To calculate the margin of error (MOE) for a proportion, you can use the formula:
\[ \text{MOE} = z \times \sqrt{\frac{p(1-p)}{n}} \]
Where:
- \( z \) is the z-score (1.96 for a 95% confidence level),
- \( p \) is the sample proportion,
- \( n \) is the sample size.
In this case:
- The sample size \( n = 110 \),
- The number of shoppers using coupons regularly \( x = 23 \).
First, we calculate the sample proportion \( p \):
\[ p = \frac{x}{n} = \frac{23}{110} \approx 0.2091 \]
Now, calculate \( 1-p \):
\[ 1-p = 1 - 0.2091 \approx 0.7909 \]
Next, we can substitute the values into the MOE formula:
\[ \text{MOE} = 1.96 \times \sqrt{\frac{0.2091 \times 0.7909}{110}} \]
Calculating the component inside the square root:
\[ 0.2091 \times 0.7909 \approx 0.1653 \] \[ \frac{0.1653}{110} \approx 0.001503 \] \[ \sqrt{0.001503} \approx 0.0388 \]
Now, substitute back into the MOE formula:
\[ \text{MOE} = 1.96 \times 0.0388 \approx 0.0761 \]
Convert the margin of error from a proportion to a percentage by multiplying by 100:
\[ \text{MOE} \approx 0.0761 \times 100 \approx 7.61% \]
Rounding to the nearest tenth of a percent:
\[ \text{MOE} \approx 7.6% \]
Therefore, the margin of error is 7.6%.