From random sample of 200 college students, 40 students indicated that they smoke cigarettes. Use the information from the sample to calculate a 95% confidence interval for the population proportion of college students that smoke cigarettes (round to nearest hundredth).
A. (0.14, 0.26)
B. (38, 42)
C. (0.17, 0.23)
D. (0.16, 0.24)
Formula:
CI95 = p ± (1.96)[√(pq/n)]
...where p = x/n, q = 1 - p, and n = sample size.
Note: ± 1.96 represents 95% confidence interval using a z-table.
For p in your problem: 40/200 = 0.2
For q: 1 - p = 1 - 0.2 = 0.8
n = 200
I let you take it from here to calculate the interval and determine your answer from the selections given.
To calculate a confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
The sample proportion is calculated by dividing the number of students who indicated they smoke cigarettes (40) by the total number of students in the sample (200):
Sample Proportion = 40/200 = 0.2
The margin of error is calculated by multiplying the critical value associated with the desired confidence level by the standard error of the proportion. For a 95% confidence level, the critical value is approximately 1.96.
The standard error of the proportion is calculated as the square root of (sample proportion * (1 - sample proportion)) divided by the square root of the sample size:
Standard Error = √((0.2 * (1 - 0.2))/200) ≈ 0.024
Multiplying the standard error by the critical value gives us the margin of error:
Margin of Error = 1.96 * 0.024 ≈ 0.047
Finally, we can construct the confidence interval:
Confidence Interval = 0.2 ± 0.047 = (0.153, 0.247)
Rounding to the nearest hundredth, the correct answer is D. (0.16, 0.24)