17M.3.AHL.TZ0.HSP_1

pestleMathematicsAIHLPaper 317M· ahl-4-18-t-and-z-test-type-i-and-ii-errorssource ↗

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.

[1]

Find unbiased estimates of $\mu$ and ${\sigma ^2}$ .

[3]

Carry out an appropriate test and state the $p$ -value obtained.

[4]

c.i.

Using a 10% significance level and justifying your answer, state your conclusion in context.

[2]

c.ii.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

${H_0}:\mu = 7,{\text{ }}{H_1}:\mu < 7$ 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]

c.i.

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]

c.ii.

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.