Researchers studying anthropometry collected body girth measurements and skeleta

Question

Researchers studying anthropometry collected body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender for 507 physically active individuals. You can load these data into R by typing library(openintro) data(bdims) Read about the bdims dataset: ?bdims (a) Write the equation of the regression line for predicting height (hgt) from shoulder girth sho.gi (b) Interpret the slope and the intercept in this context (c) Calculate R2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application (d) A randomly selected student from your class has a shoulder girth of 100 cm. Predict the height of this student using the model. (e) Construct a 95% prediction interval for the height of this new student. (f) Below are a few possible interpretations for the interval you constructed in part (e). Which are correct and which are incorrect interpretations? (i) (Correct/Incorrect) We used a procedure that would capture the true mean height 95% of the time in repeated samples (ii) (Correct/Incorrect) We used a procedure that would capture a new student's height, given that he/she has a shoulder girth of 100 cm, 95% of the time in repeated samples (iii) (Correct/Incorrect) 95% of all students have a height within the interval constructed in part (e) (g) The student from part (d) is 160 cm tall. Calculate the residual, and explain what this residlal means (h) A one year old has a shoulder girth of 56 cm. Would it be appropriate to use this linear model to predict the height of this child? Explain your reasoning.

Explanation / Answer

R- Code

library(openintro)
d=(bdims)
dim(d)
fit=lm(d[,24]~d[,10])
summary(fit)
predict(fit,level=0.95,interval="confidence")
resid(fit)
fitted(fit)

___________________________________________________________________________________________

Out puts-

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> utils:::menuInstallLocal()
* installing *source* package 'openintro' ...
** package 'openintro' successfully unpacked and MD5 sums checked
** R
** data
*** moving datasets to lazyload DB
** preparing package for lazy loading
** help
*** installing help indices
*** copying figures
** building package indices
** testing if installed package can be loaded
*** arch - i386
*** arch - x64
* DONE (openintro)
> library(openintro)
Please visit openintro.org for free statistics materials

Attaching package: ‘openintro’

The following objects are masked from ‘package:datasets’:

cars, trees

> d=(bdims)
> dim(d)
[1] 507 25
> fit=lm(d[,24]~d[,10])
> summary(fit)

Call:
lm(formula = d[, 24] ~ d[, 10])

Residuals:
Min 1Q Median 3Q Max
-19.2297 -4.7976 -0.1142 4.7885 21.0979

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 105.83246 3.27245 32.34 <2e-16 ***
d[, 10] 0.60364 0.03011 20.05 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.026 on 505 degrees of freedom
Multiple R-squared: 0.4432, Adjusted R-squared: 0.4421
F-statistic: 402 on 1 and 505 DF, p-value: < 2.2e-16

