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Date May 2013 Marks available 3 Reference code 13M.3sp.hl.TZ0.1
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Find Question number 1 Adapted from N/A

Question

The random variable X is normally distributed with unknown mean μ and unknown variance σ2. A random sample of 20 observations on X gave the following results.

x=280, x2=3977.57

Find unbiased estimates of μ and σ2.

[3]
a.

Determine a 95 % confidence interval for μ.

[3]
b.

Given the hypotheses

H0:μ=15; H1:μ15,

find the p-value of the above results and state your conclusion at the 1 % significance level.

[4]
c.

Markscheme

ˉx=14     A1

s2n1=3977.57192802380     (M1)

=3.03     A1

[3 marks]

Note: Accept any notation for these estimates including μ and σ2.

Note: Award M0A0 for division by 20.

a.

the 95% confidence limits are

ˉx±ts2n1n     (M1)

Note: Award M0 for use of z.

 

ie, 14±2.0933.0320     (A1)

Note:FT their mean and variance from (a).

 

giving [13.2, 14.8]     A1

Note: Accept any answers which round to 13.2 and 14.8.

 

[3 marks]

b.

Use of t-statistic (=14153.0320)     (M1)

Note:FT their mean and variance from (a).

 

Note: Award M0 for use of z.

 

Note: Accept 15143.0320.

 

=2.569     (A1)

Note: Accept 2.569

 

p - value=0.009392×2=0.0188     A1

Note: Accept any answer that rounds to 0.019.

 

Note: Award (M1)(A1)A0 for any answer that rounds to 0.0094.

 

insufficient evidence to reject H0 (or equivalent, eg accept H0 or reject H1)     R1

Note:FT on their p-value.

 

[4 marks]

c.

Examiners report

In (a), most candidates estimated the mean correctly although many candidates failed to obtain a correct unbiased estimate for the variance. The most common error was to divide x2 by 20 instead of 19. For some candidates, this was not a costly error since we followed through their variance into (b) and (c).

a.

In (b) and (c), since the variance was estimated, the confidence interval and test should have been carried out using the t-distribution. It was extremely disappointing to note that many candidates found a Z-interval and used a Z-test and no marks were awarded for doing this. Candidates should be aware that having to estimate the variance is a signpost pointing towards the t-distribution.

b.

In (b) and (c), since the variance was estimated, the confidence interval and test should have been carried out using the t-distribution. It was extremely disappointing to note that many candidates found a Z-interval and used a Z-test and no marks were awarded for doing this. Candidates should be aware that having to estimate the variance is a signpost pointing towards the t-distribution.

c.

Syllabus sections

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

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