17N.3.AHL.TZ0.HSP_2

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

Anne is a farmer who grows and sells pumpkins. Interested in the weights of pumpkins produced, she records the weights of eight pumpkins and obtains the following results in kilograms.

${\text{7.7}}\quad {\text{7.5}}\quad {\text{8.4}}\quad {\text{8.8}}\quad {\text{7.3}}\quad {\text{9.0}}\quad {\text{7.8}}\quad {\text{7.6}}$

Assume that these weights form a random sample from a $N(\mu ,{\text{ }}{\sigma ^2})$ distribution.

Anne claims that the mean pumpkin weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis ${{\text{H}}_0}:\mu = 7.5$ .

Determine unbiased estimates for $\mu$ and ${\sigma ^2}$ .

[3]

Use a two-tailed test to determine the $p$ -value for the above results.

[3]

b.i.

Interpret your $p$ -value at the 5% level of significance, justifying your conclusion.

[2]

b.ii.

Markscheme / solution

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

UE of $\mu$ is $8.01{\text{ }}( = 8.0125)$ A1

UE of ${\sigma ^2}$ is 0.404 (M1)A1

Note: Accept answers that round correctly to 2 sf.

Note: Condone incorrect notation, ie, $\mu$ instead of UE of $\mu$ and ${\sigma ^2}$ instead of UE of ${\sigma ^2}$ .

Note: M0 for squaring $0.594 \ldots$ giving 0.354, M1A0 for failing to square $0.635 \ldots$

[3 marks]

attempting to use the $t$ -test (M1)

$p$ -value is 0.0566 A2

Note: Accept any answer that rounds correctly to 2 sf.

[3 marks]

b.i.

$0.0566 > 0.05$ R1

we accept the null hypothesis (mean pumpkin weight is 7.5 kg) A1

Note: Apply follow through on the candidate’s $p$ -value.

Note: Do not award A1 if R1 is not awarded.

[2 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.