In a simple random sample of 90 patients who saw a certain dentist, 8 patients had their teeth whitened. What is the 95% confidence interval for the percent of all the dentist's patients who had their teeth whitened? show work
(5.9, 11.9)
(4.0, 13.8)
(3.0, 14.8)
(1.2, 16.6
To calculate the confidence interval for the percent of all the dentist's patients who had their teeth whitened, you can use a formula based on the sample proportion.
Let's define:
n = sample size = 90 (the number of patients in the sample)
x = number of patients who had their teeth whitened = 8
First, calculate the sample proportion (p-hat) by dividing the number of patients who had their teeth whitened (x) by the sample size (n):
p-hat = x / n = 8 / 90 ≈ 0.0889
Next, determine the margin of error (E) using the formula:
E = Z * sqrt((p-hat * (1 - p-hat)) / n)
where Z is the Z-score corresponding to the desired confidence level. For a 95% confidence level, Z ≈ 1.96.
Calculate E:
E = 1.96 * sqrt((0.0889 * (1 - 0.0889)) / 90) ≈ 0.0342
Finally, calculate the confidence interval:
Lower bound = p-hat - E
Upper bound = p-hat + E
Lower bound = 0.0889 - 0.0342 ≈ 0.0547
Upper bound = 0.0889 + 0.0342 ≈ 0.1231
Therefore, the 95% confidence interval for the percent of all the dentist's patients who had their teeth whitened is approximately (5.5%, 12.3%).
None of the answer options provided match the calculated confidence interval, so it seems there might be an error in the answer choices.