A random sample of 16 mid-sized cars tested for fuel consumption gave a mean of 26.4 kilometers per litre with a standard deviation of 2.3 kilometers per litre.
(a)Assuming that the kilometers per litre given by all mid-sized cars have a normal distribution,find a 99% confidence interval for the population mean μ.
(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?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.005) and its Z score.
a) 99% = mean ± Z SEm
SEm = SD/√n
b) increase n, use different confidence interval
my answer is 0.575
To find a confidence interval for the population mean μ, we can use the following formula:
Confidence Interval = sample mean ± (critical value * standard error)
(a)
Step 1: Find the critical value.
Since the sample size is 16 and the confidence level is 99%, we need to find the critical value associated with a 99% confidence level and 15 degrees of freedom (n-1 for a sample size of 16).
We can use a t-distribution table or a statistical software for this calculation. For simplicity, let's assume the critical value to be approximately 2.96.
Step 2: Calculate the standard error.
The standard error is the standard deviation of the sample divided by the square root of the sample size.
Standard Error = standard deviation / √(sample size)
Standard Error = 2.3 / √(16)
Standard Error ≈ 0.575
Step 3: Calculate the confidence interval.
Confidence Interval = sample mean ± (critical value * standard error)
Confidence Interval = 26.4 ± (2.96 * 0.575)
Confidence Interval ≈ (25.357, 27.443)
Therefore, the 99% confidence interval for the population mean μ is approximately (25.357, 27.443) kilometers per litre.
(b)
To reduce the width of the confidence interval, we have a few alternatives:
1. Increase the sample size:
By increasing the sample size, the standard error decreases, resulting in a narrower confidence interval. However, this may be time-consuming and expensive.
2. Decrease the variability:
By reducing the standard deviation of the population or sample, the standard error decreases, leading to a narrower confidence interval. This can be achieved by using more precise measurement instruments or selecting a more homogeneous sample.
3. Change the confidence level:
If a narrower confidence interval is desired, we can lower the confidence level. However, this comes at the expense of reduced confidence in capturing the true population mean.
Among these alternatives, increasing the sample size or minimizing the variability are generally the preferred options. Increasing the sample size provides more information about the population, while reducing variability makes the data more consistent. The best alternative depends on the specific situation and available resources.