S4 June 2011 Q3
3. Manuel is planning to buy a new machine to squeeze oranges in his cafe and he has two models, at the same price, on trial. The manufacturers of machine \(B\) claim that their machine produces more juice from an orange than machine \(A\). To test this claim Manuel takes a random sample of 8 oranges, cuts them in half and puts one half in machine \(A\) and the other half in machine \(B\). The amount of juice, in ml, produced by each machine is given in the table below.
| Orange | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Machine \(A\) | 60 | 58 | 55 | 53 | 52 | 51 | 54 | 56 |
| Machine \(B\) | 61 | 60 | 58 | 52 | 55 | 50 | 52 | 58 |
Stating your hypotheses clearly, test, at the 10% level of significance, whether or not the mean amount of juice produced by machine \(B\) is more than the mean amount produced by machine \(A\). (8)
| Scheme | Marks |
|---|---|
| \(d = B - A\): 1, 2, 3, −1, 3, −1, −2, 2 | M1 |
| \(\bar{d} = 0.875\) | M1 |
| \(s_d^{\,2} = \dfrac{33 - 8 \times 0.875^2}{7} = (3.8392\ldots)\) | M1 |
| \(\mathrm{H}_0 : \mu_d = 0 \qquad \mathrm{H}_1 : \mu_d \gt 0\) | B1 |
| \(t_7 = \dfrac{0.875}{\dfrac{s_p}{\sqrt{8}}} = 1.263\ldots\) awrt 1.26 | M1A1 |
| \(t_7(10\%)\) one tail critical value is 1.415 | B1 |
| Not significant. There is insufficient evidence to support the claim of manufacturer \(B\) or machine \(B\) does not produce more juice (than machine \(A\)) | A1 |
| (8 marks) |
Notes
1st M1 for attempting the \(d\)s
2nd M1 for attempting \(\bar{d}\)
3rd M1 for attempting \(s_d\) or \(s_d^{\,2}\)
4th M1 for attempting the correct test statistic
3rd A1 contextual statement only required.
Allow The juice provided by machine \(A\) is the same as by machine \(B\)
NB 2 sample test can score 3/8
M0 M0
M1 \(\dfrac{7 \times 9.27 + 7 \times 16.79}{14}\)
B1 for \(\mathrm{H}_0 : \mu_A = \mu_B \quad \mathrm{H}_1 : \mu_A \lt \mu_B\)
M0 A0
B1 1.345
A0