CAIE Further Paper 4 2021 June — Question 3 8 marks

Exam BoardCAIE
ModuleFurther Paper 4 (Further Paper 4)
Year2021
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicConfidence intervals
TypeUnbiased estimates then CI
DifficultyStandard +0.8 This is a two-sample confidence interval problem requiring calculation of sample means, variances using the computational formula, pooled variance, standard error of difference, and appropriate t-critical value. While the individual steps are standard, the multi-stage calculation with potential for arithmetic errors and the need to correctly apply the pooled two-sample procedure makes this moderately challenging for Further Maths students.
Spec5.05d Confidence intervals: using normal distribution
The heights, \(x\) m, of a random sample of 50 adult males from country A were recorded. The heights, \(y\) m, of a random sample of 40 adult males from country B were also recorded. The results are summarised as follows. $$\sum x = 89.0 \qquad \sum x^2 = 159.4 \qquad \sum y = 67.2 \qquad \sum y^2 = 113.1$$ Find a 95% confidence interval for the difference between the mean heights of adult males from country A and adult males from country B. [8]
Question 3:
AnswerMarks
389 67.2
x = =1.78 y = =1.68
AnswerMarks Guidance
50 40B1 Implied by 0.1 in the CI formula.
σ x 2 = 4 1 9    159.4− 8 5 9 0 2    = 0.02    = 5 1 0   M1 One variance correct unsimplified.
σ y 2 = 3 1 9    113.1− 67 4 . 0 22    = 0.0052308    = 3 1 2 7 50   A1 Both correctly calculated.
0.02 0.0052308  69 
σ2 = + =  
AnswerMarks Guidance
50 40 130000M1 Unsimplified expression must be seen, 40 and 50 in correct
places.
AnswerMarks Guidance
[σ2 =] 0.0005308A1 May be implied by correct final answer.
CI : 1.78 – 1.68 ± z × 0.0005308M1 Correct form for CI with a z-value.
with z = 1.96A1
0.10 ± 0.0452 or [0.0548, 0.145]A1 Allow either form.
8SC Assuming equal variances award B1M0A0, M1A1 (for use
of pooled variance formula, gives 0.0135), M1A1A0.
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
3 | 89 67.2
x = =1.78 y = =1.68
50 40 | B1 | Implied by 0.1 in the CI formula.
σ x 2 = 4 1 9    159.4− 8 5 9 0 2    = 0.02    = 5 1 0    | M1 | One variance correct unsimplified.
σ y 2 = 3 1 9    113.1− 67 4 . 0 22    = 0.0052308    = 3 1 2 7 50    | A1 | Both correctly calculated.
0.02 0.0052308  69 
σ2 = + =  
50 40 130000 | M1 | Unsimplified expression must be seen, 40 and 50 in correct
places.
[σ2 =] 0.0005308 | A1 | May be implied by correct final answer.
CI : 1.78 – 1.68 ± z × 0.0005308 | M1 | Correct form for CI with a z-value.
with z = 1.96 | A1
0.10 ± 0.0452 or [0.0548, 0.145] | A1 | Allow either form.
8 | SC Assuming equal variances award B1M0A0, M1A1 (for use
of pooled variance formula, gives 0.0135), M1A1A0.
Question | Answer | Marks | Guidance
The heights, $x$ m, of a random sample of 50 adult males from country A were recorded. The heights, $y$ m, of a random sample of 40 adult males from country B were also recorded. The results are summarised as follows.

$$\sum x = 89.0 \qquad \sum x^2 = 159.4 \qquad \sum y = 67.2 \qquad \sum y^2 = 113.1$$

Find a 95% confidence interval for the difference between the mean heights of adult males from country A and adult males from country B. [8]

\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q3 [8]}}
This paper (7 questions)
View full paper
Q1 6 Q2 7 Q3 8 Q4 5 Q4 3 Q5 10 Q6 14