We test 5 patients for a new drug. That is we give each patient the drug and count how many get cured with the drug. Without the drug the patients have a 10% probability to cure.We found that 4 out of five got cured. Test on the 95%-confidence level, the hypothesis that the new drug has no effect.
My Answer:
Let $X_i$ denote the $i^{th}$ person chosen is cured, then $E(X_i) = p$. Now let Z denote the total number of people in our experiment, so $Z = X_1 + ... + X_5$. We have that 10% is the probabilty for the patient to be cured without any drugs and we found out through experiment that 20% was the actual probability that a person will be cured without any treatment. I found 20% by, $(1 - \frac {4}{5})$. Now let $P_{20\%}$ denote the probability of actual percentage. Now testing that the drug has no effect we have that $P_{20\%} (X_1 + ... X_5 \ge 10\%)$ so$P_{20\%} (\frac {Z - n \mu}{\sigma \sqrt n} \ge \frac {10- n \mu}{\sigma \sqrt n} )$ , thus $P(N(0,1) \ge \frac {10- n \mu}{\sigma \sqrt n})$.
My question is how can I look up the probability using the z table or will $P(N(0,1) \ge \frac {10- n \mu}{\sigma \sqrt n})$ suffice?