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Answer :
To calculate the 80% confidence interval for the population mean based on the given survey data, follow these steps:
1. Identify the given information:
- Sample mean ([tex]\(\bar{x}\)[/tex]): [tex]$1.69
- Population standard deviation (\(\sigma\)): $[/tex]0.657
- Sample size ([tex]\(n\)[/tex]): 50
- Confidence level: 80%
2. Find the Z-score for the 80% confidence level:
The Z-score represents the number of standard deviations a data point is from the mean. For an 80% confidence level, the Z-score is approximately 1.2816. This value is typically found using a standard normal distribution Z-score table.
3. Calculate the standard error:
The standard error of the mean is calculated using the formula:
[tex]\[
\text{Standard Error} (SE) = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the values:
[tex]\[
SE = \frac{0.657}{\sqrt{50}} \approx 0.0929
\][/tex]
4. Calculate the margin of error:
The margin of error is calculated by multiplying the Z-score by the standard error:
[tex]\[
\text{Margin of Error} = Z \times SE = 1.2816 \times 0.0929 \approx 0.1191
\][/tex]
5. Determine the confidence interval:
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower limit} = \bar{x} - \text{Margin of Error} = 1.69 - 0.1191 \approx 1.5709
\][/tex]
[tex]\[
\text{Upper limit} = \bar{x} + \text{Margin of Error} = 1.69 + 0.1191 \approx 1.8091
\][/tex]
The 80% confidence interval for the population mean is approximately [tex]\(1.69 \pm 0.119\)[/tex]. This matches the answer choice: [tex]\(1.69 \pm 0.119\)[/tex].
1. Identify the given information:
- Sample mean ([tex]\(\bar{x}\)[/tex]): [tex]$1.69
- Population standard deviation (\(\sigma\)): $[/tex]0.657
- Sample size ([tex]\(n\)[/tex]): 50
- Confidence level: 80%
2. Find the Z-score for the 80% confidence level:
The Z-score represents the number of standard deviations a data point is from the mean. For an 80% confidence level, the Z-score is approximately 1.2816. This value is typically found using a standard normal distribution Z-score table.
3. Calculate the standard error:
The standard error of the mean is calculated using the formula:
[tex]\[
\text{Standard Error} (SE) = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the values:
[tex]\[
SE = \frac{0.657}{\sqrt{50}} \approx 0.0929
\][/tex]
4. Calculate the margin of error:
The margin of error is calculated by multiplying the Z-score by the standard error:
[tex]\[
\text{Margin of Error} = Z \times SE = 1.2816 \times 0.0929 \approx 0.1191
\][/tex]
5. Determine the confidence interval:
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower limit} = \bar{x} - \text{Margin of Error} = 1.69 - 0.1191 \approx 1.5709
\][/tex]
[tex]\[
\text{Upper limit} = \bar{x} + \text{Margin of Error} = 1.69 + 0.1191 \approx 1.8091
\][/tex]
The 80% confidence interval for the population mean is approximately [tex]\(1.69 \pm 0.119\)[/tex]. This matches the answer choice: [tex]\(1.69 \pm 0.119\)[/tex].
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