Final answer:
To estimate the necessary sample size, use the formula: n = (Z^2 * p * q) / E^2. For a 95% confidence interval and a margin of error of 0.10, the sample size required is at least 385 teachers. To calculate a 95% confidence interval for the proportion of teachers who think the semester system is suitable, use the formula: CI = p-hat ± Z * √( (p-hat * q-hat) / n ). The 95% confidence interval is (0.511, 0.689).
Step-by-step explanation:
In order to determine the necessary sample size to estimate the proportion with 95% confidence and a margin of error of 0.10, we can use the formula:
n = (Z^2 * p * q) / E^2
Where n is the required sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated proportion, q is 1 - p, and E is the desired margin of error.
Plugging in the given values, we get:
n = (1.96^2 * 0.5 * 0.5) / (0.10^2) = 384.16
Therefore, a sample size of at least 385 teachers is necessary to estimate the proportion with 95% confidence and a margin of error of 0.10.
To calculate a 95% confidence interval for the proportion of teachers who think the semester system is suitable, we can use the formula:
CI = p-hat ± Z * √( (p-hat * q-hat) / n )
Where CI is the confidence interval, p-hat is the sample proportion, Z is the z-score corresponding to the desired confidence level, q-hat is 1 - p-hat, and n is the sample size.
Plugging in the given values, we get:
CI = 0.6 ± 1.96 * √( (0.6 * 0.4) / 120) = 0.6 ± 1.96 * √( 0.002 ) = 0.6 ± 1.96 * 0.0447 = (0.511, 0.689)
Therefore, the 95% confidence interval for the proportion of teachers who think the semester system is suitable is (0.511, 0.689).