To calculate the 95% confidence interval for the proportion of State College students who smoke, you can use the formula:
Confidence Interval = Sample Proportion ± (Z * √(Sample Proportion * (1 - Sample Proportion) / Sample Size))
Where:
- Sample Proportion is the proportion of students who smoke in the sample,
- Z is the z-score corresponding to the desired confidence level (e.g., 1.96 for a 95% confidence level),
- √ represents the square root,
- Sample Size is the number of students in the sample.
In this case, the sample proportion is 41/112 = 0.3661.
The z-score for a 95% confidence level is 1.96 (you can find this value from a standard normal distribution table or use a statistical calculator).
Now, plug these values into the formula:
Confidence Interval = 0.3661 ± (1.96 * √(0.3661 * (1 - 0.3661) / 112))
Calculating the expression inside the square root first:
√(0.3661 * (1 - 0.3661) / 112) = 0.0604
Now, substitute this value into the formula:
Confidence Interval = 0.3661 ± (1.96 * 0.0604)
Next, calculate the two bounds of the confidence interval:
Upper Bound = 0.3661 + (1.96 * 0.0604) = 0.4327
Lower Bound = 0.3661 - (1.96 * 0.0604) = 0.2995
Therefore, the 95% confidence interval for the proportion of State College students who smoke is approximately 0.2995 to 0.4327.