Making a hypothesis about a population parameter

Before we go any further, probably the best way to make this situation more concrete is to see some examples of the sorts of hypotheses we could see in a hypothesis test.

Hypotheses that can be tested

batteries and coin

  1. A consumer watchdog team is testing a claim made by a battery manufacturer. The battery manufacturer claims that their batteries last an average of 60 hours. The watchdog collects a sample in order to conduct a hypothesis test. In this example, the hypothesis that is being tested is:

    The average lifetime of a battery is 60 hours.

  2. Fred has a coin that he thinks is weighted, such that the proportion of times that it lands on heads is not 50%. (He does not know which way the coin is weighted, he just suspects that it is not fair.) He decides to flip the coin several hundred times and conduct a hypothesis test. In this example, the hypothesis that is being tested is:

    The coin is fair and lands on heads 50% of the time.

    Is this the hypothesis that you would expect Fred to test?

Note in the second example that the claim that is being tested is not the claim that Fred is interested in proving. He seems to already suspect that the coin is not fair. However, the claim that is being tested is that the coin is fair. This can be generalized to a rule for hypothesis testing:

In any hypothesis test, the claim that you are testing is always a claim that a population parameter will assume some specific value.

So, even though Fred would like to prove that the coin isn't fair, the claim that he is 'testing' is that the coin is fair because this is the claim that assigns a specific value to a population parameter. To be more precise, we can let X represent the categorical variable that represents which side of the coin lands after a flip. This variable has two values: heads and tails. If the coin is fair then π, the population proportion of times that X assumes a value of heads, is equal to 50%. And this is the claim that Fred is testing: he is testing the claim that π is equal to 50%.

The battery example above also demonstrates this rule. There, we can let Y be the numerical variable that represents the lifetime of a battery. The watchdog is testing the claim that μ, the population mean of Y, is equal to 60 hours.