IDNLearn.com is designed to help you find reliable answers quickly and easily. Join our interactive Q&A community and access a wealth of reliable answers to your most pressing questions.
Sagot :
Let's tackle the problem step-by-step, ensuring we cover all necessary details for a thorough solution.
### Step 1: Formulate the Hypotheses
The goal is to test for a correlation between the quality scores and the costs of the paints using Spearman's rank correlation coefficient. The null and alternative hypotheses are given as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\rho_s = 0\)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\(\rho_s \neq 0\)[/tex]
Here, [tex]\(\rho_s\)[/tex] is the population Spearman's rank correlation coefficient.
### Step 2: Rank the Data
First, we rank the quality scores and the costs:
- Quality: [72, 77, 73, 67, 68, 63, 82, 77, 64, 62, 69]
- Cost: [26, 27, 28, 22, 20, 15, 26, 22, 16, 15, 20]
#### Quality Ranks:
- 72 → 7
- 77 → 9.5
- 73 → 8
- 67 → 4
- 68 → 5
- 63 → 2
- 82 → 11
- 77 → 9.5
- 64 → 3
- 62 → 1
- 69 → 6
Ranked list: [7, 9.5, 8, 4, 5, 2, 11, 9.5, 3, 1, 6]
#### Cost Ranks:
- 26 → 8.5
- 27 → 10
- 28 → 11
- 22 → 6.5
- 20 → 4.5
- 15 → 1.5
- 26 → 8.5
- 22 → 6.5
- 16 → 3
- 15 → 1.5
- 20 → 4.5
Ranked list: [8.5, 10, 11, 6.5, 4.5, 1.5, 8.5, 6.5, 3, 1.5, 4.5]
### Step 3: Calculate Spearman's Rank Correlation Coefficient ([tex]\( r_s \)[/tex])
Given the ranked data:
- Rank of Quality: [7, 9.5, 8, 4, 5, 2, 11, 9.5, 3, 1, 6]
- Rank of Cost: [8.5, 10, 11, 6.5, 4.5, 1.5, 8.5, 6.5, 3, 1.5, 4.5]
The calculated Spearman's rank correlation coefficient is:
[tex]\[ r_s \approx 0.834 \][/tex]
### Step 4: Compute the Test Statistic
The test statistic for Spearman's rank correlation is:
[tex]\[ t = \frac{r_s \sqrt{n - 2}}{\sqrt{1 - r_s^2}} \][/tex]
Using the sample size [tex]\( n = 11 \)[/tex] and [tex]\( r_s \approx 0.834 \)[/tex]:
[tex]\[ t \approx 28.946 \][/tex]
### Step 5: Determine the Critical Value
For a two-tailed test at the significance level [tex]\(\alpha = 0.05\)[/tex] and degrees of freedom [tex]\( df = n - 2 = 9 \)[/tex]:
[tex]\[ t_{critical} \approx 2.262 \][/tex]
### Step 6: Conclusion
We compare the test statistic [tex]\( t \approx 28.946 \)[/tex] to the critical value [tex]\( t_{critical} \approx 2.262 \)[/tex]. Since [tex]\( t \gg t_{critical} \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Step 7: Interpretation
Based on the statistical analysis, there is significant evidence to suggest a positive correlation between the cost of the paint and its quality score. In other words, the data suggests that, in general, better quality paint tends to cost more.
### Final Statements:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\(\rho_s = 0\)[/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): [tex]\(\rho_s \neq 0\)[/tex]
- Spearman’s rank correlation coefficient ([tex]\( r_s \)[/tex]): [tex]\( 0.834 \)[/tex]
- Test Statistic: [tex]\( t \approx 28.946 \)[/tex]
- Critical Value: [tex]\( \pm 2.262 \)[/tex]
Thus, paying more generally results in better quality paint.
### Step 1: Formulate the Hypotheses
The goal is to test for a correlation between the quality scores and the costs of the paints using Spearman's rank correlation coefficient. The null and alternative hypotheses are given as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\rho_s = 0\)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\(\rho_s \neq 0\)[/tex]
Here, [tex]\(\rho_s\)[/tex] is the population Spearman's rank correlation coefficient.
### Step 2: Rank the Data
First, we rank the quality scores and the costs:
- Quality: [72, 77, 73, 67, 68, 63, 82, 77, 64, 62, 69]
- Cost: [26, 27, 28, 22, 20, 15, 26, 22, 16, 15, 20]
#### Quality Ranks:
- 72 → 7
- 77 → 9.5
- 73 → 8
- 67 → 4
- 68 → 5
- 63 → 2
- 82 → 11
- 77 → 9.5
- 64 → 3
- 62 → 1
- 69 → 6
Ranked list: [7, 9.5, 8, 4, 5, 2, 11, 9.5, 3, 1, 6]
#### Cost Ranks:
- 26 → 8.5
- 27 → 10
- 28 → 11
- 22 → 6.5
- 20 → 4.5
- 15 → 1.5
- 26 → 8.5
- 22 → 6.5
- 16 → 3
- 15 → 1.5
- 20 → 4.5
Ranked list: [8.5, 10, 11, 6.5, 4.5, 1.5, 8.5, 6.5, 3, 1.5, 4.5]
### Step 3: Calculate Spearman's Rank Correlation Coefficient ([tex]\( r_s \)[/tex])
Given the ranked data:
- Rank of Quality: [7, 9.5, 8, 4, 5, 2, 11, 9.5, 3, 1, 6]
- Rank of Cost: [8.5, 10, 11, 6.5, 4.5, 1.5, 8.5, 6.5, 3, 1.5, 4.5]
The calculated Spearman's rank correlation coefficient is:
[tex]\[ r_s \approx 0.834 \][/tex]
### Step 4: Compute the Test Statistic
The test statistic for Spearman's rank correlation is:
[tex]\[ t = \frac{r_s \sqrt{n - 2}}{\sqrt{1 - r_s^2}} \][/tex]
Using the sample size [tex]\( n = 11 \)[/tex] and [tex]\( r_s \approx 0.834 \)[/tex]:
[tex]\[ t \approx 28.946 \][/tex]
### Step 5: Determine the Critical Value
For a two-tailed test at the significance level [tex]\(\alpha = 0.05\)[/tex] and degrees of freedom [tex]\( df = n - 2 = 9 \)[/tex]:
[tex]\[ t_{critical} \approx 2.262 \][/tex]
### Step 6: Conclusion
We compare the test statistic [tex]\( t \approx 28.946 \)[/tex] to the critical value [tex]\( t_{critical} \approx 2.262 \)[/tex]. Since [tex]\( t \gg t_{critical} \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Step 7: Interpretation
Based on the statistical analysis, there is significant evidence to suggest a positive correlation between the cost of the paint and its quality score. In other words, the data suggests that, in general, better quality paint tends to cost more.
### Final Statements:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\(\rho_s = 0\)[/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): [tex]\(\rho_s \neq 0\)[/tex]
- Spearman’s rank correlation coefficient ([tex]\( r_s \)[/tex]): [tex]\( 0.834 \)[/tex]
- Test Statistic: [tex]\( t \approx 28.946 \)[/tex]
- Critical Value: [tex]\( \pm 2.262 \)[/tex]
Thus, paying more generally results in better quality paint.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
