S3 June 2017 Q6
6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, \(x\) hours for the new batteries and \(y\) hours for the old batteries. The results are summarised in the table below.
| \(n\) | Sample mean | \(s^2\) | |
|---|---|---|---|
| New battery | 50 | \(\bar{x} = 83\) | 7 |
| Old battery | 40 | \(\bar{y} = 74\) | 6 |
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_{\text{new}} - \mu_{\text{old}} = 8\) | B1 |
| \(\mathrm{H}_1 : \mu_{\text{new}} - \mu_{\text{old}} \gt 8\) | B1 |
| \(z = \dfrac{\pm(83 - 74 - 8)}{\sqrt{\dfrac{7}{50} + \dfrac{6}{40}}}\) | M1 |
| \(z = \dfrac{\pm 1}{0.5385\ldots} = 1.86\) | dM1A1 |
| cv \(z = 1.6449\) | B1 |
| (1.86>1.6449) so reject \(\mathrm{H}_0\) | |
| Evidence to support engineer’s claim (that the new battery will last more than 8 hours longer than the old battery). | A1cso |
| (7) |
Notes
1st B1 Accept equivalent rearranged equation. Definitions of parameters must be clear e.g. use of 1 and 2 without definitions scores B0. Accept ‘\(x\)’ for ‘new’ and ‘\(y\)’ for ‘old’ as defined in the question.
2nd B1 Accept equivalent rearranged strict inequality. Definitions of parameters must be clear e.g. use of 1 and 2 without definitions scores B0. Accept ‘\(x\)’ for ‘new’ and ‘\(y\)’ for ‘old’ as defined in the question.
M1 for attempting standard error. Condone swapping 7 and 6. Accept 6.86 and 5.85 for 7 and 6.
dM1 for 1/ “their standard error”
A1 for awrt 1.86
NB -1.86 is A0.
If 8 missing from \(\mathrm{H}_1\) then accept \(z = \dfrac{9}{0.5385\ldots} =\) awrt 16.7 and must be consistent with their \(\mathrm{H}_1\).
B1 Accept \(\pm\) or probability of 0.9686
(Or 0.95<0.9686)
A1cso Correct comment in context. Must mention “engineers claim” or “battery”, “old”, “new” and “8”.
| Scheme | Marks |
|---|---|
| Sample sizes are large | B1 |
| CLT guarantees sample means ( \(\bar{X}\) and \(\bar{Y}\) ) are approximately normally distributed. | B1 |
| (2) | |
| (9 marks) |
Notes
2nd B1 Must mention means and normal. No assumptions are being made so B0 if key to answer.