The makers of the MAGNETIZER Engine Energizer System (EES) claim that it improves gas mileage and reduces emissions in automobiles by using magnetic free energy to increase the amount of oxygen in the fuel for greater combustion efficiency. Following are test results, performed under International and U.S. Government agency standards, on a random sample of 14 vehicles. The data give the carbon monoxide (CO) levels, in parts per million, of each vehicle tested, both before installation of EES and after installation. [SOURCE: Global Source Marketing.]
Suppose that you want to perform a hypothesis test to determine whether, on average, EES reduces CO emissions.
What is the upper and lower bound for the 99% confidence interval for the change in emissions following the installation of the MAGENTIZER system (before - after)? Show two decimal places in your answer
Thank you in advance!
99% = mean ± 2.575 SEm
SEm = SD/√n
To calculate the upper and lower bounds for the 99% confidence interval for the change in emissions following the installation of the MAGNETIZER system, you will need the sample mean, sample standard deviation, sample size, and the t-table for a 99% confidence level with (n-1) degrees of freedom.
Let's assume that 'x̄' represents the sample mean difference, 's' represents the sample standard deviation, and 'n' represents the sample size.
Given that you have performed tests on a random sample of 14 vehicles, you should have the relevant statistics from the data.
Using these values, you can calculate the confidence interval using the following formula:
Lower Bound = x̄ - (t * (s / √n))
Upper Bound = x̄ + (t * (s / √n))
Now, you need to find the critical value 't' from the t-table for a 99% confidence level with (n-1) degrees of freedom. For a sample size of 14, the degrees of freedom will be (n-1) = 14-1 = 13.
Looking up the critical value 't' for a 99% confidence level with 13 degrees of freedom from the t-table, we find that t = 2.650.
Now, substitute the values into the formula to calculate the upper and lower bounds:
Lower Bound = x̄ - (2.650 * (s / √n))
Upper Bound = x̄ + (2.650 * (s / √n))
Please note that without specific data or further information regarding the mean difference and standard deviation, I am unable to provide the exact values for the upper and lower bounds in this case. You will need to use the specific sample mean difference, sample standard deviation, and sample size from the data to calculate these values accurately.
To find the upper and lower bound for the 99% confidence interval for the change in emissions following the installation of the MAGNETIZER system, you'll need to perform a hypothesis test.
Here are the steps:
1. State the hypotheses:
- Null hypothesis (H0): The average change in emissions before and after installation of the MAGNETIZER system is zero.
- Alternative hypothesis (Ha): The average change in emissions before and after installation of the MAGNETIZER system is not zero.
2. Select the significance level, α. For a 99% confidence level, α = 0.01.
3. Calculate the sample mean (x̄) and sample standard deviation (s) of the differences in emission levels before and after installation.
4. Calculate the standard error of the mean difference (SE):
SE = s / √n
where n is the sample size.
5. Find the critical value corresponding to a 99% confidence level. Since the sample size is small (n < 30), you'll need to use a t-distribution. Look up the critical value in the t-table using n - 1 degrees of freedom and α/2 (0.01/2) as the area in the tails.
6. Calculate the margin of error (ME):
ME = critical value * SE
7. Calculate the upper and lower bounds of the confidence interval:
Lower bound = x̄ - ME
Upper bound = x̄ + ME
By following these steps, you can determine the upper and lower bounds for the 99% confidence interval for the change in emissions following the installation of the MAGNETIZER system.