Chapter B.2, Problem 45E

Hour of Birth. In the paper increased Frequency of Births in the Morning Hours” (Science, 123. pp. 985-986). P. D. King obtained records on the number of births at various hospitals over a period of several years. The table that follows shows the proportion of births at a hospital in Jamestown. New York, during each of the 24 hours of the day, where Hour 1 goes from midnight to 1:00AM. Hour 2 goes from 1:00AM to 2:00AM. and so on.

Hour	Proportion of births	Hour	Proportion of births
1	0.0447	13	0.0412
2	0.0440	14	0.0358
3	0.0461.	15	0.0378
4	0.0474	16	0.0318
5	0.0467	17	0.0375
6	0.0503	18	0.0365
7	0.0453	19	0.0351
8	0.0476	20	0.0321
9	0.0478	21	0.0360
10	0.0499	22	0.0387
11	0.0440	23	0.0403
12	0.0446	24	0.0387

The scatterplot of proportion of births versus hour, shown in Output B.36, exhibits curvature. We will use polynomial regression in an attempt to find a suitable regression equation to tit the data.

1. a. Based on the scatterplot in Output B.36, what degree polynomial would you use to lit the data?

OUTPUT B.36 Scatterplot of proportion of births versus hour for Exercise B.45

1. b. We obtained Output B.37 on pages B-57 and B-58, showing the results of fitting a sequence of polynomial regression equations in the centered predictor variable $hou{r}_c=hour-\overline{hour}$. We start with a first-degree equation in hour_c, then a second-degree equation, and so on, up to a seventh-degree equation. Apply the forward selection method to determine a polynomial regression equation for these data. Use a 5% level of significance.

OUTPUT B.37 Output of polynomial regressions for Exercise B.45

OUTPUT B.37 (cont) Output of polynomial regressions for Exercise B.45

1. c. Refer to Output B.37 Starting with a seventh degree polynomial regression equation, apply the backward elimination method to select a polynomial regression equation for these data Use a 5% level of significance.
2. d. Do the forward selection and backward elimination methods yield the same results? If so, is that always the ease?
3. e. Outputs B.38(a)-(d) on page B-59 show plots of residuals versus fitted values for the quadratic, cubic, quartic (fourth-degree), and quintic (fifth-degree) polynomial fits of the data. Interpret these residual plots with regard to the appropriateness of each of the four polynomial regression equations. Do your interpretations agree with the regression models obtained in parts (b) and (c)? Which regression model would you select?
4. f. Using Output B.38, assess the assumption of equal conditional standard deviations.