Histograms: WJEC GCSE Numeracy Higher November 2021 Unit 1 Q7

Numeracy Higher November 2021 Unit 1 Q7

WJECHigherCurrent spec6 marksHistograms

7. PowrUp is a company that makes a certain type of battery.
To find out how long these batteries last, it tests a sample of new batteries.
The lifetime of each battery in the sample is recorded.
The results are shown in the histogram below.

Histogram: Lifetime (hours) from 300 to 1000 against Frequency density 0 to 4. Bars: 300 to 550 height 0.8; 550 to 650 height 2; 650 to 700 height 2.8; 700 to 750 height 4; 750 to 950 height 0.7

Use the histogram to answer the following questions:

(a) Calculate the total number of batteries that were in the sample. [2]

(b) PowrUp can advertise the lifetime of these batteries based on the results of these tests.
75% of the sample must exceed this lifetime.

Calculate an estimate of the lifetime of these batteries that PowrUp should use in its adverts.
You must show all your working. [4]

Mark scheme (a)

Scheme	Marks	Comments
250×0.8 + 100×2 + 50×2.8 + 50×4 + 200×0.7 (= 200 + 200 + 140 + 200 + 140)	M1	Allow M1 for the sum of 5 products with any 3 correct
= 880 (batteries)	A1	CAO

Mark scheme (b)

Scheme	Marks	Comments
		FT ‘their 880’ ÷ 4 and ‘their 250×0.8’ for all marks provided their lower quartile is in the 2^nd group, i.e. ‘their 250×0.8’ < ‘their 880’ ÷ 4 < ‘their 250×0.8’+200
Search for lower quartile in the 550 – 650 group 2x = 880÷4 – 250×0.8 OR \(\dfrac{880÷4 - 250×0.8}{200}\) × 100 or 2x = 20 OR \(\dfrac{20}{200}\) × 100	S1 M1	May be seen on the diagram
(x =) 10	A1
(lower quartile = 550 + 10 =) 560 (hours)	A1	If no marks awarded, SC2 for an answer of 730 (hours) from using the upper quartile e.g. 750 – 80/200×50 OR 700 + 120/200×50 SC1 for searching in the 700 – 750 group provided sight of 880 ÷ 4 (×3)

7(b) Alternative method:

Scheme	Marks	Comments
		FT ‘their 880’ ÷ 4 and ‘their 250×0.8’ for all marks provided their lower quartile is in the 2^nd group, i.e. ‘their 250×0.8’ < ‘their 880’ ÷ 4 < ‘their 250×0.8’+200
Search for lower quartile in the 550 – 650 group 2x = (250×0.8+200) – 880÷4 OR \(\dfrac{(250×0.8+200) - 880÷4}{200}\) × 100 or 2x = 180 OR \(\dfrac{180}{200}\) × 100	S1 M1	May be seen on the diagram
(x =) 90	A1
(lower quartile = 650 – 90 =) 560 (hours)	A1	If no marks awarded, SC2 for an answer of 730 (hours) from using the upper quartile e.g. 750 – 80/200×50 OR 700 + 120/200×50 SC1 for searching in the 700 – 750 group provided sight of 880 ÷ 4 (×3)