Let's carefully analyze each given combination to determine which set of [tex]\( z \)[/tex]-values, standard deviations ([tex]\( s \)[/tex]), and sample sizes ([tex]\( n \)[/tex]) result in a margin of error (ME) of 0.95.
The formula for the margin of error is:
[tex]\[ ME = \frac{z \cdot s}{\sqrt{n}} \][/tex]
### 1. [tex]\( z = 2.14 \)[/tex], [tex]\( s = 4 \)[/tex], [tex]\( n = 9 \)[/tex]
Plugging in these values:
[tex]\[ ME = \frac{2.14 \cdot 4}{\sqrt{9}} = \frac{2.14 \cdot 4}{3} = \frac{8.56}{3} \approx 2.85 \][/tex]
### 2. [tex]\( z = 2.14 \)[/tex], [tex]\( s = 4 \)[/tex], [tex]\( n = 81 \)[/tex]
Plugging in these values:
[tex]\[ ME = \frac{2.14 \cdot 4}{\sqrt{81}} = \frac{2.14 \cdot 4}{9} = \frac{8.56}{9} \approx 0.9511 \][/tex]
### 3. [tex]\( z = 2.14 \)[/tex], [tex]\( s = 16 \)[/tex], [tex]\( n = 9 \)[/tex]
Plugging in these values:
[tex]\[ ME = \frac{2.14 \cdot 16}{\sqrt{9}} = \frac{2.14 \cdot 16}{3} = \frac{34.24}{3} \approx 11.4133 \][/tex]
### 4. [tex]\( z = 2.14 \)[/tex], [tex]\( s = 16 \)[/tex], [tex]\( n = 81 \)[/tex]
Plugging in these values:
[tex]\[ ME = \frac{2.14 \cdot 16}{\sqrt{81}} = \frac{2.14 \cdot 16}{9} = \frac{34.24}{9} \approx 3.8044 \][/tex]
By evaluating each of these combinations for [tex]\( ME \)[/tex], we see that:
- The combination [tex]\( z = 2.14 \)[/tex], [tex]\( s = 4 \)[/tex], [tex]\( n = 81 \)[/tex] yields a margin of error of approximately 0.9511, which is closest to 0.95.
- None of the other combinations result in a margin of error of 0.95.
Therefore, the combination that produces a margin of error closest to 0.95 is:
[tex]\[ z = 2.14, s = 4, n = 81 \][/tex]
However, since we need an exact match for 0.95, and recognizing the slight variance in our manual computations might still suggest verifying, we conclude none of these precisely match the exact value of 0.95.
Thus, the answer is:
None
