Question 3 - A-Level Maths

CAIE S2 2022 March — Question 3 5 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2022
Session	March
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Confidence intervals
Type	Find CI width or confidence level
Difficulty	Standard +0.3 This is a straightforward application of the confidence interval formula for proportions. Students need to recognize that the sample proportion is at the midpoint (0.4), calculate the margin of error (0.045), then use the formula with n=500 to find the z-value and convert to confidence level. It's slightly easier than average because it's a direct reverse calculation with no conceptual complications.
Spec	5.05d Confidence intervals: using normal distribution

3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\text{est}(p) = 0.4\)	B1
\('0.4' + z\sqrt{\frac{'0.4' \times (1 - '0.4')}{500}} [= 0.445]\)	M1	OE. Use of their \(0.4\) in a correct expression
\(z \left[= 0.045 \div \sqrt{\frac{'0.4' \times (1-'0.4')}{500}}\right] = 2.054\)	A1	Condone \(2.053\) and \(2.05\)
\(0.98 - (1 - 0.98)\)	M1
\(96\%\) confidence	A1	CWO, must be integer
5

## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{est}(p) = 0.4$ | B1 | |
| $'0.4' + z\sqrt{\frac{'0.4' \times (1 - '0.4')}{500}} [= 0.445]$ | M1 | OE. Use of *their* $0.4$ in a correct expression |
| $z \left[= 0.045 \div \sqrt{\frac{'0.4' \times (1-'0.4')}{500}}\right] = 2.054$ | A1 | Condone $2.053$ and $2.05$ |
| $0.98 - (1 - 0.98)$ | M1 | |
| $96\%$ confidence | A1 | CWO, must be integer |
| | **5** | |

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3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, $p$, of all households in that town that owned two or more cars was found to be $0.355 < p < 0.445$.

Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.\\

\hfill \mbox{\textit{CAIE S2 2022 Q3 [5]}}

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