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 $X$ is $\dfrac{1}{\lambda}$.
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
Not quite. $0.05$ is the probability between $15$ and $17$ seconds: $e^{-1}-e^{-17/15}\approx0.046$. The $15$ is the mean, which sets $\lambda$. The interval runs from $8$ to $17$ seconds. In 2022, 1.7% of Queensland students chose (A).
Not quite. With $\lambda=\frac{1}{15}$ the area from $8$ to $17$ seconds is $e^{-8/15}-e^{-17/15}\approx0.26$, more than twice $0.12$. If you got $0.12$, check the value of $\lambda$ you used and the two limits you entered on your GDC. In 2022, 7.1% of Queensland students chose (B).
Not quite. $0.22$ is the probability between $8$ and $15$ seconds: $e^{-8/15}-e^{-1}\approx0.219$. The $15$ is the mean, which sets $\lambda$. The interval runs from $8$ to $17$ seconds. In 2022, 12.1% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2022, 78.2% of Queensland students chose (D).
This is the interval in the question. The shaded area is about $0.26$.
Starting at the mean instead of at $8$ seconds leaves only a thin strip, about $0.05$. That is option (A).
Stopping at the mean instead of at $17$ seconds gives about $0.22$. That is option (C).
The shaded area is the probability for the interval you chose. Use the chips to jump back to $8$ to $17$ seconds.
The mean is $\frac{1}{\lambda}$, and the mean is $15$ seconds.
$$\frac{1}{\lambda}=15\quad\Rightarrow\quad\lambda=\frac{1}{15}$$
The probability is the area under the density curve between $8$ and $17$ seconds.
$$P(8\le X\le17)$$
$$=\int_{8}^{17}\frac{1}{15}e^{-x/15}\,dx$$
Evaluate with your GDC, or integrate directly.
$$\Big[-e^{-x/15}\Big]_{8}^{17}=e^{-8/15}-e^{-17/15}$$
$$\approx0.5866-0.3220\approx0.2647$$
The mean is $15$ seconds, so $\lambda=\frac{1}{15}$. The probability is the area under the density from $8$ to $17$.
$$\int_{8}^{17}\frac{1}{15}e^{-x/15}\,dx=e^{-8/15}-e^{-17/15}$$
$$\approx0.2647\approx0.26$$
How Queensland students answered in 2022
From the QCAA Specialist Mathematics subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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