S3 June 2014 Q7

EdexcelOld spec10 marksIncludes hypothesis testingNormal Distribution

7. A machine fills packets with \(X\) grams of powder where \(X\) is normally distributed with mean \(\mu\). Each packet is supposed to contain 1 kg of powder.

To comply with regulations, the weight of powder in a randomly selected packet should be such that \(\mathrm{P}(X \lt \mu - 30) = 0.0005\)

(a) Show that this requires the standard deviation to be 9.117 g to 3 decimal places. (3)

A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows

\[999.8 \quad 991.6 \quad 1000.3 \quad 1006.1 \quad 1008.2 \quad 997.0 \quad 993.2 \quad 1000.0 \quad 997.1 \quad 1002.1\]

(b) Assuming that the standard deviation of the population is 9.117 g, test, at the 1% significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg. State your hypotheses clearly. (7)

Scheme	Marks
\(z = \pm 3.2905\)	B1
\(\sigma = \frac{30}{3.2905}\)	M1
\(\sigma = 9.117\) **	A1cso
	(3)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H_0}: \mu = 1000 \quad \mathrm{H_1}: \mu \lt 1000\)	B1
mean weight = 999.54	B1
\(z = \dfrac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} = \dfrac{(999.54 - 1000)}{\frac{9.117}{\sqrt{10}}} = -0.160\) or \(\dfrac{c - 1000}{\sqrt{83.12/10}} = -2.3263 \therefore \text{CR } c \lt 993.29\)	M1A1
1% critical value = \(-2.3263\)	B1
\(-2.3263 \lt -0.160\) Accept \(\mathrm{H_0}\) / not in critical region	dM1
There is no evidence that that the machine is delivering packets of mean weight less than 1 kg	A1ft
	(7)
	(10 marks)

Notes

1st B1 both hypotheses correct.
Accept 1kg in hypotheses if consistent units used in working usually either kg or g.

2nd B1 999.54 (g) or 0.99954 (kg)

1st M1 for standardising using their mean allow \(\pm\), 1000 and \(\frac{9.117}{\sqrt{10}}\) o.e. in kg

1st A1 awrt -0.160 unless clearly using \(|z|\) (stated) then accept 0.160 or CR awrt 993
Condone -0.16 if fully correct expression seen.

3rd B1 \(\pm 2.3263\) sign consistent with test statistic or \(p = 0.4364 \gt 0.01\) NB \(p = 0.5636 \lt 0.99\)

2nd dM1 dependent upon 1st M for a correct statement linking their test statistic and their cv
Contradictory statements score M0 e.g. “significant, do not reject \(\mathrm{H_0}\)”

2nd A1 for correct conclusion in context. Must mention ‘machine’ and ‘packets’.