Math 305/505 – STATISTICAL METHODS September 14, 2009

Comparison of Two Central Values

  1. [Example 6.1 (p. 294)] Company officials were concerned about the length of time a particular drug product retained its potency. A random sample of 10 bottles of the product was drawn from the production line and analyzed for potency. A second sample of 10 bottles was obtained and stored in a regulated environment for a period of 1 year. The readings obtained from each sample are given in the table below:

Fresh / 18 / 43 / 28 / 50 / 16 / 32 / 13 / 35 / 38 / 33
Stored / 40 / 54 / 26 / 63 / 21 / 37 / 39 / 23 / 48 / 58

Things to consider:

  1. Objective – To see if there is a difference in potency between fresh and stored.
  2. Hypotheses – Null: No difference vs. Alternative: There is a difference

Ho: mu1=mu2 vs. H1: mu1 not equal mu2.

  1. Level of significance or confidence level: alpha=.05 or 95% confidence int.
  2. Statistical method – Two sample t-test.
  3. Assumptions – 1. Two samples are independent (by design of study)

Because the two samples were taken independently from the production line.

2. Two samples come from normal population

To check for normality and presence of outliers, we use boxplots and qqplots.

The boxplots indicate that both samples do not have any potential outliers and the distribution of values is more or less symmetric.

For both samples, the points in the normal qq-plotsshow linear patterns close to the reference line.The p-values from the Shapiro-Wilk test for normalityare 0.70 and 0.49, which are bigger than alpha. Hence, we didnot find evidence on non-normality, and therefore, the t-test can be used to compare the two means.

  1. Summary measures – Create a table that will include the sample means and standard deviations. Based on the sample standard deviation, determine whether it is reasonable to assume equal variance or not.
  1. Results – The p-value of the t-test is 0.0005 (<.001). Since this is less than alpha, we reject the null hypothesis and accept the alternative.

The 95% confidence interval for the difference in means is [0.272, 0.808]. Therefore, we are 95% confident that mu1-mu2 is between 0.272 and 0.808.

  1. Conclusion

Because the p-value is < alpha, therefore, we found enough evidence to conclude that there is a statistically significant difference in potency.

  1. [Example 6.5 (p. 307)] Many states are considering lowering the blood-alcohol level at which a driver is designated as driving under the influence (DUI) of alcohol. An investigator for a legislative committee designed the following test to study the effect of alcohol on reaction time. Ten participants consumed a specified amount of alcohol. Another group of ten participants consumed the same amount of a nonalcoholic drink, a placebo. The two groups did not know whether they were receiving alcohol or the placebo. The twenty participants’ average reaction times (in seconds) to a series of simulated driving situations are reported in the table below. Does it appear that alcohol consumption increases reaction time?

Placebo / 0.90 / 0.37 / 1.63 / 0.83 / 0.95 / 0.78 / 0.86 / 0.61 / 0.38 / 1.97
Alcohol / 1.46 / 1.45 / 1.76 / 1.44 / 1.11 / 3.07 / 0.98 / 1.27 / 2.56 / 1.32

Things to consider:

  1. Objective – To determine if alcohol consumption increases reaction time.
  2. Hypotheses – Null: alcohol consumption do not increase reaction time.

Alternative: Alcohol consumption increases reaction time.

Ho: mu_alc=mu_pla vs.H1: mu_alc>mu_pla

  1. Level of significance or confidence level – Use alpha = 0.05
  2. Statistical method – Use t-test if possible, otherwise use the Wilcoxon test.
  3. Assumptions
  4. The samples are independent – Yes by the design of the study.
  5. The samples are normaly distributed.

The normal qq-plots for both samples show significant deviation from the reference line, indicating evidence of non-normality. The Shapiro-Wilk test resulted with a p-value of 0.084 for the placebo group and 0.022 for the alcohol group. These are both lower than 0.10, which indicates that we have strong evidence of non-normality of the data. Therefore, the t-test will not be appropriate for this study, and hence, we will use the Wilcoxon-MannWhitney test.

  1. Summary measures – Create a table that will include the sample means and standard deviations.
  1. Results – The p-value from the Wicoxon test is 0.0068. Since this is less the alpha, we reject the null hypothesis and accept the alternative.
  1. Conclusion – Therefore, we found sufficient evidence that alcohol consumption statistically increases the reaction time of drivers.
  1. [Homework] This data are from the U.S. Department of Health and Human Services, National Center for Health Statistics, Third National Health and Nutrition Examination Survey. It has a total of 80 cases (40 males and 40 females) with each case having values for 14 variables. These variables are listed below. For this assignment, you will only work with 3 variables (Gender, Height, and Weight)
  1. Gender
  2. Age (in years)
  3. Height (in inches)
  4. Weight (in pounds)
  5. Waist (circumference in cm.)
  6. Pulse (pulse rate in beats per minute)
  7. SysBP (systolic blood pressure in mmHg)
  8. DiasBP (diastolic blood pressure in mmHg)
  9. Cholesterol (in mg)
  10. BodyMass (body mass index)
  11. Leg (upper leg length in cm)
  12. Elbow (elbow breadth in cm)
  13. Wrist (wrist breadth in cm)
  14. Arm (arm circumference in cm)