Hypothesis Tests for Correlation Coefficient Using TI-84 : Videos & Practice Problems

Testing the population correlation coefficient $\rho$ involves hypothesis testing to determine if a linear relationship exists beyond sample data. The null hypothesis assumes $\rho =0$, indicating no linear correlation, while the alternative hypothesis tests for $\rho \ne 0$ (two-tailed) or directional correlation. Using tools like the TI-84's linreg t-test, one calculates the sample correlation coefficient $r$ and p-value to compare against significance level $\alpha$. A p-value less than $\alpha$ leads to rejecting the null, confirming statistically significant linear correlation in the population.

Hypothesis Tests for Correlation Coefficient Using TI-84

Understanding the correlation coefficient r is essential for analyzing the linear relationship between two variables. When r is close to zero, it indicates a weak or no linear correlation, whereas values of r far from zero suggest a strong linear correlation. However, determining how far from zero r must be to confidently assert a true linear relationship in the population requires hypothesis testing for the population correlation coefficient, denoted as ρ.

To test whether a linear relationship exists beyond the sample data, we set up a hypothesis test where the null hypothesis (H₀) states that there is no linear correlation between the variables, meaning ρ = 0. The alternative hypothesis (H_a) depends on the claim: it can be two-sided (ρ ≠ 0) if we are testing for any association, or one-sided (ρ > 0 or ρ < 0) if we are testing for a positive or negative correlation specifically.

For example, when investigating whether poor air quality (measured by the Air Quality Index, AQI) is associated with asthma-related emergency room visits, the hypotheses would be:

H₀: ρ = 0 (no linear correlation between AQI and ER visits)

H_a: ρ ≠ 0 (there is a linear correlation between AQI and ER visits)

Using statistical software or a graphing calculator like the TI-84, the LinRegTTest function can perform this hypothesis test efficiently. After inputting the paired data sets (e.g., AQI in list L1 and ER visits in list L2), the test calculates the sample correlation coefficient r and the p-value for the test.

In this scenario, a strong positive correlation was found with r ≈ 0.99, indicating a very strong linear relationship in the sample data. The p-value, which measures the probability of observing such a correlation if the null hypothesis were true, was approximately 3.9 × 10⁻⁸. Since this p-value is much smaller than the significance level α = 0.01, the null hypothesis is rejected. This provides strong evidence that the population correlation coefficient ρ is not zero, confirming a statistically significant linear association between poor air quality and asthma-related ER visits.

It is important to note that this test only determines whether a linear correlation exists, not the exact value of ρ, nor does it imply that the sample correlation r equals the population correlation ρ. Instead, it confirms that the observed linear relationship in the sample likely reflects a true relationship in the population.

By mastering hypothesis testing for the population correlation coefficient, you can confidently assess the strength and significance of linear relationships in various datasets, enhancing your ability to make data-driven conclusions about real-world phenomena.

An economist wonders if the inflation rate is linearly correlated with the unemployment rate and is looking to use the results of their analysis for further study. They take a random sample of recent months and record the unemployment rate and inflation rate. They find $r=0.23$ and run a hypothesis test, getting a $P-value$ of $0.35$. Interpret the value of $r$ and results of the test.

• A. $r=0.23$ suggests weak positive linear correlation; fail to reject $H_0(p=0)$ since not enough evidence to support nonzero linear correlation between inflation and unemployment.
• B. $r=0.23$ suggests weak positive linear correlation; reject $H_0\left(p=0\right)$ since there is enough evidence to support nonzero linear correlation between inflation and unemployment.
• C. $r=0.23$ suggests strong positive linear correlation; fail to reject $H_0\left(p=0\right)$ since not enough evidence to support nonzero linear correlation between inflation and unemployment.
• D. $r=0.23$ suggests strong positive linear correlation; reject $H_0\left(p=0\right)$ since there is enough evidence to support nonzero linear correlation between inflation and unemployment.

Answer: (A)

Hypothesis Tests for Correlation Coefficient Using TI-84 Example 1

A psychologist investigates whether increased time spent on hobbies reduces reported stress levels by surveying 15 adults, recording their weekly hours spent on hobbies and their stress levels rated out of 10. To analyze the relationship between these two variables, the population correlation coefficient, denoted as ρ (rho), is tested through hypothesis testing.

The null hypothesis (H₀) assumes no correlation between time spent on hobbies and stress level, meaning ρ = 0. The alternative hypothesis (H_a) reflects the psychologist’s expectation that increased hobby time correlates with decreased stress, implying a negative correlation: ρ < 0.

Using statistical software or a graphing calculator, the data for time spent on hobbies (x-values) and stress levels (y-values) are entered into separate lists. The correlation test is then performed by selecting the appropriate test for correlation with the alternative hypothesis set to "less than zero" to detect a negative correlation.

The test output provides the sample correlation coefficient r and the p-value. In this case, r ≈ -0.93, indicating a very strong negative linear correlation since the value is close to -1. This means that as time spent on hobbies increases, stress levels tend to decrease significantly.

