To determine which value is outside the 99% confidence interval for the population mean, let's follow these steps:
1. Identify the given data:
- Sample size ([tex]\( n \)[/tex]) = 85
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 146
- Sample standard deviation ([tex]\( s \)[/tex]) = 34
- [tex]\( z^* \)[/tex]-score for a 99% confidence level = 2.58
2. Calculate the margin of error (ME):
The margin of error is given by the formula:
[tex]\[
ME = \frac{z^* \cdot s}{\sqrt{n}}
\][/tex]
Plugging in the values:
[tex]\[
ME = \frac{2.58 \cdot 34}{\sqrt{85}}
\][/tex]
The margin of error (ME) equals approximately 9.5146.
3. Determine the confidence interval:
The confidence interval is calculated as:
[tex]\[
\text{Lower bound} = \bar{x} - ME \approx 146 - 9.5146 = 136.4854
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME \approx 146 + 9.5146 = 155.5146
\][/tex]
4. Evaluate the provided values against the confidence interval:
- The value 135: Check if 135 is outside the interval [136.4854, 155.5146]
[tex]\[
135 < 136.4854 \quad (\text{True, so 135 is outside the interval})
\][/tex]
- The value 137: Check if 137 is outside the interval [136.4854, 155.5146]
[tex]\[
136.4854 < 137 < 155.5146 \quad (\text{False, so 137 is inside the interval})
\][/tex]
- The value 138: Check if 138 is outside the interval [136.4854, 155.5146]
[tex]\[
136.4854 < 138 < 155.5146 \quad (\text{False, so 138 is inside the interval})
\][/tex]
- The value 154: Check if 154 is outside the interval [136.4854, 155.5146]
[tex]\[
136.4854 < 154 < 155.5146 \quad (\text{False, so 154 is inside the interval})
\][/tex]
5. Conclusion:
The value 135 is outside the 99% confidence interval for the population mean. The other values (137, 138, and 154) are within the interval.
Therefore, the correct conclusion is:
- The value of 135 is outside the 99% confidence interval for the population mean.
