Edexcel S4 2016 June — Question 7 9 marks

Exam BoardEdexcel
ModuleS4 (Statistics 4)
Year2016
SessionJune
Marks9
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Mark schemeDownload PDF ↗
TopicF-test and chi-squared for variance
TypeRecover parameters from given CI
DifficultyChallenging +1.8 This question requires students to work backwards from a confidence interval for the mean to extract sample statistics, then construct a chi-squared confidence interval for variance—a multi-step process requiring understanding of both t-distribution and chi-squared distribution properties. The reverse-engineering aspect and the need to apply chi-squared tables correctly for variance intervals makes this substantially harder than routine S4 questions.
Spec5.05d Confidence intervals: using normal distribution
7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
(9)
Question 7:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\bar{x} - 2.262\frac{s}{\sqrt{10}} = 28.5\)B1 awrt 2.262
\(\bar{x} + 2.262\frac{s}{\sqrt{10}} = 48.7\)M1 \(\bar{x} - t\text{ value}\frac{s}{\sqrt{10}} = 28.5\)
Both equations correctA1 Both equations correct
\(2\bar{x} = 48.7 + 28.5\) or \(2.262\frac{s}{\sqrt{10}} = \frac{1}{2}(48.7 - 28.5)\)M1 Solving simultaneously leading to a value for \(\bar{x}\) or \(s\)
\(s = 14.1198\ldots\) (\(s^2 = 199.36\))A1 awrt 14.1 or awrt 199
\(\left\{\frac{9(14.1198^2)}{23.589},\ \frac{9(14.1198^2)}{1.735}\right\}\)M1 \(\frac{9(s^2)}{\chi^2 \text{ value}}\)
B123.589
B11.735
\(= (76.0659\ldots,\ 1034.19\ldots)\)A1 awrt 76.1 and awrt 1030
Total9 
## Question 7:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x} - 2.262\frac{s}{\sqrt{10}} = 28.5$ | B1 | awrt 2.262 |
| $\bar{x} + 2.262\frac{s}{\sqrt{10}} = 48.7$ | M1 | $\bar{x} - t\text{ value}\frac{s}{\sqrt{10}} = 28.5$ |
| Both equations correct | A1 | Both equations correct |
| $2\bar{x} = 48.7 + 28.5$ or $2.262\frac{s}{\sqrt{10}} = \frac{1}{2}(48.7 - 28.5)$ | M1 | Solving simultaneously leading to a value for $\bar{x}$ or $s$ |
| $s = 14.1198\ldots$ ($s^2 = 199.36$) | A1 | awrt 14.1 or awrt 199 |
| $\left\{\frac{9(14.1198^2)}{23.589},\ \frac{9(14.1198^2)}{1.735}\right\}$ | M1 | $\frac{9(s^2)}{\chi^2 \text{ value}}$ |
| | B1 | 23.589 |
| | B1 | 1.735 |
| $= (76.0659\ldots,\ 1034.19\ldots)$ | A1 | awrt 76.1 and awrt 1030 |
| **Total** | **9** | |
7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school.

Based on this sample, the $95 \%$ confidence interval for the mean time taken to travel to school for sixth form students from her school is\\[0pt]
[28.5, 48.7]

Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.\\
(9)\\

\hfill \mbox{\textit{Edexcel S4 2016 Q7 [9]}}
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Q1 9 Q2 13 Q3 6 Q4 9 Q5 14 Q6 15 Q7 9