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)

Verified step by step guidance

1. Understand the correlation coefficient $r=0.23$ as a measure of 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. Set up the hypothesis test for the population correlation coefficient $\mathrm{\backslash rho}$: the null hypothesis $H_0:\mathrm{\backslash rho}=\; 0$ (no linear correlation), and the alternative hypothesis $H_{\mathrm{a}}:\mathrm{\backslash rho}\ne \; 0$ (there is a linear correlation).
3. Use the given P-value of 0.35 to determine the statistical significance. The P-value represents the probability of observing a correlation as extreme as 0.23 (or more) if the null hypothesis is true.
4. Compare the P-value to the significance level (commonly 0.05). Since 0.35 is greater than 0.05, there is not enough evidence to reject the null hypothesis, meaning we fail to conclude a significant linear correlation exists.
5. Interpret the results: the weak positive correlation $r=0.23$ is not statistically significant based on the P-value, so we conclude there is insufficient evidence to support a nonzero linear correlation between inflation and unemployment rates.