S4 June 2009 Q5

EdexcelOld spec14 marksConfidence Intervals

5. A machine fills jars with jam. The weight of jam in each jar is normally distributed. To check the machine is working properly the contents of a random sample of 15 jars are weighed in grams. Unbiased estimates of the mean and variance are obtained as

\[\hat{\mu} = 560 \quad s^2 = 25.2\]

Calculate a 95% confidence interval for,

(a) the mean weight of jam, (4)

(b) the variance of the weight of jam. (5)

A weight of more than 565 g is regarded as too high and suggests the machine is not working properly.

(c) Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of jars that weigh too much. (5)

Scheme	Marks
95% confidence interval for \(\mu\) is 2.145	B1
\(560 \pm t_{14}(2.5\%)\sqrt{\dfrac{25.2}{15}} = 560 \pm 2.145\sqrt{\dfrac{25.2}{15}} = (557.2,\ 562.8)\)	M1 A1 A1
	(4)

Mark scheme (b)

Scheme	Marks
95% confidence interval for \(\sigma^2\) is \(5.629 \lt \dfrac{14 \times 25.2}{\sigma^2} \lt 26.119\)	B1, M1, B1
\(\sigma^2 \lt 62.675\ \ \sigma^2 \gt 13.507\) \(13.507 \lt \sigma^2 \lt 62.675\) awrt 13.5, 62.7	A1, A1
	(5)

Mark scheme (c)

Scheme	Marks
Require \(\mathrm{P}(X \gt 565) = \mathrm{P}\left(Z \gt \dfrac{565 - \mu}{\sigma}\right)\) to be as large as possible OR \(\dfrac{565 - \mu}{\sigma}\) to be as small as possible; both imply highest \(\sigma\) and \(\mu\).	M1
\(\dfrac{565 - 562.8}{\sqrt{62.675}} = 0.28\)	M1A1
\(\mathrm{P}(Z \gt 0.28) = 1 - 0.6103 = 0.3897\) awrt 0.39 – 0.40	M1 A1
	(5)
	(14 marks)

Notes