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In hypothesis testing, the null hypothesis, denoted as \(H_0\), serves as the default assumption or claim that we are testing against. It often states that there is no effect or no difference, and is assumed true until evidence suggests otherwise.
For example, in the exercise, \(H_0: \, \mu = 120\) means we start by assuming the population mean \(\mu\) is 120.
The null hypothesis is key because it sets the groundwork for how we interpret the results of our test.

The alternative hypothesis, denoted as \(H_1\), is what you want to prove. It suggests that there is an effect or a difference from the null hypothesis. In this exercise, \(H_1: \, \mu < 120\) indicates that the population mean \(\mu\) is less than 120.
The alternative hypothesis is essential for defining the outcome we test against the null.
There are three types of alternative hypotheses:

• Left-tailed: \(H_1\) suggests a value less than the null (\(\mu < 120\)).
• Right-tailed: \(H_1\) suggests a value greater than the null.
• Two-tailed: \(H_1\) suggests a value different from the null (either less or more).

A left-tailed test is used when the alternative hypothesis suggests that the parameter is less than the null hypothesis value. In the given exercise, since \(H_1: \mu < 120\), we perform a left-tailed test.
In such tests, the critical region, where we would reject the null hypothesis, is situated in the left tail of the probability distribution. This means we are focusing on values that are smaller than what the null hypothesis states.
The left-tailed test is particularly useful when there is an interest in identifying if a measure, like the population mean, falls below a certain threshold.

The population mean, denoted by \(\mu\), is a measure of central tendency indicating the average value of a variable across an entire population. In the context of the exercise, we are testing the hypothesis about this mean.
It's important to differentiate between the population mean and the sample mean (denoted as \(\bar{x}\)). The population mean refers to the average of a variable for the whole population, whereas the sample mean is the average calculated from a subset (sample) of that population.
In hypothesis testing, our goal is often to make inferences about the population mean based on the sample data. Thus, understanding the concept of the population mean is crucial for interpreting the results of your hypothesis test.