Investigation 12: Watching Paint Dry (due Monday, November 1)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones. You may use Minitab or a Java applet to perform calculations involving probability distributions, but be sure to specify what you asked the computer to calculate when you provide the answer.

Suppose that the drying time for a certain type of paint under specified test conditions is known to be normally distributed with mean 75 minutes and standard deviation 5 minutes.  Suppose that chemists have devised a new additive that is hoped will reduce the mean drying time (without changing the standard deviation).  Suppose that a test is conducted to measure the drying time for a test specimen, and suppose that company executives decide that they will be convinced that the additive is effective only if the drying time on this specimen is less than 70 minutes.

a) If the additive actually has no effect at all on the drying time, what is the probability that the company executives will mistakenly conclude that it is effective?  Include a sketch with your calculation.  [It's fine to draw these by hand, or you can copy the picture from the applet by clicking the "print screen" button on your keyboard and then pasting the picture into Word.  If you do this, please use the "crop" tool to select only the relevant parts of the screen to submit.]

Now suppose that the additive really is effective and that it reduces the mean drying time to 65 minutes, without changing the standard deviation of 5 minutes.

b) Draw a sketch of the two normal curves on the same scale.  [Again it's fine to sketch these by hand, or you can copy from the applet.]

c) What is the probability that this test will fail to convince the executives that the additive is effective, even though it actually is?

d) If you want alter the cut-off value from 70 in order to reduce the error probability in a) to .05, what cut-off value should you choose?

e) Using this new cut-off value in d), what is the probability that that the test will fail to convince the executives that the additive is effective, even though it actually is?

f) How does the probability in e) compare to that in c)?  Explain why this makes sense.

g) Suppose that the additive not only reduced the mean drying time to 65 minutes but also reduced the standard deviation to 2 minutes.  Re-calculate the error probability in e).  Comment on how it has changed, and explain why this makes sense.