Question

Assume that a sample is used to estimate a population mean mu. Find the 99% confidence interval for a sample of size 45 with a mean of 78.2 and a standard deviation of 10.7. Enter your answer as an open-interval (i.e., parentheses) accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

99% C.I. =

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

Answer 1

To calculate the 99% confidence interval for the population mean using a sample, we'll need the following components:

1. Sample size \( n = 45 \)
2. Sample mean \( \bar{x} = 78.2 \)
3. Sample standard deviation \( s = 10.7 \)

Since the sample size is larger than 30, we can apply the Central Limit Theorem and use the z-distribution for the confidence interval.

### Step 1: Find the critical value
For a 99% confidence level, the critical z-value (denoted as \( z^* \)) can be found using the standard normal distribution. The area in each tail at 99% confidence level is \( \frac{1 - 0.99}{2} = 0.005 \).

Using the standard normal distribution table, we find:
- \( z^* \approx 2.576 \) (you may round to three decimal places)

### Step 2: Calculate the standard error (SE)
The standard error of the sample mean is calculated with the formula:
\[
SE = \frac{s}{\sqrt{n}} = \frac{10.7}{\sqrt{45}} \approx \frac{10.7}{6.7082} \approx 1.5912
\]

### Step 3: Calculate the margin of error (ME)
The margin of error is given by:
\[
ME = z^* \times SE = 2.576 \times 1.5912 \approx 4.0995
\]

### Step 4: Calculate the confidence interval
The confidence interval is given by:
\[
(\bar{x} - ME, \bar{x} + ME)
\]

So we calculate:
- Lower limit: \( \bar{x} - ME = 78.2 - 4.0995 \approx 74.1005 \)
- Upper limit: \( \bar{x} + ME = 78.2 + 4.0995 \approx 82.2995 \)

### Final Step: Write the confidence interval
Now rounding to one decimal place, we have:
- Lower limit: \( 74.1 \)
- Upper limit: \( 82.3 \)

Thus, the 99% confidence interval is:
\[
\text{99% C.I.} = (74.1, 82.3)
\]