To predict the $y$-value when $x = 7.4$, we will use the regression equation provided and also determine if the correlation is significant based on the critical value of 0.497.

Steps to Predict the $y$-Value:

1. Regression Equation: $\hat{y} = 7.8x - 20.06$ This is the equation that predicts $y$ based on $x$.

2. Substitute $x = 7.4$ into the regression equation: $\hat{y} = 7.8(7.4) - 20.06$ $\hat{y} = 57.72 - 20.06$ $\hat{y} = 37.66$

   Therefore, the predicted $y$-value when $x = 7.4$ is 37.66.

Significance of the Correlation:

We are given a correlation coefficient of 0.388, and the critical value is 0.497.

• Since the correlation coefficient (0.388) is less than the critical value (0.497), we can conclude that the correlation is not statistically significant. This means that while the regression equation provides a predicted value, the relationship between $x$ and $y$ might not be strong enough to rely on for predictions with high confidence.

Summary:

• The predicted $y$-value for $x = 7.4$ is 37.66.
• However, the correlation between $x$ and $y$ is not significant at the given critical value, so the prediction may not be highly reliable.

Would you like further details on how to interpret the significance of the correlation, or do you have additional questions about this analysis?

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Related Questions:

1. What are the implications of having a correlation coefficient that is not statistically significant?
2. How do you interpret the slope and intercept in the regression equation?
3. What steps would you take if you wanted to improve the correlation between $x$ and $y$?
4. How do you calculate the critical value of a correlation for a given sample size?
5. What are the assumptions of linear regression, and how can they be tested?

Tip:

Always check the statistical significance of the correlation before relying on regression predictions. A weak or insignificant correlation means predictions may be unreliable.