B) A random sample of 16 mid-sized cars tested for fuel consumption gave a mean of 26.4Kilometers 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
99% = mean ± Z (SEm)
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.005) in the smaller area and its Z score. Insert data into equations above.
To find a 99% confidence interval for the population mean μ, we can use the formula:
Confidence Interval = (sample mean) ± (critical value) * (standard deviation / √(sample size))
First, we need to find the critical value associated with a 99% confidence level. This critical value corresponds to the area under the standard normal distribution curve outside the confidence interval. For a 99% confidence level, the remaining area is (1 - 0.99) / 2 = 0.005.
Using a standard normal distribution table or a calculator, we can find the z-score associated with this remaining area of 0.005. The z-score is approximately 2.576.
Now, we can calculate the confidence interval:
Confidence Interval = (26.4) ± (2.576) * (2.3 / √16)
Step 1: Calculate the standard error
Standard Error = (standard deviation) / √(sample size)
Standard Error = 2.3 / √16
Standard Error = 2.3 / 4
Standard Error = 0.575
Step 2: Calculate the confidence interval
Confidence Interval = (26.4) ± (2.576) * (0.575)
Confidence Interval = 26.4 ± 1.4816
Confidence Interval ≈ (24.92, 27.88)
Therefore, the 99% confidence interval for the population mean μ is approximately (24.92, 27.88) kilometers per liter.