A statistician is testing the null hypothesis that exactly half of all engineers will still be in the profession 10 years after receiving their bachelor's. She took a random sample of 200 graduates from the class of 1979 and determined their occupations in 1989. She found that 111 persons were still employed primarily as engineers. Construct a 95% confidence interval estimate for the proportion of engineers remaining in the profession. lower limit=? and upper limit=?
To construct a 95% confidence interval estimate for the proportion of engineers remaining in the profession, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
The sample proportion is the proportion of engineers in the sample, which is given by:
Sample Proportion = 111/200 = 0.555
To calculate the margin of error, we need to determine the critical value for a 95% confidence interval. Since the sample size is large (n > 30), we can use the standard normal distribution.
The critical value for a 95% confidence interval is approximately 1.96 (obtained from the standard normal distribution table).
Margin of Error = Critical Value * Standard Error
The standard error (SE) is the standard deviation of the sampling distribution, which is given by:
Standard Error = sqrt(p * (1 - p) / n)
where p is the sample proportion and n is the sample size.
Standard Error = sqrt(0.555 * (1 - 0.555) / 200) ≈ 0.034
Now, we can calculate the margin of error:
Margin of Error = 1.96 * 0.034 ≈ 0.067
Finally, we can construct the confidence interval estimate:
Confidence Interval = Sample Proportion ± Margin of Error
Confidence Interval = 0.555 ± 0.067
To find the lower and upper limits of the confidence interval, we subtract and add the margin of error from the sample proportion:
Lower Limit = 0.555 - 0.067 ≈ 0.488
Upper Limit = 0.555 + 0.067 ≈ 0.622
Therefore, the 95% confidence interval estimate for the proportion of engineers remaining in the profession is approximately 0.488 to 0.622.