Final answer:
To estimate the current proportion of homes with a stay-at-home parent in which the father is the stay-at-home parent, the sample size needed at a 95% confidence level is 379. At a 99% confidence level, the sample size needed is 1007.
Step-by-step explanation:
To determine the sample size needed to estimate the current proportion of homes with a stay-at-home parent in which the father is the stay-at-home parent, we can use the formula:
n = (Zα/2)2 * p * (1-p) / E2
Where:
- n is the required sample size
- Zα/2 is the critical value of the standard normal distribution corresponding to the desired confidence level. For a 95% confidence level, Zα/2 is approximately 1.96. For a 99% confidence level, Zα/2 is approximately 2.58.
- p is the estimated proportion of homes with a stay-at-home parent in which the father is the stay-at-home parent. In this case, p = 0.26
- E is the desired margin of error. In this case, E = 0.04
For a 95% confidence level:
n = (1.96)2 * 0.26 * (1-0.26) / 0.042
n = 0.9604 * 0.26 * 0.74 / 0.0016
n ≈ 378.2283
Rounding up to the nearest whole number, the sample size needed is 379.
For a 99% confidence level:
n = (2.58)2 * 0.26 * (1-0.26) / 0.042
n = 0.025874 * 0.26 * 0.74 / 0.0016
n ≈ 1006.2308
Rounding up to the nearest whole number, the sample size needed is 1007.