## StClowers one year ago A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 512 babies were born, and 256 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective? ___ <p > ____

1. douglaswinslowcooper

p = 312/624 = 0.500 best estimate mean is p N = (0.500) 624 = 312 obvious standard deviation is sqrt[Np(1-p)] = 12.5 If we approximate the binomial by a Normal we would be using mu = 312, s.d = 12.5 and the 99% C.I. is mean +- 2.57 s.d we convert the mean to p by dividing by 624 so corresponding C.I. for p would be 0.500+ - (2.57)(12.5)/624 or 0.500+- 0.051

2. StClowers

what about plugging in the numbers 256/512...I cant figure out how you did the standard deviation

3. douglaswinslowcooper

Formula for the standard deviation of a binomial distribution with N data and probability p is sqrt(Npq), where q = 1-p

4. StClowers

ok so what about the 99% part.....here is what I have so far....I got a different set of numbers because i got the first one wrong....so: p = 256/512 = .500 n= (,500)512) = 256 $\sqrt{(256)(.500)(.5)}$ mean = 256 sd = .5 at this point I am stuck...whats the rest?

5. douglaswinslowcooper

For a Normal distribution the 99.5%tile is mean + 2.57 s.d and the 0.5%tile is mean - 2.57 s.d., so the 99% confidence interval (0.5% to 99.5%) is the mean plus or minus 2.57 times the s.d.

6. StClowers

OK....I come up with .500 < p < .0502 but I dont think that is right :-/

7. douglaswinslowcooper

(mean - 2.57 s.d)/624 < p < (mean + 2.57 s.d)/624 I got 0.495 < p < 0.505

8. StClowers

oh wait....try using these numbers....the 624 is not the right one here is the problem again... A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 512 babies were born, and 256 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective?

9. StClowers

so that makes it .494 < p < .505...did you get that? I am doing something wrong I think

10. douglaswinslowcooper

".494" vs. ".495" we seem to agree. Check for rounding, yours or mine!

11. douglaswinslowcooper

The 512 and 256 values give the same p= 0.500 and almost the same s.d. when you work it out as we did for the 624 and 312 case. Just recalculate mean and s.d. and then divide by 512, not 624.

12. StClowers

ok thats what I did...so lets see if that works...well......it says its not right :( I rounded it correctly...

13. StClowers

ok so i think where i am going wrong is in the sqrt part.....I am calculating n=256 p= .500 q= .5 ...do you see where I am confused?

14. StClowers

if I do the sqrt equation like this: $\sqrt{(256)(.5)(.5)}$ then I come up with 8...but where do I plug that in?

15. douglaswinslowcooper

Wrong square root, need sqrt(512 x 0.5 x 0.5) = s.d. =11.31 p limits are (mean +- 2.57 s.d)/512 as done above. Good night.

16. StClowers

Ok great!! Thank YOU!!