Suppose you use the observed proportion p of successes on 20

Suppose you use the observed proportion, p?, of successes on 20 trials to estimate the true probability of success. Also assume that the true probability is .3

a.) What is the mean for the sampling distribution of the sample proportion p? ? Give answer to one place past the decimal.

b.) What is the variance of the sampling distribution of the sample proportion p??
(Give decimal answer to three places past decimal.)

c.) Use exact binomial probabilities to calculate the probability that the observed proportion is within .15 of the true probability, that is P( |p? - p|< .15)?
Give your answer to three places after the decimal, without the leading zero.

Solution

a.) What is the mean for the sampling distribution of the sample proportion p? ? Give answer to one place past the decimal.

mean=n*p=20*0.3 = 6

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b.) What is the variance of the sampling distribution of the sample proportion p??
(Give decimal answer to three places past decimal.)

variance= n*p*(1-p)

=20*0.3*0.7

=4.2

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c.) Use exact binomial probabilities to calculate the probability that the observed proportion is within .15 of the true probability, that is P( |p? - p|< .15)?
Give your answer to three places after the decimal, without the leading zero.

P( |p? - p|< .15) = P(-0.15<phat-p <0.15)

=P(-0.15/sqrt(0.3*0.7/20) <(phat-p)/sqrt(p*(1-p)/n) <0.15/sqrt(0.3*0.7/20) )

=P(-1.46<Z<1.46)

=0.8557 (from standard normal table)