When we run a hypothesis test, we are testing whether or not a claim made is valid from a statistical standpoint. We break up our region into two regions, one is called the null hypothesis (H0), the other is called the alternative hypothesis (HA). Again, we will only look at hypothesis tests for the mean for this module.

There are some rules we have to follow when we make our hypotheses. First, the null hypothesis must always include some level of equality. This means our possible symbols are , , or =. The alternative will be the complement of the null hypothesis (opposite region) and must be a strict inequality. This means the symbols will be >, <, or .

Based on this, we can break our hypotheses up into three different sets:

Left Tailed Hypotheses: H0:0 vs HA:<0

Right Tailed Hypotheses: H0:0 vs HA:>0

Two Tailed Hypotheses: H0:=0 vs HA:0

We can make only two different conclusions from our hypothesis test: Reject H0, or fail to reject H0. This is an important matter in statistics and basically says this: We can confirm using statistics that the null hypothesis is incorrect, but we cannot confirm the null hypothesis as being the truth.

We determine our results from a hypothesis test by using the p-value. The p-value of our test is "the probability of seeing the result we saw, or more extreme, by random chance given that the null hypothesis is true." If our p-value is small, we would say seeing something like this by random chance is very unlikely. If our p-value is larger, then it is more plausible to see something like this by random chance.

The value of alpha, , is the probability we see a type I error, which means we reject the null hypothesis when the null hypothesis is in fact the truth. As the ones doing the research, we are able to select this value ourselves. We want to choose an alpha value that appropriately identifies the ramifications of making a mistake. If the results of our test have major outcomes associated with them, we may want to have a more strict (or smaller) value chosen for alpha. If the results of our test do not have major ramifications about them, we may want to choose a less strict (or larger) value for alpha. Common alpha values are = 0.1, 0.05, and 0.01.

Our conclusion will be based on how our computed p-value related to our value for alpha. If our p-value is less than alpha, we reject the null hypothesis. If our p-value is larger than alpha, we fail to reject the null hypothesis. In the infamous words of a brilliant man, "If p is small, reject them all."

Here is the process we follow when performing a hypothesis test:

Step 1: Write out your hypotheses based on the words given in the problem

Step 2: Determine your value of alpha

Step 3: Compute your Z or T statistic

Step 4: Compute your P-value

Step 5: Make your conclusion

Now, let's discuss how we compute our Z-value. If we know the population standard deviation, then our Z value will be Z=x/n.

If we do not know the population standard deviation, we may want to use the T distribution. The T-value is found as follows: T=xs/n..

Now, let's take a look at an example. Suppose the local car dealership claims their brand new mid-sized sedan gets at least 32 mpg on average with a standard deviation of 4 mpg. We want to test to see whether or not this claim is true. So, we collect a sample of 35 of these mid-sized sedans and see the average mpg was 30 mpg. Do we have enough evidence to dispute this claim?

Step 1: H0:32mpg vs HA:<32mpg

Step 2: = 0.05

Step 3: Z=x/n=30324/35=2.958

Step 4: P-value = 0.0015

Step 5: Our p-value is smaller than our alpha value, so we will reject the null hypothesis and have statistical evidence to dispute the claim

we are going to run a hypothesis test based on a claim made by an insurance provider. Read the following:

An insurance provider states that their customers save at least, on average, 300 dollars per year by switching to them, with a standard deviation of 150 dollars. Before we decide to switch to the new company and go through all of the hassle, we want to test the claim. So, we go out and sample 64 individuals who switched to the new insurance company and found them to have saved an average of 255 dollars per year. Do we have enough evidence at the = 0.05 level to state that the insurance provider is false in their claim?

1. What are the hypotheses based on the words given in the problem?
2. Should we use a Z or T distribution in this case?
3. What is our Z or T statistic?
4. What is the P-value?
5. Based on your p-value and alpha, what conclusion will we make?
6. Based on your results, would you switch to this company? Explain why or why not (Note: this can go beyond the use of statistics, but statistical analysis can help our decisions)