A car manufacturing company is interested in finding the average fuel consumption of one of its new models Leopard F-type. Hundred test drivers are chosen and asked to drive the car in urban conditions for 8 hours. The average fuel consumption of the 100 cars is found to be 30 miles per gallon (mpg) with a standard deviation of 6.2 mpg. Find the 99% confidence interval for the average fuel consumption of Leopard F-type. If the management says that the length of the confidence interval should not be greater than 1 mpg’s, what is the minimum number of test drivers required? If it is not possible to hire new drivers, how confident would you be about a CI with length 1 mpg?
To find the 99% confidence interval for the average fuel consumption of the Leopard F-type, we can use the formula:
Confidence interval = Mean ± (Z-score * Standard deviation / √n)
Where:
- Mean is the sample mean fuel consumption (30 mpg in this case)
- Z-score is the critical value corresponding to the desired confidence level (99% in this case)
- Standard deviation is the sample standard deviation (6.2 mpg in this case)
- √n is the square root of the sample size
First, let's calculate the Z-score for 99% confidence level. The Z-score can be determined using a standard normal distribution table or a statistical calculator. For a 99% confidence level, the Z-score is approximately 2.576.
Now let's plug in the values into the formula:
Confidence interval = 30 ± (2.576 * 6.2 / √n)
To ensure that the length of the confidence interval is not greater than 1 mpg, we need to set the upper bound and lower bound of the confidence interval at a maximum difference of 1 mpg. Therefore:
1 = 2.576 * 6.2 / √n
Solving for n:
√n = 2.576 * 6.2 / 1
√n = 16
n = 256
Therefore, the minimum number of test drivers required to achieve a confidence interval length of no more than 1 mpg is 256.
If it is not possible to hire new drivers, the confidence interval cannot be adjusted. In this case, the original confidence interval, with a length greater than 1 mpg, would need to be used.
For a confidence interval length of 1 mpg, the margin of error would be 0.5 mpg. This means that we are confident that the true average fuel consumption of the Leopard F-type falls within 0.5 mpg above or below the calculated sample mean of 30 mpg. However, the confidence level of this interval would not be 99%, as the length of the confidence interval is greater than 1 mpg (margin of error). The actual confidence level would depend on the specific sample and population from which it was drawn.