Answered

Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

A recent poll of 80 randomly selected Californians showed that [tex]$38 \%(\hat{p}=0.38)$[/tex] believe they are doing all that they can to conserve water.

The state government would like to know, within a [tex]$99 \%$[/tex] confidence level, the margin of error for this poll. The [tex]$99 \%$[/tex] confidence level [tex]$z^\ \textless \ em\ \textgreater \ $[/tex]-score is 2.58.

Remember, the margin of error, [tex]$E$[/tex], can be determined using the formula [tex]$E=z^\ \textless \ /em\ \textgreater \ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$[/tex].

To the nearest whole percent, the margin of error for the poll is [tex]$\square \%$[/tex].

Sagot :

GinnyAnswer
To determine the margin of error for the given poll at a 99% confidence level, we will use the provided formula for margin of error:

[tex]\[ E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]

Given values:
- \(\hat{p} = 0.38\) (proportion of Californians who believe they are doing all they can to conserve water)
- \(n = 80\) (sample size)
- \(z^* = 2.58\) (z-score for 99% confidence level)

Step-by-Step Solution:

1. Calculate the standard error:

[tex]\[ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.38 \times (1 - 0.38)}{80}} \][/tex]

2. Simplify inside the square root:

[tex]\[ 0.38 \times (1 - 0.38) = 0.38 \times 0.62 = 0.2356 \][/tex]

[tex]\[ \frac{0.2356}{80} = 0.002945 \][/tex]

3. Find the square root:

[tex]\[ \sqrt{0.002945} \approx 0.05426 \][/tex]

4. Multiply by the z-score to find the margin of error:

[tex]\[ E = 2.58 \times 0.05426 \approx 0.14001 \][/tex]

5. Convert the margin of error to a percentage:

[tex]\[ 0.14001 \times 100 \approx 14.001 \][/tex]

6. Round to the nearest whole percent:

[tex]\[ 14.001 \approx 14 \][/tex]

Therefore, to the nearest whole percent, the margin of error for the poll is [tex]\(14\%\)[/tex].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.