GCSE Statistics · Higher · Chapter 9: Time series · Lesson 2

Seasonal variation and prediction

About 50 minutesCalculator allowed41 marks

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Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Over 7 school days in a row, starting on a Monday, the visits to a school nurse were $71$ , $76$ , $62$ , $60$ , $70$ , $76$ and $80$ . Work out the three 5-point moving averages.
[2 marks]
Answer
A2
From chapter 4
A bar chart of the price of a ticket (in pence) has a vertical axis that starts at 300. The 2022 bar is 375 and the 2024 bar is 450, so the 2024 bar looks twice as tall. Work out the real percentage increase from 2022 to 2024.
[2 marks]
Answer
A3
From chapter 7
Pia scored 40 in Maths (mean 41, standard deviation 11) and 44 in English (mean 54, standard deviation 4). In which subject did Pia do better compared with the rest of the group? Give a reason.
[2 marks]
A4
Maths skill
A length, $L$ cm, is $80$ cm correct to the nearest 10. Write down the error interval for $L$ .
[2 marks]
Answer

Examples, then your turn

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

Example 1

A cafe's sales in Q3 were £58 thousand, and the trend value for that quarter was £49 thousand. In Q1 sales were £32 thousand and the trend value was £45 thousand. Work out the seasonal variation for each quarter.

Hint Seasonal variation = actual value

-

trend value. Keep the sign.

B1
A hotel had 245 bookings in Q2, when the trend value was 230. It had 196 bookings in Q4, when the trend value was 236. Work out the seasonal variation for each quarter.
[2 marks]
Answer

Example 2

The seasonal variations for Q4 in three years were $- 12$ , $- 15$ and $- 9$ . Work out the mean seasonal variation for Q4, and say what it means.

Hint Add the three and divide by 3. Keep the signs.

B2
The seasonal variations for Q1 in four years were $8$ , $11$ , $6$ and $7$ . Work out the mean seasonal variation for Q1, and say what it means.
[2 marks]

Example 3

A 4-point moving average of $150$ is plotted between Q2 and Q3 of year 1, and one of $166$ between Q2 and Q3 of year 3. Assume the trend is a straight line. Estimate the trend value for Q3 of year 4.

Hint The two averages are 8 quarters apart. How much does the trend change each quarter?

B3
A 4-point moving average of $312$ is plotted between Q2 and Q3 of year 1, and one of $288$ between Q2 and Q3 of year 2. Assume the trend is a straight line. Estimate the trend value for Q1 of year 3.
[3 marks]
Answer

Example 4

The trend value for Q4 of year 3 is $175$ , and the trend rises by $2$ each quarter. The mean seasonal variation for Q2 is $- 11$ . Predict the value for Q2 of year 4.

Hint Find the trend value for Q2 of year 4 first, then add the mean seasonal variation.

B4
The trend value for Q2 of year 4 is $96$ , and the trend rises by $1.5$ each quarter. The mean seasonal variation for Q4 is $7$ . Predict the value for Q4 of year 4.
[2 marks]
Answer

Example 5

A firm uses its trend line and mean seasonal variations to predict its sales for Q3 of next year. Give two reasons why the prediction might not be accurate.

Hint What does the method assume about the future?

B5
A gym predicts how many members it will have in Q1 two years from now, using its trend line and mean seasonal variations. Give two reasons why this prediction might not be reliable.
[2 marks]

Practice

C1
A town used $412$ thousand litres of water in Q3, when the trend value was $437$ thousand litres. Work out the seasonal variation for Q3 and say what it means.
[2 marks]
C2
The seasonal variations for Q2 in three years were $- 4$ , $- 7$ and $- 1$ . Work out the mean seasonal variation for Q2.
[2 marks]
Answer

C3
A 4-point moving average of $86$ is plotted between Q2 and Q3 of year 1, and one of $102$ between Q2 and Q3 of year 3. Assume the trend is a straight line. Estimate the trend value for Q4 of year 3.
[3 marks]
Answer
C4
The trend value for Q4 of year 2 is $120$ , and the trend rises by $3$ each quarter. The mean seasonal variation for Q2 is $18$ . Predict the value for Q2 of year 3.
[3 marks]
Answer

Exam-style questions

D1
For a shop's quarterly sales, the trend estimate for Q1 next year is £71 thousand and the mean seasonal variation for Q1 is $- 8$ (£ thousand). Last year's Q1 sales were £56 thousand. Omar says, "Sales in Q1 next year will be £56 thousand, the same as last year."
Is Omar correct? Give a better prediction.
[3 marks]

D2
The graph shows the gas used by a school (MWh) each quarter for three years. The crosses are the 4-point moving averages, and the dashed trend line has been extended to year 4. The trend value for Q3 of year 3 is $67$ . The seasonal variations for Q3 in years 1 and 2 were $10$ and $15$ .
(a)
In Q3 of year 3 the school used $81$ MWh. Work out the seasonal variation for Q3 of year 3.
[1 mark]
Answer
(b)
Work out the mean seasonal variation for Q3.
[2 marks]
Answer
(c)
Use the trend line and your answer to (b) to predict the gas used in Q3 of year 4.
[2 marks]
Answer
(d)
Give one assumption your prediction makes.
[1 mark]
Total for D2: 6 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
For a firm's quarterly sales, the mean seasonal variations for Q1, Q2 and Q3 are $- 9$ , $4$ and $12$ . The trend estimate for Q4 next year is $120$ . Predict the sales for Q4 next year.
[3 marks]
Hint 1 · What to try
The trend is the average of the four quarters, so the four seasonal variations balance out.
Hint 2 · The first line
The four mean seasonal variations add up to $0$ , so Q4's is $- (- 9 + 4 + 12) = - 7$ .
Hint 3 · The full method
Prediction $= 120 + (- 7)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out a seasonal variation from the trend and say what it means.
I can predict a value from a trend estimate and a mean seasonal variation.
I can say what a prediction assumes and why it may not be reliable.

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