Following is a sample of five matched pairs.
$\begin{array}{llllll}
\hline \text { Sample 1 } & 20 & 20 & 23 & 18 & 22 \
\text { Sample 2 } & 23 & 16 & 15 & 14 & 18 \
\hline
\end{array}$

Let $\mu _ { 1 }$ and $\mu _ { 2 }$ represent the population means and let $\mu _ { \mathrm { d } } = \mu _ { 1 } - \mu _ { 2 }$. A test will be made of the hypotheses $H _ { 0 } : \mu _ { \mathrm { d } } = 0$ versus $H _ { 1 } : \mu _ { \mathrm { d } } 0$. Can you reject $H _ { 0 }$ at the $\alpha = 0.01$ level of significance?
A) No
B) Yes

Question

Following is a sample of five matched pairs.
$\begin{array}{llllll}
\hline \text { Sample 1 } & 20 & 20 & 23 & 18 & 22 \
\text { Sample 2 } & 23 & 16 & 15 & 14 & 18 \
\hline
\end{array}$

Let $\mu _ { 1 }$ and $\mu _ { 2 }$ represent the population means and let $\mu _ { \mathrm { d } } = \mu _ { 1 } - \mu _ { 2 }$. A test will be made of the hypotheses $H _ { 0 } : \mu _ { \mathrm { d } } = 0$ versus $H _ { 1 } : \mu _ { \mathrm { d } } > 0$. Can you reject $H _ { 0 }$ at the $\alpha = 0.01$ level of significance?
A) No
B) Yes

Quizplus · Accepted Answer

To determine if $H_0$ can be rejected, we calculate the differences between the pairs, the mean of the differences, and the standard deviation of the differences. Then, we use a t-test for paired samples to find the t-value and compare it with the critical t-value at $\alpha = 0.01$.Differences (Sample 1 - Sample 2): $-3, 4, 8, 4, 4$Mean of differences ($\bar{d}$): $3.4$Standard deviation of differences (s): $4.51$Sample size (n): $5$Standard error of $\bar{d}$ (SE): $s / \sqrt{n} = 4.51 / \sqrt{5} = 2.02$t-value: $\bar{d} / SE = 3.4 / 2.02 = 1.68$Degrees of freedom (df): $n - 1 = 4$At $\alpha = 0.01$, the critical t-value for a one-tailed test with 4 degrees of freedom is approximately $3.747$.Since the calculated t-value $1.68$ is less than the critical t-value $3.747$, we cannot reject $H_0$ at the $\alpha = 0.01$ level of significance.

Following Is a Sample of Five Matched Pairs Sample 1 2020231822 Sample 2 2316151418\begin{array}{llllll}\hline \text { Sample 1 } & 20 & 20 & 23 & 18 & 22 \\\text { Sample 2 } & 23 & 16 & 15 & 14 & 18 \\\hline\end{array} Sample 1 Sample 2 ​2023​2016​2315​1814​2218​​

Following Is a Sample of Five Matched Pairs $\begin{array}{llllll}\hline \text { Sample 1 } & 20 & 20 & 23 & 18 & 22 \\\text { Sample 2 } & 23 & 16 & 15 & 14 & 18 \\\hline\end{array}$