AQA S3 2013 June — Question 4 8 marks

Exam BoardAQA
ModuleS3 (Statistics 3)
Year2013
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicZ-tests (known variance)
TypeTest using proportion
DifficultyStandard +0.8 This is a two-sample proportion hypothesis test requiring students to formulate a non-standard null hypothesis (difference = 10%, not 0%), calculate a pooled proportion under this specific null, compute the test statistic, and interpret at a given significance level. The conceptual setup is more demanding than routine one-sample tests, requiring careful handling of the 'exceeds by more than 10%' condition, but the calculations follow standard procedures once correctly formulated.
Spec5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean
4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the \(5 \%\) level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.
Question 4:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Let \(p_M\) = proportion at medical centre, \(p_D\) = proportion at dental practice
\(H_0: p_M - p_D = 0.10\)B1 Correct \(H_0\)
\(H_1: p_M - p_D > 0.10\)B1 Correct \(H_1\) (one-tailed)
\(\hat{p}_M = 0.38\), \(\hat{p}_D = 0.21\)
Under \(H_0\): \(SE = \sqrt{\frac{p_M(1-p_M)}{n_M} + \frac{p_D(1-p_D)}{n_D}}\)M1 Method for standard error
\(= \sqrt{\frac{0.38 \times 0.62}{250} + \frac{0.21 \times 0.79}{100}}\)A1 Correct substitution
\(= \sqrt{0.000942 + 0.001659} = \sqrt{0.002601} = 0.05100\)A1 Correct SE
\(z = \frac{(0.38 - 0.21) - 0.10}{0.05100} = \frac{0.07}{0.05100} = 1.373\)M1 A1 Correct test statistic
Critical value \(z = 1.6449\)B1 Correct critical value
\(1.373 < 1.6449\), do not reject \(H_0\)A1 Correct conclusion
Insufficient evidence that the percentage at the medical centre exceeds that at the dental practice by more than 10%A1 Conclusion in context 
# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $p_M$ = proportion at medical centre, $p_D$ = proportion at dental practice | | |
| $H_0: p_M - p_D = 0.10$ | B1 | Correct $H_0$ |
| $H_1: p_M - p_D > 0.10$ | B1 | Correct $H_1$ (one-tailed) |
| $\hat{p}_M = 0.38$, $\hat{p}_D = 0.21$ | | |
| Under $H_0$: $SE = \sqrt{\frac{p_M(1-p_M)}{n_M} + \frac{p_D(1-p_D)}{n_D}}$ | M1 | Method for standard error |
| $= \sqrt{\frac{0.38 \times 0.62}{250} + \frac{0.21 \times 0.79}{100}}$ | A1 | Correct substitution |
| $= \sqrt{0.000942 + 0.001659} = \sqrt{0.002601} = 0.05100$ | A1 | Correct SE |
| $z = \frac{(0.38 - 0.21) - 0.10}{0.05100} = \frac{0.07}{0.05100} = 1.373$ | M1 A1 | Correct test statistic |
| Critical value $z = 1.6449$ | B1 | Correct critical value |
| $1.373 < 1.6449$, do not reject $H_0$ | A1 | Correct conclusion |
| Insufficient evidence that the percentage at the medical centre exceeds that at the dental practice by more than 10% | A1 | Conclusion in context |
4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years.

An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years.

Assume that each of these two samples has been randomly selected.\\
Investigate, at the $5 \%$ level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.

\hfill \mbox{\textit{AQA S3 2013 Q4 [8]}}
This paper (7 questions)
View full paper
Q1 8 Q2 14 Q3 9 Q4 8 Q5 10 Q6 11 Q7 15