William H. Knapp III

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This homework was due on Monday, November 26 at 06:00 a.m. Turkish time. Late submissions receive half credit.

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1. For the rest set of questions, please download this data, save it, and load it into R.
These data are structured differently than you've seen before. There are two columns of interest: sample and observations. Each row contains one observations from a particular sample. To get the data into a format that you can use easily for these questions, I recommend you use the following code. You are free to change the variable names, but you should leave what comes after the equals sign alone.
s1=data$observations[data$sample==1]
s2=data$observations[data$sample==2]
s3=data$observations[data$sample==3]
Basically I'm getting each observation from a particular sample and assigning it to an appropriately named variable.
Perform a one-sample t-test on the sample one data. Use a two-tailed test with an alpha of .05 against a hypothesized mean of 65. What's the p-value?

2. What's power for the test you just performed? I recommend you always specify strict=TRUE when performing power analyses for t-tests.

3. How many participants would you have needed to have a power of .9 to observe the effect you did? Remember, you can only have whole participants!

4. If the null hypothesis was that the mean was greater than or equal to 65, what would the p-value for your t-test have been?

5. What's power for the test you just performed?

6. What's alpha level would have given you a power of .85? You'll probably get an error the first time you try. If you didn't specify sig.level=NULL, power.t.test will assume sig.level=.05. Keep this in mind if you try to change other parameters and get the same error. If you get the output with a warning, you're probably ok!

7. Please perform a t-test to determine whether or not the means of sample 1 and sample 3 are equal. You should use an alpha of .01 and assume that the variances are equal. What should you do? Hint: if you're trying to figure out what test to use, notice that the sample sizes are different!
Fail to reject the null.
Reject the null.
Not enough information to tell.

8. What was the value of the observed t for the previous test?

9. What's the probability of making a Type II error for the previous test?

10. How much power would you have had if the null hypothesis was that difference of the means was greater than or equal to 0?

11. How much power would you have had if the null hypothesis was that the population mean estimated by sample 3 was greater than or equal to 0 more than the population mean estimated by sample 1? Before you try answering the question, I recommend trying to figure out what the difference of the means is under the null hypothesis. Notice, the previous question gave you the expected difference under the null and this didn't. I'm not likely to be this helpful with my hints on the exam, so pay attention to the difference.

12. Perform a paired t-test on the sample 1 and sample 2 data with an alpha level of .001 for a null hypothesis of no difference. What's the value of the t-statistic?

13. What should you do?
Fail to reject the null.
Reject the null.
Not enough information to tell.

14. What's the power you had for your test?

15. If you were expecting the second mean to be less than or equal to 3.2 more than the first, what would your t-statistic have been? Again, try writing the null in terms of the difference between the means to figure out the appropriate null and alternative.

16. What's the power you had for your test in the previous question?

17. What would your power have been if you did a two tailed test for the previous question?

18. How many participants would you have needed to have a power of .8 for the two-tailed test in the previous queston?

19. What's the upper limit for the smallest 99.9 percent confidence interval of the difference between sample 1 and sample 2?

20. What's the lower limit for the smallest 99.9 percent confidence interval of the difference between sample 1 and sample 2?