# of successes = 155, # of observations = 405, confidence interval level = .95, what is the confidence interval?

To find the confidence interval, we can use the formula for a confidence interval for a proportion:

Confidence interval = sample proportion ± Z * sqrt((sample proportion * (1 - sample proportion)) / sample size)

Given data:
Number of successes (x) = 155
Number of observations (n) = 405
Confidence level = 0.95

First, we calculate the sample proportion:
sample proportion = x / n = 155 / 405 = 0.3827

Next, we calculate the Z-score for the confidence level of 0.95:
Since the confidence level is 0.95, the critical Z value (Zα/2) is 1.96 for a two-tailed test.

Now, we calculate the margin of error:
margin of error = Z * sqrt((sample proportion * (1 - sample proportion)) / sample size)
margin of error = 1.96 * sqrt((0.3827 * (1 - 0.3827)) / 405) = 0.0473

Finally, we calculate the confidence interval:
Lower bound = sample proportion - margin of error
Upper bound = sample proportion + margin of error

Confidence interval = [0.3827 - 0.0473, 0.3827 + 0.0473] = [0.3354, 0.4300]

Therefore, the confidence interval is \boxed{[0.3354,0.4300]} .