Question

Consider the following hypotheses and the sample data drawn independently from two normally

distributed populations:

H0 : 1 

Answer 1

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations.

$H_0:\mu_1-\mu_2\geq0$

$H_1:\mu_1-\mu_2<0$

$$\begin{matrix} \bar{x}_1=249 & & \bar{x}_2=262\\ s_1=35 & & s_2=23\\ n_1=10 & & n_2=10\\ \end{matrix}$$

1.  Calculate the value of the test statistic under the assumption that the population variances are unknown but equal. 

$$df=df_1+df_2=9+9=18$$

$$t=\dfrac{ \bar{x}_1- \bar{x}_2}{\sqrt{(\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2})(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}$$

$$=\dfrac{ 249-262}{\sqrt{(\dfrac{(10-1)35^2+(10-1)23^2}{10+10-2})(\dfrac{1}{10}+\dfrac{1}{10})}}$$

$$\approx-0.981586\approx-0.98$$

2. Calculate the critical value at the 5% level of significance.

$df=18, \alpha=0.05,$ left-tailed

$$t_c=-1.734$$

3. Do you reject the null hypothesis at the 5% level?

$$t=-0.98\geq-1.734=t_c$$

No, since the value of the test statistic is not less than the critical value.

4. Calculate the value of the test statistic under the assumption that the population variances are unknown and are not equal.

$$df=\dfrac{(\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2})^2}{\dfrac{(s_1^2/n_1)^2}{n_1-1}+\dfrac{(s_2^2/n_1)^2}{n_2-1}}\approx15.55$$

$$t=\dfrac{ \bar{x}_1- \bar{x}_2}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_1^2}{n_1}}}\approx-0.981586\approx-0.98$$

5. Calculate the critical value at the 5% level of significance.

$df=15.55, \alpha=0.05,$ left-tailed

$$t_c=-1.749$$

6. Do you reject the null hypothesis at the 5% level?

$$t=-0.98\geq-1.749=t_c$$

No, since the value of the test statistic is not less than the critical value.