> predict(fit,level=0.95,interval="confidence")
fit lwr upr
1 169.9395 169.3151 170.5638
2 172.5351 171.9071 173.1632
3 175.3119 174.5752 176.0486
4 168.9133 168.2624 169.5642
5 170.7242 170.1097 171.3387
6 178.1490 177.2287 179.0694
7 180.3825 179.2891 181.4759
8 178.5112 177.5641 179.4584
9 172.8370 172.2018 173.4721
10 177.9679 177.0607 178.8752
11 176.5192 175.7109 177.3275
12 180.3825 179.2891 181.4759
13 176.1570 175.3714 176.9426
14 174.0443 173.3685 174.7200
15 170.7242 170.1097 171.3387
16 173.4406 172.7875 174.0937
17 173.5613 172.9041 174.2186
18 178.2698 177.3405 179.1990
19 171.6297 171.0147 172.2446
20 177.3643 176.4997 178.2288
21 175.8552 175.0877 176.6227
22 172.8370 172.2018 173.4721
23 176.8814 176.0495 177.7132
24 180.6240 179.5109 181.7370
25 178.6320 177.6758 179.5881
26 184.0044 182.6029 185.4058
27 175.2515 174.5181 175.9850
28 175.8552 175.0877 176.6227
29 170.9053 170.2918 171.5188
30 166.3176 165.5433 167.0919
31 174.0443 173.3685 174.7200
32 177.0021 176.1622 177.8420
33 173.7424 173.0786 174.4063
34 172.5351 171.9071 173.1632
35 173.4406 172.7875 174.0937
36 168.6115 167.9501 169.2729
37 175.1308 174.4038 175.8578
38 171.0260 170.4128 171.6392
39 172.9577 172.3194 173.5960
40 177.2436 176.3873 178.0998
41 175.3723 174.6323 176.1123
42 184.2458 182.8231 185.6686
43 173.9839 173.3106 174.6572
44 173.5613 172.9041 174.2186
45 176.5192 175.7109 177.3275
46 177.4850 176.6121 178.3580
47 171.7504 171.1344 172.3664
48 183.1593 181.8318 184.4868
49 174.2253 173.5419 174.9088
50 171.2675 170.6543 171.8807
51 177.4850 176.6121 178.3580
52 182.0727 180.8387 183.3067
53 180.8051 179.6772 181.9330
54 182.3142 181.0596 183.5688
55 168.1889 167.5109 168.8669
56 166.9213 166.1816 167.6609
57 168.7926 168.1376 169.4475
58 174.1650 173.4842 174.8458
59 179.4167 178.4003 180.4331
60 174.1046 173.4263 174.7829
61 171.3279 170.7145 171.9412
62 174.5875 173.8877 175.2874
63 173.8028 173.1366 174.4690
64 172.3541 171.7296 172.9785
65 171.4486 170.8348 172.0624
66 168.6115 167.9501 169.2729
67 175.3723 174.6323 176.1123
68 173.3802 172.7292 174.0313
69 182.4953 181.2251 183.7654
70 172.9577 172.3194 173.5960
71 179.4771 178.4559 180.4982
72 174.9497 174.2321 175.6673
73 171.9315 171.3136 172.5494
74 172.7162 172.0841 173.3484
75 177.5454 176.6682 178.4225
76 171.0260 170.4128 171.6392
77 174.8290 174.1175 175.5405
78 169.4566 168.8216 170.0916
79 175.2515 174.5181 175.9850
80 177.9679 177.0607 178.8752
81 184.3062 182.8781 185.7343
82 174.2253 173.5419 174.9088
83 174.1650 173.4842 174.8458
84 170.3620 169.7442 170.9799
85 174.5272 173.8302 175.2242
86 176.6399 175.8239 177.4560
87 180.8051 179.6772 181.9330
88 180.0807 179.0117 181.1497
89 176.9417 176.1059 177.7776
90 177.5454 176.6682 178.4225
91 173.4406 172.7875 174.0937
92 174.0443 173.3685 174.7200
93 175.8552 175.0877 176.6227
94 173.9235 173.2526 174.5944
95 181.2880 180.1201 182.4559
96 171.2071 170.5940 171.8202
97 171.1468 170.5337 171.7599
98 174.0443 173.3685 174.7200
99 175.2515 174.5181 175.9850
100 180.0807 179.0117 181.1497
101 178.3905 177.4523 179.3287
102 174.6479 173.9452 175.3506
103 167.9474 167.2590 168.6359
104 173.7424 173.0786 174.4063
105 168.9133 168.2624 169.5642
106 173.0181 172.3781 173.6581
107 176.5796 175.7674 177.3917
108 173.9235 173.2526 174.5944
109 169.0944 168.4492 169.7395
110 176.8814 176.0495 177.7132
111 170.8449 170.2312 171.4587
112 181.4087 180.2307 182.5867
113 171.6900 171.0746 172.3055
114 179.8392 178.7895 180.8890
115 176.6399 175.8239 177.4560
116 171.6900 171.0746 172.3055
117 176.2777 175.4847 177.0708
118 181.2276 180.0647 182.3905
119 177.4247 176.5559 178.2934
120 171.5090 170.8948 172.1231
121 179.4167 178.4003 180.4331
122 176.6399 175.8239 177.4560
123 175.2515 174.5181 175.9850
124 187.2037 185.5147 188.8926
125 174.5272 173.8302 175.2242
126 177.9076 177.0047 178.8105
