A2 June 2019 Q3

EdexcelCurrent spec8 marksIncludes hypothesis testingt-Tests & Further Tests

3. Yin grows two varieties of potato, plant \(A\) and plant \(B\). A random sample of each variety of potato is taken and the yield, \(x\) kg, produced by each plant is measured. The following statistics are obtained from the data.

	Number of plants	\(\sum x\)	\(\sum x^2\)
\(A\)	25	194.7	1637.37
\(B\)	26	227.5	2031.19

(a) Stating your hypotheses clearly, test, at the 10% significance level, whether or not the variances of the yields of the two varieties of potato are the same. (7)

(b) State an assumption you have made in order to carry out the test in part (a). (1)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0 : \sigma_A^2 = \sigma_B^2,\ \mathrm{H}_1 : \sigma_A^2 \neq \sigma_B^2\)	B1	2.5
\(s_A^2 = \dfrac{1}{24}\left(1637.37 - 25 \times \left(\dfrac{194.7}{25}\right)^2\right) = 5.0436\)	M1 A1	2.1 1.1b
\(s_B^2 = \dfrac{1}{25}\left(2031.19 - 26 \times \left(\dfrac{227.5}{26}\right)^2\right) = 1.6226\)	A1	1.1b
\(\dfrac{s_A^2}{s_B^2} = 3.108\ldots\)	M1	3.4
critical values upper tail \(\mathrm{F}_{24,25} = 1.96\)	B1	1.1b
There is evidence that the two variances are different.	A1ft	2.2b
	(7)

Notes

B1: both hypotheses correct using \(\sigma\) or \(\sigma^2\)

M1: Using a correct method for either \(s_A^2\) or \(s_B^2\). May be implied by a correct value

A1: awrt 5.04

A1: awrt 1.62

M1: Using the F-distribution as the model eg \(\dfrac{s_A^2}{s_B^2}\) \(\left(\text{allow } \dfrac{s_B^2}{s_A^2}[= 0.321\ldots]\right)\)

B1: awrt 1.96 or 0.506 must match their method

A1ft: Drawing a correct inference following through their CV and value for \(\dfrac{s_B^2}{s_A^2}\) or \(\dfrac{s_A^2}{s_B^2}\)
Allow \(\sigma_B^2 \neq \sigma_A^2\) Allow standard deviation instead of Var. Do not allow \(\sigma_B^2 = \sigma_A^2\)

Scheme	Marks	AO
The yields are normally distributed.	B1	1.2
	(1)
	(8 marks)