Sure, let's break down the given problem and the solution step-by-step.
1. Identify the number of surveyed players who prefer Saturdays:
- Total surveyed players ([tex]\(n\)[/tex]) = 49
- Players who prefer Saturdays ([tex]\(x\)[/tex]) = 22
2. Calculate the sample proportion ([tex]\(\hat{\rho}\)[/tex]):
[tex]\[
\hat{\rho} = \frac{x}{n} = \frac{22}{49}
\][/tex]
3. Given the margin of error:
- Margin of error ([tex]\(E\)[/tex]) = 0.18 (18%)
4. Determine the confidence interval:
The confidence interval is calculated as:
[tex]\[
\hat{\rho} \pm E
\][/tex]
First, we convert the sample proportion to a percentage:
[tex]\[
\hat{\rho} \approx 0.4489795918367347
\][/tex]
Converting to a percentage:
[tex]\[
\hat{\rho}_{\%} = 0.4489795918367347 \times 100 = 44.89795918367347 \%
\][/tex]
Now, calculate the lower and upper bounds of the confidence interval:
[tex]\[
\text{Lower bound} = \hat{\rho}_{\%} - E \times 100 = 44.89795918367347\% - 18\% \approx 26.89795918367347 \%
\][/tex]
[tex]\[
\text{Upper bound} = \hat{\rho}_{\%} + E \times 100 = 44.89795918367347\% + 18\% \approx 62.89795918367347 \%
\][/tex]
5. Interpreting the result:
- Therefore, the [tex]\(99 \%\)[/tex] confidence interval for the proportion of players who prefer the games to be played on Saturdays is approximately between [tex]\( 26.9 \% \)[/tex] and [tex]\( 62.9 \% \)[/tex].
Given the options:
- Between [tex]\( 4 \% \)[/tex] and [tex]\( 40 \% \)[/tex]
- Between [tex]\( 1 \% \)[/tex] and [tex]\( 55 \% \)[/tex]
- Between [tex]\( 27 \% \)[/tex] and [tex]\( 63 \% \)[/tex]
- Between [tex]\( 31 \% \)[/tex] and [tex]\( 67 \% \)[/tex]
The correct answer is:
[tex]\[
\text{Between \( 27 \% \) and \( 63 \% \)}
\][/tex]
