Note that: The conclusion that the sampling distribution has the same mean as Y does not involve either of the model assumptions. The independence assumption is needed for both of the other two conclusions (that the sampling distribution is normal and that the sampling distribution has standard deviation ). Forming the confidence interval proceeds by the following steps: we specify some high degree of probability; this called the confidence level. (We’ll use 0.95 to illustrate; so we’ll say “95% confidence level.”) The two conclusions of the theorem (that the sampling distribution of is normal with mean µ) imply that there is number a so that (*) The probability that lies between µ - a and µ + a is 0.95: P(µ - a < < µ + a) 0.95 [Draw a picture of the sampling distribution to help see why!] Caution: It’s important to keep in mind what is a random variable and what is a constant: Is µ a constant or a random variable? _______________ Is a a constant or a random variable? ________________ Is a constant or a random variable? ________________ trial
Here, the random variable is in the middle of the interval, which is what makes the most sense in a probability statement. All is good! Robustness again: In practice we can’t find a exactly for this procedure, since we don’t know . But using the sample standard deviation s to approximate will give a good approximation. And, once we actually have a sample, we will know the value of n, so we can use a t-distribution to find a more precise value for a for that sample size. Many procedures are (that is, don’t require an approximation), but the additional complications they involve make this procedure better for explaining the basic idea. A little algebraic manipulation (which can be stated in words as, “If the estimate is within a units of the mean µ, then µ is within a units of the estimate”) allows us to restate (*) as (**) The probability that µ lies between - a and + a is approximately 0.95: P( - a < µ < + a) is approximately0.95
Case 1: If the t-statistic lies on the horizontal axis at the small bar shown in red (around 0.5) in the picture, nothing is unusual; our data are consistent with the null hypothesis. Case 2: If the t-statistic lies on the horizontal axis at the small bar shown in green (around 2.5), then the data would be fairly unusual -- assuming the null hypothesis is true. So a t-statistic of about 2.5 would cast some reasonable doubt on the null hypothesis but a t-statistic of about 0.5 would not cast reasonable doubt on the null hypothesis. A t-statistic even further to the right would cast even more doubt on the null hypothesis. Note: A little algebra will show that if t = is unusually large, then so is , and vice-versa p-Values
Comparing p-values for samples of different size is a common mistake. Example B: Researcher 2 from Example A does everything as described above, but for convenience, his patients are all from the student health center of the prestigious university where he works. He cannot claim that his result applies to patients other than those of the age and socio-economic background, etc. of the ones he used in the study, because his sample was taken from a smaller population. Example C: Researcher 2 proceeds as in Example A, with a sample carefully selected from the population to which he wishes to apply his results, but he is testing for equality of the means of an outcome variable for the two groups. The hypothesis test he uses requires that the variance of the outcome variable for each group compared is the same.