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Lecture17_test_problem.htm |
Michael Hallstone, Ph.D. hallston@hawaii.edu |
How do I choose a
theoretical mean for the test problem covering lecture 17?
Lecture 17 covers one sample hypothesis test of means also known as a one-sample t test and you do one of these on the take home exam.
I often get asked this great question in the in-person class. I hope it’s in the videos of lecture too. But if not, this is a great example of the limitations of online education!
Basically, “What number do I test for the population mean?” This confusion is quite understandable.
The short answer is you make up a theoretical population mean before you look at your data.
But…why in the heck would you want to prove your own theory wrong? It makes no sense, “Right?” Yes, it makes no sense.
But yes you will do this for this problem. Simply make up a theoretical mean for your population before you look at your sample mean. I give you an example below.
In the real world there would be a theory out there you wanted to update. If you watched lecture videos I probably talk about how Europeans thought all they saw in the night sky revolved around them because they were “God’s chosen people.” Then Galileo Galilei proved that theory wrong. Then the Pope forced him to recant and put him in house arrest for the rest of his life. https://en.wikipedia.org/wiki/Galileo_Galilei - Controversy_over_heliocentrism
Eventually it was finally agreed that Earth revolves around the Sun. Well…that means the Sun is the center of the universe because we are God’s chosen people right? Then that theory was proven wrong. The Sun is a minor star in the Milky Way Galaxy. Well then the theory was the Milky Way was the center of the universe. Proven wrong eventually. So that’s how science and social science advances. The accepted theory is the accepted theory until there is enough evidence to overturn it.
Not so long ago boys were thought to be smarter than girls in math - just because they were boys. You may laugh at that now, but my Mom and Mom-in-law certainly suffered because of the prejudice. When they were going to school, girls back then were encouraged to choose careers in things that “fit their gender” like home economics, dietetics, art, and teaching and other lower paying careers. My smart as a whip Grandmother was not allowed to go to college because college was for boys. All of this occurred less than 100 years ago! There were studies that showed that boys scored better in math than girls too.
Well we now know that there are no inherent gender differences in math, because the social scientific observations were done in mixed gender classrooms. I think, but am not certain, that data from Sacred Hearts School – an all girl’s school on Oahu -- helped overthrow the theory. Long story short, the differences were social. Look at girls who do math in classrooms separated from boys and the differences disappear. In class, I’ve got a great comedy bit I do about how us dumb elementary aged boys act like idiots in math class. I hope it’s in the lecture videos. So science advances by proving the existing theory wrong.
Right now, the absolutely undeniable overwhelming evidence is that vaccines do not cause autism. The guy who did the study lost his license to practice medicine and the journal has retracted the article. https://en.wikipedia.org/wiki/MMR_vaccine_controversy. But that is just the current theory. We need to be open to the notion that evidence might be produced to prove that the theory is wrong. [Recall there was a ton of data showing boys did better in math, certain races were “smarter” than others, etc.]. If evidence comes along to prove that wrong – that vaccines really do cause autism -- the person will win a Nobel Prize and be famous for all of eternity in the text books.
You can do this for any of your ratio variables, but I’ll use age here because all have it and it’s simple. Imagine the age of your population. When you designed your survey you had a population in mind – a group of people you wanted to study. In this class we pretend your sample is representative of your population when it’s really not.
So imagine the mean age of your population. I'm not asking you to imagine the mean age of your sample! Pretend your imaginary population is "adults on Oahu." Now come up with a theory about what the mean age of that population is. Then test that theory.
DO NOT look at your sample mean and use that number as your theory! That will make TR =0 in step 6. In fact you can try it once on SPSS. Make your "test value" exactly your sample mean and you will see TR=0 and p=1.0.
So pretend you think the mean age of adults on Oahu is 40 and want to do the easiest two-tailed test.
For step 1:
Ho: pop mean =40
H1: pop mean not = 40
So in SPSS you use 40 as the “test value.”
If you still have questions, you are normal, but we need to speak. This is the best I can do with typing.