Question

question 4
2 pts

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

slr_model <- lm(dissolved_oxygen ~ temperature)
slr_model

##

call:

lm(formula = dissolved_oxygen ~ temperature)

##

coefficients:

(intercept) temperature

14.4636 -0.2227

identify the estimated \\(\hat{\beta}_0\\) and \\(\hat{\beta}_1\\) and complete the fitted simple linear regression equation.

dissolved oxygen = + temperature

question 5
1 pts

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

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.

Explanation:

🆕 New Concept Discovered: Simple Linear Regression (R Output Interpretation)
Reading coefficients and interpreting intercepts in regression models.

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Step 1: Identify the coefficients from the R output

From the provided R output:

• The coefficient labeled (Intercept) is the estimated y-intercept, \(\hat{\beta}_0\):

$$ \hat{\beta}_0 = 14.4636 $$

• The coefficient labeled temperature is the estimated slope, \(\hat{\beta}_1\):

$$ \hat{\beta}_1 = -0.2227 $$

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Step 2: Complete the regression equation (Question 4)

The general form of a fitted simple linear regression equation is:

$$ \text{Response} = \hat{\beta}_0 + \hat{\beta}_1 \times \text{Predictor} $$

Substituting the identified values:

$$ \text{Dissolved Oxygen} = 14.4636 + (-0.2227) \times \text{Temperature} $$

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Step 3: Interpret the intercept \(\hat{\beta}_0\) (Question 5)

The intercept \(\hat{\beta}_0\) represents the predicted value of the response variable (Dissolved Oxygen) when the predictor variable (Temperature) is equal to zero.

• Here, \(\hat{\beta}_0 = 14.4636\).
• Therefore, when the water temperature is zero degrees, the predicted dissolved oxygen level is \(14.4636\).

This matches the first option:
"When water temperature is zero degrees, the dissolved oxygen level is 14.4636."

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Answer:

Question 4:

• First box (\(\hat{\beta}_0\)): 14.4636
• Second box (\(\hat{\beta}_1\)): -0.2227

$$\text{Dissolved Oxygen} = 14.4636 + (-0.2227) \text{ Temperature}$$

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Question 5:

• Correct Option:

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