Question

A random sample of 100 automobile owners in the state of Virginia shows that an automobile is driven on average 23,500 kilometers per year with a Standard deviation of 3900 kilometers. Assume the distribution of measurements to be approximately normal. Construct a 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia.

Answer 1

The 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia is calculated using the normal distribution, yielding an interval of (22,495.36, 24,504.64) kilometers.

Step-by-step explanation:

Confidence Interval Calculation

To construct a 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia, we need to use the sample mean, standard deviation, and the sample size. Since the sample size is 100, we would use the Z-distribution for our confidence interval calculation.

The sample mean is 23,500 kilometers, the standard deviation of the sample is 3,900 kilometers, and the sample size is 100. Using the Z-value associated with a 99% confidence level (which is roughly 2.576), we can calculate the margin of error. The formula for the margin of error (ME) is:

ME = Z * (SD/sqrt(n))

Calculating the ME:

ME = 2.576 * (3900/sqrt(100))

ME = 2.576 * 390

ME = 1004.64 kilometers

Now, we add and subtract the ME from the sample mean to find the confidence interval:

Lower limit = 23,500 - 1004.64 = 22,495.36 kilometers

Upper limit = 23,500 + 1004.64 = 24,504.64 kilometers

Therefore, the 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia is (22,495.36, 24,504.64) kilometers.