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.
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.
To find the upper limit of the 90% confidence interval for the number of satisfied workers, we can use the formula for a proportion confidence interval.
The formula for the confidence interval is:
CI = p̂ ± Z * √(p̂(1-p̂)/n)
Where:
- p̂ is the sample proportion
- Z is the z-score corresponding to the desired confidence level
- n is the sample size
In this case, we have:
- p̂ (sample proportion) = 90/150 = 0.6
- Z (z-score for 90% confidence level) = 1.645 (obtained from a z-table)
- n (sample size) = 150
Substituting these values into the formula, we get:
CI = 0.6 ± 1.645 * √(0.6(1-0.6)/150)
Calculating the values inside the square root:
√(0.6(1-0.6)/150) ≈ √(0.24/150) ≈ 0.052
Substituting back into the confidence interval formula:
CI = 0.6 ± 1.645 * 0.052
Calculating the bounds of the confidence interval:
Lower limit = 0.6 - 1.645 * 0.052 ≈ 0.515
Upper limit = 0.6 + 1.645 * 0.052 ≈ 0.685
Therefore, the upper limit of the 90% confidence interval for the number of satisfied workers is approximately 0.685.