First, let's outline our approach step-by-step to solve this problem:
1. Formulate the Hypotheses:
- The null hypothesis ([tex]$H_0$[/tex]) is that there is no correlation between the quality scores and the cost of the paints, i.e., [tex]$H_0: \rho_s = 0$[/tex].
- The alternative hypothesis ([tex]$H_1$[/tex]) is that there is a correlation between the quality scores and the cost of the paints, i.e., [tex]$H_1: \rho_s \neq 0$[/tex].
- So the correct choice is D: [tex]$H_0: \rho_s = 0$[/tex] and [tex]$H_1: \rho_s \neq 0$[/tex].
2. Rank the Data:
- The quality scores: [72, 77, 73, 67, 68, 63, 82, 77, 64, 62, 69]
- The cost scores: [26, 27, 28, 22, 20, 15, 26, 22, 16, 15, 20]
The ranks for the quality scores are:
[tex]\[
[ 7.0, 9.5, 8.0, 4.0, 5.0, 2.0, 11.0, 9.5, 3.0, 1.0, 6.0 ]
\][/tex]
The ranks for the cost scores are:
[tex]\[
[ 8.5, 10.0, 11.0, 6.5, 4.5, 1.5, 8.5, 6.5, 3.0, 1.5, 4.5 ]
\][/tex]
3. Compute the Spearman Rank Correlation Coefficient ([tex]$r_s$[/tex]):
- The Spearman rank correlation coefficient, denoted as [tex]$r_s$[/tex], measures the strength and direction of association between two ranked variables.
[tex]\[
r_s = 0.834
\][/tex]
We round to three decimal places as needed.
4. Evaluate the Test Statistic and p-value:
- For the correlation calculation, we also have a p-value:
[tex]\[
\text{p-value} = 0.0014
\][/tex]
5. Decision Rule:
- Using a significance level of [tex]\(\alpha = 0.05\)[/tex], we compare the p-value to [tex]\(\alpha\)[/tex].
- Since the p-value (0.0014) is less than the significance level [tex]\(\alpha = 0.05\)[/tex], we reject the null hypothesis.
Conclusion:
- Given that we reject the null hypothesis, there is significant evidence to suggest a correlation between quality scores and cost of the paints. Therefore, based on these results, it does appear that one might get better quality paint by paying more.
