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key term - $ ho$

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Definition

$ ho$ is the symbol used to represent the correlation coefficient, which is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a crucial parameter in the context of 12.3 Testing the Significance of the Correlation Coefficient (Optional), as it allows researchers to determine whether the observed correlation in a sample is statistically significant and can be generalized to the population.

5 Must Know Facts For Your Next Test

  1. $ ho$ can take values between -1 and 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship.
  2. The null hypothesis in the context of testing the significance of the correlation coefficient is that the true population correlation coefficient is zero, meaning there is no linear relationship between the two variables.
  3. The test statistic used to determine the significance of $ ho$ is the t-statistic, which follows a t-distribution with $n-2$ degrees of freedom, where $n$ is the sample size.
  4. The p-value associated with the test statistic is used to determine the probability of observing a correlation coefficient as extreme as the one in the sample, given that the null hypothesis is true.
  5. The decision to reject or fail to reject the null hypothesis is based on the comparison of the p-value to the chosen significance level, typically $eta = 0.05$ or $eta = 0.01$.

Review Questions

  • Explain the interpretation of the correlation coefficient $ ho$ and its possible values.
    • The correlation coefficient $ ho$ is a measure of the strength and direction of the linear relationship between two variables. It can take values between -1 and 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship. The magnitude of $ ho$ reflects the strength of the linear association, while the sign indicates the direction of the relationship.
  • Describe the hypothesis testing procedure for determining the significance of the correlation coefficient $ ho$.
    • The hypothesis testing procedure for determining the significance of the correlation coefficient $ ho$ involves formulating a null hypothesis that the true population correlation coefficient is zero, meaning there is no linear relationship between the two variables. The test statistic used is the t-statistic, which follows a t-distribution with $n-2$ degrees of freedom, where $n$ is the sample size. The p-value associated with the test statistic is then compared to the chosen significance level, typically $eta = 0.05$ or $eta = 0.01$, to determine whether to reject or fail to reject the null hypothesis.
  • Analyze the implications of the significance of the correlation coefficient $ ho$ for the relationship between the two variables.
    • If the null hypothesis of $ ho = 0$ is rejected, it suggests that there is a statistically significant linear relationship between the two variables. The magnitude of $ ho$ indicates the strength of this relationship, with larger absolute values of $ ho$ corresponding to stronger linear associations. The sign of $ ho$ determines the direction of the relationship, with positive values indicating a positive linear relationship and negative values indicating a negative linear relationship. The significance of $ ho$ is crucial in understanding the nature and strength of the relationship between the variables, which can have important implications for decision-making, prediction, and further research.

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$ ho$ - (Honors Statistics) - Vocab, Definition, Explanations | Fiveable | Fiveable