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 kilometers per liter.
i) 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 μ.
ii) Suppose the confidence interval obtained in (b)(i) is too wide. How can the width of this
interval be reduced? Describe all possible alternatives. Which alternative is the best and
why?
1. 99% = mean ± 2.575 SEm
SEm = SD/√n
2. Only thing I can think of is getting a larger sample.
To find the 99% confidence interval for the population mean μ, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / √n)
i) Critical value: We use the Z-score for a 99% confidence level. The Z-score can be obtained from the Z-table or using a statistical software. In this case, the critical value for a 99% confidence level is approximately 2.576.
Sample mean = 26.4 kilometers per liter
Standard deviation = 2.3 kilometers per liter
Sample size (n) = 16
Confidence Interval = 26.4 ± (2.576) * (2.3 / √16)
Calculating the Confidence Interval:
Confidence Interval = 26.4 ± (2.576) * (0.575)
Confidence Interval = 26.4 ± 1.4822
The confidence interval for the population mean μ is (24.9178, 27.8822) kilometers per liter.
ii) To reduce the width of the confidence interval, we have a few alternatives:
1. Increase the sample size (n): By increasing the sample size, we reduce the standard error, resulting in a narrower confidence interval. However, this may require more resources and time.
2. Decrease the confidence level: By choosing a lower confidence level (e.g., 95% instead of 99%), the critical value will be smaller, resulting in a narrower interval. However, this reduces the certainty of the estimation.
3. Decrease the variability (standard deviation): If we can reduce the variability in the data, the standard deviation will be smaller, leading to a narrower confidence interval. However, this might not always be within our control.
The best alternative depends on the practical considerations and constraints in the study. Increasing the sample size is typically the most preferred option as it provides a more precise estimate with a narrower interval. However, it may not always be feasible due to constraints such as time and resources.