A researcher constructs a confidence interval for a population proportion using a sample of size 50. The value of is .3, and the resulting confidence interval is determined to be (.1323, .4677). What's the level of confidence for this interval?
A. 95%
B. Not enough information to answer.
C. 99%
D. 80%
E. 90%
To determine the level of confidence for a confidence interval, we need to look at the range of values within the interval. In this case, the confidence interval is (.1323, .4677).
To do this, we can calculate the margin of error by finding the difference between the upper and lower bounds of the confidence interval. In this case, the margin of error is:
Margin of error = (Upper bound - Lower bound) / 2
= (.4677 - .1323) / 2
= .1677
Next, we calculate the critical value, which is the number of standard deviations from the mean that corresponds to the desired level of confidence. The standard deviation for a proportion is given by the formula:
Standard deviation = sqrt((p * (1-p)) / n)
where p is the proportion in the sample and n is the sample size. In this case, p = .3 and n = 50, so the standard deviation is:
Standard deviation = sqrt((.3 * (1-.3)) / 50)
= sqrt(.21/50)
= sqrt(.0042)
≈ .0649
To find the critical value, we need to look up the z-score corresponding to the desired level of confidence in a standard normal distribution table. The z-score is the number of standard deviations away from the mean. For example, for a 95% confidence level, we need to find the z-score that leaves 2.5% in each tail of the distribution.
The z-score corresponding to a 95% confidence level is approximately 1.96.
Finally, we can use the formula for the margin of error to find the level of confidence:
Margin of error = z * (Standard deviation / sqrt(n))
Rearranging the formula:
z = (Margin of error * sqrt(n)) / Standard deviation
Substituting the values we have:
1.96 = (.1677 * sqrt(50)) / .0649
1.96 ≈ 7.526
Therefore, the level of confidence for this interval is approximately 95%. So, the correct answer is A. 95%.