A ramdom sample of 150 workers showed that 90 workers are satisfied with the new medical scheme. Find a 90% confidence interval for all satisfied workers.Give the upper limit.

Answer 1

To find the 90% confidence interval for all satisfied workers, we first need to calculate the sample proportion of workers who are satisfied.

Sample proportion (p̂) = Number of satisfied workers / Total number of workers
= 90/150
= 0.6

Next, we calculate the standard error (SE) using the formula:
SE = sqrt((p̂ * (1 - p̂)) / n)
= sqrt((0.6 * (1 - 0.6)) / 150)
= sqrt((0.6 * 0.4) / 150)
= sqrt(0.24 / 150)
= sqrt(0.0016)
≈ 0.04

Since we want a 90% confidence interval, we need to find the critical value for a 90% confidence level.

A 90% confidence level corresponds to a 5% significance level (α/2 = 0.05/2 = 0.025) and a critical value of z = 1.645.

Now, we can calculate the margin of error (ME) using the formula:
ME = Critical value * Standard error
= 1.645 * 0.04
= 0.0658

Finally, the confidence interval can be calculated as:
Confidence interval = Sample proportion ± Margin of error
= 0.6 ± 0.0658
≈ 0.5342 to 0.6658

Therefore, the upper limit of the 90% confidence interval for all satisfied workers is approximately 0.6658.