A researcher wishes to estimate the proportion of college students who cheat on exams. A poll of 490 college students showed that 33% of them had, or intended to, cheat on examinations. Find the margin of error for the 95% confidence interval.
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To find the margin of error for a confidence interval, you need to use the formula:
Margin of Error = Critical Value * Standard Error
First, you need to calculate the critical value, which depends on the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.
Next, calculate the standard error. In this case, the proportion of students who cheat is given as 33%, which can be expressed as 0.33. The standard error is then calculated using the formula:
Standard Error = sqrt((p * (1-p)) / n)
where p is the proportion of students who cheat, and n is the sample size.
Substituting the values, we get:
Standard Error = sqrt((0.33 * (1-0.33)) / 490)
Calculating this gives:
Standard Error = sqrt(0.2211 / 490) ≈ 0.021
Finally, we can calculate the margin of error:
Margin of Error = 1.96 * 0.021 ≈ 0.041
Therefore, the margin of error for the 95% confidence interval is approximately 0.041 (or 4.1%).