1. The data below are from n = 111 hospitals in the USA. The Y-variable is

Infct = percentage of patients who get an infection while in the hospital. The four predictor variables are:

- Stay = average length of patient stay in each hospital,

- Age= age of the patient,

- Culture= culture,

- Xray = a measure of how often X-rays are given in each hospital.

1. Fit the multiple linear regression model:

Report the SSE (sum of squared errors) and degrees of freedom (df) for SSE.

1. Do a general linear F-test, at a 5% significance level, to see whether Age and Xray provide significant information about the response, Infct, beyond the information provided by the other two predictor variables. Write down H₀, H_a, the test statistic, p-value, and conclusion.

[In Minitab, you can find the information for the F-statistic either by selecting Sequential sums of squares in the regression options for the model in the previous part, remembering to be careful about the order of the predictors, or by fitting a reduced model without Age and Xray. For the p-value for the F-statistic: select Calc > Probability Distributions > F. . . . Select Cumulative probability and leave the Noncentrality parameter set to 0.0. Next, enter in the respective Numerator degrees of freedom and Denominator degrees of freedom. Finally, enter the F value of interest into the box that says Input constant. For the output, you will see the F value (given as x) and the probability P(X x). The p-value for this problem is 1 - P(X x).]

1. Fit the multiple linear regression model:

Check the LINE model assumptions for this model.

[In Minitab: Click Graphs in the Minitab regression dialog box. Then select Histogram of residuals, and Residuals versus fits. Also click in the Residuals versus the variables box and select Stay and Culture. Also click Storage in the regression dialog box and select "Residuals." Finally click the Storage button to save the residuals from the model. The resulting plots can be used as follows:

• Use the Histogram, and the Anderson-Darling test (this also gives a normal probability plot) to assess the N condition. [Anderson-Darling test in Minitab:

Stat > Basic Statistics > Normality Test, enter RESI for "variable" and click "Anderson-Darling" for Tests for Normality.]

• Use the scatterplot of Residuals versus fits and the plots of Residuals versus each of Stay and Culture to assess the L and E conditions.]

Include all the plots in your write-up and briefly describe the dominant patterns in each plot and your conclusions.

1. For the modelperform the Breusch-Pagan test, usingfor constancy of error variance, assuming

That is testCalculate the test statistic, critical value, and your conclusion.

[Hint: Read the last item on page 234 of the textbook. Remember that you will need to do two regressions: (i) Y with the two predictors (Stay and Culture) and (ii) e² with the two predictors (where e is obtained from (i)).]

1. Obtain a 95% confidence interval estimate of E(Y) whenStay = 10, Culture = 7. Interpret your interval estimate.
2. Obtain a 95% prediction interval for a new observation Y(new) whenStay = 10, Culture = 7. Interpret your interval estimate and say how and why it differs from the interval in the previous part.

