In this video we discuss one tailed, two tailed hypothesis tests and we also cover p values for hypothesis testing. We discuss when to use one and two tailed tests, and the decision rule for p values.
Transcript/notes (partial)
In hypothesis testing, after stating the null and alternative hypotheses and choosing a level of significance, next you would obtain a random sample from the population and calculate a sample statistic, which is called the test statistic, such as the mean, x bar, the variance, s squared, or the proportion, p hat, in relation to the parameter in the null hypothesis, such as mew, sigma squared, or p.
This sample statistic, or test statistic is then converted to a standardized test statistic, such as z, t, or chi square. And the standardized test statistic is then used to make the decision about the null hypothesis.
In determining if we should reject the null hypothesis, one of the methods is to look at the probability of getting a standardized test statistic that is less than the level of significance, and this is where a P-value comes in.
A p-value or probability value of a hypothesis test is the probability of getting a test statistic, such as x bar, s squared or p hat, from a sample, that’s value is as extreme or more extreme than the one determined from the sample data you are using. Lets look at a visual of this.
So, lets say this box represents a population. And you pull a random sample from this population, and from your sample, you calculate the mean, or the variance or a proportion. In this box, or population, there are an infinitesimal number of random samples that can be pulled. The p value is the probability of getting a sample from this population, that has an extreme, or more extreme value that the one you got from your sample, and this value can be the mean, the variance or a proportion.
The p value depends on the test, and there are 3 hypothesis tests, a left tailed test, a right tailed test, and a 2 tailed test.
For a left tailed test, h sub a, the alternative hypothesis contains the less than inequality, so we could have for h sub 0, mew greater than or = to k, where k represents a claim value, and h sub a would be mew less than k. On a graph this looks like this, where the red line represents the standardized test statistic, and the area to the left of the line is the p value.
A right tailed test is basically the opposite of a left tailed test, where the alternative hypothesis contains the greater than inequality. For instance, for h sub 0 mew is less than or equal to k, and h sub a is mew greater than k. The graph looks like this, again the red line represents the standardized test statistic, and the area to the right of the line is the p value.
And for a two tailed test, the alternative hypothesis contains the does not = sign. For instance, for h sub 0 mew is equal to k, and h sub a is mew does not = k. The graph looks like this, the red lines represent the standardized test statistics, and the area outside of the lines is the p value. In this case, each of these areas outside of the lines would be = to one half of the p value.
It is important to remember, if the alternative hypothesis has less than inequality, it’s a left tailed test, if it has a greater than inequality, it’s a right tailed test, and if it has a does not equal sign, it’s a 2 tailed test. And you can also look at the inequalities as arrowheads, pointing to the direction of the test needed.
For each of these, we are going to write the claim out mathematically, then write out the null and alternative hypotheses, then determine the appropriate test to use.
Example 1. A recent report stated that less than 29% of people exercise twice a week. So, the claim is that the proportion, p, is less than 0.29. We know that the alternative hypothesis contains the inequality, so, h sub a is p less than 0.29, and the complement of that is p is greater than or equal to 0.29, and that is our null hypothesis, with the equality in the statement.
Since we have a less than inequality in the alternative hypothesis, this will require a left tailed test.
Lets say that research firm released data that stated that people check their phones more than 14 times an hour. So, the claim is that mew, the population mean is greater than 14. We know that the alternative hypothesis contains the inequality, so, h sub a is mew greater than 14, and the complement of that is mew is less than or equal to 0.29, and that is our null hypothesis, with the equality in the statement.
Since we have a greater than inequality in the alternative hypothesis, this will require a right tailed test.
Timestamps
0:00 What Is A Sample Test Statistic?
0:40 What Is A P Value?
1:36 What Is A Left, Right Or 2 Tailed Hypothesis Test?
3:16 Example Problem Of Which Test To Perform
4:34 What Is The Decision Rule For A P Value?