Question

consider the following r output, which models dissolved oxygen levels based on water temperature.

r
slr_model <- lm(dissolved_oxygen ~ temperature)
slr_model

##

call:

lm(formula = dissolved_oxygen ~ temperature)

##

coefficients:

(intercept) temperature

14.4636 -0.2227

which of the following options correctly interprets \\(\hat{\beta}_0\\)?

• when water temperature is zero degrees, the dissolved oxygen level is 14.4636.
• when water temperature is zero degrees, the dissolved oxygen level is -0.2227.
• when water temperature increases by one degree, the dissolved oxygen level decreases by 14.4636 units.
• when water temperature increases by one degree, the dissolved oxygen level decreases by 0.2227 units.

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question 6

consider the following r output, which models dissolved oxygen levels based on water temperature.

r
slr_model <- lm(dissolved_oxygen ~ temperature)
slr_model

##

call:

lm(formula = dissolved_oxygen ~ temperature)

##

coefficients:

(intercept) temperature

14.4636 -0.2227

which of the following options correctly interprets \\(\hat{\beta}_1\\)?

• when water temperature is zero degrees, the dissolved oxygen level is 14.4636.
• when water temperature is zero degrees, the dissolved oxygen level is -0.2227.
• when water temperature increases by one degree, the dissolved oxygen level decreases by 14.4636 units.
• when water temperature increases by one degree, the dissolved oxygen level decreases by 0.2227 units.

Explanation:

⚡ Using what you learned: regression, residuals, and model fit

Step 1: Identify the regression equation

From the R output, the linear regression model is:

$$ \text{dissolved\_oxygen} = \hat{\beta}_0 + \hat{\beta}_1 \times \text{temperature} $$

The coefficients from the output are:

• Intercept (\(\hat{\beta}_0\)): \(14.4636\)
• Slope (\(\hat{\beta}_1\)) for temperature: \(-0.2227\)

Thus, the fitted equation is:

$$ \widehat{\text{dissolved\_oxygen}} = 14.4636 - 0.2227 \times \text{temperature} $$

Step 2: Interpret \(\hat{\beta}_0\) (Intercept)

The intercept \(\hat{\beta}_0 = 14.4636\) represents the predicted value of the dependent variable (dissolved oxygen) when the independent variable (temperature) is equal to zero.

• Interpretation: When water temperature is zero degrees, the dissolved oxygen level is \(14.4636\).

Step 3: Interpret \(\hat{\beta}_1\) (Slope)

The slope \(\hat{\beta}_1 = -0.2227\) represents the change in the predicted value of the dependent variable for each one-unit increase in the independent variable.

• Interpretation: When water temperature increases by one degree, the dissolved oxygen level decreases by \(0.2227\) units.

Answer:

• For the first question (interpreting \(\hat{\beta}_0\)):

"When water temperature is zero degrees, the dissolved oxygen level is 14.4636."

• For Question 6 (interpreting \(\hat{\beta}_1\)):

"When water temperature increases by one degree, the dissolved oxygen level decreases by 0.2227 units."