Question

8. The interpersonal Reactivity Index is a survey designed to asses four different types of empathy. One type of empathy, called Empathetic Concern, measures the tendency to feel sympathy and compassion for people who are less fortunate. The index ranges from 0 (less empathetic) to 28 (more empathetic). The following data representing random samples of 16 males and 16 males, are consistent with results reported in psychological studies. Boxplots show that is reasonable to assume that the populations are approximately normal MALES 21 17 23 13 15 26 23 13 8 20 15 12 18 16 25 FEMALES 19 25 16 19 24 26 25 28 20 26 23 2123 15 20 a) Can you conclude that there is a difference in mean empathy scores between men and women? Use 5% significance level. Construct a 95% confidence level for the mean difference in empathy scores between men and women. b)

Answer 1

For males

$\bar x1$ = 17.9375, s1 = 5.17, n1 = 16

For females

$\bar x2$ = 22, s2 = 3.69, n2 = 16

a) H₀:

μ1 μ2 =

    H₁: $\mu1 \neq \mu2$

The test statistic t = ($\bar x1 - \bar x2$)/sqrt(s1^2/n1 + s2^2/n2)

                            = (17.9375 - 22)/sqrt((5.17)^2/16 + (3.69)^2/16)

                            = -2.558

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((5.17)^2/16 + (3.69)^2/16)^2/(((5.17)^2/16)^2/15 + ((3.69)^2/16)^2/15)

    = 27

At 5% significance level, the critical values are t^* = +/- 2.052

Since the test statistic value is less than the negative critical value(-2.558 < -2.052), so we should reject the null hypothesis.

So at 5% significance level we can conclude that there is a difference in mean empathy scores between men and women.

b) At 95% confidence interval, the critical values is t^* = 2.052

The 95% confidence interval for $\mu1 - \mu2$ is

($\bar x1 - \bar x2$) +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

   = (17.9375 - 22) +/- 2.052 * sqrt((5.17)^2/16 + (3.69)^2/16)

= -4.0625 +/- 3.2585

= -7.321, -0.804