To determine the critical values for the linear correlation coefficient at a significance level of 0.05, we first need to find the critical value using the t-distribution, as the correlation coefficient follows a t-distribution with \( n - 2 \) degrees of freedom, where \( n \) is the number of paired observations.
In this case, you have \( n = 7 \), which gives us \( n - 2 = 5 \) degrees of freedom.
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Determine the critical t-value for a one-tailed test at the significance level of 0.05 with 5 degrees of freedom. Using a t-table or calculator, the critical t-value is approximately 2.571.
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Calculate the critical correlation coefficient (r) using the following formula:
\[ r_{critical} = \sqrt{\frac{t^2}{t^2 + n - 2}} \]
Plugging in our critical t-value:
\[ r_{critical} = \sqrt{\frac{2.571^2}{2.571^2 + 5}} \approx \sqrt{\frac{6.607}{11.607}} \approx \sqrt{0.569} \approx 0.754 \]
Thus, the critical values for the correlation coefficient at the 0.05 significance level are approximately ±0.754.
- Evaluate the linear correlation coefficient of 0.445 against the critical values:
Since 0.445 is less than 0.754, we do not reject the null hypothesis and conclude that there is insufficient evidence to support a linear correlation between the size and revenue of department stores.
Therefore, the correct answer is:
B) Critical values = ±0.754; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.