Question

A researcher wanted to study the effect of a newly developed gasoline additive (Additive X) on automobile mileage (miles per gallon, MPG). To gather information, a random sample of cars has been selected. For each car, the MPG was measured both when gasoline with Additive X is used and when gasoline without Additive X is used. The order of the two treatments (with Additive X versus without Additive X) was randomized and care was taken so that there was no carry over effect between the two MPG measurements The 95% confiden ce interval for the mean mileage improvement* Hd for the population of all such cars was found to be (1 MPG, 4 MPG). The 99% confidence interval for the mean mileage improvement* Hd for the population of all such cars was found to be (0.5 MPG, 4.5 MPG). improvement is defined as the difference in MPG for cars using gasoline with Additive X less gasoline without Additive X. The researcher knows that you have some statistical background and she would like for you to revise the following 2 statement that she has prepared for her report. Please read the statements carefully and share with us your opinion. For each statement, the researcher will need the following two items: 1. Decide if the provided statement 2. If appropriate, explain why this is a correct statement. If not appropriate, provide a correct statement Statement 1: TBD appropriate or not appropriate. points We estimate the population mean improvement in mileage (gasoline with Additive X less without Additive X) to be 2.5 MPG with a 99% margin of error of 2 MPG. this statement is not appropriate. corrected, the statement should read: we estimate the sample mean improvement (dbar) in mileage to be 2.5 MPG with a 99% margin of error 2 .b For each statement, the researcher will need the following two items: 1. Decide if the provided statement is appropriate or not appropriate. 2. If appropriate, explain why this is a correct statement. If not appropriate, provide a correct statement. TBD points Statement 2: If we repeated this sampling procedure 300 times and for each repetition we computed the corresponding 99% confidence interval, we would expect 297 of the resulting intervals would each contain the corresponding sample mean difference in automobile mileage (gasoline with Additive X less without Additive X). this statement is not appropriate. the correct statment should read: If we repeated this sampling procedure 300 times and for each repetition we computed the corresponding 99% confidence interval, we would expect 297 of the resulting intervals would each contain the corresponding population mean difference in automobile mileage (gasoline with Additive less without Additive X). .C A competitor's fuel additive had an established average improvement in mileage of 0.75 MPG. The researcher used the study results to test the following theories: TBD points Ho: Hd 0.75 MPG versus Ha: Pd # 0.75 MPG where the parameter Hd represents the mean mileage improvement (with Additive X less without Additive X) for the population of all such cars. According to her calculations, the correspoding p-value for this test is 0.028. The researcher has checked her calculations multiple times and believes that her p-value is correct. Use the two confidence intervals to explain the researcher why the p-value of 0.028 is reasonable p-value for this test. the p value of .028 is reasonable for this test given the confidence intervals. The 95% confidence interval does not include the.75 value. Because the.75 value is not included in the 95% confidence interval but it is included in the not extremely small but wider 99% confidence interval a p value of .28 seems reasonable because the p value also not extremely large which shows there is not STRONG evidence against the null but there is significnt enough evidence to reject the null because a 99% confidence interval is very wide and contains man values, more than the 95% confidence interval. 1.d We were told the results are based on a random sample of cars. State the remaining assumption (in context) needed TBD to perform the confidence intervals and the hypothesis test. points we must also assume that the data are observations from a normally distributed population mean mileage improvement

Answer 1

1. a) We always estimate the population mean and not the sample mean. Hence, the statement is correct as it tells us about the population mean improvement in mileage and gives the margin of error.

1. b) The written answer is correct.

1. c) The corresponding p-value for the test is 0.028. The p-value is very small which means that there is a very low probability (2.8%) that the result we got was due to just chance. Hence, we can say that this is a reasonable p-value as it is less than 0.05 or less than 5% of probability that this result occured due to chance.

1. d) The written answer is corre