Final answer:
Plugging in the values, the 95% confidence interval for the proportion is 0.1607 to 0.2815. To find the confidence interval for the number of teachers, multiply the lower and upper limits by the total number of teachers in Arkansas.
Step-by-step explanation:
To find a 95% confidence interval for the proportion of Arkansas teachers who were thinking of retiring in the next five years, you can use the formula:
In this case, the sample proportion is 42/190 = 0.2211.
From the appendix table, the z-score corresponding to a 95% confidence level is approximately 1.96.
Plugging in the values, we get:
Lower Limit = 0.2211 - 1.96 * √((0.2211(1-0.2211))/190) = 0.1607
Upper Limit = 0.2211 + 1.96 * √((0.2211(1-0.2211))/190) = 0.2815
Therefore, the 95% confidence interval for the proportion of Arkansas teachers who were thinking of retiring in the next five years is 0.1607 to 0.2815.
To determine the 95% confidence interval for the number of teachers thinking of retiring in the next five years in the entire state, you can multiply the lower and upper limits of the proportion by the total number of teachers in Arkansas. In this case, there are 38,000 teachers in Arkansas.
Lower Limit = 0.1607 * 38,000 = 6,112
Upper Limit = 0.2815 * 38,000 = 10,712
Therefore, the 95% confidence interval for the number of teachers thinking of retiring in the next five years in the entire state is 6,112 to 10,832 teachers.