Standard +0.3 This is a standard F-test for equality of variances with clearly stated sample sizes and variances. Students need to set up hypotheses, calculate F = 225/36 = 6.25, find critical values from tables for a two-tailed test at 10% level with (10,8) degrees of freedom, and compare. While it requires knowledge of the F-distribution and two-tailed testing, it's a routine application of a bookwork procedure with no conceptual complications, making it slightly easier than average.
The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is 36 mm\(^2\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is 225 mm\(^2\). 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]
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 marks total)
Guidance notes:
- B1 both correct. Must use \(\sigma\). May use different notation to A and B
- M1 \(\frac{225}{36}\) or \(\frac{36}{225}\) allow \(\frac{15}{6}\) or \(\frac{6}{15}\)
- A1 either 6.25 or 0.16
- B1 CR must match their method
- A1 context must include "lengths of slabs"
| $H_0: \sigma_A^2 = \sigma_B^2$; $H_1: \sigma_A^2 \neq \sigma_B^2$ | B1 |
| $S_A^2 / S_B^2 = \frac{225}{36} = 6.25$ $\left(\frac{36}{225} = 0.16\right)$ | M1A1 |
| CR: $F_{10,8} > 3.35$ $\left(\frac{1}{F_{10,8}}\right)$ | 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 marks total)** |
**Guidance notes:**
- B1 both correct. Must use $\sigma$. May use different notation to A and B
- M1 $\frac{225}{36}$ or $\frac{36}{225}$ allow $\frac{15}{6}$ or $\frac{6}{15}$
- A1 either 6.25 or 0.16
- B1 CR must match their method
- A1 context must include "lengths of slabs"
---
The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is 36 mm$^2$. The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is 225 mm$^2$. 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]
\hfill \mbox{\textit{Edexcel S4 2012 Q3 [5]}}