Question

3. A study was conducted to relate birthweight (measured in hundreds of grams) and estriol level (measured in mg/24hr) in 11 pregnant women. A simple linear regression model was fit to the sample data. Use the output from this page to answer the following questions.

   THE REG PROCEEDURE

ANALYSIS OF VARIANCE

   SOURCE DF SUM OF SQUARE MEAN SQUARE F-VALUE PR > F

MODEL 1 228.38    228.381 22.557 0.001045

   ERROR    9    91.12 10.124

TOTAL 10 319.5

  

   PARAMETER ESTIMATES

   Variable    DF    PARAMETER ESTIMATE STANDARD ERROR    T-VALUE Pr > ltl

   Intercept 1 21.0123 2.4751    8.489 1.5E-05

   estriol    1 0.6306    0.1328    4.749 0.00104

R-square    0.7148   

(a)10 Use the output above to fill in the following blanks to test whether there is a significant linear relationship between estriol level in pregnant women and birthweight:
We are interested in the hypotheses: H0 :___ vs HA :____ ,
where [define any parameters used above] ____
.
We can test this hypothesis using the test statistic ,
which follows a distribution with degrees of freedom if H0 is true.
The p-value for this test is_____ .
So, at a significance level of ↵ =0 .05, we would (circle one):
accept the null hypothesis / reject the null hypothesis / fail to reject the null hypothesis
and conclude that [give your conclusion in the context of the problem of interest and be as specific as possible about the direction and magnitude of any effect]
.

(b)1 Based on above output, what birthweight would you expect for a child born to a mother with an estriol level of 15 mg/24hr?
(c)1 Construct a 95% Confidence Interval for the average birthweight at an estriol level of 15 mg/24hr. (Note that SE(E[Y]) = 0.98).
(d)2 A 95% Prediction Interval corresponding to an estriol level of 15 mg/24hr is (22.92, 38.02). Interpret this interval and contrast it with the interval found in (c).
(e)2 Can estriol be used to explain a large proportion of the variability in birthweight? Justify your answer.
(f)2 Find Pearson’s correlation coefficient for this sample. Interpret this value.
(g)2 Will increasing a mother’s estriol level increase the birthweight of her child? Justify your answer.

(h)4 Explain the assumptions underlying this regression modelling.
(i)4 Use the information from the next page to assess, as much as possible, the suitability of this regression model.
(j)2 Based on all of these considerations, what can you conclude about the relationship between birthweight and estriol level in pregnant women.

Answer 1

Here the response variable is birthweight (denoted by y) and the predictor variable is estriol level (x),both measured in their corresponding units.

From the given information ,the fitted regression equation is as follows:

y=alpha+beta*x,

where alpha(intercept)=21.0123 & beta( slope coefficient )=0.6306.

Thus the fitted regression equation is :

birthweight=21.0123+0.6306*estriol_level

a) We are interested in the hypothesis :H0: beta=0 vs H1:beta is not equal to 0,

where beta is the slope coefficient which gives us the amount of  change in birthweight with unit change in estriol_level.

The p-value for this test is= 0.00104 which is less than the significance level =0.05.

So we reject the null hypothesis at 5% level of significance and conclude on the basis of the given data that there is evidence which suggests that the effect of estriol is significant on determining the birth weight of babies.

b) If we put estriol_level=15mg/(24 hrs) in the fitted regression equation given in the data then the predicted birth weight is: birthweight= 21.0123+0.6306*15=30.4713(measured in hundreds of grams)

c) The 95% Confidence Interval for the average birthweight at an estriol level of 15 mg/24hr is given by:

[E(Y) - talpha/2,n-2*SE(E(Y)) , E(Y) + talpha/2,n-2*SE(E(Y)) ]

Here E(Y)=E(Y given X=15)=30.4713, t0.025,0=2.262.(here alpha=level of significance=0.05), SE(Y)=0.98 (given).

Therefore required confidence interval is: [28.25454,32.68806]

d)A prediction interval tells us where we can expect to see the next data point being sampled to fall. Here a 95% Prediction Interval corresponding to an estriol level of 15 mg/24hr is (22.92, 38.02).This means that we can be 95% confident that a next new value will fall within this interval given the estriol_level is at 15mg/24 hours.

A confidence interval tells us about the likely location of the true population parameter. Here it is likely that the expected value of birthweight will fall in the interval [28.25454,32.68806] when the estriol_level is at 15mg/24 hours.

e) Yes, estriol level can be used to explain a large proportion of the variability in birthweight. It is because the value of R-square=0.7148. That means that 71%(approx) of the total variation in birthweight can be explained by the variability in estriol level.

f)Pearson's correlation coefficient is given as 0.7148, which implies high positive correlation between birthweight and estriol level.

g)Yes ,increasing the mother's estriol level will increase the birthweight of the baby since there is positive correlation between the birthweight and the estriol level.

h)Assumptions : i) The errors are normally distributed.

ii)The model is homoscedastic.

iii) The relationship between the response and the predictor is linear.

iv) There are no outliers.