To express the given confidence interval [tex]$51.3\% \pm 5.7\%$[/tex] in interval form with the resulting values as decimals rounded to three places, follow these steps:
1. Convert the given percentages to decimals:
- The confidence level [tex]\(51.3\%\)[/tex] should be converted to a decimal by dividing it by 100:
[tex]\[
51.3\% = \frac{51.3}{100} = 0.513
\][/tex]
- Similarly, the margin of error [tex]\(5.7\%\)[/tex] should be converted to a decimal:
[tex]\[
5.7\% = \frac{5.7}{100} = 0.057
\][/tex]
2. Calculate the lower bound of the interval:
- Subtract the margin of error (in decimal form) from the confidence level (in decimal form):
[tex]\[
\text{Lower bound} = 0.513 - 0.057 = 0.456
\][/tex]
3. Calculate the upper bound of the interval:
- Add the margin of error (in decimal form) to the confidence level (in decimal form):
[tex]\[
\text{Upper bound} = 0.513 + 0.057 = 0.570
\][/tex]
4. Express the result in interval form:
- The confidence interval in decimal form, rounded to three decimal places, is:
[tex]\[
(0.456, 0.570)
\][/tex]
Therefore, the confidence interval [tex]\(51.3\% \pm 5.7\%\)[/tex] expressed in interval form with values rounded to three decimal places is:
[tex]\[
(0.456, 0.570)
\][/tex]
