Example 3. Some people seem to believe that you can fix anything with duct tape. Even so, many wereskeptical when researchers announced that duct tape may be a more effective and less painfulalternative to liquid nitrogen, which doctors routinely use to freeze warts. A 2002 article described astudy conducted at Madigan Army Medical Center. A sample of 204 patients with warts were randomlyassigned to either the duct tape treatment or the more traditional freezing treatment. The 104 in theduct tape group wore duct tape over the wart for 6 days, then removed the tape, soaked the area inwater, and used an emery board to scrape the area. This process was repeated for a maximum of 2months or until the wart was gone. Data consistent with values in the article are as follows:Number with WartnSuccessfully RemovedLiquid nitrogen freezing10060Duct tape10488Do the data give statistical evidence that duct tape works better than freezing? Exercise 3. Revisit Example 3. Our goal is to assess whether the data give statistical evidence that duct tapeworks better than freezing.(a) Carefully describe the inference target. Hint: It will be the difference of two probabilities. Describe theseprobabilities as, for example, p1 = PRe, ("success"), where "success“ is . and RE, is.(b) Compute and interpret a 95% confidence interval for the inference target.(c) Carry out a test at level 0.001 of the relevant hypotheses. Include a PV report.

Given : n1 = 100 x1 = 60 n2 = 104 x2 = 88Solution : A) Significance level, α =1-0.95 = 0.05 Sample proportion p̂1 = x1/n1 =60/100 =0.6 p̂2 = x2/n2 =88/104 = 0.8462 95% confidence interval for population proportion p1 - p2 : Difference of two probabilities : p̂1 - p̂2 =0.6 - 0.8462 =-0.2462 B) 95% confidence interval for population proportion p1 - p2 : Point estimate: p̂1 - p̂2 =0.6 - 0.8462 =-0.2462 critical value critical value at α=0.05 is Zα/2 = 1.96 ....from the standard normal distribution table Margin of error: ME=Zα/2 * √(p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2) =1.96 × √0.6(1-0.6)/100 +0.8462(1-0.8462)/104 = 0.1184 Margin of error is 0.1184 Now 95% confidence interval is CI=p̂ ± ME =-0.2462 ± 0.1184 =( -0.2462 - 0.1184 , -0.2462 + 0.1184 ) = (-0.3646, -0.1278) Interpretation : We are 95% confident that the difference between the population proportion is between -0.3646 and -0.1278C) x1= 60.0 n1= 100 x2= 88.0 n2= 104 significance level, 0.01 sample proportion p̂1 = x1 / n1 = 60.0 / 100 = 0.6 p̂2 = x2 / n2 = 88.0 / 104 = 0.84615 Pooled Proportion =( 60.0 + 88.0 )/ 100 + 104 = 0.7255 Hypothesis test : The null and alternative hypothesis is Ho : p1 = p2 Ha : p1 < p2 The hypothesis test is a left tailed test Test statistic test statistic for two proportion z test is = ( 0.6 - 0.84615 )/(v( 0.7255 (1- 0.7255 ))/(1/ 100 )+/(1/ 104 )) = -3.94 Test statistic is -3.94 P-value : p-value with test statistic z= -3.94 is P-value =P(Z < Zobs ) = P(z < -3.94 ) = 0.0 ....... (from standard normal distribution table) ∴ P-value is 0.0 Decision : since P-value is less than significance level,α = 0.01 ,we reject the null hypothesis Decision is reject Ho Conclusion : There is sufficient evidence to conclude that duct tape works better than freezing.