A researcher wishes to estimate the proportion of college students who cheat on exams. A poll of 400 college students showed that 33% of them had or intended to, cheat on examinations. Find the margin of error for the 95% confidence interval.
To find the margin of error for a proportion estimate, we need to use the formula:
Margin of Error = Critical Value * Standard Error
Here, the critical value refers to the number of standard deviations we need to account for to achieve a desired level of confidence. In this case, we want a 95% confidence interval, so our critical value corresponds to the z-score at a 95% confidence level.
To find the critical value, we can use a standard normal distribution table or a calculator. For a 95% confidence level, the critical value is approximately 1.96.
Next, we need to calculate the standard error, which measures how much the sample estimate may vary from the true population proportion. The formula for the standard error of a proportion is:
Standard Error = sqrt((p * (1-p)) / n)
Here, p represents the sample proportion and n represents the sample size.
Given that the sample proportion (p) is 33% or 0.33, and the sample size (n) is 400, we can substitute these values into the formula:
Standard Error = sqrt((0.33 * (1-0.33)) / 400)
Calculating this expression, we get:
Standard Error ≈ sqrt(0.2211 / 400) ≈ sqrt(0.00055275) ≈ 0.0235
Finally, we can substitute the critical value (1.96) and the standard error (0.0235) into the margin of error formula:
Margin of Error = 1.96 * 0.0235 ≈ 0.0461
Therefore, the margin of error for the 95% confidence interval is approximately 0.0461, or 4.61%.