A random sample of 100 automobile owners in a particular city shows that an automobile is driven on average 23,500 km/year. The population standard deviation is 3900 km/year. The distribution of measurements is approximately normal. Construct a 99% confidence interval for the average number of km an automobile is driven annually in the particular city. What is the lower Confidence Interval boundary? What is the upper Confidence Interval boundary?


We are given the following information to construct a 99% confidence interval for the average number of kilometers driven by an automobile annually:

- Sample mean \((\bar{x}) = 23,500 \text{ km/year}\)
- Population standard deviation \((\sigma) = 3900 \text{ km/year}\)
- Sample size \((n) = 100\)
- Confidence level = 99%

### Step 1: Determine the z-score for a 99% confidence level
For a 99% confidence interval, the z-score corresponding to the tail area of 0.005 (which is \( \frac{1 - 0.99}{2} = 0.005\)) is approximately \( z = 2.576 \).

### Step 2: Compute the standard error (SE)
The standard error of the mean is calculated as:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{3900}{\sqrt{100}} = \frac{3900}{10} = 390 \text{ km/year}
\]

### Step 3: Calculate the margin of error (MOE)
The margin of error is given by:
\[
MOE = z \times SE = 2.576 \times 390 = 1004.64 \text{ km/year}
\]

### Step 4: Calculate the confidence interval
The 99% confidence interval for the population mean is:
\[
\text{Confidence interval} = \bar{x} \pm MOE = 23,500 \pm 1004.64
\]
This gives:
- Lower boundary: \( 23,500 - 1004.64 = 22,495.36 \)
- Upper boundary: \( 23,500 + 1004.64 = 24,504.64 \)

Thus, the 99% confidence interval is approximately:

\[
(22,495.36 \text{ km/year}, 24,504.64 \text{ km/year})
\]

### Answers:
- **Lower Confidence Interval boundary**: 22,495.36 km/year
- **Upper Confidence Interval boundary**: 24,504.64 km/year

Would you like further details on how to interpret these results, or any follow-up questions?

### 5 Related Questions:
1. How would the confidence interval change if we increased the sample size?
2. What is the effect of changing the confidence level to 95% on the interval?
3. How would the confidence interval be affected if the population standard deviation were larger?
4. Why do we use the z-score instead of the t-score in this calculation?
5. What are the assumptions required for constructing this confidence interval?

### Tip:
A higher confidence level results in a wider confidence interval, meaning greater certainty but less precision in the estimate.

A random sample of 100 automobile owners in a particular city shows that an automobile is driven on average 23,500 km/year. The population standard deviation is 3900 km/year. The distribution of measurements is approximately normal. Construct a 99% confidence interval for the average number of km an automobile is driven annually in the particular city. What is the lower Confidence Interval boundary? What is the upper Confidence Interval boundary?

Learn how to construct a 99% confidence interval for the average kilometers driven by an automobile annually in a city. Given a sample mean of 23,500 km/year, a population standard deviation of 3900 km/year, and a sample size of 100, this example shows detailed calculations for finding the lower and upper confidence interval boundaries.

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