In this lab, there are three problems involving probability.
Last semester, on the class survey in the first lab, there were two questions:
What do you think is the probability that the world will be destroyed in the next 50 years?
and
What do you think is the probability that the world will be destroyed by a nuclear war or a massive meteorite in the next 50 years?
In similar courses to STA102, roughly 1/3 of the students believed that the probability of the second event was higher than the probability of the first event. Also, roughly 1/3 of the students believed that first event had no chance of occurring. Using what you now know about probabilities, explain why the answer to the second question cannot be more than the first question. Why do you think this occurs? Should the order that the questions are asked in matter?
This is a form of incoherence, and in betting games incoherent assessment of probabilities means that someone else can make money off of you!
Before there was Who Wants to be a Millionaire, there was a game show called Let's Make A Deal, hosted by Monty Hall. Contestants were offered a choice of three doors. Open the correct one, and you won a car or a grand vacation package. Open either of the others and you got a donkey or a gag prize. There was a twist to the game; after the contestant had chosen, but before the door was opened, Monty would reveal the contents of one of the other doors to reveal a donkey or a gag gift and then ask the contestant if he or she wanted to stick with the first choice or switch to the other unopened door. Question: is it a good idea to change your mind, a bad idea, or does it make no difference. Use your new knowledge of probability theory to find the answer, and if you get stuck, try out the game on the
(Not really a biostatistics problem, but have fun!)
A good source for medical information is the Ovid Web pages, via the Ovid Web Gateway.
Note: The test that you report on does not necessarily have to be a medical test, although these often have all the necessary information available.
Turn in a brief (1 page max!) typed report. In your report, you should
Your report should be written so that someone like your grandmother could understand the results. Be sure to explain why a positive test result does not mean that the person actually has the disease and why the probability that a person has the disease given that they found out that they had a positive test result is not the same as the prevalence in the population.
Start up S-Plus 2000.
To construct a plot of "Probability of Disease given a Positive Test Result" versus "Disease Rate in the Population" we will need to create the two variables. To start we need to create a new Dataframe; go to the File menu, select New, and then Data Set.
To create a sequence of Prevalence in column 1, go to the Data menu, and select Fill....This will create a sequence of numbers. In the dialog, specify 1 for the column. For the length field enter how many prevalence values that you would like to plot, say 25. Enter in the smallest prevalence for Start and then choose a value for Increment so that the last value in the column is the maximum prevalence that you wish to plot. i.e. Start + (n-1)*Increment. Click on Apply to create the column; if the results are what you want click on OK. Otherwise change the options and Apply again. Rename the column Prevalence or something more intuitive than V1.
To create the sequence of probabilities, go to the Data menu and select Transform... Enter the name for the new column, Probability, in the Target Column. In the Expression field, enter in the formula for updating probabilities as a function of Prevalence (or the name of your column 1) and the specificity and sensitivity. Make sure that you include parentheses so that the calculations are carried out correctly! Click on Apply, and verify by hand that you have the correct results. Then click on OK.
You should have a dataframe now with two columns, Prevalence and Probability. To create a plot of these, go to the Graph menu and select 2D Plot. Choose Line Plot and then click on OK. In the popup dialog box, select Prevalence for the X column (the X-axis) and Probability for the Y-column (Y-axis). Click on OK.
If you would like to modify the labels on the Y or X axes, simply double-click on the text, and then replace the text with a more descriptive label. To add a title, select the Insert menu, and Titles... then Main. Enter in a descriptive title. Repeat to add a subtitle but this time enter in the source for your information. Your figure should be able to stand on its own!
Describe the resulting plot.
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