Comparing Two Means

1. Comparing Two Means

2. Two-sample problems: • compare the responses in two different groups • each group is considered to be a sample from a different population • responses of each group are independent of those in the other group • samples MAY be of different sizes

3. examples: 1- randomized comparative experiment with treatment and control (placebo) groups 2- compare random samples separately collected from two populations—M/F results, 9th and 12th grade results

4. If given data, display graphically for comparison: • back-to-back stemplot for small samples • dotplots, boxplots or histograms for larger samples (scales must be the same to compare)

5. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01.

6. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 1- Identify the means to be compared and assign a  variable 1. A= the mean amount of acetaminophen in cold tablet brand A B= the mean amount of acetaminophen in cold tablet brand B

7. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 2- write out your hypothesis statements 2. H0: A - B = 0 ( A =  B) Ha: A - B0 ( A B)

8. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 3- Assumptions and Conditions • The samples were selected independently. • Told the samples were selected randomly. • There are more than 120 tablets of each brand so n < 10% N • Since the samples are not large (nA and nB <30), we need to show that the populations (of amounts of acetaminophen are both normally distributed.

9. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 4- state the significance level (alpha) 4. Significance level:= 0.01

10. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 5- find the test statistic (the t-value) 5. Test statistic:

11. 5. Test statistic:

12. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 6- find the p-value 6. P-value = 0.0025 (from calculator)

13. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 7- compare p-value to alpha 7. is p-value < ? .0025 < .0100? yes, reject null hypothesis

14. In an attempt to determine if two competing brands of cold medicine contain, on the average, the same amount of acetaminophen, twelve different tablets from each of the two competing brands were randomly selected and tested for the amount of acetaminophen each contains. The results (in milligrams) follow. State and perform an appropriate hypothesis test. Use a significance level of 0.01. 8- write out conclusion Based on a p-value of 0.0025, the data provides strong evidence that the mean amount of acetaminophen is not the same for both brands. It appears that the average amount per tablet for brand A is less than that for brand B.

15. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength.

16. 1. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength.

17. 1. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength. • Requirements • The two samples are independently chosen. • Both are random samples. • The sample sizes are both large (nA ≥ 30 and nB ≥ 30) • More than 500 pieces of each thread possible so n < 10% N. • OK to use two-sample t-procedures.

18. 1. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength. 3. 98% CI:

19. 1. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength. 3. (–11.73, –6.073)

20. 1. Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. The sample data on tensile strength is given in the following table. Construct a 98% confidence interval for the difference of the population mean tensile strength. 4. I am 98% confident that the mean difference of the tensile strengths of these two thread types is contained in the interval –11.730 to –6.073.

21. A student recorded the mileage he obtained while commuting to school in his car. He kept track of the mileage for twelve different tankfuls of fuel, involving gasoline of two different octane ratings. Compute the 95% confidence interval for the difference of mean mileages. His data follow: 87 Octane 90 Octane 26.4, 27.6, 29.7 30.5, 30.9, 29.2 28.9, 29.3, 28.8 31.7, 32.8, 29.3