Higher June 2023 Paper 2 Q14
14 A company has 123 employees.
Information about their hourly rates of pay is shown in the table.
| Hourly rate, £\(p\) | Number of employees |
|---|---|
| \(10 \leqslant p \lt 14\) | 66 |
| \(14 \leqslant p \lt 20\) | 32 |
| \(20 \leqslant p \lt 40\) | 15 |
| \(40 \leqslant p \lt 100\) | 10 |
| Total \(=\) 123 |
The owner of the company uses the data to make two statements.
| Statement A “Over 30% of employees have an hourly rate that is more than £17” |
| Statement B “The average hourly rate of pay is more than £20” |
(a) Show working that supports Statement A. [3 marks]
(b) Why might Statement A not be true? [1 mark]
(c) Work out an estimate of the mean to support Statement B. [3 marks]
(d) Why is the mean not the best average to represent the data? [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Works out best estimate of the percentage of employees with hourly rate more than £17 | ||
| \(32 \div 2\) or 16 | M1 | oe implied by 41 or 82 |
| \((15 + 10 + \text{their } 16) \div 123\) or \(41 \div 123\) or \(\dfrac{1}{3}\) or 0.33(...) or \((66 + \text{their } 16) \div 123\) or \(82 \div 123\) or \(\dfrac{2}{3}\) or 0.66(…) or 0.67 | M1dep | oe eg \((123 - 66 - \text{their } 16) \div 123\) or 13(.0…)(%) + [12, 12.2](%) + 8(.1…)(%) |
| 33(.3…)(%) | A1 | oe eg 0.33(3…) and 0.3 allow 33.2(%) from 13(%) + 12.2(%) + 8(%) SC3 37 (or 36.9) and explains that a minimum of 12 of 32 people earn more than £17 |
| Alternative method 2 Compares best estimate of the number of employees with hourly rate more than £17 with 30% of number of employees | ||
| \(32 \div 2\) or 16 | M1 | oe implied by 41 or 82 |
| \(0.3 \times 123\) or 36.9 or \(0.7 \times 123\) or 86.1 | M1 | oe accept 36 or 37 for 36.9 accept 86 or 87 for 86.1 |
| 41 and 36.9 or 82 and 86.1 | A1 | accept 36 or 37 for 36.9 accept 86 or 87 for 86.1 SC3 37 (or 36.9) and explains that a minimum of 12 of 32 people earn more than £17 |
| Alternative method 3 Shows that a value of \(x\) gives a percentage \(\gt 30\%\) | ||
| \((15 + 10 + x) \div 123\) where \(12 \leqslant x \leqslant 32\) | M2 | oe eg \((25 + x) \div 123\) must see 15 and 10 or 25 |
| \((15 + 10 + x) \div 123\) where \(12 \leqslant x \leqslant 32\) and evaluates \((15 + 10 + x) \div 123 \times 100\) correctly | A1 | evaluations rounded or truncated to nearest integer or better SC3 37 (or 36.9) and explains that a minimum of 12 of 32 people earn more than £17 |
| Alternative method 4 Shows a number of employees that gives a percentage \(\gt 30\%\) | ||
| \(0.3 \times 123\) or 36.9 | M1 | oe accept 36 or 37 for 36.9 |
| \(15 + 10 + x\) or \(25 + x\) where \(12 \leqslant x \leqslant 32\) | M1dep | must see 15 and 10 or 25 |
| 36.9 and evaluates \(15 + 10 + x\) correctly where \(12 \leqslant x \leqslant 32\) | A1 | accept 36 or 37 for 36.9 SC3 37 (or 36.9) and explains that a minimum of 12 of 32 people earn more than £17 |
Additional guidance
Up to M2 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts
16 may be seen by the table
Alt 1 67% needs further explanation to score A1
Ignore irrelevant working in an otherwise fully correct response
For the SC3, minimum of 12 may be implied by an explanation that
\(10 + 15 + x\) is at least 37 or \(25 + x\) is at least 37
Responses involving interpolation should be escalated
| Answer | Mark | Comments |
|---|---|---|
| Valid reason | B1 | eg all employees in the second interval may earn less than £17 |
Additional guidance
