Question

A sample of \(n=500(x, y)\) pairs was collected and a test of \(H_{0}: \rho=0\) versus \(H_{a}: \rho \neq 0\) was carried out. The resulting \(P\) -value was computed to be \(.00032\). a. What conclusion would be appropriate at level of significance .001? b. Does this small \(P\) -value indicate that there is a very strong linear relationship between \(x\) and \(y\) (a value of \(\rho\) that differs considerably from zero)? Explain.

Short answer

a. At the level of significance .001, the null hypothesis \(H_{0}: \rho=0\) is rejected because the \(P\) value .00032 is less than the significance level .001. Hence, there is evidence at the .001 level to suggest that the correlation coefficient \(\rho\) is not 0.\nb. No, the small \(P\) -value does not indicate a very strong linear relationship between \(x\) and \(y\). A small \(P\) -value indicates statistical significance but it does not tell anything about strength of the relationship.

Step-by-step solution

Analyze the \(P\) value

Firstly, the \(P\) value is compared with the significance level. The \(P\) value, .00032 is less than the significance level, .001. Therefore, there is a significant discrepancy between the observed data and what would be expected if \(H_{0}: \rho=0\) was true to enough to reject the null hypothesis.

Conclusion of test

Since the \(P\) value is less than the significance level, the null hypothesis is rejected. Therefore, we conclude that there is evidence at the .001 level to suggest that the population correlation coefficient \(\rho\) is not 0.

Understanding the interpretation of small \(P\) value

A small \(P\) -value indicates statistical significance and it provides strong evidence against the null hypothesis. But it does not indicate the size or importance of the observed effect and do not tell anything about strength of the relationship. Hence, the small \(P\) -value cannot be used to say there is a very strong linear relationship between \(x\) and \(y\).

Key concepts

P-value significance

Understanding the significance of a P-value is crucial in hypothesis testing. At its core, the P-value quantifies the probability of observing the obtained results, or more extreme ones, if the null hypothesis (\(H_0\)) were true. It is a tool for making decisions; a small P-value suggests that your observed data is highly unlikely under the assumption that the null hypothesis is correct. For the exercise at hand, a P-value of .00032 means that there is only a 0.032% chance that the sample data would occur if the null hypothesis of no correlation were true. Since this P-value is significantly lower than the chosen significance level of .001, it leads to the rejection of the null hypothesis, signaling that the observed correlation is statistically unlikely to have occurred by chance alone. Consequently, the smaller the P-value, the stronger the evidence against the null hypothesis.

Null hypothesis

The null hypothesis, denoted as \(H_0\), is a default statement that there is no effect or no difference; it reflects a position of skepticism or neutrality towards an observational claim. In the context of the correlation between two variables, \(H_0: \rho=0\) means there is no association between the variables. Hypothesis testing is designed to assess the validity of the null hypothesis by using sample data. If the result indicates that what we observed is highly unlikely if \(H_0\) were true—typically judged by a P-value less than a predetermined significance level—we reject \(H_0\) in favor of the alternative hypothesis, \(H_a\), which is the claim being tested. However, rejecting the null hypothesis does not mean proving the alternative hypothesis to be true beyond doubt; it merely suggests there's enough evidence against \(H_0\).

Population correlation coefficient

The population correlation coefficient, denoted by \(\rho\) (rho), measures the strength and direction of a linear relationship between two variables in an entire population. It's a parameter that ranges from -1 to 1, where 0 indicates no linear correlation, 1 or -1 indicates a perfect linear relationship, and values in between suggest varying degrees of linear association. In the exercise provided, testing whether \(\rho \) equals zero is a way of asking whether there's any linear correlation to be aware of at all. Concluding that \(\rho \) is different from zero (either positively or negatively), as the step-by-step solution confirms, means that there exists some degree of linear relationship. However, it does not specify how strong that relationship is. The actual value of the population correlation coefficient would be needed to assess the strength of the relationship.

Statistical significance

Statistical significance is a term used to express that the results of a statistical analysis or test reflect a true effect or relationship, and are not likely due to random variation. It’s often assessed using a P-value against a predetermined significance level or alpha (\(\alpha\)), such as 0.05, 0.01, or 0.001. When a P-value is lower than these thresholds, the results are considered statistically significant. In the exercise, the P-value of .00032 was lower than the significance level of .001, making it statistically significant. This result implies we have enough evidence to conclude that the actual correlation in the population is unlikely to be zero. It's important to differentiate statistical significance from practical significance; the former only talks about the likelihood of an effect not being due to chance, while the latter considers whether the effect size is large enough to be of real-world importance or relevance.

Linear relationship analysis

Linear relationship analysis involves examining how well two variables correlate in a straight-line manner. A scatterplot can provide a visual assessment of the relationship, while the correlation coefficient quantifies it. The closer the data points are to forming a straight line when plotted on a scatterplot, the stronger the linear relationship. In hypothesis testing, once we determine that there is a statistically significant linear relationship (i.e., the correlation coefficient \(\rho\) is not zero), we may then proceed to estimate \(\rho\) and create a regression model to predict one variable based on the other. It's important to note that statistical significance does not automatically mean a strong linear relationship exists. Instead, it suggests that the linear relationship is consistent enough that it cannot be attributed to random chance alone. Further analysis is necessary to understand the strength and practical significance of the relationship.