6.1 Anatomy of a Hypothesis Test

A hypothesis test begins with a claim about the value of particular a population parameter.

This statement takes the form of a null hypothesis \[H_0 : \mu = x\]

This statement gets contrasted against an alternative hypothesis

\[H_1 : \mu \neq x\]

The null hypothesis \((H_0)\) represents a belief of a population parameter that you would like to disprove, while the alternative hypothesis \((H_1)\) is the opposite of the null and represents a claim you would like to show.

A hypothesis test uses the characteristics of the sample to determine if the statement about the population parameter in the null appears consistent (or inconsistent) with the characteristics of the sample. Recall that we are still under the assumption that the characteristics of the sample are similar to the true characteristics of the population. Therefore, if the sample characteristics are inconsistent with the statement in the null hypothesis, then you are likely to reject the null hypothesis. This means that the null hypothesis does not capture the true characteristics of the population (because the sample does). If the sample characteristics are similar to those stated in the null hypothesis, then you do not have evidence to reject the null and you conclude to not reject the null hypothesis.

In other words, if you reject the null, you have statistical evidence that \(H_1\) is correct (and the null hypothesis cannot be correct). If you do not reject the null, you have failed to prove the alternative hypothesis. Note that failure to prove the alternative does NOT mean that you have proven the null. In other words, there IS a difference between do not reject and accept!

This distinction between do not reject and accept cannot be emphasized enough. First, if anyone concludes that they accept the null in this class - you will get marked incorrect. If you conclude to accept the null outside of this class - then people will suspect that you don’t fully understand what you are talking about. Second, we can never say accept the null because it is simply too strong of a statement to make regarding a population parameter.

Suppose we believe that the population mean life span of our light bulbs is \(x\) hours. A hypothesis test will give us a specific way of testing this belief, and allows us to conclude whether or not this statement is consistent with our sample.

\[H_0:\mu=x \quad versus \quad H_1:\mu\neq x\]

We will discuss how to formally conduct a hypothesis tests in a bit. For now, lets compare these hypotheses with the confidence interval we calculated in the previous section.

\[887 \leq \mu \leq 928\]

Recall that our confidence interval states that with 95% confidence, the population average life span of the light bulbs is somewhere between 887 hours and 928 hours - meaning that any value within this range is equally likely. It also states that there is only a 5% chance that the population parameter lies outside of this range.

If we were to test that the population average lifespan was 1000 hours,

\[H_0:\mu=1000 \quad versus \quad H_1:\mu\neq 1000\]

then our confidence interval would give us evidence to reject the null because there would be less than a 5% chance for the null to be true. The sample characteristics and the statement in the null hypothesis are therefore inconsistent.

If we were to test that the population average lifespan was 900 hours, then we will reach a different conclusion.

\[H_0:\mu=900 \quad versus \quad H_1:\mu\neq 900\]

Since 900 is a value inside our confidence interval, then we would not have evidence to reject the null and we therefore conclude do not reject the null. The reason why we never say accept is that while 900 is within the confidence interval, there are also a continuum of other values in there. The true population mean might be 901, 900.0001, 910, etc. If you were to accept the null, then you are explicitly stating that the population parameter is exactly 900 - we do not have enough evidence for this.

Note that while we are seeing a clear connection between hypothesis tests and confidence intervals, hypothesis tests can get more sophisticated than this. It is therefore worthwhile to consider a formal solution methodology.