Answer:
(3.01%, 14.77%)
Step-by-step explanation:
The confidence interval of a proportion is:
CI = p ± SE × CV,
where p is the proportion, SE is the standard error, and CV is the critical value (either a t-score or a z-score).
We already know the proportion: 8/90. But we need to find the standard error and the critical value.
The standard error is:
SE = √(p (1-p) / n)
SE = √((8/90) * (82/90) / 90)
SE = 0.03
To find the critical value, we must first find the alpha level and the degrees of freedom.
The alpha level for a 95% confidence interval is:
α = (1 - 0.95) / 2 = 0.025
The degrees of freedom is one less than the sample size:
df = n - 1 = 90 - 1 = 89
Since df > 30, we can approximate this with a normal distribution.
If we look up the alpha level in a z score table, we find the z-score is 1.96. That's our critical value. CV = 1.96.
Now we can find the confidence interval:
CI = 8/90 ± 0.03 * 1.96
CI = 0.0889 ± 0.0588
CI = (0.0301, 0.1477)
So we are 95% confident that the percent of patients who had their teeth whitened is between 3.01% and 14.77%.
