Confidence Intervals for One Mean: Assumptions

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” re gonna be investigating the assumptions of the one sample t procedures the one sample sample t procedures assume that we have a simple random sample from a normally distributed and here we re gonna ask ourselves if that normality assumption is violated what are the consequences so let s look at that normality assumption in a bit more detail. The good news is these t procedures tend to actually work quite well under certain violations of the normality assumption and we say the procedures are robust to certain violations of the normality assumption. But overall they re not gonna work as well if the normality assumption is violated sometimes they re gonna work quite poorly so if we d say we were doing a confidence interval and the assumptions are violated then the true confidence level of that interval is gonna be different from the stated confidence level. We might be saying it s a 95 interval.

But it might only contain the true value of the population mean let s say in 90 of the time or something along those lines and let s investigate that in a bit more detail. So suppose we have these four distributions and this one over here is simply a normal distribution. So if we re sampling from this distribution. The enormity assumption is perfectly justified and the procedures work perfectly.

But let s look at three different violations over here. We have a uniform distribution and over here. We have an exponential distribution. Which has some skewness to it and over here.

We have a distribution that looks quite a bit like the normal. But it has heavier tails. So we ve got three different violations of the normality assumption and one situation in which the normality assumption is perfectly justified and let s see what happens in a few different spots and we re gonna investigate this through simulation so the simulation is this we re drawing 100000 samples from each of these distributions. And we ll do this for different sample sizes and for each of these samples were gonna calculate a 95 interval and then we re gonna ask ourselves this question what percentage of those intervals actually contain the parameter mu and if that percentage is close to 95 the procedures are working very well in that scenario and if that percentage is quite different from 95.

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Then the procedure is not working very well in that scenario. So let s see how that pans out for those distributions that we looked at here first of all we ve got this nor distributed population. So the normality assumption is perfectly justified so theoretically this procedure is gonna work perfectly regardless of the sample size. So the true confidence level is actually 95 all the way.

But a differs here it says ninety five point one here. But that s just due to sampling. Variability there is some sampling variability or even when we draw 100000. Samples.

Now what about when that assumption is violated so first let s look over here at this uniform distribution. Remember this uniform distribution looks something like this it s got shorter tails in the normal normal distribution will truncate. It there. And when we have a sample size of 5 well that true coverage.

Probability is only around ninety three and a half percent. Now that s not too bad actually and once we start getting up here into 20 sample sizes of 20 or 50. Or that type of thing. We re actually very very close to the 95.

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So the sampling from the uniform distribution. Even though. The normality assumption is violated the procedure works. Quite well even for smaller sample sizes.

Now what about over here. When we look at an exponential distribution. Where we have some skewness well the story s a little worse here when we have a very small sample size of 5 our true coverage probability is of our 95. Interval is actually closer to 88.

So it s not working quite as well. But as our sample size gets bigger and bigger this coverage. Probability is tending towards the 95 percent once we get up to 50 or so wow. We re up to 93 and a half it s not so bad.

But for smaller sample sizes. We re gonna be overstating the case. If we actually say we have a 95 interval because that s not really true now what happens. If we look at a situation.

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Here where we ve got a distribution that looks pretty close to the normal in a sense. It s just got heavier tails. Well we re actually overstating our case. A bit here for a sample size of 5.

We re gonna be saying we have a 95 intial. But the true coverage probability is actually a little bit higher and this effect. Though similar to the others is going to be going away as the sample size increases. We re gonna be tending towards at 95 so overall when the normality assumption is violated the true coverage probability of our 95 interval will actually be different from 95.

It might be quite a bit less than 95. But as the sample size increases. It s going to be tending towards that 95 very rough guideline. If we have a sample size of at least 40 these procedures work fairly well in most situations.

But if our sample size gets a little bit lower than that the procedures might start to break down a bit and especially. If there s some skewness or outliers. The teeth procedures. Don t like skewness or outliers.

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So the procedures start to break down. Obviously if we re closer to 40. Then they might not break down quite as much. But if we start to get even smaller still once we start to creep down less than 15 or so.

Well. Now. It s starting to get a little sketchy using those t. Procedures.

And if we want to use them. We should be pretty confident that the normality assumption is reasonable before we go ahead and use them otherwise we might really be reporting misleading results. ” ..

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