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Assume that the distribuiton of Radonite is Gausian or normal (we are beginning to wonder if anything about WW is normal but that is another story!).
We have access to the target population from which these five samples were susposedly drawn; the mean of the Radonite concentration is 60.00 and its standard deviation is 10.
What is the probability that the sample WW obtained was drawn from this target population?
You should generate a set of 10,000 normally distributed values which have mean of 60 and a standard deviation of 10. Manip > Generate Random Numbers. Plot a histogram and normal probability plot. Confirm that the mean and standard deviation of your set are close to those specified.
There are several ways we can proceed. In this exercise we will look at confidence intervals on the sample mean. See section 18.2 for some background information.
The Confidence Interval Applet lets you conduct some sampling experiments which you may be able to relate to the Radonite question. Each line on the plot represents one confidence interval for the mean (assuming known variance). Note that each sample includes 5 observations. With an alpha of 0.05, 5 confidence intervals out of 100 or 50 out of 1,000 experiments should NOT include the known sample mean (0 in this case).
With an alpha of .05, draw 1,000 samples. How many do not include the sample mean?_____
Repeat with another 1,000. How many do include the sample mean _____?
Do this three more times (a total of 5,000 total). What is the average number of samples that do not include the sample mean? What should it be?
Repeat the above experiment with an alpha of 0.02. After your 5,000 trials, what is the average number of samples that do not include the sample mean? What should it be?
Do you see any differences between the two experiments?
With alpha set at 0.05 and the results showing in the figure, drag the slide bar to the left (decreasing alpha). What happens to the length of the confidence interval? What happens to the number of samples which do not include the known mean? Generalize your observations about the relationship between confidence interval and alpha.
Calc > Estimate opens a window that will allow you to experiment with confidence intervals. Select Z-interval for individual means and 90% confidence. Use the Individual sampling option. For my run I learned that with 90% confidence 59.73 < mean < 60.06. What does this mean?
In the long run (many sampling experiments) 90% of the invervals we compute from independently drawn samples will include the true mean. It does not mean that 90% of the time the mean will fall between 59.73 and 60.06!
Try the following experiment. Click on the vector containing the 10,000 samples in the Radonite target population. Manip > Sample. Select 1% of the samples and repeat 20 times. Select all of the results (20 icons) and Calc > Estimate. Use the Z values and a sigma of 10 and 90% confidence. What should we expect....90% of the sample confidence intervals should contain the known mean of 60.00. Therefore, 2 of our 20 intervals should not include 60. Keep in mind that this is a sampling experiment and that we may not find exactly what has been predicted. Check your 20 intervals. How many do not include the mean of 60?
Repeat but draw only 10 samples (0.1%). Note the approximate range of the upper and lower bounds. What happens?
Change the confidence from 90% to 95%. What should we expect to happen. Notice that we are requesting a more confident result. 95% of our sample confidence intervals should contain the known mean of 60. Note that the approximate range of the upper and lower bounds exceeds that for the 90% confidence interval.
What do you think will happen if we specify a 75% confidence interval? Should the range between upper and lower bounds increase or decrease? Try it an see. Write a paragraph in which you discuss the balance between precision and certainty.
You have probably observed that the upper bound rarely, if ever, has included 70....the mean of extractable Radonite. What should you do to increase the upper bound? Try it? What is the tradeoff?
Compute the means for the 20 samples. How many exceed 70?
Give us your evaluation of WW's good deal. Include a summary of your experiments and your paragraph discussing the balance between precision and certainty.
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I am pleased that you have accepted a position as a junior explorationist with Geological Analytical Services (GAS). Unfortunately, the person who was to be your supervisor (Dr. Percy Cambrian - good ole pC) has been replaced. We have been forced us to hire Dr. Willard Wobble, statistician supreme who is a close personal friend of the owner of GAS. To test your worth you have been assigned as his third chief assistant. For the next few months we will be sending you some of WW's finished work - it will be your task to tell the management whether you agree with his analyses. Your reports should be relatively short. Make use of computer output whenever possible and make sure that you clearly state the problem, the procedures you used and your interpretation. Assuming that you accept this assignment (after all, how many jobs are available right now - this was written some time ago!), remember that you always answer our phones with the following message:
Don't let it pass.
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