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Date May 2019 Marks available 1 Reference code 19M.3.AHL.TZ0.Hsp_3
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Write down Question number Hsp_3 Adapted from N/A

Question

In a large population of hens, the weight of a hen is normally distributed with mean μ  kg and standard deviation σ  kg. A random sample of 100 hens is taken from the population.

The mean weight for the sample is denoted by X ¯ .

The sample values are summarized by  x = 199.8 and  x 2 = 407.8  where x  kg is the weight of a hen.

It is found that σ = 0.27 . It is decided to test, at the 1 % level of significance, the null hypothesis μ = 1.95 against the alternative hypothesis μ > 1.95.

State the distribution of X ¯ giving its mean and variance.

[1]
a.

Find an unbiased estimate for μ .

[1]
b.

Find an unbiased estimate for σ 2 .

[2]
c.

Find a 90 % confidence interval for  μ .

[3]
d.

Find the p -value for the test.

[2]
e.i.

Write down the conclusion reached.

[1]
e.ii.

Markscheme

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

X ¯ N ( μ , σ 2 100 )       A1

Note: Accept n in place of 100.

[1 mark]

a.

μ ^ = x n = 199.8 100 = 1.998       A1

Note: Accept 2.00, 2.0 and 2.

[1 mark]

b.

s n 1 2 = n n 1 ( x 2 n x ¯ 2 ) = 100 99 ( 407.8 100 1.998 2 )       (M1)

= 0.086864

unbiased estimate for  σ 2 is 0.0869      A1

Note: Accept any answer which rounds to 0.087.

[2 marks]

c.

90 % confidence interval is  1.998 ± 1.660 0.0869 100         (M1)

= (1.95, 2.05)      A1A1

Note: FT their σ from (c).

Note: Condone the use of the z -value 1.645 since n is large.

Note: Accept any values that round to 1.95 and 2.05.

[3 marks]

d.

p -value is 0.0377       A2

Note: Award A1 for the 2-tail value 0.0754.

Note: Award A2 for 0.0377 and A1 for any other value that rounds to 0.038.

Note: FT their estimated mean from (b), note that 2 gives p  = 0.032(0).

[2 marks]

e.i.

accept the null hypothesis     A1

Note: FT their p -value.

[1 mark]

e.ii.

Examiners report

[N/A]
a.
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b.
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c.
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d.
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e.i.
[N/A]
e.ii.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators
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