Final answer:
The 95% confidence interval for the proportion of Iowa teachers thinking of retiring in the next five years is (0.209504, 0.326856). Applying this to the entire state with 36,000 teachers, we estimate that between approximately 7534 and 11766 teachers are considering retirement in that timeframe.
Step-by-step explanation:
To calculate the 95% confidence interval for the proportion of Iowa teachers thinking of retiring in the next five years, we use the formula for a confidence interval for a population proportion:
- ∆ = √[p'(1 - p') / n]
- CI = p' ± Z*∆
where:
- p' is the sample proportion
- n is the sample size
- Z* is the critical value for the 95% confidence level, which is approximately 1.96.
First, we calculate the sample proportion (p'):
p' = 59/220 = 0.26818 (rounded to five decimal places)
Next, we calculate the standard error (∆):
∆ = √[(0.26818 * (1 - 0.26818)) / 220] = √[(0.26818 * 0.73182) / 220] ≈ √[0.000895) ≈ 0.029916
Then, we find the error margin:
E = Z*∆ = 1.96 * 0.029916 ≈ 0.058676
Now, the 95% confidence interval is:
CI = 0.26818 ± 0.058676
CI = (0.209504 , 0.326856) (rounded to five decimal places)
To estimate the number of teachers in all of Iowa, we use the interval proportion and the total number of teachers, which is 36,000.
Number of teachers thinking of retiring:
Lower estimate = 0.209504 * 36,000 ≈ 7534
Upper estimate = 0.326856 * 36,000 ≈ 11766
Therefore, we can say with 95% confidence that between approximately 7534 and 11766 teachers in Iowa are thinking of retiring in the next five years.