Ivan surveyed 49 randomly selected players from the 134 players of the soccer club teams to see if they wanted the games to be played on Saturdays or Sundays. Twenty-two of the players said that they preferred that the games be played on Saturdays. Ivan correctly determined that the margin of error, [tex]\(E\)[/tex], of his survey using a [tex]\(99 \%\)[/tex] confidence interval ([tex]\(z^*\)[/tex] score 2.58) is approximately [tex]\(18 \%\)[/tex].

What is the [tex]\(99 \%\)[/tex] confidence level for players who prefer the games be played on Saturdays?

[tex]\[
C=\hat{\rho} \pm E
\][/tex]

A. Between [tex]\(4 \%\)[/tex] and [tex]\(40 \%\)[/tex]
B. Between [tex]\(1 \%\)[/tex] and [tex]\(55 \%\)[/tex]
C. Between [tex]\(27 \%\)[/tex] and [tex]\(63 \%\)[/tex]
D. Between [tex]\(31 \%\)[/tex] and [tex]\(67 \%\)[/tex]



Answer :

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]