Question

Assume that a sample is used to estimate a population mean mu. Find the 80% confidence interval for a sample of size 949 with a mean of 30.3 and a standard deviation of 18.1. Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

___ < mu < ___


Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer 1

To calculate the 80% confidence interval for the population mean (\(\mu\)), we will use the formula for the confidence interval when the sample size is large (in this case, \(n = 949\)), which uses the z-distribution.

The formula for the confidence interval is:

\[
\bar{x} \pm z_{c} \left(\frac{\sigma}{\sqrt{n}}\right)
\]

where:

- \(\bar{x}\) = sample mean = 30.3
- \(z_{c}\) = critical z-value for the desired confidence level
- \(\sigma\) = sample standard deviation = 18.1
- \(n\) = sample size = 949

### Step 1: Find the critical value \(z_{c}\)

For an 80% confidence interval, we are interested in the critical value that leaves 10% in each tail (100% - 80% = 20%, and half of that is 10%).

Using the standard normal distribution table, the z-value that corresponds to an area of 0.90 (which is the 80% coverage) is approximately \(z_{c} \approx 1.281\).

### Step 2: Calculate the standard error (SE)

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{18.1}{\sqrt{949}}
\]

Calculating \(\sqrt{949}\):

\[
\sqrt{949} \approx 30.8
\]

Now, calculate the standard error (SE):

\[
SE = \frac{18.1}{30.8} \approx 0.587
\]

### Step 3: Calculate the margin of error (ME)

\[
ME = z_{c} \times SE = 1.281 \times 0.587 \approx 0.752
\]

### Step 4: Calculate the confidence interval

Now we can calculate the lower and upper limits of the confidence interval:

Lower limit:

\[
\bar{x} - ME = 30.3 - 0.752 = 29.548 \approx 29.5
\]

Upper limit:

\[
\bar{x} + ME = 30.3 + 0.752 = 31.052 \approx 31.1
\]

### Final Answer

Hence, the 80% confidence interval for the population mean \(\mu\) is:

\[
29.5 < \mu < 31.1
\]