127 177.0021 176.1622 177.8420
128 182.6764 181.3907 183.9621
129 177.5454 176.6682 178.4225
130 179.7185 178.6784 180.7587
131 173.1388 172.4953 173.7823
132 174.1650 173.4842 174.8458
133 175.4326 174.6893 176.1759
134 179.7789 178.7339 180.8238
135 174.8894 174.1748 175.6039
136 173.9235 173.2526 174.5944
137 173.5613 172.9041 174.2186
138 177.9076 177.0047 178.8105
139 171.2675 170.6543 171.8807
140 179.7185 178.6784 180.7587
141 183.5818 182.2175 184.9462
142 177.0625 176.2185 177.9064
143 176.1570 175.3714 176.9426
144 178.5112 177.5641 179.4584
145 180.1411 179.0672 181.2149
146 176.5796 175.7674 177.3917
147 174.0443 173.3685 174.7200
148 171.8711 171.2539 172.4884
149 169.6377 169.0070 170.2683
150 178.0887 177.1727 179.0047
151 177.0625 176.2185 177.9064
152 168.7926 168.1376 169.4475
153 174.7083 174.0027 175.4139
154 177.6661 176.7805 178.5518
155 174.5272 173.8302 175.2242
156 177.0625 176.2185 177.9064
157 175.8552 175.0877 176.6227
158 171.2071 170.5940 171.8202
159 183.8233 182.4378 185.2088
160 171.7504 171.1344 172.3664
161 180.8654 179.7326 181.9983
162 179.8392 178.7895 180.8890
163 179.7185 178.6784 180.7587
164 175.1912 174.4610 175.9214
165 172.8973 172.2606 173.5340
166 181.8916 180.6730 183.1103
167 175.5534 174.8033 176.3034
168 181.1069 179.9540 182.2597
169 181.4691 180.2860 182.6521
170 177.1832 176.3311 178.0353
171 180.8654 179.7326 181.9983
172 177.2436 176.3873 178.0998
173 179.6581 178.6228 180.6935
174 176.1570 175.3714 176.9426
175 180.0807 179.0117 181.1497
176 176.8210 175.9931 177.6489
177 176.7607 175.9367 177.5846
178 176.6399 175.8239 177.4560
179 174.9497 174.2321 175.6673
180 180.0203 178.9561 181.0845
181 175.5534 174.8033 176.3034
182 175.9156 175.1445 176.6866
183 179.2356 178.2333 180.2379
184 185.2720 183.7579 186.7862
185 180.9862 179.8433 182.1290
186 175.6137 174.8602 176.3672
187 176.4588 175.6544 177.2633
188 171.3882 170.7747 172.0018
189 178.3905 177.4523 179.3287
190 175.9759 175.2013 176.7506
191 175.4930 174.7463 176.2397
192 178.2698 177.3405 179.1990
193 172.4144 171.7888 173.0400
194 178.0887 177.1727 179.0047
195 180.3825 179.2891 181.4759
196 170.9053 170.2918 171.5188
197 177.4850 176.6121 178.3580
198 178.6923 177.7316 179.6531
199 177.4850 176.6121 178.3580
200 177.0625 176.2185 177.9064
201 169.2151 168.5735 169.8567
202 172.5955 171.9661 173.2249
203 180.1411 179.0672 181.2149
204 178.8734 177.8989 179.8479
205 173.2595 172.6123 173.9067
206 172.8370 172.2018 173.4721
207 173.4406 172.7875 174.0937
208 175.7948 175.0309 176.5588
209 178.8734 177.8989 179.8479
210 180.4429 179.3446 181.5412
211 178.2698 177.3405 179.1990
212 183.5215 182.1624 184.8805
213 180.9862 179.8433 182.1290
214 173.6821 173.0204 174.3437
215 173.5613 172.9041 174.2186
216 182.4953 181.2251 183.7654
217 175.4326 174.6893 176.1759
218 173.1388 172.4953 173.7823
219 177.2436 176.3873 178.0998
220 177.0625 176.2185 177.9064
221 182.4953 181.2251 183.7654
222 180.9258 179.7879 182.0637
223 173.6821 173.0204 174.3437
224 181.7709 180.5625 182.9793
225 185.3324 183.8129 186.8519
226 176.6399 175.8239 177.4560
227 170.7846 170.1705 171.3987
228 179.7185 178.6784 180.7587
229 176.4588 175.6544 177.2633
230 177.6661 176.7805 178.5518
231 180.3825 179.2891 181.4759
232 179.1149 178.1219 180.1078
233 173.6217 172.9623 174.2811
234 174.1650 173.4842 174.8458
235 179.2960 178.2890 180.3030
236 176.2174 175.4280 177.0067
237 176.9417 176.1059 177.7776
238 177.0625 176.2185 177.9064
239 184.9702 183.4831 186.4574
240 179.1149 178.1219 180.1078
241 176.0363 175.2580 176.8146
242 179.3563 178.3446 180.3680
243 177.1832 176.3311 178.0353
244 182.0727 180.8387 183.3067
245 178.8734 177.8989 179.8479
246 180.1411 179.0672 181.2149
247 175.1912 174.4610 175.9214
248 163.1787 162.1861 164.1712
249 165.8951 165.0948 166.6953
250 158.9532 157.6104 160.2959
251 164.3859 163.4835 165.2884
