I had recently conducted an experiment: the moment the fuel light in my car goes on, I will find a closest gas station to fill up the tank. But the number of gallons needed to fill up are all over the place. This makes me wonder: 1) how can we verify it is actually one gallon when the pump says it pumps one gallon of gas into your car, 2) if we know some gas stations have shorted us, how to find out what the true volume of gas is.

These two problems are known as hypothesis testing (testing the claim, i.e., the hypothesis, that a supposed gallon is indeed one gallon, and estimation (estimating the true volume) in statistical inference. They are a lot more complex thatthey might appearbecause of the nature of continuous variables.

Since the volume is measured, i.e., a continuous variable, you know if you take a measure of the volume of a supposed gallon of gas pumped out the of machine, it will never be exactly 1.0000000... gallon.

So if it turns out to be 0.9823, can you say they are cheating?

You might say I will take 30 measures and look at their mean, but again if it turns out to be 0.9912? can you say they are cheating?

Outline your thoughts and share.The statistical inference on modern stat textbooks are the results of two statisticians's answer (Fisher's p-value and Neyman's critical values) to these questions, and they are not even in full agreement. So don't feel bad if you can't solve them in 1 hour. However, the more thoughts you put into this, the less challenging you will find the answers provided in the next two sections.

1. Continuing from the gas example, using the hypothesis terms, our H0: mu=1, H1: mu not equal to 1. Let us assume the volume of gas dispensed for one supposed gallon, denoted as X, follows normal distribution with a known sigma=0.02. And you decide to use X_bar out of 30 measures of one supposed gallon as the test statistic, whatare reject region and do not reject regionusing alpha=0.05,respectively?Draw the density function, identify the rejection non rejection region on the graph, and use excel for the calculation.Share your result in the discussion board. (1 point)

2. If the gas station has secretly adjust the machine to mu=0.95 gallon. The sigma stays the same at 0.02. What is the power,i.e. 1 - beta, ofyour test in part 1? Draw the density function and use excel for calculation. (1 point)

Feel free to post multiple response on this topic. This is quite involved, and no one gets them right the first time. (Similar to the jump routine in the matrix if you happen to be a fan to the movie as well.)