Infct	Stay	Age	Culture	Xray
4.1	7.13	55.7	9	39.6
1.6	8.82	58.2	3.8	51.7
2.7	8.34	56.9	8.1	74
5.6	8.95	53.7	18.9	122.8
5.7	11.2	56.5	34.5	88.9
5.1	9.76	50.9	21.9	97
4.6	9.68	57.8	16.7	79
5.4	11.18	45.7	60.5	85.8
4.3	8.67	48.2	24.4	90.8
6.3	8.84	56.3	29.6	82.6
4.9	11.07	53.2	28.5	122
4.3	8.3	57.2	6.8	83.8
7.7	12.78	56.8	46	116.9
3.7	7.58	56.7	20.8	88
4.2	9	56.3	14.6	76.4
5.5	11.08	50.2	18.6	63.6
4.5	8.28	48.1	26	101.8
6.4	11.62	53.9	25.5	99.2
4.2	9.06	52.8	6.9	75.9
4.1	9.35	53.8	15.9	80.9
4.2	7.53	42	23.1	98.9
4.8	10.24	49	36.3	112.6
5	9.78	52.3	17.6	95.9
4.8	9.84	62.2	12	82.3
4	9.2	52.2	17.5	71.1
3.9	8.28	49.5	12	113.1
4.5	9.31	47.2	30.2	101.3
3.2	8.19	52.1	10.8	59.2
4.4	11.65	54.5	18.6	96.1
4.9	9.89	50.5	17.7	103.6
5	11.03	49.9	19.7	102.1
5.2	9.84	53	17.7	72.6
5.3	11.77	54.1	17.3	56
6.1	13.59	54	24.2	111.7
6.3	9.74	54.4	11.4	76.1
5	10.33	55.8	21.2	104.3
2.8	9.97	58.2	16.5	76.5
4.6	7.84	49.1	7.1	87.9
4.1	10.47	53.2	5.7	69.1
1.3	8.16	60.9	1.9	58
3.7	8.48	51.1	12.1	92.8
4.7	10.72	53.8	23.2	94.1
3	11.2	45	7	78.9
5.6	10.12	51.7	14.9	79.1
5.5	8.37	50.7	15.1	84.8
4.6	10.16	54.2	8.4	51.5
5.5	10.9	57.2	10.6	71.9
1.8	7.67	51.7	2.5	40.4
4.2	8.88	51.5	10.1	86.9
5.6	11.48	57.6	20.3	82
4.3	9.23	51.6	11.6	42.6
7.6	11.41	61.1	16.6	97.9
7.8	12.07	43.7	52.4	105.3
3.1	8.63	54	8.4	56.2
3.9	11.15	56.5	7.7	73.9
3.7	7.14	59	2.6	75.8
4.3	7.65	47.1	16.4	65.7
3.9	10.73	50.6	19.3	101
4.5	11.46	56.9	15.6	97.7
3.4	10.42	58	8	59
5.7	11.18	51	18.8	55.9
5.4	7.93	64.1	7.5	98.1
4.4	9.66	52.1	9.9	98.3
5	7.78	45.5	20.9	71.6
4.3	9.42	50.6	24.8	62.8
4.4	10.02	49.5	8.3	93
3.7	8.58	55	7.4	95.9
4.5	9.61	52.4	6.9	87.2
3.5	8.03	54.2	24.3	87.3
4.2	7.39	51	14.6	88.4
2	7.08	52	12.3	56.4
5.2	9.53	51.5	15	65.7
4.5	10.05	52	36.7	87.5
3.4	8.45	38.8	12.9	85
4.5	6.7	48.6	13	80.8
2.9	8.9	49.7	12.7	86.9
4.9	10.23	53.2	9.9	77.9
4.4	8.88	55.8	14.1	76.8
5.1	10.3	59.6	27.8	88.9
2.9	10.79	44.2	2.6	56.6
3.5	7.94	49.5	6.2	92.3
5.5	7.63	52.1	11.6	61.1
4.7	8.77	54.5	5.2	47
1.7	8.09	56.9	7.6	56.9
4.1	9.05	51.2	20.5	79.8
2.9	7.91	52.8	11.9	79.5
4.3	10.39	54.6	14	88.3
4.8	9.36	54.1	18.3	90.6
5.8	11.41	50.4	23.8	73
2.9	8.86	51.3	9.5	87.5
2	8.93	56	6.2	72.5
1.3	8.92	53.9	2.2	79.5
5.3	8.15	54.9	12.3	79.8
5.3	9.77	50.2	15.7	89.7
2.5	8.54	56.1	27	82.5
3.8	8.66	52.8	6.8	69.5
4.8	12.01	52.8	10.8	96.9
2.3	7.95	51.8	4.6	54.9
6.2	10.15	51.9	16.4	59.2
2.6	9.76	53.2	6.9	80.1
4.3	9.89	45.2	11.8	108.7
2.7	7.14	57.6	13.1	92.6
6.6	13.95	65.9	15.6	133.5
4.5	9.44	52.5	10.9	58.5
2.9	10.8	63.9	1.6	57.4
1.4	7.14	51.7	4.1	45.7
2.1	8.02	55	3.8	46.5
5.7	11.8	53.8	9.1	116.9
5.8	9.5	49.3	42	70.9
4.4	7.7	56.9	12.2	67.9
3.1	9.41	59.5	20.6	91.7