AP Statistics (Answer according to what an AP stats or college freshman would be

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

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