The most common therapy for fingernail warts is cryotherapy, in which liquid nit

Question

The most common therapy for fingernail warts is cryotherapy, in which liquid nitrogen is applied to the area of the wart every few weeks, freezing it for a few seconds until the wart falls off. An alternative therapy is to put duct tape on the wart and leave it there for a week at a time until the wart goes away. In a randomized trial, 51 patients with fingernail warts were randomly assigned to a cryotherapy treatment or a duct-tape treatment. Of 26 patients treated with duct tape, 22 had their warts completely cured. With the liquid nitrogen treatment, 15 of 25 patients were cured of their warts. a. Conduct a hypothesis test to see if there is a difference in efficacy between cryotherapy and duct tape, in treating fingernail warts. Write the hypotheses, calculate the test statistic, show the rejection region, and make a decision. Also calculate and report the P-value for your test. Write a conclusion. b. How much more likely is a patient to be cured using duct tape compared to cryotherapy? c. Calculate the odds of being cured using duct tape. d. Calculate the odds of being cured using cryotherapy. e. Calculate and interpret the odds ratio for being cured, comparing duct tape to cryotherapy. In a test of Murphy's Law, pieces of toast were buttered on one side and then dropped. Murphy's Law predicts that they will land butter-side down. Out of 982 total slices of toast, 610 landed butter-side down. Does this finding provide evidence in favor of Murphy's Law? (You can set this up as a two-sided alternate hypothesis) a. State your hypotheses and define your parameters of interest. b. Calculate the test statistic, rejection region, and make a decision using the rejection region. c. Find the P-value and state your decision using the P-value. d. Write your conclusion.

Explanation / Answer

a) Use a chi-squared test.

Observed:

Healed

Did not heal

Row sums

Duct tape

22

4

26

Cryotherapy

15

10

25

Column sums

37

14

51

Expected:

Healed

Did not heal

Row sums

Duct tape

26*37/51=18.86

26*14/51=7.14

26

Cryotherapy

25*37/51=18.14

25*14/51=6.86

25

Column sums

37

14

51

Chi-squared test statistic:

(22 - 18.86)2/18.86 + (4 - 7.14)2/7.14 + (15 - 18.14)2/18.14 + (10 – 6.86)2/6.86

= 0.52 +1.38 + 0.54 +1.44 = 3.88

This test has (r-1)(c-1) = (2-1)(2-1)=1 degree of freedom.

The critical value for a chi-squared distribution with 1 degree of freedom at alpha = 0.05 is 3.84.

Thus, we reject the null hypothesis, and conclude that duct tape is a significantly better treatment for warts.

b) Duct tape is (22/26)/(15/25) = 1.41times more likely to cure than cryotherapy

c) p1 = x1/n1 = 22/26 = 0.846

ô1 = p1/(1- p1) = 5.50

e) p2 = x2/n2 = 15/25 = 0.6

ô2 = p2/(1- p2) = 1.5

e) ôr = ô1/ ô2 = 5.5/1.5 = 3.67(Odds ratio)

This means that the duct tape seems to work 3.67 times better than nitrogen

95% CI for the odds ratio

ln(ôr) = ln(3.67) =1.30

SE(ln(ôr)) = sqrt((1/a) + (1/b) + (1/c) + (1/d)) = sqrt((1/22) + (1/4) + (1/15) + (1/10)) = 0.678

1.30 - 0.678 < ln(ôr) < 1.30 + 0.678

e 0.622 < ôr < e 1.97

1.86 < ôr < 7.23

Healed

Did not heal

Row sums

Duct tape

22

4

26

Cryotherapy

15

10

25

Column sums

37

14

51

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