Statistical inference — Question 247

Original QCE Vault practice · 10 marks

Q247 · Practice questionTechnology-activeComplex unfamiliar10 marks

QUESTION 247 (10 marks)

Independent waiting times W are exponential, in minutes, with

P(W>3)=e^{-3/2}

. A study of 100 waits reports mean 2.1 and sample standard deviation 1.8.

Determine the rate and population mean.

[2 marks]

Use the CLT to approximate

P(\overline W>2.1)

under the model. Exponential standard deviation is

1/\lambda

[2 marks]

For inference from the sample, treat the rate as unknown. Using the reported summaries and z=1.96, find an approximate 95 percent mean interval and explain why its standard error differs from (b).

[3 marks]

Under the assumed exponential model,

P(W>3)=e^{-3/\mu}

where

\mu

is mean waiting time. Transform the interval in (c) into an approximate 95 percent interval for this survival probability, giving four decimal places. Explain the endpoint order.

[3 marks]

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WORKED SOLUTION

Practice marking scheme

10 marks

ANSWER

(a)

\lambda=0.5

per minute; mean

2

min. (b)

0.3085

. (c)

(1.7472,2.4528)

min. (d)

(e^{-3/1.7472},e^{-3/2.4528})\approx(0.1796,0.2943)

Worked solution

(a)

e^{-3\lambda}=e^{-3/2}

gives lambda=1/2. The exponential mean is its reciprocal, 2.

(b) Model standard deviation is 2, giving standard error 2/10=0.2. Thus the normal approximation gives

1-\Phi((2.1-2)/0.2)=1-\Phi(0.5)=0.3085

2.1\pm1.96(0.18)=2.1\pm0.3528

. Part (b) uses model population standard deviation, while (c) estimates it from the sample. Normality of the sample mean is approximate because individual waits are non-normal.

(d) The function

g(\mu)=e^{-3/\mu}

increases for

\mu>0

, because

g^{\prime}(\mu)=3e^{-3/\mu}/\mu^2>0

. It therefore maps the lower mean endpoint to the lower survival endpoint and the upper to the upper. Apply g to (1.7472,2.4528), giving approximately (0.1796,0.2943). Coverage is inherited from the approximate mean interval under the exponential model, rather than becoming exact.

Equivalent mathematically justified methods accepted.

Exact answers unless a decimal accuracy is specified.

Part a: Use exponential survival to find the rate.

[1 mark]

Part a: Find the mean.

[1 mark]

Part b: Use model standard error 0.2.

[1 mark]

Part b: Standardise and evaluate the upper tail.

[1 mark]

Part c: Use estimated standard error 0.18.

[1 mark]

Part c: Give both endpoints.

[1 mark]

Part c: Distinguish model and estimated standard errors.

[1 mark]

Part d: Establish monotonicity or otherwise justify endpoint order.

[1 mark]

Part d: Transform both interval endpoints.

[1 mark]

Part d: Evaluate the probability interval to four decimal places.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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Statistical inference — Question 247

QUESTION 247 (10 marks)

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