S4 June 2012 Q3
3. The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders’ merchant is 36 mm2. The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders’ merchant is 225 mm2. Test at the 10% significance level whether or not there is evidence that the lengths of paving slabs sold by these builders’ merchants differ in variability. State your hypotheses clearly.
(You may assume the lengths of paving slabs are normally distributed.) (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_A^2 = \sigma_B^2;\ \mathrm{H}_1 : \sigma_A^2 \neq \sigma_B^2\) | B1 |
| \(S_A^2 / S_B^2 = \dfrac{225}{36} = 6.25 \quad \left(\dfrac{36}{225} = 0.16\right)\) | M1A1 |
| CR: \(\mathrm{F}_{10,8} \gt 3.35 \quad \left(\dfrac{1}{F_{10,8}} = 0.299\right)\) | B1 |
| Since 6.25 is in the critical region we can assume that the lengths of paving slabs sold by the builders merchant differ in variability. | A1ft |
| (5) | |
| (5 marks) |
Notes
B1 both correct. Must use \(\sigma\). May use different notation to \(A\) and \(B\)
M1 \(\dfrac{225}{36}\) or \(\dfrac{36}{225}\) allow \(\dfrac{15}{6}\) or \(\dfrac{6}{15}\)
A1 either 6.25 or 0.16
B1 CR must match their method
A1 context must include “lengths of slabs”