Final answer:
To construct the 95% confidence interval for the population proportion of grade A in math classes, calculate the sample proportion (p), the Standard Error (SE), the margin of error (ME), and plug these into the confidence interval formula. The calculated interval of 0.205 to 0.263 is closest to Option D, which is 0.197 to 0.270.
Step-by-step explanation:
To construct a 95% confidence interval for the population proportion of grade A in math classes in Spring 2020, we can use the following formula:
Confidence Interval (CI) = p ± (z-score * Standard Error (SE))
Where:
- p is the sample proportion of students who got an A, which is calculated as the number of students with an A (193) divided by the total number of students surveyed (824).
- The z-score corresponds to the 95% confidence level, which is approximately 1.96.
- Standard Error (SE) is calculated as √(p(1-p)/n), where n is the total number of students surveyed (824).
Let's calculate the 95% CI step by step:
- Calculate the sample proportion (p): p = 193/824 ≈ 0.234.
- Calculate the Standard Error (SE): SE = √(0.234(1-0.234)/824) ≈ 0.015.
- Calculate the margin of error (ME): ME = z-score * SE = 1.96 * 0.015 ≈ 0.029.
- Calculate the confidence interval: CI = p ± ME = 0.234 ± 0.029.
This gives us a confidence interval of approximately 0.205 to 0.263.
Based on the options provided, the closest interval is Option D. 0.197 to 0.270.