High school student actually attempting this math problem, please help?

Question

I did the below problem but my AP Stats teacher told me I did them wrong. I have no idea how to do them correctly so I get the right answer. Can you please help me?

4. 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.

b. Calculate the exact binomial probability
(I got .0405)

d. Recalculate the probability in part b using a normal approximation without the continuity correction. 
(I got .0559)

e. Recalculate the probability in part b using a normal approximation with the continuity correction. 
(I got .0634)

I really need help with this, please explain how to get the right answer

Mark · Accepted Answer

a. Let x = number of women in sample. Then P(more than half are women) = b(x ≥ 26 | n=50, p = 0.4)

b. P(x ≥ 26) = 0.0573437605422 (I used http://stattrek.com/Tables/Binomial.aspx with 0.4, 50 and 26 as the parameters) Doing this by hand is absurd. You can also use the binomcdf function if you have a TI-83 or similar calculator as 1– binomcdf(25)

c. 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

...█   
─┬───────►∞
...26 ...............

Normal P(X ≥ 26) = P (Z ≥ 1.732) ≈ 0.04163

d. 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

Notice how much closer this is to the exact binomial probability.

britney · Answer

no

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High school student actually attempting this math problem, please help?

2 Answers