Hypothesis Tests for Correlation Coefficient Using TI-85: Videos & Practice Problems

Testing the population correlation coefficient $\rho$ involves hypothesis testing to determine if a linear association exists between two variables beyond sample data. The null hypothesis assumes $\rho =0$, indicating no linear correlation, while the alternative hypothesis tests for $\rho \ne 0$ (two-sided), or specifically positive or negative correlation. Using tools like the TI-84’s linreg t-test, one calculates the sample correlation coefficient $r$ and p-value. A p-value less than the alpha level (e.g., 0.01) leads to rejecting the null, confirming statistically significant linear association in the population. This process is essential for understanding relationships in data analysis and inferential statistics.

Hypothesis Tests for Correlation Coefficient Using TI-84

Understanding the correlation coefficient r is essential for analyzing the strength and direction of a 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 correlation exists beyond the sample data, we set up a hypothesis test where the null hypothesis (H₀) states that there is no linear correlation, meaning ρ = 0. The alternative hypothesis (H_a) depends on the claim: if we are testing for any association, it is ρ ≠ 0; for a positive correlation, ρ > 0; and for a negative correlation, ρ < 0. This distinction is crucial and should be guided by the context of the problem.

For example, when investigating whether poor air quality (measured by the Air Quality Index, AQI) is associated with asthma-related emergency room visits, the goal is to determine if a statistically significant linear correlation exists. Using a significance level of α = 0.01, the hypothesis test evaluates if the observed correlation in the sample reflects a true correlation in the population.

The test can be efficiently performed using statistical tools such as the TI-84 calculator’s LinRegTTest function. After inputting the AQI data and ER visit counts into separate lists, the test calculates the sample correlation coefficient r and the corresponding p-value. A very small p-value (less than α) leads to rejecting the null hypothesis, providing strong evidence that ρ ≠ 0 and confirming a significant linear relationship.

In the example, the correlation coefficient was approximately r = 0.99, indicating a strong positive correlation, and the p-value was about 3.9 × 10⁻⁸, which is much smaller than 0.01. This result means there is sufficient evidence to conclude that poor air quality and asthma-related ER visits are linearly correlated in the population.

It is important to note that this hypothesis test only determines the presence of a linear correlation, not the exact value of the population correlation coefficient ρ, nor does it imply that ρ equals the sample correlation r. Instead, it confirms whether the observed linear relationship is statistically significant and likely to exist whenever these variables interact.

By mastering hypothesis testing for the population correlation coefficient, you can confidently assess the strength and significance of linear relationships in various real-world contexts, enhancing your data analysis skills and interpretation of statistical results.

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 $H_0\left(p=0\right)$ 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

In analyzing the relationship between call length and customer satisfaction at a call center, it is essential to understand the concept of the population correlation coefficient, denoted as ρ (rho). This coefficient measures the strength and direction of a linear relationship between two variables. Here, the focus is on whether shorter call lengths correspond to higher customer satisfaction, implying a negative correlation.

To investigate this, a hypothesis test is conducted with the null hypothesis (H₀) stating that ρ = 0, meaning no linear correlation exists between call length and satisfaction. The alternative hypothesis (H_a) posits that ρ < 0, indicating a negative correlation where shorter calls lead to increased satisfaction.

Using statistical software or a TI-84 calculator, the call length data and customer satisfaction ratings are inputted as two lists. The test for the correlation coefficient is then performed, specifying the alternative hypothesis as ρ < 0. The output provides both the sample correlation coefficient r and the p-value for the test.

The sample correlation coefficient r is found to be approximately −0.80, which is close to −1, indicating a strong negative linear relationship between call length and customer satisfaction. This means that as call length decreases, customer satisfaction tends to increase significantly.

The p-value obtained is approximately 1.688 × 10⁻⁴, which is much smaller than the common significance level α = 0.05. Since the p-value is less than α, the null hypothesis is rejected, providing strong evidence to support the alternative hypothesis that a negative correlation exists.

This analysis highlights the importance of understanding correlation and hypothesis testing in evaluating relationships between variables. The correlation coefficient r quantifies the strength and direction of the linear association, while the p-value helps determine the statistical significance of the observed correlation. Together, these tools enable data-driven decisions, such as optimizing call lengths to enhance customer satisfaction in a call center environment.

