How to identify and distinguish a sample and population mean?

Question

In a village the mean rent was 1830; a rental company comes and surveys with a sample size of 25 for a hypothetical testing to test if the μ is equals 1830 or not.
This new survey gets a mean value of 1700 with a standard deviation of 200.
For the company to deduce t-value(for hypothesis testing) which is sample mean and which will be population mean? I assumed 1830 to be the population mean but it turned out to be the sample mean and 1700 to be the population mean.

My question is, is there any general way/thumb rule to find and distinguish population. and sample mean ? Thanks in advance.

Answer 1

Obviously distinguishing the population mean from the sample mean is the second most important thing just next to calculating the $t $-value.

Basically, given question such as these, you need to first identify: what are we checking the observations against? Here, the company is checking their observations against 1830. Hence, this is the population mean.

They do their observations based on a sample of a particular size. Obviously, the word sample should ring bells that they are going to find the sample mean.

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Note that in $t $- test, we check whether the mean if the sampled population is $\mu $, which is our population mean. Here, $\mu $ is 1830. Why? Because, it is the population mean.

In contrast, the company takes observations and finds the mean as 1700. Thus is the sample mean, as they are working on a sample.

Answer 2

$$t=\frac{\bar{x}-\mu}{\frac{s_{n-1}}{\sqrt{n}}}$$ Since this is a situation of unknown population variance, you must find $s_{n-1}$ as this is an unbiased estimate of the population variance. $\bar{x}$ is the sample mean, $\mu$ is the hypothesised population mean. $n$ is the sample size. $s_{n-1}\ ^2=\frac{n}{n-1}\cdot 200^2.$

Edit: Perhaps you are getting confused because this is a one-tailed test on the left side of the T-Distribution. So $-(\bar{x}-\mu)=\mu-\bar{x}$, is this why you believe that 1830 is the sample mean? Because the results of the survey are by definition the sample mean from the survey.