To determine which of the given values falls within the 95% confidence interval for the population mean, let's break down the steps needed to find the confidence interval and assess each value.
1. Determine the Margin of Error (ME):
The formula given for the margin of error is not exactly the one described in the problem. Instead, we use:
[tex]\[
ME = z \cdot \frac{S}{\sqrt{n}}
\][/tex]
Given:
- Sample size ([tex]\( n \)[/tex]) = 60
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 28
- Standard deviation ([tex]\( S \)[/tex]) = 5
- [tex]\( z \)[/tex]-score = 1.96
Let's calculate the margin of error:
[tex]\[
ME = 1.96 \times \frac{5}{\sqrt{60}}
\][/tex]
2. Calculate the Confidence Interval:
The confidence interval can be expressed as:
[tex]\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\][/tex]
Substituting the values:
[tex]\[
\text{Lower Bound} = 28 - ME
\][/tex]
[tex]\[
\text{Upper Bound} = 28 + ME
\][/tex]
Let's put the calculated margin of error values:
[tex]\[
\text{Lower Bound} = 26.73
\][/tex]
[tex]\[
\text{Upper Bound} = 29.27
\][/tex]
3. Assess Each Value:
- Value: 26
[tex]\[
\text{Is } 26 \text{ within } (26.73, 29.27)? \quad \text{No.} \quad 26 < 26.73
\][/tex]
- Value: 27
[tex]\[
\text{Is } 27 \text{ within } (26.73, 29.27)? \quad \text{Yes.} \quad 26.73 \leq 27 \leq 29.27
\][/tex]
- Value: 32
[tex]\[
\text{Is } 32 \text{ within } (26.73, 29.27)? \quad \text{No.} \quad 32 > 29.27
\][/tex]
- Value: 34
[tex]\[
\text{Is } 34 \text{ within } (26.73, 29.27)? \quad \text{No.} \quad 34 > 29.27
\][/tex]
Summary of Values within the Confidence Interval:
- 26: No, 26 is not within the interval [tex]\( (26.73, 29.27) \)[/tex]
- 27: Yes, 27 is within the interval [tex]\( (26.73, 29.27) \)[/tex]
- 32: No, 32 is not within the interval [tex]\( (26.73, 29.27) \)[/tex]
- 34: No, 34 is not within the interval [tex]\( (26.73, 29.27) \)[/tex]
Thus, the only value within the 95% confidence interval for the population mean is 27.
