1. **Problem Statement:** We are given correlation coefficients between first weekend gross, regional gross, and worldwide gross of six movies. We need to test if there is a significant linear relationship between these variables at the 0.05 significance level. 2. **Hypotheses:** The correct hypotheses for testing correlation significance are: $$H_0: \rho = 0$$ $$H_1: \rho \neq 0$$ This means the null hypothesis states no linear correlation, and the alternative states there is a correlation. 3. **Critical Values:** For a two-tailed test with $\alpha = 0.05$ and degrees of freedom $df = n-2 = 6-2=4$, the critical t-values are: $$\pm t_{\alpha/2, df} = \pm 2.7764$$ 4. **Test Statistic Formula:** The test statistic for correlation $r$ is: $$t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}$$ where $n=6$. 5. **Given Correlations and Calculations:** - First weekend gross and regional gross: $r = -0.1409$ $$t = \frac{-0.1409 \times \sqrt{4}}{\sqrt{1 - (-0.1409)^2}} = \frac{-0.1409 \times 2}{\sqrt{1 - 0.0199}} = \frac{-0.2818}{0.9900} = -0.2846$$ - First weekend gross and worldwide gross: $r = -0.3116$ $$t = \frac{-0.3116 \times 2}{\sqrt{1 - 0.3116^2}} = \frac{-0.6232}{\sqrt{1 - 0.0971}} = \frac{-0.6232}{0.9515} = -0.6558$$ - Regional gross and worldwide gross: $r = 0.9491$ $$t = \frac{0.9491 \times 2}{\sqrt{1 - 0.9491^2}} = \frac{1.8982}{\sqrt{1 - 0.9008}} = \frac{1.8982}{0.3123} = 6.0775$$ 6. **Decisions:** - For first weekend vs regional gross: $|t|=0.2846 < 2.7764$, do not reject $H_0$. Insufficient evidence of linear relationship. - For first weekend vs worldwide gross: $|t|=0.6558 < 2.7764$, do not reject $H_0$. Insufficient evidence of linear relationship. - For regional vs worldwide gross: $|t|=6.0775 > 2.7764$, reject $H_0$. Significant linear relationship exists. **Final answers:** - Hypotheses: $H_0: \rho=0$, $H_1: \rho \neq 0$ - Critical values: $-2.7764, 2.7764$ - Test statistics: $t_{STAT} = -0.2846$, $-0.6558$, $6.0775$ - Decisions: No significant linear relationship for first weekend vs regional and worldwide gross; significant relationship for regional vs worldwide gross.