In a sample of seven cars, each car was tested for nitrogen-oxide emissions (in grams per mile) and the following results were obtained:0.18, 0.12, 0.05, 0.19, 0.18,0.13, 0.15. Assuming that this sample is representative of the cars in use, construct a 98% confidence interval estimate of the mean amount of nitrogen-oxide emissions for all cars. If the EPA requires that nitrogen-oxide emissions be less than 0.165 g/mi,can we safely conclude that this requirement is being met?
I don't need the answer, I just need to know how to do the steps to get to the answer. This is my last class and I hopefully graduate. This class is extremely hard because I never took anything beyond basic math
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.01) and its Z score.
98% = mean ± Z SEm
98% = mean ± 2.333 SEm
I hope this helps.
To construct a 98% confidence interval estimate of the mean amount of nitrogen-oxide emissions for all cars, you can follow these steps:
Step 1: Calculate the sample mean (x̄) of the nitrogen-oxide emissions. Add up all the values and divide by the sample size (n).
Step 2: Calculate the sample standard deviation (s) of the nitrogen-oxide emissions. This measures the spread of the data. You can use the formula:
s = √(Σ(xi - x̄)² / (n - 1))
where Σ represents the summation of the squared differences between each value (xi) and the sample mean (x̄).
Step 3: Determine the critical value (z*) from the standard normal distribution table for a 98% confidence level. This value corresponds to the desired level of confidence and sample size. In this case, the critical value is approximately 2.33.
Step 4: Calculate the standard error of the mean (SE), which is the standard deviation divided by the square root of the sample size:
SE = s / √n
Step 5: Calculate the margin of error (ME) by multiplying the critical value (z*) by the standard error of the mean (SE):
ME = z* * SE
Step 6: Calculate the lower and upper bounds of the confidence interval:
Lower bound = x̄ - ME
Upper bound = x̄ + ME
Step 7: Interpret the confidence interval. The confidence interval gives you a range of values within which the true population mean is likely to fall with a certain level of confidence. If the requirement of the EPA falls within this range, you can conclude that the requirement is being met.
Remember to consult a standard normal distribution table or use statistical software to find the critical value (z*) corresponding to the desired confidence level.
Good luck with your studies, and congratulations on almost graduating!
To construct a 98% confidence interval estimate of the mean amount of nitrogen-oxide emissions for all cars, follow these steps:
Step 1: Calculate the sample mean (x̄) and sample standard deviation (s) of the nitrogen-oxide emissions.
Step 2: Determine the critical value corresponding to a 98% confidence level. Since the sample size is small (n < 30), we need to use a t-distribution. The critical value can be found using a t-table or a calculator. For a 98% confidence level, with a sample size of 7-1=6 degrees of freedom, the critical value would be t*.
Step 3: Calculate the margin of error (E) using the formula: E = t* * (s / √n), where t* is the critical value, s is the sample standard deviation, and n is the sample size.
Step 4: Compute the lower and upper bounds of the confidence interval by subtracting and adding the margin of error from the sample mean: Lower bound = x̄ - E and Upper bound = x̄ + E.
Step 5: The confidence interval estimate is the range between the lower and upper bounds obtained in step 4. It represents the range within which we are 98% confident the true mean nitrogen-oxide emissions for all cars would fall.
Now, to determine whether the EPA requirement is being met, evaluate if the requirement value (0.165 g/mi) falls within the confidence interval. If the requirement value is within the confidence interval, we cannot safely conclude that it is being met. However, if the requirement value is outside the confidence interval, we can safely conclude that it is not being met.
Remember, it is important to consult with your course materials or instructor to check if there are any specific procedures or adjustments required.