Question

A sample dataset of 567 has a mean of 12.3 with a standard deviation of 0.7. What is the 95 percent confidence interval? Round the answer to the nearest hundredth.
Use the table to find the critical value.
Confidence Level c Critical Value z
c
​

0.95 1.96
0.98 2.33
0.99 2.58

Answer 1

To calculate the 95 percent confidence interval for the mean, you can use the following formula:

\[
\text{Confidence Interval} = \text{Mean} \pm (z \times \frac{\sigma}{\sqrt{n}})
\]

Where:
- \(\text{Mean} = 12.3\)
- \(z\) is the critical value for a 95% confidence level, which is \(1.96\)
- \(\sigma = 0.7\) (standard deviation)
- \(n = 567\) (sample size)

First, calculate the standard error (SE):

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{567}}
\]

Calculating \(\sqrt{567}\):

\[
\sqrt{567} \approx 23.811
\]

Now substitute this back into the formula for standard error:

\[
SE \approx \frac{0.7}{23.811} \approx 0.0294
\]

Now, calculate the margin of error (ME):

\[
ME = z \times SE = 1.96 \times 0.0294 \approx 0.0576
\]

Finally, calculate the confidence interval:

\[
\text{Lower Limit} = \text{Mean} - ME = 12.3 - 0.0576 \approx 12.2424
\]

\[
\text{Upper Limit} = \text{Mean} + ME = 12.3 + 0.0576 \approx 12.3576
\]

Rounding to the nearest hundredth, the confidence interval is:

\[
\text{Confidence Interval} \approx (12.24, 12.36)
\]

So, the final 95 percent confidence interval is **(12.24, 12.36)**.