$H_0$

Definition

$H_0$, also known as the null hypothesis, is a statistical term that represents the initial or default assumption about a population parameter or the relationship between variables. It is the hypothesis that is tested in a statistical significance test, such as the one used in the context of 12.4 Testing the Significance of the Correlation Coefficient.

5 Must Know Facts For Your Next Test

1. The null hypothesis, $H_0$, represents the assumption that there is no relationship or difference between the variables being studied.
2. In the context of testing the significance of the correlation coefficient, $H_0$ states that the population correlation coefficient is zero, meaning there is no linear relationship between the two variables.
3. The goal of the statistical test is to determine whether the sample correlation coefficient provides sufficient evidence to reject the null hypothesis and conclude that there is a significant linear relationship between the variables.
4. The test statistic used to evaluate the significance of the correlation coefficient follows a t-distribution, and the p-value is used to determine the strength of the evidence against the null hypothesis.
5. If the p-value is less than the chosen significance level (e.g., 0.05), the null hypothesis is rejected, indicating that the observed correlation coefficient is statistically significant.

Review Questions

• Explain the role of the null hypothesis, $H_0$, in the context of testing the significance of the correlation coefficient.
  • In the context of testing the significance of the correlation coefficient, the null hypothesis, $H_0$, represents the assumption that there is no linear relationship between the two variables being studied. The statistical test is designed to determine whether the sample correlation coefficient provides sufficient evidence to reject the null hypothesis and conclude that there is a significant linear relationship between the variables. The test statistic and p-value are used to evaluate the strength of the evidence against the null hypothesis, with a p-value less than the chosen significance level indicating that the null hypothesis should be rejected.
• Describe how the alternative hypothesis, $H_1$, relates to the null hypothesis, $H_0$, in the context of testing the significance of the correlation coefficient.
  • The alternative hypothesis, $H_1$, is the complement of the null hypothesis, $H_0$, in the context of testing the significance of the correlation coefficient. While the null hypothesis states that there is no linear relationship between the variables (i.e., the population correlation coefficient is zero), the alternative hypothesis represents the researcher's belief that there is a significant linear relationship between the variables (i.e., the population correlation coefficient is not equal to zero). The statistical test is designed to determine whether the evidence from the sample data is strong enough to reject the null hypothesis and support the alternative hypothesis.
• Analyze the importance of the p-value in the context of testing the significance of the correlation coefficient and its relationship to the null hypothesis, $H_0$.
  • The p-value is a critical component in the context of testing the significance of the correlation coefficient and its relationship to the null hypothesis, $H_0$. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the chosen significance level (e.g., 0.05), it provides strong evidence against the null hypothesis, indicating that the observed correlation coefficient is statistically significant and the null hypothesis should be rejected. Conversely, a p-value greater than the significance level suggests that the sample data does not provide sufficient evidence to reject the null hypothesis, and the researcher cannot conclude that there is a significant linear relationship between the variables.

Related terms

$H_1$: The alternative hypothesis, which is the hypothesis that the researcher believes to be true and is the complement of the null hypothesis.

Statistical Significance: The likelihood that the observed relationship or difference between variables is due to chance, as determined by a statistical test.

p-value: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.