Well, today we want to investigate a distribution that will come up later in the lecture. It is called the $t$-distribution. Then we want to see how MINITAB helps us find confidence intervals. If not specified, assume we want at 95% confidence level for an answer.
We want to investigate the $t$-distribution. We can do this a few ways. First, MINITAB will gladly generate random samples from the $t$ distribution if we ask it. Second, we can create our own by looking at
$$t = \frac{\bar{X} - \mu}{s/\sqrt{n}}$$
First, create 100 values of the $t$ distribution for various values of degrees of freedom. Try d.f. = 5, 20,50 and 100. Look at boxplots or normal plots to decide if the distribution looks symmetric or not, long-tailed on not and normal or not.
Next, theory states that if the $X_1,X_2,\dots X_n$ are normal, then the statistic above has the $t$-distribution. with $n-1$ degrees of freedom. Investigate with $n=6$ and $mu =0$ and $\sigma=1$.
Finally, let’s investigate how much this changes of the $X_i$ are no longer assumed to be normal. Will the distribution of $t$ change dramatically? Investigate with $n=6$ and $X_i$ a uniform on the interval $[0,1]$ and again with $X_i$ exponential with mean 4.
The $z$-test for 1-sample is easy to implement in MINITAB. We just need some data. Download the data setAnesthet from Kitchen’s data set. Then do the following
Look at the data to see if it is approximately normal?
We want to assume the data has a standard deviation of 3. Does this look correct?
Perform a 1-sample z-test for a confidence interval. To do so find the dialog under Stat->Basic Statistics->1-Sample z Select the proper variable, confidence interval and fill in the desired level. For us we want a 80% confidence level, and assume the s.d. is 3 if it looked good above.
Report back
Create 15 random numbers that are normally distributed with mean 10 and s.d. 5. Use MINITAB to find a 1-sample z test at the 95% level. Did it get it right?
The $t$-test is just as easy to do in MINITAB. Do a $t$-test on the same data. Is it correct now? Comment on the relationship between the confidence intervals.
Download the data set Rat from Kitchen’s site. Do a $t$ test for mean if the data suggests it is appropriate. If not, say why not.
Do the same for the data set Puerto (weekly incomes of Puerto Ricans in Miami.).
The median may be the appropriate measure of center. If so, you might want to have a confidence interval for it too. MINITAB has the 1-Sample sign test available. TO find it go to Stat->Nonparamterics->1-sample sign and go from there. Try it out on the data set Malpract (on the size of malpractice awards).