The p-value obtained is approximately 3.56 × 10^-7, which is much smaller than the common significance level α = 0.05. Because the p-value is less than α, the null hypothesis is rejected, providing strong evidence to support the alternative hypothesis that there is a significant negative linear correlation between time spent on hobbies and reported stress level.

This analysis highlights the importance of understanding correlation coefficients and hypothesis testing in statistics. The correlation coefficient r quantifies the strength and direction of a linear relationship between two variables, while the p-value helps determine the statistical significance of this relationship. Together, they allow researchers to draw meaningful conclusions about how one variable may influence another.

Hypothesis Tests for Correlation Coefficient Using TI-84 Example 2

An HR manager is investigating whether the length of employee training, measured in days, affects the score employees receive on their first performance review, which is rated out of 10. To explore this relationship, a scatter plot is created by plotting training length on the x-axis and performance scores on the y-axis. This visual representation helps determine if a linear correlation between the two variables is plausible before conducting formal statistical tests.

Using a TI-84 calculator, the training data is entered into list L1 and the performance scores into list L2. The scatter plot is generated by enabling the stat plot feature, selecting the scatter plot type, and setting the x-list to L1 and the y-list to L2. Adjusting the viewing window ensures all data points are visible. The resulting scatter plot suggests a positive linear trend: as training length increases, performance scores tend to increase as well, indicating a potential linear relationship worth further investigation.

To quantify this relationship, the population correlation coefficient, denoted as ρ (rho), is tested through hypothesis testing. The null hypothesis (H₀) states that ρ = 0, meaning no linear correlation exists between training length and performance score. The alternative hypothesis (H_a) posits that ρ ≠ 0, indicating some linear correlation, without specifying direction.

Using the TI-84 calculator’s statistical test menu, the correlation test is performed with the training data as the independent variable and the scores as the dependent variable. The calculator outputs the sample correlation coefficient r and the p-value for the test. In this case, r ≈ 0.5, which represents a moderate positive linear correlation, as r ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation).

The p-value obtained is approximately 0.055. Comparing this to a common significance level of α = 0.05, the p-value is slightly greater, meaning there is insufficient evidence to reject the null hypothesis. Therefore, the data does not provide strong enough evidence to conclude a statistically significant linear correlation between the length of training and employee performance scores.

In summary, while the scatter plot and correlation coefficient suggest a moderate positive relationship, the hypothesis test indicates that this observed correlation could be due to random chance at the 5% significance level. This highlights the importance of combining visual data analysis with formal statistical testing to draw reliable conclusions about relationships between variables.

Here’s what students ask on this topic:

How do you perform a hypothesis test for the population correlation coefficient using a TI-84 calculator?

To perform a hypothesis test for the population correlation coefficient (ρ) using a TI-84, first enter your paired data into lists (e.g., L1 for x-values and L2 for y-values). Then, press the STAT button, navigate to the TESTS menu, and select the LinRegTTest function. Input the correct lists for X and Y, set frequency to 1, and choose the alternative hypothesis (e.g., ρ ≠ 0 for two-tailed). Calculate to obtain the sample correlation coefficient (r) and the p-value. Compare the p-value to your significance level (α). If p-value < α, reject the null hypothesis (ρ = 0), indicating significant linear correlation in the population.

What are the null and alternative hypotheses when testing for correlation using the TI-84?

When testing for correlation, the null hypothesis (H₀) states that there is no linear correlation between the two variables, meaning the population correlation coefficient is zero: $\rho =0$. The alternative hypothesis (H_a) depends on the claim: for a two-tailed test, it is $\rho \ne 0$, indicating any nonzero correlation. For directional tests, it could be positive correlation $\rho >0$ or negative correlation $\rho <0$. The choice depends on the research question and should be carefully read from the problem statement.

How do you interpret the p-value from a correlation hypothesis test on the TI-84?

The p-value from a correlation hypothesis test on the TI-84 indicates the probability of observing the sample data if the null hypothesis (ρ = 0) is true. A small p-value (less than the significance level α, often 0.05 or 0.01) suggests strong evidence against the null hypothesis. This means you reject H₀ and conclude there is statistically significant linear correlation between the variables in the population. Conversely, a large p-value means insufficient evidence to reject H₀, so you cannot claim a significant linear relationship.

What does the sample correlation coefficient (r) tell you in a hypothesis test for correlation?

The sample correlation coefficient (r) measures the strength and direction of the linear relationship between two variables in your sample data. Values close to 1 or -1 indicate strong positive or negative linear correlation, respectively, while values near 0 suggest weak or no linear correlation. In a hypothesis test, r is used to calculate the test statistic and p-value, but the test itself determines if the observed correlation is statistically significant for the population. Thus, r alone does not confirm significance; the p-value must be considered.

Can the TI-84 hypothesis test for correlation determine the exact population correlation coefficient?

No, the TI-84 hypothesis test for correlation does not determine the exact population correlation coefficient (ρ). Instead, it tests whether there is enough evidence to conclude that ρ is different from zero, indicating a significant linear relationship. The test provides the sample correlation coefficient (r) and a p-value to assess significance, but it does not estimate the precise value of ρ. For estimating ρ, confidence intervals or other estimation methods are used.