Statistical inference — Question 4

QCAA 2022, Paper 2 · 1 mark

Q4 · 2022 · Technology-activeSimple familiar1 mark

QUESTION 4

The time taken for students to answer questions in a class is assumed to be a random variable

X

with an exponential distribution that has the probability density function

f(x)=\begin{cases}\lambda e^{-\lambda x},&x\ge0\\0,&\text{otherwise}\end{cases}

. The mean of

X

\dfrac1\lambda

. The mean length of time taken for students to answer questions in this class is 15 seconds. The probability that the next question in this class is answered between 8 seconds and 17 seconds is

(A)

0.05

(B)

0.12

(C)

0.22

(D)

0.26

ABCD

Question linkOriginal paper

WORKED SOLUTION

Answer D

1 mark

ANSWER

Worked solution

The mean is

1/\lambda=15

, so

\lambda=1/15

. Therefore

P(8<X<17)=F(17)-F(8)=e^{-8/15}-e^{-17/15}\approx0.264

, which rounds to

0.26

Selects option D.

[1 mark]

Worked explanation by QCE Vault; correct option verified against the QCAA answer key.

View the full QCAA marking guide

Related questions

Q4 · 2025 QCAA · Paper 1 · 1 mark
$X$ is a random variable with mean $\mu$ and standard deviation $\sigma$ . From random samples of $X$ values, each of size $n$ , the sample mean is calculated. This sampling and calculation is repeated a large number of times. The mean of the distribution of the sample means would be approximately
Statistical inference
Q9 · 2025 QCAA · Paper 1 · 1 mark
An approximate confidence interval for a population mean is calculated based on a sample size of 16 and is found to have width, $w$ . A second confidence interval is calculated from another sample with the same sample standard deviation. Given that the same $z$ -value was used for both intervals and the width of the second interval is $2w$ , what is the size of…
Statistical inference
Q14 · 2025 QCAA · Paper 1 · 5 marks
A factory produces cans of juice. Each can has a labelled volume of 500 mL.
The factory manager conducted a random sample of 100 cans to assess whether the labelled volume was being met satisfactorily in production. The mean volume of the sample was 498.9 mL with a sample standard deviation of 4.0 mL.
Statistical inference
Q5 · 2025 QCAA · Paper 2 · 1 mark
The height (cm) of people in a certain population is normally distributed with a standard deviation of $7.42$ cm.
A researcher takes repeated random samples of $15$ people and calculates the mean height for each sample.
The expected standard deviation (cm) of the distribution of these sample mean heights would be approximately
Statistical inference