chain base index number for 2023. Give your answer to 1 decimal place.
[3 marks]
Hint 1 · What to try
The chain base numbers for 2023 and 2024 multiply to give the change from 2022 to 2024.
Hint 2 · The first line
So (chain 2023) $\times 1.04 = 1.08$ .
Hint 3 · The full method
Chain 2023 $= 1.08 \div 1.04 \times 100$ .
Answer

E2
A town's death rates are $2$ , $10$ and $70$ per 1000 for people aged under 40, 40 to 69, and 70 and over. Its standardised death rate is $13.6$ per 1000. The standard population has $30%$ aged under 40. What percentage is aged 40 to 69?
[3 marks]
Hint 1 · What to try
Call the percentage aged 40 to 69 $x$ . Then $70 - x$ per cent are aged 70 and over.
Hint 2 · The first line
$2 \times 0.3 + 10 \times \frac{x}{100} + 70 \times \frac{70 - x}{100} = 13.6$ .
Hint 3 · The full method
This gives $49.6 - 0.6 x = 13.6$ .
Answer
E3
A control chart for boxes of cereal has warning limits of $81.2$ g and $88.8$ g (above 500 g). A sample mean is $89.9$ g. What should be done? You must show your working.
[3 marks]
Hint 1 · What to try
The target is halfway between the warning limits, and each is 2 standard deviations from it.
Hint 2 · The first line
The target is $85$ and the standard deviation is $1.9$ , so the action limits are $85 \pm 5.7$ .
Hint 3 · The full method
Compare $89.9$ with $88.8$ and with $90.7$ .
E4
The chain base index numbers for a price over four years have a geometric mean of $104$ . The first three are $102$ , $106$ and $103$ . Work out the fourth. Give your answer to 1 decimal place.
[3 marks]
Hint 1 · What to try
The product of the four, as decimals, is $1.0 4^{4}$ .
Hint 2 · The first line
$1.0 4^{4} = 1.16985856$ and $1.02 \times 1.06 \times 1.03 = 1.113636$ .
Hint 3 · The full method
Divide one by the other.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out simple, chain base and weighted index numbers, and an average rate with the geometric mean.
I can work out crude and standardised rates and the general fertility rate, and explain why crude rates mislead.
I can work out warning and action limits and decide what to do after a sample.

Answers and mark scheme

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