Question

You may need to use the appropriate appendix table or technology to answer this question.

A population proportion is 0.40. A sample of size 300 will be taken and the sample proportion
p
will be used to estimate the population proportion. (Round your answers to four decimal places.)
(a)
What is the probability that the sample proportion will be within ±0.03 of the population proportion?
(b)
What is the probability that the sample proportion will be within ±0.05 of the population proportion?

Answer 1

To determine the probability that the sample proportion will be within ±0.03 of the population proportion, we use the standard error formula and the normal distribution. The probability is 0.978. For ±0.05, the probability is 0.962.

Step-by-step explanation:

To determine the probability that the sample proportion will be within ±0.03 of the population proportion, we need to use the standard error formula:

SE = sqrt((p * (1-p)) / n)

SE = sqrt((0.40 * (1-0.40)) / 300)

SE = 0.0292

Using a normal distribution, we can calculate the probability by finding the area under the curve between the range of ±0.03 around the mean.

The probability is given by:

1 - 2 * P(Z < 0.03)

Using technology or an appendix table, we can find that P(Z < 0.03) is approximately 0.511.

Therefore, the probability that the sample proportion will be within ±0.03 of the population proportion is 1 - (2 * 0.511) = 0.978.

To calculate the probability that the sample proportion will be within ±0.05 of the population proportion, we follow the same steps but use a range of ±0.05 instead of ±0.03. Using technology or an appendix table, we find that P(Z < 0.05) is approximately 0.519.

Therefore, the probability that the sample proportion will be within ±0.05 of the population proportion is 1 - (2 * 0.519) = 0.962.