Let's break down the problem and solve each part step by step. The given data is:

$\sum x = 20, \, \sum x^2 = 90, \, \sum y = 44, \, \sum y^2 = 430, \, \sum xy = 195, \, n = 5$

(a) Find the value of the linear correlation coefficient $r$.

The formula for the linear correlation coefficient $r$ is:

$r = \frac{n \sum xy - \sum x \sum y}{\sqrt{\left(n \sum x^2 - (\sum x)^2\right)\left(n \sum y^2 - (\sum y)^2\right)}}$

Plugging in the values:

$r = \frac{5(195) - (20)(44)}{\sqrt{(5)(90) - (20)^2} \sqrt{(5)(430) - (44)^2}}$

Let me compute that for you.The linear correlation coefficient $r$ is approximately $0.918$, rounded to three decimal places.

(b) Find the value of the coefficient of determination.

The coefficient of determination $r^2$ represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It's calculated as:

$r^2 = (r)^2$

Let me calculate that and round to the nearest whole percent.The coefficient of determination $r^2$ is approximately 84%, meaning that 84% of the variance in $y$ can be explained by the variance in $x$.

(c) Find the equation of the regression line.

The equation of the regression line is given by:

$y = mx + b$

where $m$ (the slope) and $b$ (the y-intercept) are calculated as follows:

$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$ $b = \frac{\sum y - m \sum x}{n}$

I'll compute the values of $m$ and $b$ for the equation.The equation of the regression line is:

$y = 1.9x + 1.2$

(d) Predict the $y$-value if $x = 4.5$ assuming the linear correlation is not significant.

Using the regression equation $y = 1.9x + 1.2$, we substitute $x = 4.5$:

$y = 1.9(4.5) + 1.2$

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can substitute $x = 4.5$ into the regression equation $y = 1.9x + 1.2$ to get the predicted value manually:

$y = 1.9(4.5) + 1.2 = 8.55 + 1.2 = 9.75$

Thus, the predicted $y$-value is approximately $9.75$.

Let me know if you'd like further details or have any questions! Here are five related problems you can explore:

1. What does a correlation coefficient of 0.918 indicate about the relationship between $x$ and $y$?
2. How would the coefficient of determination change if the correlation were weaker?
3. What happens to the regression line if the data points show no correlation?
4. How does increasing the number of data points affect the accuracy of the regression equation?
5. If the slope of the line was negative, what would that imply about the relationship?

Tip: When interpreting regression lines, always consider the slope and correlation coefficient to understand the strength and direction of the relationship between variables.