Answer:
Explanation:
To calculate the probability that the sample proportion will differ from the population proportion by more than 0.03, we can use the formula for the standard error of the sample proportion:
Standard Error (SE) = sqrt[(p * (1 - p)) / n]
Where:
- p is the population proportion (0.06 in this case).
- n is the sample size (314 in this case).
Now, we want to find the probability that the sample proportion will differ from the population proportion by more than 0.03. This means we're interested in finding the probability of:
|Sample Proportion - Population Proportion| > 0.03
To do this, we can use the z-score formula:
z = (Sample Proportion - Population Proportion) / SE
We want to find the probability that |z| > (0.03 / SE). This probability can be found using a standard normal distribution table or calculator.
First, calculate SE:
SE = sqrt[(0.06 * (1 - 0.06)) / 314]
SE ≈ 0.0246 (rounded to four decimal places)
Now, find the z-score for |z| > (0.03 / SE):
z = |0.03 / 0.0246| ≈ 1.2195 (rounded to four decimal places)
Next, find the probability that |z| > 1.2195 using a standard normal distribution table or calculator. In this case, you're looking for the probability in the tails of the distribution, so you'll find P(|z| > 1.2195).
The probability depends on the specific normal distribution table or calculator you're using, but it will be a value between 0 and 1. Round the answer to four decimal places as requested.