In a simple random sample of 80 patients that go to a certain dentist, 9 patients had their teeth whitened. which interval is the 95% confidence interval for the percent of all the dentist's patients who had their teeth whitened?
(4.4, 18.2)
(8.4, 14.2)
(5.5, 17.1)
(7.8, 14.8)
The first one is the answer. Just took the test.
To find the 95% confidence interval for the percent of all the dentist's patients who had their teeth whitened, we can use the formula for confidence interval for proportions.
The formula for the confidence interval is:
CI = p̂ ± z * √[ (p̂ * (1 - p̂)) / n]
Where:
- p̂ is the sample proportion (number of patients who had their teeth whitened divided by the total sample size).
- z is the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of 1.96).
- n is the sample size.
In this case, the sample proportion is 9/80 = 0.1125.
Using the formula, we can calculate the confidence interval:
CI = 0.1125 ± 1.96 * √[(0.1125 * (1 - 0.1125)) / 80]
Calculating the values within the confidence interval, we get:
CI = 0.1125 ± 1.96 * √[0.09969375 / 80]
CI = 0.1125 ± 1.96 * 0.03542378
CI = 0.1125 ± 0.06944137
CI ≈ (0.0431, 0.1819)
To convert this interval into percentages, we can multiply it by 100:
CI ≈ (4.31%, 18.19%)
The correct answer is therefore (4.4, 18.2).