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Date None Specimen Marks available 3 Reference code SPNone.3sp.hl.TZ0.1
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Determine 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]
a.

Determine a 99 % confidence interval for \(\mu \) .

[2]
b.

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]
c.

Markscheme

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]

a.

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

[2 marks]

b.

(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]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 7 - Option: Statistics and probability » 7.3 » Unbiased estimators and estimates.

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