1) You test a new drug to reduce blood pressure. A group of 15 patients with high blood pressure report the following systolic pressures (measured in mm Hg): y S before medication: 178 111 142 134 151 159 172 188 124 119 121 186 121 138 184 148.53 27.409 after medication: 175 105 140 129 148 160 170 187 121 122 119 185 119 137 180 146.47 27.67 change: 3 6 2 5 3 -1 21 3-3 2 12 14 2.067 2.219 a) Calculate a 90% CI for the change in blood pressure. b) Calculate a 99.9% CI for the change in blood pressure. c) Does either interval (the one you calculated in (a) or (b)) include 0? Why is this important? d) Now conduct a one sample t-test using = 0, and a = .10. Are the results consistent with (a)? Why or why not? e) Finally, conduct a one sample t-test using Why or why not? = 0, and a .001. Are the results consistent with (b)? (Make sure you answer the part in bold for (d) and (e). See also problem 5) 2) Here are the heights of seven male giraffes (in m): 5.5 5.1 6.5 5.2 6.4 6.2 5.9 (a) Test Ho: u=5.828571. Use a = 0.05. (b) Repeat, but now test Ho: = 5.2. Again, use = 0.05 (c) Finally repeat (b) except use a = 0.01. (d) What happened as you went from b to c (explain!). (e) As you make a smaller does it become easier or more difficult to reject Ho? Why? Make sure you understand this! 4) Let's examine the relationship between CI's and hypothesis tests: Hint: you need to think about how/when confidence levels and hypothesis tests are equivalent. In particular, what happens to a CI as you change the confidence level? (a) You calculate a 99% confidence interval for u and come up with (-10, -26). If you test Ho: = -27 and use a = .01, will you reject Ho? Why or why not? (b) Now you calculate a 95% CI for and come up with (-5, 3). If you test Ho: = -7 and use a = .10, will you reject Ho? Why or why not? (c) Finally, you calculate a 95% CI for for and come up with (-24, 2). If you test Ho: = - 14 and use a = .01, will you reject Ho? Why or why not? 5) (a) Suppose you test a new medication and reject the null hypothesis (you conclude the medicine works). What kind of error could you have made? (b) Again, you test a new medication and you fail to reject the null hypothesis (you conclude it does not work). What kind of error could you have made? 6) (a) If you make a too big (e.g., a = 0.2 (you should never use a > 0.1)), what kind of error are you more likely to make (type I or type II)? Why? (b) If you make a too small (e.g., a = 0.0000000001), what kind of error are you more likely to make? Why? 3) Let's examine the levels of levels of sodium in the blood. 20 people yield the following results (in mEq/L): y = 131.95 s = 2.544 (a) Test Ho: u=129 using a = 0.001 (that's not a typo). (b) Figure out the p-value. To do this, take the t* value that you calculated in (a) and use R: pt (absolute-value-of-your-t*, df, lower.tail FALSE) *2 Since it's a two sided test, you need to multiply by 2 at the end. We'll talk more about this in lecture when we do one sided tests; for now just realize that without the "*2" at the end R gives you a one sided p-value. For example, if your t* = 4.65 and you have 67 degrees of freedom you would do: pt (4.65, 67, lower.tail = FALSE) *2 and you would get back 1.609352e-05 (c) Is the p-value smaller than a? (d) Why is this (c) important (see also part (a))? It is essential that you understand question (c)! (e) What is the smallest value of a for which you would reject the Ho? Hint: this will not be a "standard" value of a. The correct answer here is a value that is not only small, but totally absurd). Another hint: think about the relationship of p-values to . As mentioned in (d) it is very important that you understand the connection between a and p-values. (f) For which of the following values of a would you reject? Why (see (c))? i).10 ii) .05 iii) .01 iv) .001 v) .00000001 (note that no one in their right mind would actually use a = .00000001)