Construct a two-way scatter plot of birth weight versus

Question

Question 1.

The data set lowbwt contains information for the sample of 100 low birth weight infants born in Boston, Massachusetts [3] (Appendix B, Table B.7).

The variable grmhem is a dichotomous random variable indicating whether an infant experienced a germinal matrix hemorrhage. The value 1 indicates that a hemorrhage occurred and 0 that it did not. The infants' five-minute apgar scores are saved under the name apgar5, and indicators of toxemia-where 1 represents a diagnosis of toxemia during pregnancy for the child's mother and 0 no such diagnosis-under the variable name tox.

a) Using germinal matrix hemorrhage as the response, fit a logistic regression model of the form ln[p^{^}/(1-p^)] = α^ + β^{^}₁x₁ Where X₁ is five-minute apgar score. Interpret β₁ the estimated coefficient of Apgar score.

b) If a particular child has a five-minute apgar score of 3, what is the predicted probability that this child will experience a brain hemorrhage? What is the probability if the child's score is 7?

c) At the 0.05 level of significance, test the null hypothesis that the population parameter PI is equal to 0. What do you conclude?

d) Now fit the regression model.

ln[p^{^}/(1-p^)] = α^ + β^{^}₁x₁ Where x₂ represents toxemia status.

Interpret β₂ the estimated coefficient of toxemia.

e) For a child whose mother was diagnosed with toxemia during pregnancy, what is the predicted probability of experiencing a germinal matrix hemorrhage? What is the probability for a child whose mother was not diagnosed with toxemia?

f) What are the estimated odds of suffering a germinal matrix hemorrhage for children whose mothers were diagnosed with toxemia relative to children whose mothers were not?

g) Construct a 95% confidence interval for the population odds ratio. Does this interval contain the value 1? What does this tell you?

TABLE- Data set lowbwt;

Variables sbp, sex, tox, grmhem, gestage, and apgar5

43	Male	No	No	29	7	62	Female	No	No	27	7	67	Female	No	No	27	8
51	Male	No	No	31	8	59	Female	No	No	27	8	40	Female	No	Yes	31	8
42	Female	No	No	33	0	36	Male	No	No	27	9	48	Female	No	No	26	8
39	Female	No	No	31	8	47	Female	No	No	32	8	36	Male	No	No	27	5
48	Female	Yes	No	30	7	45	Male	No	Yes	31	2	44	Male	No	No	27	6
31	Male	No	Yes	25	0	62	Female	No	Yes	28	5	53	Female	Yes	No	35	9
31	Male	Yes	No	27	7	75	Male	Yes	No	30	7	45	Female	Yes	No	28	6
40	Female	No	No	29	9	44	Male	No	No	29	0	54	Male	No	No	30	8
57	Female	No	No	28	6	39	Male	No	No	28	8	44	Male	Yes	No	31	2
64	Female	Yes	No	29	9	48	Female	No	Yes	31	7	42	Male	No	No	30	5
46	Female	No	No	26	7	43	Female	Yes	No	27	6	50	Female	No	No	27	0
47	Female	No	Yes	30	6	19	Female	No	Yes	25	4	48	Female	No	No	25	5
63	Female	No	No	29	8	63	Male	No	No	30	7	29	Female	No	Yes	25	5
56	Female	No	No	29	1	42	Male	No	No	28	6	30	Female	No	Yes	26	2
49	Male	No	No	29	8	44	Female	No	No	28	9	36	Female	No	No	29	0
87	Male	No	No	29	7	25	Female	No	No	25	8	44	Female	No	No	29	0
46	Female	No	No	29	8	26	Female	No	No	23	8	46	Female	Yes	No	34	9
66	Female	No	No	33	8	27	Male	No	No	27	9	51	Male	Yes	No	30	4
42	Female	Yes	No	33	8	35	Male	No	No	28	8	51	Male	No	No	29	5
52	Female	No	No	29	7	40	Male	No	No	27	7	43	Male	Yes	No	33	7
51	Male	No	No	28	7	44	Female	No	No	27	6	48	Male	No	No	30	5
47	Female	No	No	30	9	66	Male	No	No	26	8	52	Male	No	No	29	8
54	Male	No	No	27	4	59	Female	No	No	25	3	43	Male	No	No	24	6
64	Male	No	No	33	9	24	Female	No	No	23	7	42	Male	Yes	No	33	8
37	Female	No	No	32	7	40	Female	No	Yes	26	3	48	Male	No	Yes	25	5
36	Male	Yes	No	28	3	49	Female	No	No	24	5	49	Female	Yes	No	32	8
45	Female	No	Yes	29	7	53	Male	Yes	No	29	9	62	Male	Yes	No	31	7
39	Male	No	No	28	7	45	Female	No	No	29	9	45	Male	No	No	31	9
29	Female	No	No	29	4	50	Male	No	Yes	27	8	51	Female	Yes	Yes	31	6
61	Female	No	No	30	3	64	Male	No	No	30	7	52	Male	No	No	29	8
53	Male	No	No	31	7	48	Female	No	No	30	6	47	Male	Yes	No	32	5
64	Female	No	No	30	7	48	Female	No	Yes	32	4	40	Female	Yes	No	33	8
35	Female	No	No	31	6	58	Female	Yes	No	33	7	50	Female	No	No	28	7
34	Male	No	No	29	9

Question 2. Measurements of length and weight for a sample of 20 low birth weight infants are contained in the data set twenty (Appendix B, Table B.23).

The length measurements are saved under the variable name length, and the corresponding birth weights underweight.

(a) Construct a two-way scatter plot of birth weight versus length for the 20 infants in the sample. Without doing any calculations, sketch your best guess for the least-squares regression line directly on the scatter plot.

(b) Now compute the true least-squares regression line. Draw this line on the scatter plot. Does the actual least-squares line concur with your guess? Based on the two-way scatter plot, it is clear that one point lies outside the range of the remainder of the data. This point corresponds to the ninth infant in the sample of size 20. To illustrate the effect that the outlier has on the model, remove this point from the data set.

(c) Compute the new least-squares regression line based on the sample of size 19, and sketch this line on the original scatter plot. How does the least-squares line change? In particular, comment on the values of the slope and the intercept.

(d) Compare the coefficients of determination (R2) and the standard deviations from regression (sylx) for the two least-squares regression lines. Explain how these values changed when you removed the outlier from the original data set. Why did they change?

Length	weight
41	1360
40	1490
38	1490
38	1180
38	1200
32	680
33	620
38	1060
30	1320
34	830
32	880
39	1130
38	1140
39	1350
37	950
39	1220
38	980
42	1480
39	1250
38	1250