S4 June 2018 Q4
4. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5% significance level,
- the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail
and - the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.
A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded.
| Sample size (\(n\)) | Sample mean (\(\bar{x}\)) | Standard deviation (\(s\)) | |
|---|---|---|---|
| Tackfast (\(T\)) | 6 | 5.27 | 0.31 |
| Goglue (\(G\)) | 5 | 10.12 | 0.66 |
It can be assumed that the force at which each glue fails is normally distributed.
The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_G^{\,2} = \sigma_T^{\,2}\) against \(\mathrm{H}_1 : \sigma_G^{\,2} \gt \sigma_T^{\,2}\) | B1 |
| Test stat, \(F_{4,5} = \dfrac{0.66^2}{0.31^2} = 4.53 \quad \left(\dfrac{1}{\mathrm{F}_{4,5}} = \dfrac{0.31^2}{0.66^2} = 0.221\right)\) | M1A1 |
| Critical value, \(F_{4,5} = 5.19\ (0.1927)\) | B1 |
| Not in critical region, therefore no evidence to reject \(\mathrm{H}_0\) | |
| No evidence of difference in standard deviation (allow variance) | A1cso |
| (5) |
Notes
B1 both hypotheses. allow \(\mathrm{H}_0 : \sigma_T = \sigma_G\) against \(\mathrm{H}_1 : \sigma_G \gt \sigma_T\). Must use \(\sigma\) or \(\sigma^2\) and make clear which is \(\mathrm{H}_0\) and which is \(\mathrm{H}_1\). Do not allow in words
M1 allow 0.31 and 0.66 rather than 0.312 and 0.662 if they write the formula down
B1 correct CV for their \(F\) or a correct comparison if use \(p\)
Final A1: – All previous marks must be awarded. Variances are the same or var are not different
| Scheme | Marks |
|---|---|
| \(s_p^{\,2} = \dfrac{5 \times 0.31^2 + 4 \times 0.66^2}{5 + 4}\) | M1 |
| \(s_p^{\,2} = 0.24698\ldots\) or \(s_p = 0.49697\ldots\) awrt 0.247 or 0.497 | A1 |
| \(\mathrm{H}_0 : \mu_G = \mu_T + 4 \quad \mathrm{H}_1 : \mu_G \gt \mu_T + 4\) | B1 |
| critical value CR: \(t_9(0.05) \gt \pm 1.833\) | B1 |
| \(t = \pm\dfrac{10.12 - 5.27 - 4}{\sqrt{0.24698\left(\frac{1}{5} + \frac{1}{6}\right)}} = \pm 2.8245\ldots\) or \(p =\) awrt 0.0099549 awrt 2.82, 2.825 | M1 A1 |
| There is evidence to reject \(\mathrm{H}_0\) \(\mu_G\) is greater than \(\mu_T + 4\). The suppliers claim is supported. | A1 |
| (7) |
Notes
M1 Allow use of 0.31 and 0.66. May be seen in part (a)
B1 both hypotheses using \(\mu\). Do not allow \(\geqslant\) sign instead of >. May use different letters eg \(A\) and \(B\) but they must be defined.
B1 correct CV but must match \(t\)-value or a correct comparison if use \(p\)
M1 use of correct formula with their \(s_p\) – condone missing 4
M1 use of correct formula with their \(s_p\). (which must have been attempted)
A1 A correct statement or longhand of suppliers claim with the word force and mean and is more than 4 greater oe Do not allow contradicting statements.
| Scheme | Marks |
|---|---|
| \(\dfrac{\bar{X}_G - \bar{X}_T - 4}{\sqrt{0.24698\left(\frac{1}{5} + \frac{1}{6}\right)}} \gt 1.833\) | M1 |
| \(\bar{X}_G - \bar{X}_T \gt 1.833 \times \sqrt{0.24698\left(\tfrac{1}{5} + \tfrac{1}{6}\right)} + 4\) | |
| \(\bar{X}_G - \bar{X}_T \gt 4.55\) | A1 |
| (2) |
Notes
M1 correct LHS with 1.833 ( or their CV used in (b)) NB subst in 4.55 for is M0
| Scheme | Marks |
|---|---|
| No change to standard deviation | B1 |
| \(\bar{X}_G - \bar{X}_T = 4.35\) | M1 |
| Previously they would have changed to Goglue, now they will remain with Tackfast or they will no longer change, or they would have changed but now they will not oe | A1 |
| (3) | |
| (17 marks) |