A simple random sample of 400 individuals provides 124 Yes responses.
(a) What is the point estimate of the proportion of the population that would provide Yes responses?
b) What is your estimate of the standard error of the proportion, ? (Round your answer to four decimal places.)
(c) Compute the 95% confidence interval for the population proportion. (Round your answers to four decimal places.) to
The Correct Answer and Explanation is :
Solution:
(a) The point estimate of the proportion of the population that would provide “Yes” responses is:
[
\hat{p} = \frac{x}{n} = \frac{124}{400} = 0.31
]
(b) The standard error of the proportion is:
[
SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{(0.31)(1-0.31)}{400}} = 0.0231
]
(c) The 95% confidence interval for the population proportion is given by:
[
\hat{p} \pm Z_{\alpha/2} \cdot SE_{\hat{p}}
]
Using the standard normal table, the critical value for a 95% confidence level is:
[
Z_{\alpha/2} = 1.96
]
[
\text{Margin of Error} = 1.96 \times 0.0231 = 0.0453
]
[
\text{Lower Bound} = 0.31 – 0.0453 = 0.2647
]
[
\text{Upper Bound} = 0.31 + 0.0453 = 0.3553
]
Thus, the 95% confidence interval for the population proportion is (0.2647, 0.3553).
Explanation:
- Point Estimate: The best estimate of the proportion of the population that would respond “Yes” is calculated by dividing the number of “Yes” responses by the total sample size.
- Standard Error: This measures the variability of the sample proportion and tells us how much we expect the sample proportion to fluctuate from sample to sample.
- Confidence Interval: It provides a range of values in which the true population proportion is likely to fall, with 95% confidence. The margin of error depends on both the standard error and the confidence level (z-score).
The graph above visually represents the normal distribution of the sample proportion, with the confidence interval shaded in gray.