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Answer :
To construct a 99.8% confidence interval for the population mean [tex]\( \mu \)[/tex] based on the information provided, follow these steps:
1. Identify the given information:
- Sample size ([tex]\( n \)[/tex]) = 57
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 58.5
- Sample standard deviation ([tex]\( s \)[/tex]) = 9.5
- Confidence level = 99.8%
2. Calculate the z-score:
- Since the confidence level is 99.8%, the level of significance ([tex]\( \alpha \)[/tex]) is 0.2% or 0.002 in decimal.
- Find the z-score that corresponds to [tex]\((1 - \alpha/2) = 0.999\)[/tex] using a standard normal distribution table. For a 99.8% confidence interval, the z-score is approximately 3.090.
3. Calculate the margin of error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = z \times \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
- Substituting the values:
[tex]\[
\text{ME} = 3.090 \times \left(\frac{9.5}{\sqrt{57}}\right)
\][/tex]
- The margin of error is approximately 3.9.
4. Construct the confidence interval:
- The confidence interval is given by:
[tex]\[
\bar{x} - \text{ME} < \mu < \bar{x} + \text{ME}
\][/tex]
- Use the sample mean and margin of error calculated:
[tex]\[
58.5 - 3.9 < \mu < 58.5 + 3.9
\][/tex]
- Simplifying this, the confidence interval is approximately:
[tex]\[
54.6 < \mu < 62.4
\][/tex]
So, the 99.8% confidence interval for the population mean [tex]\( \mu \)[/tex] is [tex]\( 54.6 < \mu < 62.4 \)[/tex].
1. Identify the given information:
- Sample size ([tex]\( n \)[/tex]) = 57
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 58.5
- Sample standard deviation ([tex]\( s \)[/tex]) = 9.5
- Confidence level = 99.8%
2. Calculate the z-score:
- Since the confidence level is 99.8%, the level of significance ([tex]\( \alpha \)[/tex]) is 0.2% or 0.002 in decimal.
- Find the z-score that corresponds to [tex]\((1 - \alpha/2) = 0.999\)[/tex] using a standard normal distribution table. For a 99.8% confidence interval, the z-score is approximately 3.090.
3. Calculate the margin of error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = z \times \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
- Substituting the values:
[tex]\[
\text{ME} = 3.090 \times \left(\frac{9.5}{\sqrt{57}}\right)
\][/tex]
- The margin of error is approximately 3.9.
4. Construct the confidence interval:
- The confidence interval is given by:
[tex]\[
\bar{x} - \text{ME} < \mu < \bar{x} + \text{ME}
\][/tex]
- Use the sample mean and margin of error calculated:
[tex]\[
58.5 - 3.9 < \mu < 58.5 + 3.9
\][/tex]
- Simplifying this, the confidence interval is approximately:
[tex]\[
54.6 < \mu < 62.4
\][/tex]
So, the 99.8% confidence interval for the population mean [tex]\( \mu \)[/tex] is [tex]\( 54.6 < \mu < 62.4 \)[/tex].
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