AP Statistics (Answer according to what an AP stats or college freshman would be
ID: 3205031 • Letter: A
Question
AP Statistics (Answer according to what an AP stats or college freshman would be expected to know)
1) In a particular faculty 60% of students are men and 40% are women. In a random sample of 50 students what is the probability that more than half are women?
Let the random variable X = number of women in the sample.
Assume X has the binomial distribution with n = 50 and p = 0.4.
(a) Calculate the exact binomial probability.
(b) Recalculate the probability in part (a) using a normal approximation without the continuity correction.
(c) Recalculate the probability in part (a) using a normal approximation with the continuity correction.
Please label your answers clearly so I can follow. Thanks
Explanation / Answer
a. Let us denote as given x = number of women in sample.
now P(more than half are women) = b(x 26 | n=50, p = 0.4)
. P(x 26) = 0.0573437605422 from stastical distribution table
We can also use the binomial cdf function if you have a TI-83 or similar calculator as 1– binomcdf(25)
B) Variance = np(1–p) = 50(.4)(.6) = 12. So SD = 12 3.4641
Mean = np = 50(.4) = 20
If X = 26, then Z = (X–mean)/SD = (26–20)/12 = 6/12 =3 1.732
Normal P(X 26) = P (Z 1.732) 0.04163
C ) Continuity correction: use X = 25.5 (back off half a unit to get entire bar.
Normal P(X 25.5) = P (Z 1.5877) = 0.056177