A simple random sample of 85 is drawn from a normally distributed population, and the mean is found to be 146, with a standard deviation of 34. Which of the following values is outside the [tex][tex]$99 \%$[/tex][/tex] confidence interval for the population mean? Use the table below to help you answer the question.

\begin{tabular}{|c|c|c|c|}
\hline Confidence Level & [tex][tex]$90 \%$[/tex][/tex] & [tex][tex]$95 \%$[/tex][/tex] & [tex][tex]$99 \%$[/tex][/tex] \\
\hline [tex][tex]$z^*$[/tex][/tex]-score & 1.645 & 1.96 & 2.58 \\
\hline
\end{tabular}

Remember, the margin of error, [tex][tex]$ME$[/tex][/tex], can be determined using the formula [tex][tex]$ME = \frac{z \cdot s}{\sqrt{n}}$[/tex][/tex].

A. The value of 135 because it is not greater than 136.5.
B. The value of 137 because it is greater than 136.5.
C. The value of 138 because it is less than 153.2.
D. The value of 154 because it is greater than 153.2.



Answer :

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.