PHANTOMS: A Method of Testing Hypotheses

P: Parameter of Interest If you haven’t noticed yet, the parameter of interest is kind of a big deal. First, and foremost, we always want to consider who or what the research is trying to generalize to.

H: Hypotheses These are broken up into the Ho (null hypothesis) and Ha (alternative hypothesis). The null hypothesis usually a neutral statement. Often times this will result in having a small equation representing the current common belief. The alternative hypothesis is a declarative statement which is counter to the null hypothesis.

A: Assumptions and Conditions These are the things that must be true in order to carry out a confidence interval or hypothesis test. Examples of this are the rules of thumb for proportions, checking for a normal distribution with sampling distributions, or that the sample size is large enough for t-tests. In every situation, we must either evidence or assume that the design from which the data was gathered contains an appropriate element of randomness to reduce bias.

N: Name the Test It seems silly…but they really like it when you name the test. It lets them know that you know that they know you know what you’re talking about. Some people think this is optional for our exams…some people are wrong. We have several tests that we will look at. Z-Tests, Two-Sample Z-Tests, T-Tests, Two-Sample T-Tests, 1-Proportion Z-Test, 2-Proportion Z-Test, Chi-Square Test, Chi-Square Goodness of Fit Test, and LinReg T-Test This seems like a lot…remember that all of them will follow this same exact format.

T: Test Statistic This is where all of the math happens. For our Z and T tests, this is where we calculate Z and T scores based on our hypotheses. For Chi-Square and LinReg T-Tests, this is where we said we use a calculator. The whole purpose of these calculations is to find a probability, the chances that an event or pair of events occurred.

O: Obtain the P-Value Once we’ve done our calculations, we must state what the P-Value is. Simple as that.

M: Make a Decision The P-Value is then compared to a set level based on the importance of the study. The more important the study, the smaller the probability (P-Value) needs to be. Commons alpha ( ) levels are 0.01, 0.05, 0.10. For example, if the study isn’t very important (studying candy preference), we would generaly use an alpha of 0.10. If the P-Value is less than the alpha, then we Reject the null hypothesis. If the P-value is not less than the alpha, then we Fail to Reject the null hypothesis. We always site the comparison between the P-Value and the alpha as why we choose to reject or fail to reject. Never…ever…ever…accept a null hypothesis. To accept is to claim an absolute truth, and unfortunately, those are few and far between.

S: Statement of the Conclusion in Context Finally, we wrap everything up by putting everything together. We note if we have rejected or failed to reject the null hypothesis. We restate the null hypothesis as a part of this statement to keep the conclusion in context. We site the comparison between the P-Value and the alpha to evidence why we have chosen to reject or to fail to reject.

Example Time! A manufacturer claims that a particular automobile model will get 50 miles per gallon on the highway. The researcher at a consumer-oriented magazine believes that this claim is high and plan on a test with a simple random sample of 30 cars. Assuming the standard deviation between individual cars is 2.3 miles per gallon, what should the researchers conclude if the sample mean is 49 miles per gallon?

PH Parameter of Interest: Hypotheses: The fuel efficiency, in miles per gallon, of a particular model of automobile. Hypotheses: Ho: The automobile does get 50 mpg Ha: The automobile gets less than 50 mph Notice that we did not use the value of 49. We consider what is happening with the parameter, the sample either helps to reject or fail to reject that parameter.

AN Assumptions: Name the Test: The problem states that it is a random collection of cars. We must assume that the distribution of the fuel efficiency of this particular model of car is approximately normal to continue. Name the Test: We will conduct a Z-Test

TO Test Statistic: N( ) Obtain a P-Value: The P-Value associated with is 0.0087

MS Make a Decision: Statement of Conclusion in Context: We will reject Ho based on the evidence that the P-Value is 0.0087. This is less than an acceptable alpha of 0.05, and even less than a more extreme alpha of 0.01. Statement of Conclusion in Context: We reject the Ho, that the fuel efficiency of this particular model of car is indeed 50 mpg, as evidenced by a P-Value of 0.0087 which is less than any reasonable alpha level for this situation.

It’s a Process Get comfortable with the process. Notice this is an extension of the process we should always do for Confidence Intervals. From now on, you should do PATS with confidence intervals. Parameter of Interest, Assumptions and Conditions, Test Statistic, and Statement of Conclusion in Context. This process is good for Every test, and Every confidence interval we do. There will be small changes along the way, but if you get use to PHANTOMS, you can more easily score an E on every question about hypothesis testing.