Margin of error
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population. Margin of error is positive whenever a population is incompletely sampled and the outcome measure has positive variance (that is, it varies).
Step 1: Find the critical value. The critical value is either a t score or a z-score. If you aren’t sure, see: t score vs z score. In general, for small sample size (under 30) or when you don’t know the population standard deviation, use a t score. Otherwise, use a z score.
Step 2: Find the standard deviation or the standard error . These are essentially the same thing, only you must know your population parameters in order to calculate standard deviation. Otherwise, calculate the standard error .
Step 3 :Multiply the critical value from Step 1 by thestandard deviation or the standard error from Step 2. For example, if your CV is 1.95 and your SE is 0.019, then:
1.95 * 0.019 = 0.03705
Sample question: 900 students were surveyed and had an average GPA of 2.7 with a standard deviation of 0.4. Calculate the margin of error for a 90% confidence level:
- The critical value is 1.645
- The standard deviation is 0.4 (from the question), but as this is a sample, we need the standard error for the mean. The formula for the SE of the mean is standard deviation / √(sample size), so: 0.4 / √(900)=0.013.
- 1.645 * 0.013 = 0.021385
Formula to calculate margin of error of proportion
\[z* \sqrt{\frac{p(1-p)}{n}}\]
Where:
p = sample proportion (“P-hat”).
n = sample size
z = z-score
Example question: 1000 people were surveyed and 380 thought that climate change was not caused by human pollution. Find the MoE for a 90% confidence interval.
Step 1: Find P-hat by dividing the number of people who responded positively. “Positively” in this sense doesn’t mean that they gave a “Yes” answer; It means that they answered according to the statement in the question. In this case, 380/1000 people (38%) responded positively.
Step 2: Find the z-score that goes with the given confidence interval. A 90% confidence interval has a z-score (a critical value) of 1.645.
step 3 Insert the values in \[z* \sqrt{\frac{p(1-p)}{n}}\]
\[1.645* \sqrt{\frac{0.38(1-0.38)}{1000}}\]=0.0252 0r 2.52%
