To calculate the margin of error (ME) for a proportion, you can use the following formula:
\[ \text{Margin of Error} (ME) = z \times \sqrt{\frac{p(1-p)}{n}} \]
Where:
- \( z \) is the z-score corresponding to the desired confidence level (1.96 for 95% confidence level)
- \( p \) is the sample proportion
- \( n \) is the sample size
First, we need to find the sample proportion \( p \). Given that 23 out of 110 shoppers use coupons regularly, we have:
\[ p = \frac{23}{110} \approx 0.2091 \]
Next, we calculate \( 1 - p \):
\[ 1 - p \approx 1 - 0.2091 = 0.7909 \]
Now we can plug these values into the margin of error formula:
\[ ME = 1.96 \times \sqrt{\frac{0.2091 \times 0.7909}{110}} \]
Calculating the value inside the square root:
\[ 0.2091 \times 0.7909 \approx 0.1655 \]
\[ \frac{0.1655}{110} \approx 0.001505 \]
Now take the square root:
\[ \sqrt{0.001505} \approx 0.0388 \]
Finally, calculate the margin of error:
\[ ME = 1.96 \times 0.0388 \approx 0.0762 \]
To express the margin of error as a percentage, we multiply by 100:
\[ ME \approx 0.0762 \times 100 \approx 7.62% \]
Rounding to the nearest tenth of a percent:
\[ \text{Margin of Error} \approx 7.6% \]
Thus, the margin of error is 7.6%.