2.5.4 Hypothesis testing for correlation

• What does Pearson's correlation coefficient measure?

  Linear relationship strength
• If the absolute value of Pearson's correlation coefficient exceeds the critical value, we reject the null hypothesis.
• If the p-value is less than the significance level, we reject the null hypothesis
• What does the null hypothesis in correlation testing assume?

  No correlation exists
• Match the type of alternative hypothesis with its mathematical expression:

  Two-sided ↔️ $\rho \neq 0$

  One-sided (Positive) ↔️ $\rho > 0$

  One-sided (Negative) ↔️ $\rho < 0$
• What does a correlation coefficient of $r =$$1$ indicate?


  Perfect positive correlation
• The formula for Pearson's correlation coefficient involves summing the product of (x_{i} - \bar{x})(y_{i} - \bar{y})</latex>.
• The significance level is the probability of rejecting the null hypothesis when it is actually true
• What are two common significance levels used in hypothesis testing?

  0.05 and 0.01
• Steps to calculate Pearson's correlation coefficient

  1️⃣ Find the means $\bar{x}$ and $\bar{y}$

  2️⃣ Calculate $(x_{i} - \bar{x})$ and $(y_{i} - \bar{y})$

  3️⃣ Compute $(x_{i} - \bar{x})(y_{i} - \bar{y})$

  4️⃣ Sum $(x_{i} - \bar{x})(y_{i} - \bar{y})$

  5️⃣ Calculate $(x_{i} - \bar{x})^{2}$ and $(y_{i} - \bar{y})^{2}$

  6️⃣ Sum $(x_{i} - \bar{x})^{2}$ and $(y_{i} - \bar{y})^{2}$

  7️⃣ Calculate the square root of the product

  8️⃣ Divide the sum of $(x_{i} - \bar{x})(y_{i} - \bar{y})$ by the square root of the product
• What does $\alpha =$$0.05$ represent in hypothesis testing?


  5% significance level
• If the p-value is less than $\alpha$, we reject
• If the p-value is greater than or equal to $\alpha$, we fail to reject the null hypothesis.
• What is summarized in the table regarding significance levels, p-value ranges, and decision criteria?

  Hypothesis testing for correlation
• Match the significance level with the corresponding p-value range to reject the null hypothesis:

  0.05 (5%) ↔️ $p < 0.05$

  0.01 (1%) ↔️ $p < 0.01$
• What does hypothesis testing for correlation assess?

  Linear relationship between variables
• The null hypothesis in correlation testing states there is no correlation between variables.
• To test correlation, we use the correlation coefficient r
• What is the mathematical notation for the null hypothesis in correlation testing?

  $\rho =$$0$
• The alternative hypothesis in correlation testing states that a correlation exists
• If $r =$$0.75$, exceeds a critical value of $0.6$, and the p-value is less than $0.05$, we reject the null hypothesis.
• What does the null hypothesis assume in correlation testing?

  No correlation exists
• A one-sided alternative hypothesis can posit either a positive or negative correlation
• Match the type of hypothesis with its corresponding statement:

  Two-sided ↔️ $\rho \neq 0$

  One-sided (Positive) ↔️ $\rho > 0$

  One-sided (Negative) ↔️ $\rho < 0$
• What does Pearson's correlation coefficient (r) measure?

  Strength and direction of linear relationship
• A value of $r =$$1$ indicates a perfect positive correlation
• A value of $r =$$0$ means there is no linear correlation between variables.
• Steps to calculate Pearson's correlation coefficient (r)

  1️⃣ Find the means $\bar{x}$ and $\bar{y}$

  2️⃣ Calculate $(x_{i} - \bar{x})$ and $(y_{i} - \bar{y})$

  3️⃣ Compute $(x_{i} - \bar{x})(y_{i} - \bar{y})$

  4️⃣ Sum $(x_{i} - \bar{x})(y_{i} - \bar{y})$

  5️⃣ Calculate $(x_{i} - \bar{x})^{2}$ and $(y_{i} - \bar{y})^{2}$

  6️⃣ Sum $(x_{i} - \bar{x})^{2}$ and $(y_{i} - \bar{y})^{2}$

  7️⃣ Calculate $\sqrt{\sum (x_{i} - \bar{x})^{2} \sum (y_{i} - \bar{y})^{2}}$

  8️⃣ Compute $r =$$\frac{\sum (x_{i} - \bar{x})(y_{i} - \bar{y})}{\sqrt{\sum (x_{i} - \bar{x})^{2} \sum (y_{i} - \bar{y})^{2}}}$
• What is the significance level ($\alpha$)?


  Probability of Type I error
• If the p-value is less than $\alpha$, we reject
• The t-distribution is used for small sample sizes.
• How are the degrees of freedom ($df$) calculated in the t-distribution?


  $df =$$n - 1$
• Match the significance level with its corresponding critical values for $df =$$20$:


  0.05 ↔️ $\pm 2.086$

  0.01 ↔️ $\pm 2.845$
• What does the test statistic (t-value) measure?

  Deviation from null hypothesis
• What does the test statistic (t-value) measure?

  Deviation from null hypothesis
• The formula for calculating the t-value is t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^{2}}}
• What does $r$ represent in the t-value formula?


  Pearson's correlation coefficient
• The variable $n$ in the t-value formula represents the sample size
• The calculated t-value in the example is 5.07
• What is the purpose of hypothesis testing for correlation?

  Assess linear relationship