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Date November 2018 Marks available 4 Reference code 18N.3.AHL.TZ0.Hsp_3
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number Hsp_3 Adapted from N/A

Question

Mr Sailor owns a fish farm and he claims that the weights of the fish in one of his lakes have a mean of 550 grams and standard deviation of 8 grams.

Assume that the weights of the fish are normally distributed and that Mr Sailor’s claim is true.

Kathy is suspicious of Mr Sailor’s claim about the mean and standard deviation of the weights of the fish. She collects a random sample of fish from this lake whose weights are shown in the following table.

Using these data, test at the 5% significance level the null hypothesis  H 0 : μ = 550  against the alternative hypothesis  H 1 : μ < 550 , where  μ  grams is the population mean weight.

Kathy decides to use the same fish sample to test at the 5% significance level whether or not there is a positive association between the weights and the lengths of the fish in the lake. The following table shows the lengths of the fish in the sample. The lengths of the fish can be assumed to be normally distributed.

Find the probability that a fish from this lake will have a weight of more than 560 grams.

[2]
a.i.

The maximum weight a hand net can hold is 6 kg. Find the probability that a catch of 11 fish can be carried in the hand net.

[4]
a.ii.

State the distribution of your test statistic, including the parameter.

[2]
b.i.

Find the p-value for the test.

[2]
b.ii.

State the conclusion of the test, justifying your answer.

[2]
b.iii.

State suitable hypotheses for the test.

[1]
c.i.

Find the product-moment correlation coefficient  r .

[2]
c.ii.

State the p-value and interpret it in this context.

[3]
c.iii.

Use an appropriate regression line to estimate the weight of a fish with length 360 mm.

[3]
d.

Markscheme

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

X N (550, 82)        (M1)

P ( X > 560 ) 0.10564 = 0.106        A1

 

[2 marks]

a.i.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

X i N (550, 82),  i = 1 ,…, 11

let  Y = i = 1 11 X i

E ( Y ) = 11 × 550 ( 6050 )        A1

Var ( Y ) = 11 × 8 2 ( 704 )        (M1)A1

P ( Y 6000 ) = 0.02975 = 0.0298        A1

 

[4 marks]

a.ii.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

t distribution with 7 degrees of freedom       A1A1

 

[2 marks]

b.i.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

p = 0.25779…= 0.258       A2

 

[2 marks]

b.ii.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

p > 0.05      R1

therefore we conclude that there is no evidence to reject  H 0        A1

Note: FT their p-value.

Note: Only award A1 if R1 awarded.

 

[2 marks]

b.iii.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

H 0 : ρ = 0 H 1 : ρ > 0        A1

Note: Do not accept  r in place of ρ .

 

[1 mark]

c.i.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

r = 0.782       A2

 

[2 marks]

c.ii.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

0.01095… = 0.0110      A1

since 0.0110 < 0.05      R1

there is positive association between weight and length      A1

Note: FT their p-value.

Note: Only award A1 if R1 awarded.

Note: Conclusion must be in context.

 

[3 marks]

c.iii.

Note: Accept all answers that round to the correct 2sf answer in (a), (b) and (c) but not in (d).

 

regression line of y (weight) on x (length) is       (M1)

y = 0.8267…  x + 255.96…       (A1)

x = 360  gives  y = 554       A1

Note: Award M1A0A0 for the wrong regression line, that is y = 0.7393… x – 51.62….
 

[3 marks]

d.

Examiners report

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

Syllabus sections

Topic 4—Statistics and probability » SL 4.4—Pearsons, scatter diagrams, eqn of y on x
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Topic 4—Statistics and probability

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