https://file.aiquickdraw.com/mathbot/file/17270750263844pgequn7.jpg

### a) Scatter Plot
The scatter plot of \(x\) vs \(y\) is shown above. It displays how the two variables are related visually.

### b) Description of the Correlation
The scatter plot does not show a clear linear relationship. The points are fairly scattered, and the trend appears to be slightly negative, meaning that as \(x\) increases, \(y\) tends to decrease, but the correlation is not strong.

### c) Pearson's Product-Moment Correlation Coefficient
The calculated Pearson's correlation coefficient is approximately \(r = -0.480\). This suggests a weak negative correlation between the two variables.

### d) Significance of the Correlation
The critical value for the correlation coefficient for nine items is 0.521. Since the absolute value of our calculated \(r\) (0.480) is less than the critical value (0.521), we conclude that the data does not suggest a statistically significant correlation between the two variables.

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Would you like any further explanation or details?

Here are five related questions to expand on this topic:

1. How is Pearson's correlation coefficient interpreted in general?
2. What are some limitations of using correlation to infer relationships?
3. How can outliers affect the correlation coefficient?
4. What is the difference between correlation and causation?
5. How can the significance of correlation be tested using different methods?

**Tip:** Always ensure that you have enough data points and that they are well-distributed when interpreting correlation results to avoid misleading conclusions.

The values of two variables, measured for nine different items, are shown in the table. (a) Use technology to plot the data on a scatter graph. (b) Describe the correlation suggested by the graph. (c) Calculate Pearson's product-moment correlation coefficient. (d) The critical value of the correlation coefficient for nine items is 0.521. Do the data suggest that there is significant correlation between the two variables?

This problem involves plotting two variables on a scatter graph and calculating Pearson's correlation coefficient. It compares the correlation with a critical value to determine statistical significance. Suitable for Grades 10-12 students studying statistics.

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