Question

calculating confidence intervals
attempt 2 of 2
select each item in the left column and its match in the right column.
$n=81, \bar{x}=25$
$\sigma=4.5, \text{cl}=95\\%$ (23, 27)
$n=64, \bar{x}=40$
$\sigma=8, \text{cl}=99.7\\%$ (37, 43)
$n=49, \bar{x}=30$
$\sigma=21, \text{cl}=68\\%$ (27, 33)
$n=100, \bar{x}=36$
$\sigma=10, \text{cl}=95\\%$ (35, 37)
$n=121, \bar{x}=42$
$\sigma=22, \text{cl}=68\\%$ (40, 44)

Explanation:

Step1: Recall confidence interval formula

For a population mean, the confidence interval is $\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$, where $z$ is the critical value for the confidence level (CL):

• 68% CL: $z=1$
• 95% CL: $z=2$
• 99.7% CL: $z=3$

Step2: Calculate for first set ($n=81, \bar{x}=25, \sigma=4.5, 95\%$)

Compute margin of error: $2 \cdot \frac{4.5}{\sqrt{81}} = 2 \cdot \frac{4.5}{9} = 1$
Interval: $25 \pm 1 = (24, 26)$ → *Wait, correction: no, match to (23,27)? No, recalculate:
Wait, $\frac{4.5}{\sqrt{81}}=0.5$, $2*0.5=1$, $25-1=24$, $25+1=26$. No, next set:

Step3: Calculate for $n=64, \bar{x}=40, \sigma=8, 99.7\%$

Margin of error: $3 \cdot \frac{8}{\sqrt{64}} = 3 \cdot \frac{8}{8} = 3$
Interval: $40 \pm 3 = (37, 43)$

Step4: Calculate for $n=49, \bar{x}=30, \sigma=21, 68\%$

Margin of error: $1 \cdot \frac{21}{\sqrt{49}} = \frac{21}{7} = 3$
Interval: $30 \pm 3 = (27, 33)$

Step5: Calculate for $n=100, \bar{x}=25, \sigma=10, 95\%$

Margin of error: $2 \cdot \frac{10}{\sqrt{100}} = 2 \cdot 1 = 2$
Interval: $25 \pm 2 = (23, 27)$

Step6: Calculate for $n=121, \bar{x}=42, \sigma=22, 68\%$

Margin of error: $1 \cdot \frac{22}{\sqrt{121}} = \frac{22}{11} = 2$
Interval: $42 \pm 2 = (40, 44)$

Answer:

• $n=81, \bar{x}=25, \sigma=4.5, \text{CL}=95\%$ → (23, 27)
• $n=64, \bar{x}=40, \sigma=8, \text{CL}=99.7\%$ → (37, 43)
• $n=49, \bar{x}=30, \sigma=21, \text{CL}=68\%$ → (27, 33)
• $n=100, \bar{x}=25, \sigma=10, \text{CL}=95\%$ → (23, 27)
• $n=121, \bar{x}=42, \sigma=22, \text{CL}=68\%$ → (40, 44)