How do you set up the null and alternative hypotheses for testing correlation?

When testing the population correlation coefficient, the null hypothesis (H 0 ) always states that there is no linear correlation: ρ = 0 . The alternative hypothesis (H a ) depends on the research question: for a two-tailed test, it is ρ ≠ 0 (testing for any correlation); for a one-tailed test, it could be ρ > 0 (positive correlation) or ρ < 0 (negative correlation). The choice depends on whether you want to detect any correlation or a specific direction.

11. Correlation

Inferences for the Correlation Coefficient - Excel

11. Correlation

Inferences for the Correlation Coefficient - Excel: Videos & Practice Problems

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Performing a hypothesis test for the population correlation coefficient $ρ$ involves testing if a linear relationship exists between two variables beyond sample data. The null hypothesis assumes $ρ = 0$ , indicating no linear correlation, while the alternative hypothesis tests for $ρ \neq 0$ . The test statistic $t$ is calculated using the sample correlation coefficient $r$ and degrees of freedom $n - 2$ . A p-value less than the significance level $α$ leads to rejecting the null hypothesis, confirming significant linear correlation. This method enhances inferential statistics by extending sample findings to populations.

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Inferences for Correlation Coefficient - Excel

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Inferences for Correlation Coefficient - Excel Video Summary

The correlation coefficient, denoted as r, measures the strength and direction of a linear relationship between two variables in a dataset. When r is close to zero, it indicates a weak or no linear correlation, while values far from zero suggest a strong linear correlation. However, r only describes the sample data, and to determine if this linear relationship extends to the entire population, hypothesis testing for the population correlation coefficient, represented by ρ (rho), is essential.

To test whether a linear correlation exists between two variables in the population, we start by setting up hypotheses. The null hypothesis (H₀) states that there is no linear correlation, meaning ρ = 0. The alternative hypothesis (H₁) depends on the claim: if testing for any linear correlation (positive or negative), it is ρ ≠ 0; if testing specifically for positive or negative correlation, it would be ρ > 0 or ρ < 0, respectively.

Once the hypotheses are established, the test statistic t is calculated using the formula:

\[t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}\]>

where r is the sample correlation coefficient and n is the sample size. The degrees of freedom for this test are n - 2. This t-score follows a Student's t-distribution, allowing us to find the p-value, which quantifies the probability of observing the data assuming the null hypothesis is true.

For a two-tailed test, where the alternative hypothesis is ρ ≠ 0, the p-value is calculated based on the absolute value of the t-score and the degrees of freedom. If the p-value is less than the chosen significance level α (commonly 0.05), we reject the null hypothesis, concluding that there is statistically significant evidence of a linear correlation in the population.

For example, if a game company collects data from 13 players to examine the relationship between playtime and enjoyment score, and calculates a sample correlation coefficient of approximately 0.74, the degrees of freedom would be 11. Plugging these values into the formula yields a t-score around 3.68. The corresponding p-value for this two-tailed test might be about 0.004, which is less than 0.05, leading to rejection of the null hypothesis. This indicates strong evidence that playtime and enjoyment score are linearly correlated in the population.

Understanding how to perform hypothesis testing for the population correlation coefficient is crucial for determining whether observed relationships in sample data reflect true associations in the broader population. This process integrates key statistical concepts such as correlation, degrees of freedom, t-distribution, and p-values, providing a robust framework for analyzing linear relationships between variables.

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Inferences for Correlation Coefficient - Excel Example 1

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Inferences for Correlation Coefficient - Excel Example 1 Video Summary

To determine if there is a positive linear relationship between humidity level and heat index on hot days, we test the population correlation coefficient, denoted as ρ (rho). The null hypothesis (H₀) states that there is no linear relationship, meaning ρ = 0. The alternative hypothesis (H_a) proposes a positive linear relationship, so ρ > 0.

The test statistic used is the t-score, calculated from the sample correlation coefficient r and the sample size n. The formula for the t-score is:

\[ t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} \]

Here, r is the sample correlation coefficient, and the degrees of freedom (df) equal n - 2. For example, with a sample size of 23 days, df = 21. The correlation coefficient is computed from the paired data of humidity and heat index.

After calculating the t-score, the p-value is found using the right-tailed t-distribution because the alternative hypothesis is one-sided (ρ > 0). The p-value represents the probability of observing a test statistic as extreme as the calculated t-score under the null hypothesis.

In this case, the t-score was approximately 32.9, leading to a p-value around 7.3 × 10⁻²⁰, which is significantly smaller than the significance level α = 0.05. Since the p-value < α, the null hypothesis is rejected, providing strong evidence that a positive linear relationship exists between humidity level and heat index on hot days.

This process highlights the importance of hypothesis testing in statistics to assess relationships between variables, using correlation coefficients, t-distributions, and p-values to make informed conclusions about population parameters.

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To perform a hypothesis test for the population correlation coefficient (ρ) in Excel, start by stating the null hypothesis (H₀: ρ = 0) which assumes no linear correlation, and the alternative hypothesis (H_a) depending on the claim (e.g., ρ ≠ 0 for two-tailed). Calculate the sample correlation coefficient (r) using the formula =CORREL(array1, array2). Determine degrees of freedom as n - 2, where n is the sample size. Compute the test statistic t using the formula: $t = \frac{r}{\sqrt{\frac{1 - r^{2}}{n}}}$ . Then, find the p-value with =T.DIST.2T(ABS(t), degrees_freedom). If the p-value is less than the significance level α (commonly 0.05), reject H₀ and conclude a significant linear correlation exists.

The test statistic t for testing the population correlation coefficient is calculated using the formula: $t = \frac{r}{\sqrt{\frac{1 - r^{2}}{n}}}$ , where r is the sample correlation coefficient and n is the sample size. This formula accounts for the degrees of freedom (n - 2) and measures how far the observed correlation is from zero in terms of standard error. The resulting t-value is then used to find the p-value for hypothesis testing.

In a correlation hypothesis test, the p-value represents the probability of observing the sample correlation coefficient or one more extreme if the null hypothesis (ρ = 0) is true. Using Excel, after calculating the test statistic t, you find the p-value with =T.DIST.2T(ABS(t), degrees_freedom) for a two-tailed test. If the p-value is less than the chosen significance level α (commonly 0.05), you reject the null hypothesis, indicating there is significant evidence of a linear correlation between the variables. If the p-value is greater than α, you fail to reject the null, suggesting insufficient evidence to conclude a linear relationship exists.

To calculate the degrees of freedom (df) for testing the population correlation coefficient, use the formula: $df = n - 2$ , where n is the sample size. This adjustment accounts for estimating two parameters (means of the two variables) when calculating the correlation. For example, if you have data from 13 pairs of observations, the degrees of freedom would be 13 - 2 = 11. This df value is used in the t-distribution to find the p-value for the hypothesis test.

When testing the population correlation coefficient, the null hypothesis (H₀) always states that there is no linear correlation: $ρ = 0$ . The alternative hypothesis (H_a) depends on the research question: for a two-tailed test, it is $ρ \neq 0$ (testing for any correlation); for a one-tailed test, it could be $ρ > 0$ (positive correlation) or $ρ < 0$ (negative correlation). The choice depends on whether you want to detect any correlation or a specific direction.