| Date | May 2017 | Marks available | 4 | Reference code | 17M.3sp.hl.TZ0.1 |
| Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
| Command term | State | Question number | 1 | Adapted from | N/A |
Question
A farmer sells bags of potatoes which he states have a mean weight of 7 kg . An inspector, however, claims that the mean weight is less than 7 kg . In order to test this claim, the inspector takes a random sample of 12 of these bags and determines the weight, \(x\) kg , of each bag. He finds that \[\sum {x = 83.64;{\text{ }}\sum {{x^2} = 583.05.} } \] You may assume that the weights of the bags of potatoes can be modelled by the normal distribution \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\).
State suitable hypotheses to test the inspector’s claim.
Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).
Carry out an appropriate test and state the \(p\)-value obtained.
Using a 10% significance level and justifying your answer, state your conclusion in context.
Markscheme
\({H_0}:\mu = 7,{\text{ }}{H_1}:\mu < 7\) A1
[1 mark]
\(\bar x = \frac{{83.64}}{{12}} = 6.97\) A1
\(s_{n - 1}^2 = \frac{{583.05}}{{11}} - \frac{{{\text{ }}{{83.64}^2}}}{{132}} = 0.0072\) (M1)A1
[3 marks]
\(t = \frac{{6.97 - 7}}{{\sqrt {\frac{{0.0072}}{{12}}} }} = - 1.22(474 \ldots )\) (M1)(A1)
\({\text{degrees of freedom}} = 11\) (A1)
\(p{\text{ - value}} = 0.123\) A1
Note: Accept any answer that rounds correctly to 0.12.
[4 marks]
because \(p > 0.1\) R1
the inspector’s claim is not supported (at the 10% level)
(or equivalent in context) A1
Note: Only award the A1 if the R1 has been awarded
[2 marks]
