A researcher wishes to estimate the proportion of college students who cheat on exams. A poll of 410 college students showed that 15% of them had, or intended to, cheat on examinations. Find the 95% confidence interval for the population proportion. Round to four decimal places.
To calculate the 95% confidence interval for the population proportion, we can use the formula:
Confidence Interval = sample proportion ± margin of error
First, let's calculate the sample proportion. Given that 15% of the 410 college students have or intend to cheat on exams, the sample proportion would be 0.15.
Next, we need to calculate the margin of error. The margin of error can be determined using the following formula:
Margin of Error = z * sqrt((p̂ * (1 - p̂)) / n)
Where:
- z represents the z-score corresponding to the desired confidence level (in this case, 95% confidence level).
- p̂ represents the sample proportion.
- n represents the sample size.
The z-score associated with a 95% confidence level is approximately 1.96. Substituting the values into the formula, we get:
Margin of Error = 1.96 * sqrt((0.15 * (1 - 0.15)) / 410)
Calculating this expression will give us the value of the margin of error. Let's compute the margin of error using a calculator or software.
Margin of Error ≈ 0.0279 (rounded to four decimal places)
Finally, we can construct the confidence interval by subtracting and adding the margin of error from the sample proportion:
Confidence Interval = 0.15 ± 0.0279
Calculating the confidence interval, we get:
Confidence Interval ≈ (0.1221, 0.1779) (rounded to four decimal places)
Therefore, the 95% confidence interval for the population proportion of college students who cheat on exams is approximately 0.1221 to 0.1779.