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Date	None Specimen	Marks available	5	Reference code	SPNone.3sp.hl.TZ0.1
Level	HL only	Paper	Paper 3 Statistics and probability	Time zone	TZ0
Command term	Calculate and State	Question number	1	Adapted from	N/A

Question

A shopper buys 12 apples from a market stall and weighs them with the following results (in grams).

117, 124, 129, 118, 124, 116, 121, 126, 118, 121, 122, 129

You may assume that this is a random sample from a normal distribution with mean $\mu$ and variance ${\sigma ^2}$ .

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

[3]

Determine a 99 % confidence interval for $\mu$ .

[2]

The stallholder claims that the mean weight of apples is 125 grams but the shopper claims that the mean is less than this.

(i) State suitable hypotheses for testing these claims.

(ii) Calculate the p-value of the above sample.

(iii) Giving a reason, state which claim is supported by your p-value using a 5 % significance level.

[5]

unbiased estimate of $\mu = 122$ A1

unbiased estimate of ${\sigma ^2} = 4.4406{ \ldots ^2} = 19.7$ (M1)A1

Note: Award (M1)A0 for 4.44.

[3 marks]

the 99 % confidence interval for $\mu$ is [118, 126] A1A1

[2 marks]

(i) ${{\text{H}}_0}:\mu = 125;{\text{ }}{{\text{H}}_1}:\mu < 125$ A1

(ii) p-value = 0.0220 A2

(iii) the shopper’s claim is supported because $0.0220 < 0.05$ A1R1

[5 marks]

[N/A]

${H_0}$ and

${H_1}$ .