Cactuses are succulents, meaning they have thickened, fleshy parts adapted to store water. While in Arizona investigating natural supplies of cactuses, Grace described a random sample of 22 cactuses which showed the following results regarding the relationship between root length and root diameter:

- The mean root length is 10.79 feet with a standard deviation of 0.96 feet.
- The mean root diameter is 5.04 cm with a standard deviation of 1.14 cm.
- The coefficient of correlation is 0.8911.

Tasks:
a. Determine the Least Squares Regression Line (LSRL) equation and label the variables in context.
b. Describe the slope and its meaning in the context of this sample.
c. Use the LSRL to predict the root diameter of a cactus whose root length is 9.40 feet.
d. Calculate the residual for the point (9.90, 4.80).
e. Is a linear model the most appropriate? How do you know?

The Least Squares Regression Line (LSRL) is calculated using the provided sample data, and the slope's meaning within the context is explained. Predictions and residuals are computed using the LSRL, and the appropriateness of a linear model is considered based on the strength of correlation and biological considerations.

Explanation:

To address the student's inquiry, we need to determine the Least Squares Regression Line (LSRL) that models the relationship between root length (independent variable, x) and root diameter (dependent variable, y), given the sample data from the cactuses. We use the formula y = a + bx, where b is the slope and a is the intercept. The slope (b) is given by r (sy/sx), where r is the coefficient of correlation, sy is the standard deviation of the dependent variable, and sx is the standard deviation of the independent variable. In this case:

b = 0.8911 * (1.14 cm/0.96 feet)

To find a, use the means of y and x along with the slope: a = mean(y) - b * mean(x). This provides the intercept.

For part b, the slope indicates how much the root diameter (in cm) is expected to increase for each additional foot of root length. Given the high correlation coefficient, there's a strong positive relationship between root length and diameter.

For part c, to predict the root diameter for a cactus with a root length of 9.40 feet, we substitute x=9.40 into the LSRL equation.

For part d, the residual is the difference between the observed value and the predicted value from the LSRL: residual = observed y - (a + bx), where x is the root length and y is the root diameter.

For part e, to determine if a linear model is appropriate, we consider the strength of the correlation coefficient (0.8911), which suggests a strong linear relationship. However, we should also examine the residuals for any patterns that indicate non-linearity, along with the context of the data regarding cactus roots. Typically, root length and diameter may follow a non-linear pattern due to biological factors.

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