Final answer:
The sample proportion for females is 0.428, the standard error of the proportion is 0.0313, the margin of error at a 95% confidence level is 0.0614, and the 95% confidence interval for the proportion of females is approximately (0.37, 0.49).
Step-by-step explanation:
To answer the student's question, we will perform a step-by-step calculation for each part:
- Sample proportion for females (p'): This is found by dividing the number of females by the total sample size. p' = 107 / 250 = 0.428.
- Standard error of the proportion (SE): The standard error can be calculated using the formula SE = sqrt[p'(1 - p') / n], where p' is the sample proportion and n is the sample size. SE = sqrt[0.428(1 - 0.428) / 250] = sqrt[0.428 * 0.572 / 250] = sqrt[0.000977856] = 0.0313.
- Margin of Error (E): The margin of error for a 95% confidence interval can be found using the formula E = Z * SE, where Z is the Z-score corresponding to the confidence level. For 95% confidence, Z is typically 1.96. E = 1.96 * 0.0313 = 0.0614.
- 95% confidence interval: The confidence interval can be calculated as p' ± E. Thus, the interval is 0.428 ± 0.0614, which gives us (0.3666, 0.4894) when rounded to the nearest hundredth.
All the calculations are completed assuming the sampling distribution of the proportion is approximately normal, which is reasonable given the large sample size.