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. (2 points)
-

Answer 1

To calculate the 95% confidence interval for the mean, we use the formula:

\[
\text{Confidence Interval} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right)
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(z\) is the critical value for the desired confidence level.
- \(\sigma\) is the standard deviation.
- \(n\) is the sample size.

Given:
- \(\bar{x} = 12.3\)
- \(\sigma = 0.7\)
- \(n = 567\)

First, we need to find the critical value \(z\) for a 95% confidence level. For a 95% confidence interval, the critical z-value is approximately 1.96 (you can find this in the standard normal distribution table).

Now, we calculate the standard error (SE):

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{567}} \approx \frac{0.7}{23.811} \approx 0.0294
\]

Next, we calculate the margin of error (ME):

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

Now we can compute the confidence interval:

\[
\text{Confidence Interval} = 12.3 \pm 0.0576
\]

Calculating the lower and upper bounds:

\[
\text{Lower bound} = 12.3 - 0.0576 \approx 12.2424 \approx 12.24 \text{ (rounded to the nearest hundredth)}
\]

\[
\text{Upper bound} = 12.3 + 0.0576 \approx 12.3576 \approx 12.36 \text{ (rounded to the nearest hundredth)}
\]

Thus, the 95% confidence interval is:

\[
\boxed{(12.24, 12.36)}
\]