Question 18
The mass of a certain species of kangaroo is known to be normally distributed with a mean mass of $\mu$ kg and standard deviation of $\sigma$ kg.
When one of the kangaroos is randomly selected, the probability that its mass is greater than 83.2 kg is 0.145.
When a sample of 12 kangaroos is randomly selected, the probability that the sample mean mass is less than 74.1 kg is 0.079.
A 90% approximate confidence interval for $\mu$ is calculated using a random sample of $n$ of the kangaroos that has a sample mean mass of 79.1 kg and a sample standard deviation equal to $\sigma$.
Determine the possible range of values that $n$ could have been, given that the confidence interval did not contain $\mu$.
Two probabilities pin down μ, then the interval has to miss it
Watch the two shaded tails give you two equations. Then watch the 90% interval shrink as $n$ grows, until it no longer reaches $\mu$.
The interval is centred on 79.1, which is above $\mu$. So it can only miss $\mu$ by having its lower bound above 76.63.
Find the smallest sample that misses μ
The interval is $79.1\pm1.645\times\frac{6.21}{\sqrt{n}}$. Change $n$ and watch where the lower bound sits compared with $\mu\approx76.63$.
Work through the solution one mark at a time
Try each step yourself before you reveal it. There are three situations here: one kangaroo, a sample of 12, and a sample of $n$.
For one kangaroo, $X\sim N\left(\mu,\sigma^{2}\right)$. Use inverse normal on your calculator to find the z-score with 0.145 above it.
$$P(X>83.2)=0.145\quad\Rightarrow\quad\frac{83.2-\mu}{\sigma}=1.058$$
$$\mu=83.2-1.058\sigma\quad\ldots(1)$$
Marker: correctly uses the sample of 1 to determine an equation in terms of $\mu$ and $\sigma$
For the sample of 12, the sample mean $\bar{X}$ has the same mean but a standard deviation of $\frac{\sigma}{\sqrt{12}}$.
$$P\left(\bar{X}<74.1\right)=0.079\quad\Rightarrow\quad\frac{74.1-\mu}{\frac{\sigma}{\sqrt{12}}}=-1.412$$
$$\mu=74.1+1.412\frac{\sigma}{\sqrt{12}}\quad\ldots(2)$$
The z-score is negative because 74.1 is in the lower tail.
Marker: correctly uses the sample of 12 to determine an equation in terms of $\mu$ and $\sigma$
Solve (1) and (2) together on your calculator, for example by graphing both and finding where they meet.
$$\mu\approx76.63\text{ kg}\qquad\sigma\approx6.21\text{ kg}$$
You keep follow-through marks from here if you slipped earlier.
Marker: solves simultaneous equations to determine the values of $\mu$ and $\sigma$
The sample mean of 79.1 is above $\mu$, so the interval can only miss $\mu$ if its lower bound is above 76.63. Find the $n$ where the lower bound equals $\mu$. For 90%, $z\approx1.645$.
$$79.1-1.645\times\frac{6.21}{\sqrt{n}}=76.63\quad\Rightarrow\quad n\approx17.1$$
Use 1.645 here, not 1.64. With 1.64 you get $n\approx17.0$, which sits right on the boundary, so you cannot tell whether $n=17$ works. The QCAA solution writes 1.64, but its answer of 17.1 only comes from 1.645.
Marker: determines the solution of $n$
$n$ has to be a whole number. A bigger $n$ gives a narrower interval, which pulls the lower bound up and away from $\mu$. So you round up.
$$n\ge18,\quad n\in\mathbb{Z}$$
You can check it: $n=17$ gives a lower bound of about 76.62, which still contains $\mu$, but $n=18$ gives about 76.69, which does not. Writing $n=18,\ 19,\ 20,\ \ldots$ is also accepted.
Marker: evaluates the reasonableness of the solution to the equation to determine suitable integer values of $n$
The last mark is for how you set out your working. Use proper notation such as $P\left(\bar{X}<74.1\right)$, label your equations, and say why you round $n$ up.
The sample size was $n\ge18$.
Marker: shows logical organisation communicating key steps
Each phrase of the question gave you something
You are not given $\mu$ or $\sigma$. You have to find both from two different probability statements, one about a single kangaroo and one about a sample mean, before you can even start on the confidence interval.
The part I would check first is the direction of your final answer. The QCAA often ask for a range of $n$, and you need to decide whether bigger or smaller samples make the interval miss $\mu$.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2020 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.