A random sample of 16 mid-sized cars tested for fuel consumption gave a mean of 26.4 kilometers per liter with a standard deviation of 2.3.
(a) Assuming that the kilometers per liter given by all mid-sized cars have a normal distribution, find a 99% confidence interval for the population mean.
(b) Suppose the confidence interval obtained in (a) is too wide. How can the width of this interval be reduced? Describe all possible alternatives. Which alternative is the best and why?
To find a 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √Sample Size)
(a) Let's calculate the confidence interval for the population mean:
Sample Size (n) = 16
Sample Mean (x̄) = 26.4
Standard Deviation (σ) = 2.3
First, we need to find the critical value corresponding to a 99% confidence level. Since the sample size is small (n < 30), we should use the t-distribution rather than the standard normal distribution. For a 99% confidence level and 15 degrees of freedom (n - 1), the critical value is approximately 2.947.
Calculating the confidence interval:
Confidence Interval = 26.4 ± (2.947) * (2.3 / √16)
By substituting the values and simplifying the equation, we get:
Confidence Interval = 26.4 ± 2.18
Therefore, the 99% confidence interval for the population mean is (24.22, 28.58).
(b) To reduce the width of the confidence interval, we have a few alternatives:
1. Increase the sample size: Increasing the sample size provides more precise information about the population mean, resulting in a narrower interval. However, this may be expensive and time-consuming.
2. Decrease the standard deviation: If we can find a way to reduce the variability in the population, the width of the interval will also decrease. This can be achieved by improving the quality of the cars or using more fuel-efficient models.
3. Use a higher confidence level: If a wider margin of error is acceptable, we can use a lower confidence level, such as 95% or 90%, instead of 99%. However, this increases the chance of capturing an incorrect parameter value.
Among these alternatives, increasing the sample size is usually the best option as it provides more accurate and reliable estimates. However, it is essential to consider the feasibility and cost-effectiveness of collecting larger samples before making a decision.