1 and 2 variance HT’s and CI’s
1 and 2 variance HT’s and CI’s
So far we have done HT’s and CI’s for a population mean, population proportion, difference of 2 population means, and difference of 2 population proportions. There is another parameter that we might be interested in namely __________ ________ which measures how consistent the data is.
Sometimes instead of standard deviation we talk about variance which is the same as standard deviation without the square root, but they both measure the same thing, variability.
All HT’s and CI’s for standard deviations are considered very risky and should be only done with the help of a statistical consultant. Unlike the mean and proportions, the standard deviation is very sensitive to non-normality even for bigger sample sizes. We can still do the calculations but we should hesitate to apply to much confidence in those calculations. Unlike the sample mean the CLT does not kick in for sample standard deviation.
Question: Which seems the least important to you in general? Is it means, proportions, or standard deviations?
As an example a basketball player would like to have a high number of points on average (mean), make a high percent of free-throws (proportion), and be very consistent (low standard deviation). All are noble, but there is one that is just the frosting on the cake, which one?
By the way sometimes we like the standard deviation to be low and sometimes high. As an example if you are growing tomatoes in your garden, you would like a high standard deviation of maturity times so you can have fresh tomatoes over a long period of time. If you are a commercial farmer you would like a low standard deviation so you can harvest all at one time.
[pic] has a [pic] distribution with df = n-1 when you have sample data of size n picked at random from a normal population with standard deviation [pic].
Question: What is the total area under[pic] ?
Question: Is the [pic]distribution symmetric?
Confidence Interval for [pic] (or [pic] ) - won’t be s +/- E since [pic] is not symmetric.
Step 1:Draw a [pic]and find the middle 95% and find the 2 values from the table (this is for a 95% CI, of course it changes for different confidence levels)
Step 2: Use [pic] to get two values for [pic]. (A CI has 2 values, high and low even if symmetric, for example, 10+/-4 is 6 to 14), if the problem asks for [pic] then take the square roots.
HT for [pic] (or [pic] )
Step 1. Put what you want to show in Ha. Just like before the < or > or not = in Ha tells you what tail(s) you have for your rejection region.
Step 2. Use [pic]. .
If for example you have Ha: [pic] >10 then Ho: is [pic][pic]. Since Ho is always assumed to be true at first we use [pic]=10. The best evidence that [pic]>10 is when s is the biggest. The bigger s is the further right [pic] will be.
[pic] where [pic] and [pic] are standard deviations from two independent samples of size [pic] and [pic]from a normal population has an F distribution with [pic]degrees of freedom on top and [pic] degrees of freedom on bottom.
Question: What is the total area under F?
Question: Is F symmetric?
HT for comparing two sample variances
Step 1: Put what you want to show in Ha and rewrite with a fraction and a [pic], [pic], or [pic], but make sure the population variance with the bigger sample variance is on top.
Step 2: Use [pic]
Note since the bigger sample variance is on top, [pic] will never be in the left tail since [pic]. This is even true if the problem has a left tail involved.
If Ha: [pic] >1 then the best evidence for that is when [pic] is big, namely in the right tail.
We should comment that there are different ways of comparing numbers. By far the most common is subtraction and we did this for means and proportions. But, also by far the second most common is by a ratio and this is what we are using for variances.
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