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?
a. 99% = mean ± Z SEm
SEm = SD/√n
find the table in the back of your text labeled "areas under normal distribution to find the proportion (±.005) and its Z score. Insert into equation above and calculate.
b. One is increase sample size.
(a) To find the 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = sample mean ± (critical value * (standard deviation / √n))
1. First, we need to find the critical value. Since we want a 99% confidence interval, the alpha level (α) is (1 - 0.99) / 2 = 0.005. Looking up the critical value for a two-tailed test with an alpha of 0.005 in the z-table, we find it to be approximately 2.576.
2. Next, we substitute the given values into the formula:
Confidence Interval = 26.4 ± (2.576 * (2.3 / √16))
Simplifying this expression:
Confidence Interval = 26.4 ± (2.576 * 0.575)
Calculating the upper and lower limits:
Upper Limit = 26.4 + (2.576 * 0.575) ≈ 28.015
Lower Limit = 26.4 - (2.576 * 0.575) ≈ 24.785
Therefore, the 99% confidence interval for the population mean is approximately (24.785, 28.015) kilometers per liter.
(b) To reduce the width of the confidence interval, we have a few alternatives:
1. Increase the sample size: Increasing the sample size (n) decreases the standard error and narrows the confidence interval. However, this might not always be feasible or cost-effective.
2. Use a smaller confidence level: Choosing a smaller confidence level, such as 95% instead of 99%, will result in a narrower confidence interval. However, this comes at the cost of reducing the level of certainty.
3. Reduce the standard deviation: If we can somehow decrease the variability of the fuel consumption data, the standard deviation will be smaller, leading to a narrower confidence interval. However, this is often not under our control.
In this case, since the sample size is already fixed, and we cannot reduce the standard deviation or change the confidence level, our best option to reduce the width of the confidence interval would be to increase the sample size. This will provide more precise estimates and reduce the uncertainty associated with the mean fuel consumption of mid-sized cars.
To find the 99% confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (critical value × standard error)
First, we need to calculate the critical value. Since the sample size is small (n = 16), we will use a t-distribution instead of a normal distribution. The degrees of freedom are (n - 1) = 15. From a t-table or software, we find the critical value for a 99% confidence level for 15 degrees of freedom is approximately 2.947.
Next, we need to calculate the standard error using the formula:
Standard error = standard deviation / √(sample size)
In this case, the standard deviation is 2.3 and the sample size is 16. So the standard error is 2.3 / √(16) = 0.575.
Now we can calculate the confidence interval by substituting the values into the formula:
Confidence interval = 26.4 ± (2.947 × 0.575)
Calculating this expression, we get:
Lower bound = 26.4 - (2.947 × 0.575) ≈ 24.414
Upper bound = 26.4 + (2.947 × 0.575) ≈ 28.386
Therefore, the 99% confidence interval for the population mean is approximately 24.414 to 28.386 kilometers per liter.
(b) If the confidence interval obtained in (a) is too wide, there are several alternatives to reduce its width:
1. Increase the sample size: By collecting data from more mid-sized cars, we can reduce the standard error and, consequently, narrow the confidence interval.
2. Decrease the variability: If possible, try to reduce the standard deviation of the fuel consumption among mid-sized cars. This will result in a smaller standard error and a narrower confidence interval.
3. Change the confidence level: If a narrower confidence interval is acceptable, the confidence level can be reduced. However, a lower confidence level means being less certain about the true population mean.
Among these alternatives, increasing the sample size is usually the best option. It provides a more accurate estimate of the population mean and reduces the margin of error. Additionally, increasing the sample size helps to account for any potential outliers or unusual characteristics in the data.