Question

A researcher wants to compare the performance of students living with their parents and

the performance of these students whose parents are working. Random samples were

taken from two normally distributed groups of population. 25 students who are living

with their parents posted an average grade of 91.27  with standard deviation of 3,

whereas the 20 randomly selected students whose parents are working abroad have an

average grade of 88.72 with standard deviation of 3.4. Test whether there is a significant

difference between the two groups of students at 0.05 level of significance.

State the null and alternative hypotheses.

Identify if it is one-tailed test or two-tailed test.

Find its critical value and identify the rejection region.

Compute the pooled variance.

Compute the test statistic.

State the decision and conclusion.

Answer 1

$H_0:\mu _1=\mu _2\\H_1:\mu _1\ne \mu _2\\Two-tailed\\t-test\\\left| T \right|>t_{inv}\left( 0.975,25+20-2 \right) =2.0167-rejection\,\,region\\s_p=\sqrt{\frac{\left( n_1-1 \right) {s_1}^2+\left( n_2-1 \right) {s_2}^2}{n_1+n_2-2}}=\sqrt{\frac{24\cdot 3^2+19\cdot 3.4^2}{24+19}}=3.18295\\T=\frac{\bar{x}_1-\bar{x}_2}{s_p}=\frac{91.27-88.72}{3.18295}=0.801144\\T<2.0167\Rightarrow H_0\,\,is\,\,not\,\,rejected$

Conclusion: there is no significant difference between the performance of students living with their parents and the performance of these students whose parents are working