Final answer:
To obtain a 90% confidence interval with a margin of error of 0.08 for a population proportion estimated at 0.25, a sample size of at least 159 people is required. The p-value indicates the probability of observing a sample proportion at least as extreme as the one obtained, assuming the null hypothesis is true.
Step-by-step explanation:
To determine how large a sample should be taken to provide a 90% confidence interval with a margin of error of 0.08 when the planning value for the population proportion is p*=0.25, we use the formula for sample size in proportion studies:
n = (Z² * p * (1 - p)) / E²
Where:
- Z is the z-score corresponding to the desired confidence level (1.645 for 90% confidence).
- p is the estimated population proportion (0.25 in this case).
- E is the desired margin of error (0.08).
Plugging these values into the formula, we get:
n = (1.645² * 0.25 * (1 - 0.25)) / 0.08²
n = (2.706025 * 0.25 * 0.75) / 0.0064
n = 158.14
You would need to survey a minimum of 159 people (since we always round up to the next whole number in sample size calculations) to be 90% confident that the estimated population proportion is within a margin of error of 0.08.
The p-value in this context refers to the probability that the observed sample proportion would be at least as extreme as it is, assuming the null hypothesis is true. For instance, if the null hypothesis states the population proportion is 0.25, and the sample proportion is found to be significantly different, the p-value helps us determine the likelihood of observing such a result due to random chance alone.