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 training duration and review scores is plausible before conducting formal statistical tests.

Using a TI-84 calculator, the training data is entered into list L1 and the corresponding scores into list L2. A 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 days increase, performance scores tend to increase as well, indicating a potential linear correlation worth testing further.

To quantify this relationship, the population correlation coefficient, denoted as ρ (rho), is tested using 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 relationship, without specifying direction (positive or negative).

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). A value of 0.5 suggests a moderate association between training length and performance score.

The p-value obtained is approximately 0.055. This p-value is compared to the significance level α = 0.05. Since 0.055 is slightly greater than 0.05, the result is not statistically significant at the 5% level. Therefore, the null hypothesis is not rejected, indicating insufficient evidence to conclude a nonzero linear correlation between training duration and employee performance scores.

In summary, while the scatter plot and correlation coefficient suggest a moderate positive relationship between training length and performance review scores, the hypothesis test reveals that this observed correlation is not statistically significant at the conventional 5% level. This means that based on the given data, we cannot confidently assert that the length of training impacts employee performance scores in a linear manner.

Here’s what students ask on this topic:

How do you perform a hypothesis test for the population correlation coefficient using a TI-85 calculator?

To perform a hypothesis test for the population correlation coefficient (ρ) using a TI-85 calculator, first enter your paired data into two lists (e.g., L1 and L2). Then, access the statistics or test menu and select the linear regression t-test function (often labeled as LinRegTTest). Set the x-list and y-list to your data lists, and specify the alternative hypothesis: ρ ≠ 0 for two-tailed, ρ > 0 for positive correlation, or ρ < 0 for negative correlation. Calculate the test, and the calculator will output the sample correlation coefficient (r) and the p-value. Compare the p-value to your significance level (α). If p < α, reject the null hypothesis (ρ = 0), indicating significant linear correlation in the population.

What is the null and alternative hypothesis when testing for correlation using TI-85?

When testing for correlation using the TI-85, the null hypothesis (H₀) is that the population correlation coefficient ρ equals zero, meaning no linear correlation exists between the two variables: $H{0}_{}:\rho =0$. The alternative hypothesis (H_a) depends on the research question: for a two-tailed test, it is $H{a}_{}:\rho \ne 0$, indicating any linear correlation; for a positive correlation test, $H{a}_{}:\rho >0$; and for a negative correlation test, $H{a}_{}:\rho <0$. The TI-85 allows you to select the appropriate alternative hypothesis when performing the test.

How do you interpret the p-value from a correlation hypothesis test on the TI-85?

The p-value from a correlation hypothesis test on the TI-85 indicates the probability of observing the sample data, or something more extreme, assuming the null hypothesis (ρ = 0) is true. If the p-value is less than your chosen significance level (α, commonly 0.01 or 0.05), you reject the null hypothesis, concluding there is statistically significant evidence of a linear correlation between the variables in the population. Conversely, if the p-value is greater than α, you fail to reject the null hypothesis, meaning there is insufficient evidence to claim a linear relationship. Remember, this test only assesses the presence of linear correlation, not the strength or exact value of ρ.

What does the correlation coefficient r represent in the TI-85 hypothesis test output?

The correlation coefficient r in the TI-85 hypothesis test output represents the sample correlation between the two variables in your dataset. It measures the strength and direction of the linear relationship, ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation). A value close to zero indicates weak or no linear correlation. However, r is only a sample statistic and does not directly equal the population correlation coefficient ρ. The hypothesis test uses r to help determine if there is enough evidence to conclude that ρ is different from zero in the population.

How do you enter data into the TI-85 for correlation hypothesis testing?

To enter data into the TI-85 for correlation hypothesis testing, press the STAT button, then select the option to edit lists. Input your first variable's data into list L1 and your second variable's data into list L2. Make sure each pair of data points corresponds by row (i.e., the first value in L1 pairs with the first value in L2). After entering all data points, you can proceed to perform the LinRegTTest using these lists as your x-list and y-list inputs.