The standard deviation of a sample proportion p gets smaller as the sample size
ID: 3260366 • Letter: T
Question
The standard deviation of a sample proportion p gets smaller as the sample size n increases. If the population proportion is p = 0.52, how large a sample is needed to reduce the standard deviation of p to = 0.004? (The 689599.7 rule then says that about 95% of all samples will have p within 0.01 of the true p. Round your answer to up to the next whole number.)
What is the probability that a sample proportion p falls between 0.46 and 0.52 if the sample is an SRS of size n = 5000? (Round your answer to four decimal places.)
Combine these results to make a general statement about the effect of larger samples in a sample survey. (Which is the correct statment?)
a)Larger samples have no effect on the probability that p will be close to the true proportion p.
b)Larger samples give a larger probability that p will be close to the true proportion p.
c)Larger samples give a smaller probability that p will be close to the true proportion p.
Explanation / Answer
a)
standard deviation = 0.004
=>
sqrt( p * (1-p) /n) = 0.004
=>
sqrt( 0.52 * 0.48 /n ) = 0.004
=>
n = 15600
b)
standard error = sqrt( 0.52 * 0.48 / 5000) = 0.007065
P( 0.46 < x < 0.52)
= P( X < 0.52) - P(X < 0.46)
= P(Z < 0.52 - 0.52/0.007065) - P(Z < 0.46- 0.52 / 0.007065)
= P(Z < 0) - P(Z < -8.49)
= 0.5 - 0
= 0.5
c)
Larger samples have no effect on the probability that p will be close to the true proportion p.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.