- #1
Agent Smith
- 193
- 20
Thread moved from the technical forums to the schoolwork forums
TL;DR Summary: Sims and sample size
A statistics question I have in my notes goes like this:
Our significance level ##\alpha = 0.01##
The percentage of left-handed people in the general population is ##10\%##. Liliana is curious if this is true for her arts class and so she takes a random sample of ##8## [please note this number] students from her arts class and finds that ##1## is left-handed. That is the proportion of lefties in her class is ##0.125##.
The null hypothesis: ##H_0## is that the proportion of lefties in Liliana's class = ##10\%##
The alternative hypothesis: ##H_a## is that the proportion of lefties in Liliana's class ##> 10\%##
She then conducts a 100 simulations, each time taking a sample size of ##8## [please note this number] from a virtual population in which ##10\%## are lefties. It turns out that in ##2## of her simulations the proportion of lefties is ##\geq 0.125##. This means, I'm told, that the probability of getting a proportion of lefties ##\geq 0.125## is ##\frac{2}{100} = 0.02##.
Then, the back-of-the-book answer says, since the ##\text{P-value} = 0.02## and ##\alpha = 0.01## and ##0.02 > 0.01##, we can't reject ##H_0##.
I hope all the above is correct.
My question concerns the sample size ##8## (the number I asked be noted). This sample size is too small for the number of successes and the number of failures to be ##\geq 10## i.e. one condition for inference from the sample is unmet and yet we have made an inference. Am I supposed to conclude that with simulations like the one described above we need not bother about sample size? So for this particular question, if my sample size is ##6##, I need only ensure that the simulation consists of samples of size ##6## and I'll still be able to make legitimate inferences from the sim???
N.B. Also if we reset ##\alpha = 0.05##, since ##0.02 < 0.05##, we can reject ##H_0## and conclude that Liliana's arts class has an "unusually high number" of lefties, right?
A statistics question I have in my notes goes like this:
Our significance level ##\alpha = 0.01##
The percentage of left-handed people in the general population is ##10\%##. Liliana is curious if this is true for her arts class and so she takes a random sample of ##8## [please note this number] students from her arts class and finds that ##1## is left-handed. That is the proportion of lefties in her class is ##0.125##.
The null hypothesis: ##H_0## is that the proportion of lefties in Liliana's class = ##10\%##
The alternative hypothesis: ##H_a## is that the proportion of lefties in Liliana's class ##> 10\%##
She then conducts a 100 simulations, each time taking a sample size of ##8## [please note this number] from a virtual population in which ##10\%## are lefties. It turns out that in ##2## of her simulations the proportion of lefties is ##\geq 0.125##. This means, I'm told, that the probability of getting a proportion of lefties ##\geq 0.125## is ##\frac{2}{100} = 0.02##.
Then, the back-of-the-book answer says, since the ##\text{P-value} = 0.02## and ##\alpha = 0.01## and ##0.02 > 0.01##, we can't reject ##H_0##.
I hope all the above is correct.
My question concerns the sample size ##8## (the number I asked be noted). This sample size is too small for the number of successes and the number of failures to be ##\geq 10## i.e. one condition for inference from the sample is unmet and yet we have made an inference. Am I supposed to conclude that with simulations like the one described above we need not bother about sample size? So for this particular question, if my sample size is ##6##, I need only ensure that the simulation consists of samples of size ##6## and I'll still be able to make legitimate inferences from the sim???
N.B. Also if we reset ##\alpha = 0.05##, since ##0.02 < 0.05##, we can reject ##H_0## and conclude that Liliana's arts class has an "unusually high number" of lefties, right?