Answer :
To find the 80% confidence interval for the population mean given the sample mean, population standard deviation, and sample size, we can follow these steps:
1. Identify the Known Values:
- Sample mean: [tex]\( \bar{x} = 1.69 \)[/tex]
- Population standard deviation: [tex]\( \sigma = 0.657 \)[/tex]
- Sample size: [tex]\( n = 50 \)[/tex]
- Confidence level: 80%
2. Determine the Z-Score for the Confidence Level:
The confidence level is 80%, which corresponds to the middle 80% of the distribution. This means 10% is in each tail. Use a Z-score table or statistical software to find the Z-score that places 10% in the upper tail of a standard normal distribution. This Z-score is approximately 1.282.
3. Calculate the Standard Error (SE):
The standard error of the mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting in the known values:
[tex]\[
SE = \frac{0.657}{\sqrt{50}} \approx 0.0929
\][/tex]
4. Calculate the Margin of Error (ME):
The margin of error is determined by multiplying the Z-score by the standard error:
[tex]\[
ME = Z \times SE \approx 1.282 \times 0.0929 \approx 0.1191
\][/tex]
5. Determine the Confidence Interval:
Finally, calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower bound} = \bar{x} - ME = 1.69 - 0.1191 \approx 1.5709
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME = 1.69 + 0.1191 \approx 1.8091
\][/tex]
Therefore, the 80% confidence interval for the population mean is approximately [tex]\(1.69 \pm 0.119\)[/tex], which corresponds to the choice [tex]\(1.69 \pm 0.119\)[/tex].
1. Identify the Known Values:
- Sample mean: [tex]\( \bar{x} = 1.69 \)[/tex]
- Population standard deviation: [tex]\( \sigma = 0.657 \)[/tex]
- Sample size: [tex]\( n = 50 \)[/tex]
- Confidence level: 80%
2. Determine the Z-Score for the Confidence Level:
The confidence level is 80%, which corresponds to the middle 80% of the distribution. This means 10% is in each tail. Use a Z-score table or statistical software to find the Z-score that places 10% in the upper tail of a standard normal distribution. This Z-score is approximately 1.282.
3. Calculate the Standard Error (SE):
The standard error of the mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting in the known values:
[tex]\[
SE = \frac{0.657}{\sqrt{50}} \approx 0.0929
\][/tex]
4. Calculate the Margin of Error (ME):
The margin of error is determined by multiplying the Z-score by the standard error:
[tex]\[
ME = Z \times SE \approx 1.282 \times 0.0929 \approx 0.1191
\][/tex]
5. Determine the Confidence Interval:
Finally, calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower bound} = \bar{x} - ME = 1.69 - 0.1191 \approx 1.5709
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME = 1.69 + 0.1191 \approx 1.8091
\][/tex]
Therefore, the 80% confidence interval for the population mean is approximately [tex]\(1.69 \pm 0.119\)[/tex], which corresponds to the choice [tex]\(1.69 \pm 0.119\)[/tex].