| Fewer than 12 employees could earn more than £17 per hour | B1 |
| Only 10 might get more than £17 in second class interval (10 could be replaced by any integer from 0 to 11 inclusive) | B1 |
| More than 12 in group 2 earn less than £17 | B0 |
| Everyone in second group may earn 14 or 15 or 16 | B1 |
| 21 people may earn between £14 and £17 (21 could be replaced by any integer from 22 to 32 inclusive) | B1 |
| More people may earn between £14 and £17 | B0 |
| People in the 14 to 20 group aren’t evenly distributed | B0 |
| Not everyone in 14 – 20 earns more than £17 | B0 |
| Not many in second group may get more than £17 | B0 |
| Some of second group may get more than £17 | B0 |
| 14 to 20 includes people who get less than £17 | B0 |
| 2nd group includes some getting less than 17 and some getting more than 17 | B0 |
| We don’t know what each person earns | B1 |
| We don’t know how many of 2nd group earn less than £17 per hour | B1 |
| Under £17 isn’t in the data | B1 |
| Grouped data or it is only an estimate or using midpoints or data is wrong | B0 |
| Ignore irrelevant working but do not ignore incorrect working |
| Answer | Mark | Comments |
|---|---|---|
| \(12 \times 66\) or 792 and \(17 \times 32\) or 544 and \(30 \times 15\) or 450 and \(70 \times 10\) or 700 | M1 | oe implied by 2486 may be seen by the table allow one product or \(fx\) value to be incorrect |
| (their 792 + their 544 + their 450 + their 700) \(\div\) 123 or \(2486 \div 123\) | M1dep | oe eg \(\dfrac{792 + 544 + 450 + 700}{66 + 32 + 15 + 10}\) condone bracket error if working seen eg \(792 + 544 + 450 + 700 \div 123\) |
| 20.2(1…) | A1 | allow 20.20 if M2 seen and no errors |
Additional guidance
| Four values with three correct from 792, 544, 450, 700 can score up to M2 if they add and divide by 123 | |
| Correct products or values seen but a different method used eg \(123 \div 4\) | M0M0 |
| 20.2(1…) in working with answer given as the interval \(20 \leqslant p \lt 40\) | M2A0 |
| Ignore any references to statement B eg £20.21 which makes B wrong | M2A1 |
| Condone \(20.\dot{2}\), \(20.\dot{2}1\) etc for \(20.\dot{2}113\dot{8}\) | |
| Do not allow rounding of any of their 4 values in the second mark eg 792 544 450 700 \((800 + 544 + 450 + 700) \div 123\) | M1 M0 |
| Answer | Mark | Comments |
|---|---|---|
| Valid reason referring to the distribution | B1 | eg 98 employees earned below £20 |
Additional guidance
| Less than a half earned more than £20 | B1 |
| Over a half earned between £10 and £14 | B1 |
| Lots earned 10 to 14 | B0 |
| Only 25 people were over £20 | B1 |
| 25 people were over £20 | B0 |
| Not many earned more than the mean | B0 |
| Most earned less than £20 | B1 |
| Some earned less than the mean, some earned more | B0 |
| Mean is not a real amount of money | B0 |
| Median is between £10 and £14 | B1 |
| Median is better or mode is better | B0 |
| Modal class is \(10 \leqslant p \lt 14\) | B1 |
| The mode is between £10 and £14 (condone mode as modal class) | B1 |
| We don’t know what each person earns | B0 |
| Grouped data or it is only an estimate or using midpoints or data is wrong | B0 |
| The range is large | B0 |
| The data has extreme values or outliers or anomalous values | B1 |
| The data is (positively) skewed | B1 |
| The distribution is not symmetrical | B1 |
| The distribution is not evenly spread | B1 |
| Not representative | B0 |
| Lots of low values or high values can make the mean inaccurate | B0 |
| Ignore irrelevant working but do not ignore incorrect working |