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Date	November 2016	Marks available	7	Reference code	16N.3.AHL.TZ0.Hsp_2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Find	Question number	Hsp_2	Adapted from	N/A

Question

John rings a church bell 120 times. The time interval, ${T_i}$ , between two successive rings is a random variable with mean of 2 seconds and variance of $\frac{1}{9}{\text{ second}}{{\text{s}}^2}$ .

Each time interval, ${T_i}$ , is independent of the other time intervals. Let $X = \sum\limits_{i = 1}^{119} {{T_i}}$ be the total time between the first ring and the last ring.

The church vicar subsequently becomes suspicious that John has stopped coming to ring the bell and that he is letting his friend Ray do it. When Ray rings the bell the time interval, ${T_i}$ has a mean of 2 seconds and variance of $\frac{1}{{25}}{\text{ second}}{{\text{s}}^2}$ .

The church vicar makes the following hypotheses:

${H_0}$ : Ray is ringing the bell; ${H_1}$ : John is ringing the bell.

He records four values of $X$ . He decides on the following decision rule:

If $236 \leqslant X \leqslant 240$ for all four values of $X$ he accepts ${H_0}$ , otherwise he accepts ${H_1}$ .

Find

(i) ${\text{E}}(X)$ ;

(ii) ${\text{Var}}(X)$ .

[3]

Explain why a normal distribution can be used to give an approximate model for $X$ .

[2]

Use this model to find the values of $A$ and $B$ such that ${\text{P}}(A < X < B) = 0.9$ , where $A$ and $B$ are symmetrical about the mean of $X$ .

[7]

Calculate the probability that he makes a Type II error.

[5]

Markscheme

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

(i) ${\text{mean}} = 119 \times 2 = 238$ A1

(ii) ${\text{variance}} = 119 \times \frac{1}{9} = \frac{{119}}{9}{\text{ }}( = 13.2)$ (M1)A1

Note: If 120 is used instead of 119 award A0(M1)A0 for part (a) and apply follow through for parts (b)-(d). (b) is unaffected and in (c) the interval becomes $(234,{\text{ }}246)$ . In (d) the first 2 A1 marks are for $0.3633 \ldots$ and $0.0174 \ldots$ so the final answer will round to 0.017.

[3 marks]

justified by the Central Limit Theorem R1

since $n$ is large A1

Note: Accept $n > 30$ .

[2 marks]

$X \sim N\left( {238,{\text{ }}\frac{{119}}{9}} \right)$

$Z = \frac{{X - 238}}{{\frac{{\sqrt {119} }}{3}}} \sim N(0,{\text{ }}1)$ (M1)(A1)

${\text{P}}(Z < q) = 0.95 \Rightarrow q = 1.644 \ldots$ (A1)

so ${\text{P}}( - 1.644 \ldots < Z < 1.644 \ldots ) = 0.9$ (R1)

${\text{P}}( - 1.644 \ldots < \frac{{X - 238}}{{\frac{{\sqrt {119} }}{3}}} < 1.644 \ldots ) = 0.9$ (M1)

interval is $232 < X < 244{\text{ }}({\text{3sf}}){\text{ }}(A = 232,{\text{ }}B = 244)$ A1A1

Notes: Accept the use of inverse normal applied to the distribution of $X$ .

Alternative is to use the GDC to find a pretend $Z$ confidence interval for a mean and then convert by multiplying by 119.

Either $A$ or $B$ correct implies the five implied marks.

Accept any numbers that round to these 3sf numbers.

[7 marks]

under ${{\text{H}}_1},{\text{ }}X \sim N\left( {238,{\text{ }}\frac{{119}}{9}} \right)$ (M1)

${\text{P}}(236 \leqslant X \leqslant 240) = 0.41769 \ldots$ (A1)

probability that all 4 values of $X$ lie in this interval is

${(0.41769 \ldots )^4} = 0.030439 \ldots$ (M1)(A1)

so probability of a Type II error is 0.0304 (3sf) A1

Note: Accept any answer that rounds to 0.030.

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.15—Central limit theorem

19M.3.AHL.TZ0.Hsp_3e.ii:
Write down the conclusion reached.
22M.1.AHL.TZ1.9b:
A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than $950 g$ .

Find the probability of passing the inspection if the statistician’s suggestion is followed.
21M.1.AHL.TZ1.14a:
Find the mean weight of a bag of apples.
21N.1.AHL.TZ0.14a:
Find the probability that a sack is under its labelled weight.
21N.1.AHL.TZ0.14b:
Find the lower quartile of the weights of the sacks of potatoes.
16N.3.AHL.TZ0.Hsp_2b:
Explain why a normal distribution can be used to give an approximate model for $X$ .
19M.3.AHL.TZ0.Hsp_3d:
Find a 90 % confidence interval for $\mu$ .
16N.3.AHL.TZ0.Hsp_2d:
Calculate the probability that he makes a Type II error.
SPM.1.AHL.TZ0.10:
In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.

Two customers enter the shop together. They are served one at a time.

Find the probability that the total time taken to serve both customers will be less than 4 minutes.

Clearly state any assumptions you have made.
19M.3.AHL.TZ0.Hsp_3c:
Find an unbiased estimate for ${\sigma ^2}$ .
19M.3.AHL.TZ0.Hsp_3a:
State the distribution of $\bar X$ giving its mean and variance.
16N.3.AHL.TZ0.Hsp_2a:
Find

(i) ${\text{E}}(X)$ ;

(ii) ${\text{Var}}(X)$ .
19N.3.AHL.TZ0.Hsp_3a:
State the central limit theorem as applied to a random sample of size $n$ , taken from a distribution with mean $\mu$ and variance ${\sigma ^2}$ .
19M.3.AHL.TZ0.Hsp_3e.i:
Find the $p$ -value for the test.
19M.3.AHL.TZ0.Hsp_3b:
Find an unbiased estimate for $\mu$ .
21M.1.AHL.TZ1.14c:
Find the probability that a bag selected at random weighs more than $1 kg$ .
21M.1.AHL.TZ1.14b:
Find the standard deviation of the weights of these bags of apples.
21N.1.AHL.TZ0.14c:
The sacks of potatoes are transported in crates. There are $10$ sacks in each crate and the weights of the sacks of potatoes are independent of each other.

Find the probability that the total weight of the sacks of potatoes in a crate exceeds $500 kg$ .

Topic 4—Statistics and probability » AHL 4.18—T and Z test, type I and II errors

Topic 4—Statistics and probability

Question

Markscheme

Examiners report

Syllabus sections

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