Question

Birth weight and gestational age.  The Child Health and Development Studies considered pregnancies among women in the San Francisco East Bay area. Researchers took a random sample of 50 pregnancies and used statistical software to construct a linear regression model to predict a baby's birth weight in ounces using the gestation age (the number of days the mother was pregnant). A portion of the computer output and the scatter plot is shown below. Round all calculated results to four decimal places.

Coefficients	Estimate	Std. Error	t value	Pr(>|t|)
Intercept	14.2143	39.6523	0.3585	0.7216
gestation	0.3769	0.1428	2.6394	0.0112

---

Residual standard error: 16.1871 on 48 degrees of freedom

Multiple R-squared: 0.1267, Adjusted R-squared: 0.1086

1. Use the computer output to write the estimated regression equation for predicting birth weight from length of gestation.

Birth weight =  +  * gestation

2. Using the estimated regression equation, what is the predicted birth weight for a baby with a length of gestation of 278 days?

3. The recorded birth weight for a baby with a gestation of 278 days was 106 ounces. Complete the following sentence:

The residual for this baby is  . This means the birth weight for this baby is  ? higher than the same as lower than  the birth weight predicted by the regression model.

4. Complete the following sentence:

% of the variation in  ? Birth weight Gestation age Babies Pregnancy  can be explained by the linear relationship to  ? Birth weight Gestation age Babies Pregnancy .

Do the data provide evidence that gestational age is associated with birth weight? Conduct a t-test using the information given in the R output and the hypotheses

?0:?1=0H0:β1=0 vs. ??:?1≠0HA:β1≠0

3. Test statistic =

4. Degrees of freedom =

5. P-value =

6. Based on the results of this hypothesis test, there is  ? little evidence some evidence strong evidence very strong evidence extremely strong evidence  of a linear relationship between the explanatory and response variables.

7. Calculate a 95% confidence interval for the slope, ?1β1. (  ,   )

Answer 1

1.

Estimated regression equation for predicting birth weight from length of gestation:

Birth weight = 14.2143 + (0.3769) * gestation

2.

Predicted value of y at x = 278

ŷ = 14.2143 + (0.3769) * 278 = 118.9925

3.

Residual = y - ŷ = 106 - 118.9925 = -12.9925

The residual for this baby is -12.9925. This means the birth weight for this baby is Lower than the birth weight predicted by the regression model.

4.

12.67 % of the variation in Birth weight can be explained by the linear relationship to Gestation age.

--

H0: β1 = 0 vs. HA: β1 ≠ 0

3. Test statistic = 2.6394

4. Degrees of freedom = 48

5. P-value = 0.0112

6. Based on the results of this hypothesis test, there is strong evidence of a linear relationship between the explanatory and response variables.

7. Critical value, t_c = T.INV.2T(0.05, 48) = 2.0106

95% Confidence interval for slope:

Lower limit = b1 - tc*se(b1) = 0.3769 - 2.0106*0.1428 = 0.0898

Upper limit = b1 + tc*se(b1) = 0.3769 + 2.0106*0.1428 = 0.6640