Question 3
According to a recent census, the mean hours worked per week by all Australian workers is 35.6 hours.
A mean of 36.1 hours worked per week is calculated from a random selection of 500 Australian workers.
Based on this data, which of the following is correct?
Not quite. The symbols are the right ones, but they are swapped. The census covers all Australian workers, so 35.6 describes the population: $\mu=35.6$. The 500 workers are a sample, so their mean is $\bar x=36.1$.
Not quite. There are two slips here. 35.6 comes from the census of every worker, so it is the population mean $\mu$. The capital $\bar X$ is the random variable for the sample mean, which takes a different value for each sample. The value from this one sample is written with a lowercase $\bar x$, so $\bar x=36.1$.
That’s it. Step through the hints below to see why.
Not quite. $\bar x=36.1$ is right, but the census figure is a population parameter, written $\mu$. $\bar X$ is the random variable for the sample mean, and it changes from sample to sample. Its long-run average is $E(\bar X)=\mu$, which may be where the mix-up comes from, but $\bar X$ itself is never one fixed number like 35.6.
Hours worked per week. Each dot is the mean of one random sample of 500 workers. The question doesn’t give a standard deviation, so I simulated these samples from a population with the census mean, 35.6 hours, and a standard deviation I chose myself, $\sigma=14$ hours. The first dot is the sample in the question.
This is the sample in the question. Its mean comes from 500 workers, so it is a statistic, $\bar x=36.1$. The census counts every worker, so its mean is the parameter, $\mu=35.6$. Take another sample and watch which value moves.
The census line hasn’t moved: $\mu=35.6$ whichever sample you take. Each sample gives its own value of $\bar x$. The capital $\bar X$ names the random variable behind those values, the mean of a sample you haven’t taken yet. The dot plot is building up its distribution. As more dots arrive, they gather around $\mu$.
First decide which number describes the whole population and which comes from a sample. A census measures every member of the population. A random selection of 500 workers is a sample.
Values for a whole population are parameters, written with Greek letters. Values worked out from a sample are statistics, written with lowercase Latin letters.
$$\text{mean: }\ \mu\ \text{(population)},\quad\bar x\ \text{(sample)}$$
So the census gives $\mu=35.6$ and the sample gives $\bar x=36.1$.
That leaves the capital $\bar X$. A capital letter names a random variable. $\bar X$ is the mean of a sample you haven’t taken yet, so it has a whole distribution of possible values. Once a sample is taken, its value is written $\bar x$.
$$E(\bar X)=\mu$$
Take several samples in the interactive. Each one adds a value of $\bar x$, and the dot plot of those values is the distribution of $\bar X$.
The census covers every Australian worker, so 35.6 is the population mean, a parameter. The 500 workers are a random sample, so 36.1 is the sample mean, a statistic.
$$\bar x=36.1,\quad\mu=35.6$$
$\bar X$ is the random variable for the sample mean, so it isn’t used for a single known value.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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