If r‐represent the correlation coefficient between age and weight and N is the number of subjects. The calculated value of r = 0.510 and N = 15. The table value of r under column 0.05 is 0.514 and 0.01 is 0.561 for N - 2(=13)df. Then which of the following will be correct?
There is no significant correlation between age and weight
Understanding Correlation Significance
This question asks us to determine the significance of the calculated correlation coefficient between age and weight based on the given sample size and critical table values. The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables.
Key Information Provided:
- Calculated correlation coefficient, r = $0.510$
- Sample size, N = $15$
- Degrees of freedom, df = $N - 2 = 15 - 2 = 13$
- Critical table value of r at $\alpha = 0.05$ level for 13 df = $0.514$
- Critical table value of r at $\alpha = 0.01$ level for 13 df = $0.561$
Testing for Statistical Significance
To determine if the observed correlation ($r = 0.510$) is statistically significant, we compare the absolute value of the calculated r ($|r|$) with the critical table value for the given degrees of freedom and significance level (alpha). The significance level represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Common levels are 0.05 (5%) and 0.01 (1%).
The null hypothesis ($\text{H}_0$) for testing a correlation coefficient is typically that there is no linear correlation in the population ($\rho = 0$). The alternative hypothesis ($\text{H}_1$) is that there is a linear correlation ($\rho \ne 0$).
The decision rule is:
- If $|r_{\text{calculated}}| > r_{\text{table}}$, we reject the null hypothesis ($\text{H}_0$) and conclude the correlation is statistically significant at the chosen alpha level.
- If $|r_{\text{calculated}}| \le r_{\text{table}}$, we fail to reject the null hypothesis ($\text{H}_0$) and conclude there is no statistically significant correlation at the chosen alpha level.
Comparing Calculated r with Table Values
We compare the calculated $|r| = |0.510| = 0.510$ with the table values for 13 degrees of freedom.
| Significance Level ($\alpha$) | Critical Table Value of r (13 df) | Calculated $|r|$ | Comparison | Decision |
|---|---|---|---|---|
| 0.05 | $0.514$ | $0.510$ | $0.510 \le 0.514$ | Fail to Reject $\text{H}_0$ |
| 0.01 | $0.561$ | $0.510$ | $0.510 \le 0.561$ | Fail to Reject $\text{H}_0$ |
As shown in the table, the absolute value of the calculated correlation coefficient ($0.510$) is less than or equal to the critical table value at both the 0.05 level ($0.514$) and the 0.01 level ($0.561$).
Conclusion on Significance
Since the calculated correlation coefficient ($0.510$) does not exceed the critical value at either the 0.05 or the 0.01 significance level, we conclude that the correlation between age and weight in this sample is not statistically significant at these levels. This means we do not have enough statistical evidence to reject the null hypothesis that the true population correlation is zero.
Analyzing the Options
- Option 1: There is a significant correlation between age and weight at 0.01 level.
Incorrect. Calculated $|r|$ (0.510) is less than the table value (0.561) at $\alpha=0.01$.
- Option 2: There is a significant correlation between age and weight at 0.05 level.
Incorrect. Calculated $|r|$ (0.510) is less than or equal to the table value (0.514) at $\alpha=0.05$.
- Option 3: There is no significant correlation between age and weight.
Correct. Based on our comparison with the critical values, the calculated correlation is not statistically significant at the standard 0.05 or 0.01 levels.
- Option 4: Null hypothesis is rejected.
Incorrect. When a result is not statistically significant, we fail to reject the null hypothesis. Rejecting the null hypothesis implies significance.
Therefore, the correct statement is that there is no significant correlation between age and weight based on the provided data and critical values.
Revision Table: Correlation Significance Test Summary
| Concept | Description | Decision Rule |
|---|---|---|
| Correlation Coefficient (r) | Measures strength and direction of linear relationship in a sample. | N/A |
| Significance Level ($\alpha$) | Probability of Type I error (rejecting $\text{H}_0$ when true). Common values: 0.05, 0.01. | N/A |
| Degrees of Freedom (df) | For correlation, typically N-2. | N/A |
| Critical Table Value | Threshold value for r at specific $\alpha$ and df. | Compare $|r_{\text{calculated}}|$ to this value. |
| Statistical Significance | Is the observed correlation likely due to chance or a real relationship? | If $|r_{\text{calculated}}| > r_{\text{table}}$, significant; if $|r_{\text{calculated}}| \le r_{\text{table}}$, not significant. |
| Null Hypothesis ($\text{H}_0$) | $\rho = 0$ (no population correlation). | Rejected if significant, Failed to Reject if not significant. |
Additional Information: Interpreting Non-Significant Correlation
A non-significant correlation coefficient does not necessarily mean there is absolutely no relationship between the variables. It means that based on the sample size and the calculated correlation value, the observed relationship is not strong enough to rule out the possibility that the true correlation in the population is zero, at the chosen significance level.
Several factors influence significance:
- Magnitude of r: A stronger correlation (closer to +1 or -1) is more likely to be significant.
- Sample Size (N): Larger sample sizes provide more power to detect a significant correlation, even if r is small. With N=15, the sample size is relatively small, which requires a larger r value to reach significance compared to larger samples.
- Significance Level ($\alpha$): A higher $\alpha$ (e.g., 0.10) makes it easier to find significance, while a lower $\alpha$ (e.g., 0.01) requires stronger evidence.
In this specific case, the calculated r of 0.510 is quite close to the critical value at the 0.05 level (0.514), but it falls just below the threshold. This highlights the importance of comparing against the critical value precisely.
Important Questions from Hypothesis - Teaching
Given below are two statements: One is labeled as Assertion A and the other is labeled as Reason R.
Assertion (A):- Research Hypothesis (H1) cannot be directly verified.
Reasons (R):- Null Hypothesis (H0) is helpful in making a claim by the researcher that his/her findings are not fortuitous or by chance.
In the light of the above statements, choose the most appropriate answer from the options given below:
When a researcher rejects a true 'Null Hypothesis' (H 0) in his/her study and accepts the 'Alternate Hypothesis' (H 1), what type of error is likely?
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: A proposition is a statement about observable phenomena (concepts) that may be judged as true or false.
Reason R: When a proposition is formulated for empirical testing, it is called a hypothesis.
In light of the above statements, choose the most appropriate answer from the options given belowGiven below are two statements
Statement I: The context of discovery involves non‐rational, intuitive processes while the context of justification is based on logical processes.
Statement II: The process of hypothesis generation doesn't strictly follow rigorous logical reasoning.
In light of the above statements, choose the most appropriate answer from the options given belowMatch List I with List II :
Choose the correct answer from the options given below :List I
Statistical testList I
Application(A)
Chi-square
(I)
Is used to determine the significance between group means.
(B)
t-test
(II)
A procedure to decompose variation into two or more independent variables.
(C)
ANOVA
(III)
Analyses the relationship between two or more independent variables and a single dependent variable.
(D)
Multiple regression
(IV)
Produces a value that reflects the relationship between expected and observed frequencies.
