Question

The R output below gives the summary of the multiple regression model for birth weight based on both gestation length and smoking status:

lm(formula = Weight ~ Weeks + SmokingStatus, data = births)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1724.42 558.84 -3.086 0.00265 **

Weeks 130.05 14.52 8.957 2.39e-14 ***

SmokingStatusSmoker -294.40 135.78 -2.168 0.03260 *

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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 484.6 on 97 degrees of freedom

Multiple R-squared: 0.4636, Adjusted R-squared: 0.4525

F-statistic: 41.92 on 2 and 97 DF, p-value: 7.594e-14

(e) Based on the model output, what is the estimated birth weight for a birth at 35 weeks gestation to a non-smoking mother? [1 mark]

(f) Briefly interpret the value ‘130.05’ in the output. [1 mark]

g) Why do the residuals have 97 degrees of freedom? [1 mark]

h) Based on the multiple regression model, is there any evidence of a difference in mean birth weight between smoking and non-smoking mothers? Justify your conclusion with reference to the R output above. [2 marks]

(i) Briefly explain why the conclusion from the multiple regression model might be different to the conclusion from the two-sample t-test in (d). [2 marks]

Answer 1

The R output below gives the summary of the multiple regression model for birth weight based on both gestation length and smoking status:

lm(formula = Weight ~ Weeks + SmokingStatus, data = births)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1724.42 558.84 -3.086 0.00265 **

Weeks 130.05 14.52 8.957 2.39e-14 ***

SmokingStatusSmoker -294.40 135.78 -2.168 0.03260 *

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 484.6 on 97 degrees of freedom

Multiple R-squared: 0.4636, Adjusted R-squared: 0.4525

F-statistic: 41.92 on 2 and 97 DF, p-value: 7.594e-14

(e) Based on the model output, what is the estimated birth weight for a birth at 35 weeks gestation to a non-smoking mother? [1 mark]

The estimated regression line is

Birth weight = -1724.42+130.05*weeks-294.40*smokingstatus.

Estimated Birth weight = -1724.42+130.05*35 -294.40*0

=2827.33

(f) Briefly interpret the value ‘130.05’ in the output. [1 mark]

When weeks increases by 1 unit, the birth weight increases by 130.05.

g) Why do the residuals have 97 degrees of freedom? [1 mark]

total sample size 100, therefore total Df = 100-1 =99 and df for regression is 2

df for residuals = 99-2=97.

h) Based on the multiple regression model, is there any evidence of a difference in mean birth weight between smoking and non-smoking mothers? Justify your conclusion with reference to the R output above. [2 marks]

calculated t = -2.168, P= 0.03260 which is < 0.05 level of significance. Regression coefficient for SmokingStatus is significant. Therefore there is sufficient evidence of a difference in mean birth weight between smoking and non-smoking mothers.

(i) Briefly explain why the conclusion from the multiple regression model might be different to the conclusion from the two-sample t-test in (d). [2 marks]

Multiple regression model might be different to the conclusion from the two-sample t-test because it considers the two independent variables together. two-sample t-test considers the two independent variables one at a time.