## Question 501:

1

a). It's usually Ok to use the normal approximation to the binomial is n*p and n*q > 5. We have 24*.68 = 16.32 and 24*.32 = 7.68 so we're ok. The mean would then be n*p  = 16.32 and the standard deviation is n*p*q = SQRT (24*.68*.32) = 2.29

b). We'd find the z-score just like we do if the data were continuous (this data is discrete binary since results were agree or disagree to the reliability question). To account for the discreteness we have to apply what's called a continuity correction by adding .5 to the mean, giving us 16.82.

The z-score is (15.5-16.32)/2.29 = -.358 which has a 1-sided probability of 36%. To check our assumptions, we can use the Binomial probabilities directly. Using the Excel function =BINOMDIST(15,24,0.68,TRUE) gets us 35.1%, which is really close to the 36% approximation.  So there is between a 35 and 36% probability that at most 15 people out of 24 will agree the results are very reliable given 68% of the population agrees.  If you need just one answer then use 36%.