6. Given the following two random samples, test the hypothesis of no difference

Question

6. Given the following two random samples, test the hypothesis of no difference in population means at CL = .95.

Sample 1: {2.0, 2.5, 2.1, 2.3, 2.1, 1.6, 2.6, 2.1, 2.1, 2.4, 2.1, 2.1, 1.5, 1.7, 2.1, 2.3, 2.5, 2.3}

Sample 2: {2.9, 1.8, 1.6, 2.0, 2.5, 2.0, 2.6, 1.6, 1.7, 2.0, 2.6, 1.6, 2.5, 1.7, 1.8, 1.4, 2.3, 2.1}

6.1 An independent test

6.2 A dependent test

6.3 Discuss the criteria or rationale to determine when a dependent, versus an independent, test should (should not) be used assuming both are possible.

Explanation / Answer

Q.6

H0 : 1 = 2

Ha : 1 2

6.1 An indepenent test

Here n1 = n2 = 18

sample means

for x1 = 2.1333

x2 = 2.0389

Sample standard deviation

s1 = 0.300

s2 = 0.434

Pooled standard deviation sp = sqrt [{(n1 -1)s12 + (n2 -1)s22 }/ (n1 + n2 -2)}]

sp = 0.3735

Standard error of sample mean = sp * sqrt(1/n1 + 1/n2 ) = 0.3735 * sqrt (2/18) = 0.1245

Test statistic

t = (2.1333 - 2.0389)/ 0.1245 = 0.758

Here dF = 0.05 , dF = 18 + 18 -2 = 34

tcritical = 2.03222

so here t < tcritical so we fail to reject the null hypothesis and conclude that both population means are equal.

6.2

Here , H0 : d = 0

Ha : d 0

Difference table with mean difference and stadnard deviation of difference

Here Mean difference dbar = 0.0944

standard error of difference = sd /sqrt(n) = 0.5341/ sqrt(18) = 0.1259

HEre test statistic

t = 0.0944/0.1259 = 0.750224

Here dF = 18-1 = 17; alpha =0.05

tcritical =  2.1098

Here t < tcritical so we fail to reject the null hypothesis and conclude that both population means are equal.

6.3 The dependent--samples t test is used to compare two means for situations in which every participant is in both samples (or situation of two set of units that are matched in pairs, for example, husbands and wives). Independent sampel t test is used when participants are not matched and of differenet groups.

Sample 1 sample 2 Difference 2 2.9 -0.9 2.5 1.8 0.7 2.1 1.6 0.5 2.3 2 0.3 2.1 2.5 -0.4 1.6 2 -0.4 2.6 2.6 0 2.1 1.6 0.5 2.1 1.7 0.4 2.4 2 0.4 2.1 2.6 -0.5 2.1 1.6 0.5 1.5 2.5 -1 1.7 1.7 0 2.1 1.8 0.3 2.3 1.4 0.9 2.5 2.3 0.2 2.3 2.1 0.2 Average 0.0944 Std. Dev. 0.5341

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