An article stated, \"Surveys tell us that more than half of America\'s college g

Question

An article stated, "Surveys tell us that more than half of America's college graduates are avid readers of mystery novels." Let p denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion p that is based on a random sample of 205 college graduates.

(a) If p = 0.6, what are the mean value and standard deviation of p? (Round your answers to four decimal places.)

standard deviation = ?

If p = 0.7, what are the mean value and standard deviation of p? (Round your answers to four decimal places.)

standard deviation=?

Does p have approximately a normal distribution in both cases? Explain. (Which one is correct)

(1)Yes, because in both cases np > 10 and n(1 p) > 10.

(2)No, because in both cases np < 10 or n(1 p) < 10.    

(3)No, because when p = 0.6, np < 10.

(4) No, because when p = 0.7, np < 10.

(b) Calculate P(p 0.7) for p = 0.6. (Round your answer to four decimal places.)
Calculate P(p 0.7) for p = 0.7.

(c) Without doing any calculations, how do you think the probabilities in Part (b) would change if n were 380 rather than 205? (Which one is correct)

(1)When p = 0.6, the P(p 0.7) would decrease if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would decrease if the sample size was 380 rather than 205.

(2)When p = 0.6, the P(p 0.7) would remain the same if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would remain the same if the sample size was 380 rather than 205.   

(3)When p = 0.6, the P(p 0.7) would decrease if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would remain the same if the sample size was 380 rather than 205.

(4)When p = 0.6, the P(p 0.7) would remain the same if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would decrease if the sample size was 380 rather than 205.

standard deviation = ?

If p = 0.7, what are the mean value and standard deviation of p? (Round your answers to four decimal places.)

standard deviation=?

Does p have approximately a normal distribution in both cases? Explain. (Which one is correct)

(1)Yes, because in both cases np > 10 and n(1 p) > 10.

(2)No, because in both cases np < 10 or n(1 p) < 10.    

(3)No, because when p = 0.6, np < 10.

(4) No, because when p = 0.7, np < 10.

(b) Calculate P(p 0.7) for p = 0.6. (Round your answer to four decimal places.)
Calculate P(p 0.7) for p = 0.7.

(c) Without doing any calculations, how do you think the probabilities in Part (b) would change if n were 380 rather than 205? (Which one is correct)

(1)When p = 0.6, the P(p 0.7) would decrease if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would decrease if the sample size was 380 rather than 205.

(2)When p = 0.6, the P(p 0.7) would remain the same if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would remain the same if the sample size was 380 rather than 205.   

(3)When p = 0.6, the P(p 0.7) would decrease if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would remain the same if the sample size was 380 rather than 205.

(4)When p = 0.6, the P(p 0.7) would remain the same if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would decrease if the sample size was 380 rather than 205.

Explanation / Answer

a) p= 0.6
Mean = 0.6
sd= sqrt(pq/n) = sqrt(0.6*0.4/205) = 0.03422

If p= 0.7
Mean = 0.7
sd= sqrt(pq/n) = sqrt(0.7*0.3/205) = 0.032

Correct answer: Otpion (1) Yes, because in both cases np > 10 and n(1 p) > 10.
b)
P(p>=0.7)=P(Z >= (0.7-0.6)/0.03422) = P(Z>=2.9223) = 0.00174
P(p>=0.7)=P(Z >= (0.7-0.7)/0.032) = P(Z>=0) = 0.5

c) Correct Answer: (3)When p = 0.6, the P(p 0.7) would decrease if the sample size was 380 rather than 205. When p = 0.7, the P(p 0.7) would remain the same if the sample size was 380 rather than 205.

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