To determine if any of the given values (135, 137, 138, 154) are outside the 99% confidence interval for the population mean, we will follow these steps:
### 1. Determine the Mean and Standard Deviation
- Sample size ([tex]\(n\)[/tex]) = 85
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 146
- Sample standard deviation ([tex]\(s\)[/tex]) = 34
- Z-score for 99% confidence level = 2.58
### 2. Calculate the Margin of Error
The margin of error (ME) can be calculated using the formula:
[tex]\[ ME = z \cdot \frac{s}{\sqrt{n}} \][/tex]
Substituting the values:
[tex]\[ ME = 2.58 \cdot \frac{34}{\sqrt{85}} \][/tex]
This results in a margin of error:
[tex]\[ ME \approx 9.51456987992626 \][/tex]
### 3. Calculate the Confidence Interval
The confidence interval is given by:
[tex]\[ \text{Lower bound} = \bar{x} - ME \][/tex]
[tex]\[ \text{Upper bound} = \bar{x} + ME \][/tex]
Substituting the mean and the margin of error:
[tex]\[ \text{Lower bound} \approx 146 - 9.51456987992626 \][/tex]
[tex]\[ \text{Lower bound} \approx 136.48543012007374 \][/tex]
[tex]\[ \text{Upper bound} \approx 146 + 9.51456987992626 \][/tex]
[tex]\[ \text{Upper bound} \approx 155.51456987992626 \][/tex]
### 4. Check the Given Values
We compare each given value (135, 137, 138, 154) to the bounds of the confidence interval:
- 135: This value is less than the lower bound (136.48543012007374), so it is outside the interval.
- 137: This value is within the interval [tex]\(136.48543012007374 \leq 137 \leq 155.51456987992626\)[/tex].
- 138: This value is within the interval [tex]\(136.48543012007374 \leq 138 \leq 155.51456987992626\)[/tex].
- 154: This value is within the interval [tex]\(136.48543012007374 \leq 154 \leq 155.51456987992626\)[/tex].
### Conclusion
The value 135 is outside the 99% confidence interval for the population mean. Therefore, the value of 135 does not fall within the computed confidence interval and is the correct answer to the given question.
