Final answer:
A 95% confidence interval for the proportion of men with red-green color blindness is calculated using a formula involving the Z-score, sample proportion, and sample size. For a 96% confidence interval with a specified margin of error, the sample size is determined using a formula that includes the Z-score, estimated proportion, and margin of error squared.
Step-by-step explanation:
Constructing Confidence Interval and Determining Sample Size
To answer your questions:
Construct a 95% confidence interval estimate of the proportion of all men with red-green color blindness given that 7 out of 80 men tested have the condition.
Determine the sample size needed to estimate the proportion of male red-green color blindness with a 96% confidence interval and a specified margin of error (part of the question is cut off, but usually, this would specify an exact error value such as ±0.05).
Part a: Given the data, the point estimate (p) for the proportion is 7/80. To calculate the confidence interval, we would use the formula for a proportion:
CI = p ± Z*(√[p(1-p)/n])
Where CI is the confidence interval, Z is the Z-score corresponding to the 95% confidence level (which is about 1.96), p is the sample proportion, and n is the sample size.
First, let's calculate the error bound (EB) for the confidence interval:
EB = Z*√[p(1-p)/n]
Now, applying the values we have:
p = 7/80 = 0.0875
Z = 1.96 for 95% confidence level
n = 80
EB = 1.96*√[0.0875(1-0.0875)/80]
After calculating EB, you could find the confidence interval by:
CI = 0.0875 ± EB
This gives us the range within which we are 95% confident the true proportion of men with red-green color blindness lies.
Part b: The formula for determining the sample size (n) for a given margin of error (E), confidence level (Z), and estimated proportion (p) is:
n = (Z^2 · p(1-p)) / E^2
Without the actual margin of error specified, we cannot calculate the exact sample size needed for a 96% confidence level. However, you can substitute the Z-score for 96% confidence and your desired margin of error into the formula to find 'n'.