Question

Dean Parmalee wished to know if the year-end grades assigned to Wright State University Medical School students are predictive of their second-year medical board scores. The following table shows, for 89 students, the year-end score (AVG, in percent of 100) and the score on the second year medical board (BOARD) examination (data: medscores.mtw).

AVG	BOARD	AVG	BOARD	AVG	BOARD
95.73	257	85.91	208	82.01	196
94.03	256	85.81	210	81.86	179
91.51	242	85.35	212	81.7	207
91.49	223	85.3	225	81.65	202
91.13	241	85.27	203	81.51	230
90.88	234	85.05	214	81.07	200
90.83	226	84.58	176	80.95	200
90.6	236	84.51	196	80.92	160
90.3	250	84.51	207	80.84	205
90.29	226	84.42	207	80.77	194
89.93	233	84.34	211	80.72	196
89.83	241	84.34	202	80.69	171
89.65	234	84.13	229	80.58	201
89.47	231	84.13	202	80.57	177
88.87	228	84.09	184	80.1	192
88.8	229	83.98	206	79.38	187
88.66	235	83.93	202	78.75	161
88.55	216	83.92	176	78.32	172
88.43	207	83.73	204	78.17	163
88.34	224	83.47	208	77.39	166
87.95	237	83.27	211	76.3	170
87.79	213	83.13	196	75.85	159
87.01	215	83.05	203	75.6	154
86.86	187	83.02	188	75.16	169
86.85	204	82.82	169	74.85	159
86.84	219	82.78	205	74.66	167
86.3	228	82.57	183	74.58	154
86.13	210	82.56	181	74.16	148
86.1	216	82.45	173	70.34	159
85.92	212	82.24	185

a) Create scatterplots of BOARD vs. AVG. Assess the nature of the relationship of these variables.

2. Give the estimates of the coefficients of your model.

3. Interpret the Estimated intercept and slope?

4. What is the value of R-square? Interpret it.

5. Are there any outliers or influential observations?

6. Give the predicted BOARD score for an individual with an 85 average. Interpret it.

7. Assess the fit of the model and interpret.

8. Explain why this model is not appropriate for someone with AVG value of 60.0.

9. Use the regression model to give 95% mean and individual prediction intervals for AVG score of 95.73 and 94.03. Make a graph showing the 95% confidence bands for the mean and the 95% prediction intervals.

type in will be best：）

Answer 1

In order to solve this question I used R software.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')
> head(d)
AVG BOARD  
1 95.73 257
2 94.03 256
3 91.51 242
4 91.49 223
5 91.13 241
6 90.88 234
> attach(d)
> plot(AVG,BOARD)
> fit=lm(BOARD~AVG)
> summary(fit)

Call:
lm(formula = BOARD ~ AVG)

Residuals:
Min 1Q Median 3Q Max
-28.931 -8.150 2.397 7.193 39.441

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -191.0296 22.9342 -8.329 1.06e-12 ***
AVG 4.6815 0.2727 17.169 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12.49 on 87 degrees of freedom
Multiple R-squared: 0.7721, Adjusted R-squared: 0.7695
F-statistic: 294.8 on 1 and 87 DF, p-value: < 2.2e-16

a.

Scatter plot:

b.

Estimates of the coefficients of your model:

Slope = 4.6815

Intercept = -191.0296

c.

Interpretation:

For intercept:

When AVG score of student is zero, then their second-year medical board scores is -191.0296

For slope:

When AVG score of student is increased by 1 then second-year medical board scores is increased by 4.6815

d.

R-square = 0.7721

It means 77.21% of variation in second-year medical board scores is explained by the average score.