Answer:
[tex]z = -1.53[/tex] --- test statistic
[tex]p = 0.1260[/tex] --- p value
Conclusion: Fail to reject the null hypothesis.
Step-by-step explanation:
Given
[tex]n_1 = 80[/tex] [tex]\bar x_1= 104[/tex] [tex]\sigma_1 = 8.4[/tex]
[tex]n_2 = 70[/tex] [tex]\bar x_2 = 106[/tex] [tex]\sigma_2 = 7.6[/tex]
[tex]H_o: \mu_1 - \mu_2 = 0[/tex] --- Null hypothesis
[tex]H_a: \mu_1 - \mu_2 \ne 0[/tex] ---- Alternate hypothesis
[tex]\alpha = 0.05[/tex]
Solving (a): The test statistic
This is calculated as:
[tex]z = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} }}[/tex]
So, we have:
[tex]z = \frac{104 - 106}{\sqrt{\frac{8.4^2}{80} + \frac{7.6^2}{70} }}[/tex]
[tex]z = \frac{104 - 106}{\sqrt{\frac{70.56}{80} + \frac{57.76}{70}}}[/tex]
[tex]z = \frac{-2}{\sqrt{0.8820 + 0.8251}}[/tex]
[tex]z = \frac{-2}{\sqrt{1.7071}}[/tex]
[tex]z = \frac{-2}{1.3066}[/tex]
[tex]z = -1.53[/tex]
Solving (b): The p value
This is calculated as:
[tex]p = 2 * P(Z < z)[/tex]
So, we have:
[tex]p = 2 * P(Z < -1.53)[/tex]
Look up the z probability in the z score table. So, the expression becomes
[tex]p = 2 * 0.0630[/tex]
[tex]p = 0.1260[/tex]
Solving (c): With [tex]\alpha = 0.05[/tex], what is the conclusion based on the p value
We have:
[tex]\alpha = 0.05[/tex]
In (b), we have:
[tex]p = 0.1260[/tex]
By comparison:
[tex]p > \alpha[/tex]
i.e.
[tex]0.1260 > 0.05[/tex]
So, we fail to reject the null hypothesis.
