The 95% confidence interval for the proportion of smokers who quit after 10 days of treatment is (0.749, 0.851).
Let's calculate the 95% confidence interval for the proportion of smokers who quit after 10 days of treatment.
Step 1: Calculate the sample proportion
The sample proportion is the proportion of smokers in the sample who quit after 10 days of treatment. In this case, the sample proportion is:
p = 80/300 = 0.8
Step 2: Calculate the standard error
The standard error is a measure of the variability of the sample proportion. It is calculated as follows:
SE = sqrt(p(1-p)/n)
where:
p is the sample proportion
n is the sample size
In this case, the standard error is:
SE = sqrt(0.8(1-0.8)/300) = 0.026
Step 3: Calculate the margin of error
The margin of error is the amount by which we can be confident that the true proportion of smokers who quit is within our confidence interval. It is calculated as follows:
ME = Z*SE
where:
Z is the z-score corresponding to the desired confidence level
SE is the standard error
The z-score for a 95% confidence interval is 1.96. Therefore, the margin of error is:
ME = 1.96*0.026 = 0.051
Step 4: Calculate the confidence interval
The confidence interval is the interval of values that we are confident includes the true proportion of smokers who quit. It is calculated as follows:
CI = p +/- ME
In this case, the confidence interval is:
CI = 0.8 +/- 0.051 = (0.749, 0.851)
Therefore, the 95% confidence interval for the proportion of smokers who quit after 10 days of treatment is (0.749, 0.851).