Answered

The Pearson correlation coefficient for a relationship between two variables from a data sample of [tex]$n=9$[/tex] was found to be [tex]$r \approx -0.677$[/tex]. Is this relationship statistically significant at the [tex][tex]$\alpha=0.05$[/tex][/tex] level?

\begin{tabular}{|c|c|c|}
\hline \multicolumn{3}{|c|}{Critical Values of the Pearson Correlation Coefficient} \\
\hline [tex]$n$[/tex] & [tex]$\alpha = 0.05$[/tex] & [tex]$\alpha = 0.01$[/tex] \\
\hline 4 & 0.950 & 0.990 \\
\hline 5 & 0.878 & 0.959 \\
\hline 6 & 0.811 & 0.917 \\
\hline 7 & 0.754 & 0.875 \\
\hline 8 & 0.707 & 0.834 \\
\hline 9 & 0.666 & 0.798 \\
\hline 10 & 0.632 & 0.765 \\
\hline 11 & 0.602 & 0.735 \\
\hline 12 & 0.576 & 0.708 \\
\hline
\end{tabular}

Answer:

A. Yes, the linear relationship between the variables is statistically significant at the 0.05 level of significance, because [tex]$|r|\ \textgreater \ $[/tex] critical value.

B. Yes, the linear relationship between the variables is statistically significant at the 0.05 level of significance, because [tex]$r\ \textgreater \ 0$[/tex].

C. No, the linear relationship between the variables is not statistically significant at the 0.05 level of significance, because [tex][tex]$|r| \leq$[/tex][/tex] critical value.



Answer :

GinnyAnswer
To determine whether the linear relationship between the two variables is statistically significant at the [tex]\( \alpha = 0.05 \)[/tex] level, let's follow these steps:

1. Given Values:
- Sample size ([tex]\( n \)[/tex]) = 9
- Pearson correlation coefficient ([tex]\( r \)[/tex]) ≈ -0.677
- Level of significance ([tex]\( \alpha \)[/tex]) = 0.05

2. Critical Value:
- From the provided critical values table for [tex]\( n = 9 \)[/tex] and [tex]\( \alpha = 0.05 \)[/tex], the critical value is 0.666.

3. Absolute Value of [tex]\( r \)[/tex]:
- Calculate the absolute value of the Pearson correlation coefficient:
[tex]\[ |r| = |-0.677| = 0.677 \][/tex]

4. Compare [tex]\( |r| \)[/tex] with the Critical Value:
- [tex]\( |r| = 0.677 \)[/tex]
- Critical value = 0.666

We see that [tex]\( |r| = 0.677 \)[/tex] is greater than the critical value of 0.666.

5. Conclusion:
- Since [tex]\( |r| \)[/tex] is greater than the critical value, the relationship between the variables is statistically significant at the [tex]\( \alpha = 0.05 \)[/tex] level.

Therefore, the correct conclusion is:

Yes, the linear relationship between the variables is statistically significant at the 0.05 level of significance, because [tex]\( |r| > \)[/tex] critical value.

Other Questions