Could Tobacco be Good for you? Two Sided Rejection Regions in Hypothesis Testing
Audio Brief
Show transcript
This episode covers the concept of a two-sided rejection test within hypothesis testing, using an NBA example to illustrate its application.
There are three key takeaways from this discussion. First, a two-sided test is used when the direction of a potential change is unknown. Second, these tests split the significance level across both tails of the distribution, making rejection criteria stricter. Third, hypothesis testing offers a structured way to determine if observed differences are statistically significant, rather than random chance.
A two-sided rejection test is employed when the potential impact of an intervention, such as new rules in the NBA, could result in either an increase or a decrease in a metric. This approach ensures both possibilities are considered.
This test requires splitting the chosen significance level, for instance, 0.05, equally into two rejection regions in the tails of the distribution. This makes it more challenging to reject the null hypothesis compared to a one-sided test.
Regardless of the test type, the core process involves formulating null and alternative hypotheses, calculating a test statistic from sample data, and comparing it to critical values. The NBA example showed insufficient evidence to conclude new rules affected scoring, highlighting that small observed differences may not be statistically significant.
This framework provides a robust method for drawing data-driven conclusions about population changes.
Episode Overview
- The episode explains the concept of a two-sided rejection test within the framework of hypothesis testing.
- It uses a practical example from the NBA, analyzing whether new rules have changed the average points scored per game.
- The instructor demonstrates how to formulate the null and alternative hypotheses for a scenario where the outcome could be either an increase or a decrease.
- He calculates the Z-score (test statistic) based on sample data and compares it to the critical values of a two-sided rejection region.
- The video concludes by interpreting the results, showing that in this specific case, there is not enough statistical evidence to claim that the new rules had any effect on scoring.
Key Concepts
- Hypothesis Testing: A statistical method used to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population.
- Null Hypothesis (H₀): The default assumption that there is no effect or no change. In the example, it's that the average points per game remain the same as the previous season.
- Alternative Hypothesis (Hₐ): The claim that there is an effect or a change. For a two-sided test, this means the value is simply not equal to the original value (it could be higher or lower).
- Two-Sided Rejection Test: A test used when the effect of an intervention is unknown and could go in either direction (e.g., a rule change could either increase or decrease scoring). The rejection region is split between the two tails of the normal distribution.
- Test Statistic (Z-score): A standardized value calculated from the sample data to determine how many standard errors the sample mean is away from the population mean proposed in the null hypothesis.
- Rejection Region: The area in the tails of the distribution that is unlikely to occur if the null hypothesis were true. If the test statistic falls in this region, the null hypothesis is rejected.
- Failing to Reject the Null Hypothesis: The conclusion reached when the test statistic does not fall into the rejection region, meaning there is insufficient evidence to support the alternative hypothesis.
Quotes
- At 00:13 - "I have a really nice example of a two-sided rejection region test that I learned, you know, 20 years ago when I was taking Dr. John Quintanilla's class." - The instructor introduces the origin of the example problem, adding a personal touch.
- At 01:24 - "They might increase or decrease the average, the mean scoring. So in this case, we need a two-sided rejection region because there's a chance that we have increased or decreased the mean significantly." - The instructor explains the core reason for choosing a two-sided test: the direction of the potential change is uncertain.
- At 07:27 - "I fail to reject the null hypothesis... Essentially what that means is that we really can't say that this modification increased or decreased the scoring. We think it had no effect at all. These rules didn't change scoring." - The speaker provides the final conclusion of the hypothesis test and translates the statistical outcome into a clear, real-world interpretation.
Takeaways
- A two-sided rejection test is appropriate when you want to determine if a change has occurred, but you are unsure if the change will be an increase or a decrease.
- For a two-sided test, the significance level (e.g., 0.05) is split between the two tails of the distribution (e.g., 0.025 in each tail), making the criteria for rejection more stringent than a one-sided test.
- The fundamental steps of hypothesis testing remain the same: state the null and alternative hypotheses, calculate the test statistic from your sample, and compare it to the critical value(s) to make a decision.
- Just because a sample mean is slightly different from the population mean does not mean a statistically significant change has occurred; hypothesis testing provides a rigorous way to determine if the difference is meaningful or likely due to random chance.
