To determine which of the given values falls within the 95% confidence interval for the population mean, let's carefully follow these steps:
Step 1: Understand the Given Data
- Sample Size ([tex]\(n\)[/tex]): 60
- Sample Mean ([tex]\(\bar{x}\)[/tex]): 28
- Sample Standard Deviation ([tex]\(s\)[/tex]): 5
- [tex]\(z\)[/tex]-score for 95% confidence interval: 1.96
Step 2: Calculate the Margin of Error (ME)
The formula for the margin of error is:
[tex]\[
ME = \frac{z \cdot s}{\sqrt{n}}
\][/tex]
Plugging in the given values:
[tex]\[
ME = \frac{1.96 \cdot 5}{\sqrt{60}}
\][/tex]
Given that we have pre-calculated the values, we find:
[tex]\[
ME \approx 1.2652
\][/tex]
Step 3: Calculate the Confidence Interval
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean.
- Lower Bound:
[tex]\[
\text{Lower Bound} = \bar{x} - ME = 28 - 1.2652 \approx 26.735
\][/tex]
- Upper Bound:
[tex]\[
\text{Upper Bound} = \bar{x} + ME = 28 + 1.2652 \approx 29.265
\][/tex]
So, the 95% confidence interval is approximately (26.735, 29.265).
Step 4: Check the Given Values Against the Confidence Interval
We need to determine if each value is within the derived confidence interval (26.735, 29.265):
- Value 26:
[tex]\[
26 < 26.735 \quad \text{(Not within the interval)}
\][/tex]
- Value 27:
[tex]\[
26.735 \leq 27 \leq 29.265 \quad \text{(Within the interval)}
\][/tex]
- Value 32:
[tex]\[
32 > 29.265 \quad \text{(Not within the interval)}
\][/tex]
- Value 34:
[tex]\[
34 > 29.265 \quad \text{(Not within the interval)}
\][/tex]
Conclusion:
The only value within the 95% confidence interval is 27. Therefore, the correct interpretation is:
- The value of 27 is within the confidence interval because it is greater than 26.735 and less than 29.265.
Hence, the correct answer is:
The value of 27, because it is greater than 26.7 and less than 29.3.
