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Answer:
a) The sampling distribution for n=540 has a mean sample proportion of p=0.37 and a standard deviation of σs=0.0208.
b) probability = 0.99998
d) probability = 0.99874
e) You gain 0.12% in probability for an increase of 80% in sample size.
The increase in sample size is not justified by the increase in probability, for this margin of error (Δp=0.09).
Step-by-step explanation:
a) We have a known population proportion π=0.37 and we have to describe the sampling distribution when the sample size is n=540.
The mean sample proportion is expected to be the same as the population proportion:
The standard deviation of the sampling will be the population standard deviation divided by the square root of the sample size:
Then, we can say that the sampling distribution will have a p=0.37 and a standard deviation σs=0.0208.
b) We have to calculate the probability that the sample proportion will be within 0.09 of the population proportion.
We can calculate the z-value as:
As the distribution is symmetrical, we can calculate the probabilty that he sample proportion will be within 0.09 of the population proportion as:
probability = 0.99998
d. Now the sample is smaller (n=300), so the standard deviation of the samping distribution:
We have to recalculate the z-scores:
And the probability is:
probability = 0.99874
e. The increase in sample size is 80%
and the increase in probability is 0.12%
You gain 0.12% in probability for an increase of 80% in sample size.
The increase in sample size is not justified by the increase in probability, for this margin of error (Δp=0.09).