# Parametric vs. Non-parametric Testing

Gerber Biotech Industries was testing three different drugs namely Breath-Easy, Fresh Mint and Sober Breath, meant for neutralizing the alcohol content in the breath so that the (drunken) drivers can pass the breath analyzer test. The three drugs are tested on randomly selected volunteers and the time taken for each of the drugs to neutralize the effect is recorded. Breath-Easy and Fresh Mint were tested on 8 individuals. The data is presented in the table below:

Volunteer Number

Drug 1 2 3 4 5 6 7 8

Time Taken in Minutes

BE 35 51 56 57 60 74

FM 43 53 41 57 36 55

SB 35 32 28 44 40 34 73 48

(a) Test whether there is any significant difference between the average time taken by the three drugs using an appropriate parametric test (Use 0.10 percent significance level)

(b) Repeat the test using an appropriate non-parametric test.

(C) Should the conclusions between the two tests consistent? Why or why not? If the conclusions are not consistent, which one would you agree with and why?

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

The problem is solved using One-way ANOVA and KRUSKAL WALLIS TEST.

(a) Test whether there is any significant difference between the average time taken by the three drugs using an appropriate parametric test (Use 0.10 percent significance level)

This is a question on comparing more than 2 means and ANOVA is the parametric test to be used for this test.

Hypothesis:

H0: Three populations are identical

H1: atleast one of the 3 populations is different from others

Calculate F-statistics using the data and following degrees of freedom.

dfC=3-1=2

dfE=6+6+8-3=17

dfT=6=6+8-1=19

F=MSC/MSE=324.15/152.38=2.13

F Critical from table at DOF in numerator=2 and DOF in denominator=17, at Î±=0.1 is=2.64

If Observed F (F statistic) is greater than the critical F value, reject the null hypothesis.

As Observed F(2.13) is less than the critical F (2.64), we fail to reject the null hypothesis.

(b) Repeat the test using an appropriate non-parametric test.

Non parametric alternative to the one-way ANOVA is KRUSKAL WALLIS TEST

H0: Three populations are identical

H1: atleast one of the 3 populations is different from others

Step 1: Rank the data in all groups together. In case of a tie, an average of the two ranks is assigned to both the values.

Step 2: Calculate T_j, Total of ranks for each group:

Step 3: Calculate Kruskal-Wallis (K) Statistic using following formula:

K=5.085

Step 3: K value is approximately chi-square distributed with 2 (C-1) degrees of freedom.

Step 4: Reject the null hypothesis if K value is greater than the chi-square critical value.

As K=5.085 >Chi-square(0.1,2)=4.6052, We reject the null hypothesis and conclude that atleast one of the three populations is different from others.

(C) Should the conclusions between the two tests consistent? Why or why not? If the conclusions are not consistent, which one would you agree with and why?

No , The conclusions need not be consistent.

Reasons:

The parametric tests are based on assumptions about the population characteristics. Also, parametric test (ANOVA) uses the observations directly to conduct the test. Non-parametric tests are based on lesser assumptions about the population and in this case, is based on the ranks of the observation instead of the observations itself.

In this case, as the data is available for parametric test, the values mean something to us and the non-parametric test does not use actual data, I would agree with the result of the parametric test i.e. the One way ANOVA. The conclusion then would be to fail to reject the null hypothesis. Thus, all population means are the same.

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