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

Answer: (A)

Verified step by step guidance

1. Understand that the correlation coefficient $r=0.23$ measures the strength and direction of a linear relationship between two variables. Here, $r=0.23$ indicates a weak positive linear correlation between inflation rate and unemployment rate.
2. Recognize that the hypothesis test is conducted to determine if the observed correlation is statistically significant. The null hypothesis $H_0$ states that the population correlation coefficient $p$ is zero (no linear correlation), while the alternative hypothesis $H_{\mathrm{a}}$ states that $p\backslash neq\; 0$.
3. Interpret the P-value of 0.35 obtained from the hypothesis test. The P-value represents the probability of observing a correlation as extreme as 0.23 (or more) if the null hypothesis were true. A high P-value (commonly above 0.05) suggests insufficient evidence to reject the null hypothesis.
4. Conclude that since the P-value (0.35) is greater than the typical significance level (e.g., 0.05), we fail to reject the null hypothesis. This means there is not enough statistical evidence to support a nonzero linear correlation between inflation and unemployment rates.
5. Summarize the interpretation: the correlation coefficient indicates a weak positive linear relationship, but the hypothesis test shows that this observed correlation is not statistically significant, so we cannot confidently say there is a linear association between the two variables based on this sample.