95 % confidence interval In a random sample of 41 cyclists two years ago, 18 tested positive for drugs. A random sample of 45
is conducted this year and 15 test positive.
(a) Give a 95 percent confidence interval for the percentage who tested positive two years ago.
(b) Can you conclude that drug use (or its detection rate) has fallen? Specifically
i. What is the null hypothesis?
ii. What is the value of the test statistic and what is its distribution?
iii. What is the
p
value of the test statistic? Interpret what this means for the validity of the null
hypothesis
I dont understand how to get the Confidence Interval without any mean or standard error
 A: The 95 percent confidence interval for a percentage is classically obtained by adding and subtracting to the proportion p an interval computed using the formula 1.96 Sqrt[(p(1-p)/N], where N is the total number of observations, and 1.96 is the standard number of SDs extending from the mean of a normal distribution and encompassing 95% of the population. Also, to obtain a correction for continuity, this interval has to be increased by 0.5/N. Applying this to the 41 cyclists sampled two years ago, of whom 18 tested positive for drugs, we get a proportion of 0.44, an interval of 0.16, and a 95% CI ranging from 0.27 to 0.60. This interval includes the new proportion observed this year (15/45=0.333). Based on this simple analisys, you could roughly conclude that the proportion probably has not fallen significantly. 
However, to better explore the null hypothesis (that the two proportions are not significantly different, i.e., that the proportion has not fallen) a more appropriate way is to compute the 95% CI of the difference between proportions.  This can be calculated by adding and substracting to the observed difference p1-p2 an interval given by 1.96 Sqrt[(p1(1-p1)/N1 + p2(1-p2)/N2]. Using the data provided, we get that the difference p1-p2 is 0.44-0.33=0.11, and the interval is 0.21. Thus, the 95% CI for the difference of the two proportions ranges from -0.10 to +0.31. Since this interval encompasses zero, you can conclude that the two proportions are not significantly different. 
