S4 June 2013 (R) Q2

EdexcelOld spec7 marksConfidence Intervals

2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm{N}(\mu, \sigma^2)\).
A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma^2\) as shown below.

\[\hat{\mu} = 2.5 \qquad s^2 = 0.36\]

(a) Find a 95% confidence interval for \(\sigma^2\) (5)

(b) State with a reason whether or not the confidence interval supports the assertion that \(\sigma^2 = 0.495\) (2)

Mark scheme (a)

Scheme	Marks
\(\dfrac{29 \times 0.36}{45.722} \lt \sigma^2 \lt \dfrac{29 \times 0.36}{16.047}\)	M1B1,B1
\(0.228 \lt \sigma^2 \lt 0.651\)	M1 A1
	(5)

Notes

1st M1 use of \(\dfrac{29 \times s^2}{\chi^2}\) \(\left(\text{May use } \dfrac{s^2}{F_{29,\infty}} \text{ or } s^2 \times F_{29,\infty}\right)\) \(\left(\text{Based on } \dfrac{s^2}{\sigma^2} = F_{29,\infty}\right)\)

1st B1 45.722 \(\left(\text{using } \dfrac{s^2}{F_{29,\infty}} \text{ and } s^2 \times F_{29,\infty}\right)\)

2nd B1 16.047 (may use \(F_{29,\infty} = 1.4686\))

2nd M1 correct answer using their \(\chi^2\) value (correct using their \(F_{29,\infty}\))

A1 awrt 0.228 and awrt 0.651 (awrt 0.245 and awrt 0.529)

Scheme	Marks
Since 0.495 lies in the interval or \(0.228 \lt 0.495 \lt 0.651\)	B1ft
yes	B1ftd
	(2)
	(7 marks)