K300, Prof. Kruschke

K300, Prof. Kruschke K300 Statistics, Prof. Kruschke

Homework 8, Due at the beginning of class, Tuesday April 5, 2005.

Be sure to

write your name and ID on every page

make a copy of your homework for your own records and exam study

staple (not paper clip) your pages

write clearly

show your work as appropriate -- an unannotated sequence of numbers and derivations that mysteriously ends up with the correct numerical answer will not be given full credit

answer every part of every question (unless instructed otherwise).

15 pts. A researcher is interested in the effects of caffeine on how fast people solve a maze. She gives four randomly selected people a cup of (caffeinated) coffee, and four other randomly selected people a cup of de-caffeinated coffee. Then she has each person solve a complex maze. Here are the duractions, in minutes, for solving the maze:
Caffeine: 0.250, 0.200, 0.167, 0.143
De-caf: 1.000, 0.500, 0.333, 0.250
A. Conduct a t-test on the durations, assuming a non-directional test and a .05 significance level. Do the computations by hand and show your work, but you do not need to draw the distributions. Be sure to specify the critical t value and your conclusion.
B. Do the two groups seem to have equal variances of durations? Do the durations within each group appear to be symmetrically distributed, or are they skewed? Should we trust the conclusion from part A, and why or why not (mention the assumptions behind the critical t value)?
C. Instead of measuring each participant's duration to complete the maze, the researcher transforms the durations into speeds. Thus, instead of describing each person's performance as minutes per maze, she describes each person's performance as mazes per minute (i.e., the number of equally difficult mazes they would solve in a minute, if they kept the same pace). For example, a duration of 0.250 minutes per maze is transformed into a speed of 1/0.250 = 4.000 mazes per minute. Transform every duration above into speed by taking the reciprocal. Then conduct a t-test on the speeds, assuming a non-directional test and a .05 significance level. Do the computations by hand and show your work, but you do not need to draw the distributions. Be sure to specify the critical t value and your conclusion..
D. Do the two groups seem to have equal variances of speeds? Do the speeds within each group appear to be symmetrically distributed, or are they skewed? Should we trust the conclusion from part C, and why or why not (mention the assumptions behind the critical t value)?
15 pts. A zoologist is studying the sounds produced by two species of bats. She randomly samples 10 recordings from each of the two species, and measures the dominant frequency (in Hz) of each sound production. The data are as follows:
```
spec_1  spec_2
  4675   38177
 34544   80822
  6634    7708
   518    1212
  6974   46630
 13360    4024
   545   89322
   365     898
   314   32860
  1556    4230
```
A. Do the two species differ in the mean dominant frequency of their sound productions? Conduct a t-test (non-directional, .05 significance) in Excel. Print out a single page that shows the data in the spreadsheet and the results of test, and be sure to write your conclusion under the t-test output. Also, be sure to clearly write on the page what exercise it is for, and please put the page in order with the other exercises.
B. Can we trust the results of the t-test in part A? Why or why not? (Mention variances, symmetry, and how the critical t value is generated.)
C.The zoologist is actually interested in the effective pitch (do-re-mi...) of the sound, not the raw frequency of sound. To convert to pitch from frequency, just take the logarithm of frequency. In Excel, use the ln function to take the logarithm: For a number in cell A1, compute its logarithm by typing "=ln(A1)" in cell B1. Conduct a t-test (non-directional, .05 significance) on the pitches in Excel. Print out a single page that shows the data in the spreadsheet and the results of test, and be sure to write your conclusion under the t-test output. Also, be sure to clearly write on the page what exercise it is for, and please put the page in order with the other exercises.
D. Can we trust the results of the t-test in part C? Why or why not? (Mention variances, symmetry, and how the critical t value is generated.)
10 pts. A researcher is interested in the effects of different kinds of stress on how well people do on a test. She gives four randomly selected people some strenuous exercise, and four other randomly selected people threats of grave consequences if they do poorly. Then she has each person take the test. Here are the scores, out of 25 possible:
Exercise: 23, 22, 20, 3
Threats: 21, 4, 2, 1
The scores do not appear to be normally distributed; in fact, they appear to be bimodal, with some people responding well to stress and other people reacting badly. The researcher believes that a standard t-test is inappropriate because of the non-normality, and decides to use resampling instead.
In Oncourse, go to the Schedule tab (not the In Touch / Group Spaces tab) and get the Excel spreadsheet for resampling. Type in the scores for the two groups. Press F9 a few times so that you see how the values change with different random samplings, and stop pressing F9 when the result looks typical.
A. For the Null Hypothesis worksheet, briefly state how the numbers in cells B4 and F4 are generated.
B. Does the null hypothesis sampling distribution have a single mode like the t distributions we have seen for sampling from normal distributions?
C. What is the critical t value, in the re-sampled t distribution, on the high (positive) end of the distribution? [Hint: it will be close to, but not exactly equal to, the t value for a normal population.]
D. Does the actual data t value exceed the critical t value in the re-sampled t distribution? Should the researcher reject the null hypothesis that the two types of stress have equal effects?
E. For the Alternative Hypothesis worksheet, briefly state how the numbers in cells B4 and F4 are generated.
F. On the Alternative Hypothesis worksheet, press F9 until typical values appear. Does the alternative hypothesis sampling distribution have a single mode like the t distributions we have seen for sampling from normal distributions?
G. Under the alternative hypothesis, what is the probability that the re-sampled t will exceed the critical value from the null hypothesis?