252 168.1889 167.5109 168.8669
253 162.2732 161.2095 163.3369
254 162.1525 161.0791 163.2259
255 162.8768 161.8609 163.8928
256 165.3518 164.5163 166.1873
257 175.5534 174.8033 176.3034
258 164.9292 164.0651 165.7934
259 164.8085 163.9360 165.6810
260 166.4987 165.7351 167.2623
261 159.3757 158.0697 160.6817
262 164.1445 163.2245 165.0644
263 161.3677 160.2304 162.5051
264 160.1604 158.9218 161.3991
265 168.6115 167.9501 169.2729
266 160.2208 158.9873 161.4543
267 168.6115 167.9501 169.2729
268 160.7641 159.5765 161.9517
269 173.0784 172.4367 173.7202
270 163.9030 162.9653 164.8408
271 176.5192 175.7109 177.3275
272 167.1023 166.3725 167.8322
273 163.2390 162.2512 164.2269
274 164.4463 163.5482 165.3444
275 167.0420 166.3089 167.7751
276 168.4304 167.7621 169.0986
277 159.6775 158.3975 160.9575
278 157.7459 156.2969 159.1948
279 165.0500 164.1941 165.9058
280 174.6479 173.9452 175.3506
281 163.1787 162.1861 164.1712
282 158.3495 156.9539 159.7451
283 166.6798 165.9266 167.4329
284 173.5613 172.9041 174.2186
285 165.4121 164.5807 166.2436
286 166.2572 165.4793 167.0352
287 166.3780 165.6073 167.1487
288 166.4987 165.7351 167.2623
289 166.9213 166.1816 167.6609
290 163.7823 162.8356 164.7290
291 161.8506 160.7528 162.9484
292 163.4805 162.5111 164.4499
293 165.4121 164.5807 166.2436
294 164.3256 163.4188 165.2324
295 160.7641 159.5765 161.9517
296 169.9998 169.3766 170.6231
297 166.0762 165.2872 166.8651
298 169.2151 168.5735 169.8567
299 169.9395 169.3151 170.5638
300 164.8085 163.9360 165.6810
301 164.6274 163.7422 165.5126
302 163.9634 163.0301 164.8967
303 164.9896 164.1296 165.8496
304 162.8768 161.8609 163.8928
305 167.0420 166.3089 167.7751
306 163.2390 162.2512 164.2269
307 166.8609 166.1179 167.6039
308 164.6274 163.7422 165.5126
309 169.0944 168.4492 169.7395
310 162.8768 161.8609 163.8928
311 168.1285 167.4480 168.8091
312 172.7162 172.0841 173.3484
313 162.9372 161.9260 163.9484
314 164.1445 163.2245 165.0644
315 167.8871 167.1959 168.5783
316 164.3256 163.4188 165.2324
317 163.5408 162.5760 164.5057
318 161.6092 160.4917 162.7267
319 162.3939 161.3399 163.4480
320 163.8427 162.9005 164.7849
321 166.2572 165.4793 167.0352
322 165.4725 164.6450 166.3000
323 161.4885 160.3610 162.6159
324 164.5670 163.6775 165.4565
325 160.4623 159.2492 161.6753
326 172.5351 171.9071 173.1632
327 168.0078 167.3220 168.6936
328 171.6297 171.0147 172.2446
329 169.0944 168.4492 169.7395
330 171.7504 171.1344 172.3664
331 176.4588 175.6544 177.2633
332 162.5147 161.4702 163.5591
333 169.5169 168.8834 170.1504
334 165.8347 165.0306 166.6388
335 159.8586 158.5942 161.1231
336 165.2914 164.4519 166.1309
337 163.1183 162.1211 164.1155
338 163.9634 163.0301 164.8967
339 164.2652 163.3541 165.1764
340 167.7060 167.0064 168.4055
341 163.2994 162.3161 164.2826
342 168.0078 167.3220 168.6936
343 165.6536 164.8379 166.4693
344 168.6718 168.0126 169.3310
345 165.8951 165.0948 166.6953
346 172.3541 171.7296 172.9785
347 170.4828 169.8663 171.0993
348 163.6012 162.6409 164.5615
349 168.2493 167.5737 168.9248
350 162.9372 161.9260 163.9484
351 168.9133 168.2624 169.5642
352 164.8085 163.9360 165.6810
353 164.0238 163.0950 164.9526
354 168.0078 167.3220 168.6936
355 168.2493 167.5737 168.9248
356 171.9315 171.3136 172.5494
357 172.6559 172.0251 173.2866
358 167.5249 166.8166 168.2332
359 184.0044 182.6029 185.4058
360 165.7140 164.9022 166.5257
361 169.0340 168.3870 169.6810
362 165.5329 164.7094 166.3564
363 165.2914 164.4519 166.1309
364 167.5249 166.8166 168.2332
365 166.3176 165.5433 167.0919
366 157.6855 156.2311 159.1398
367 166.0762 165.2872 166.8651
368 162.1525 161.0791 163.2259
369 170.6035 169.9881 171.2189
370 160.1604 158.9218 161.3991
371 167.1023 166.3725 167.8322
372 169.2755 168.6356 169.9153
373 166.8005 166.0542 167.5469
374 161.3074 160.1650 162.4497
375 173.5010 172.8458 174.1561
376 165.1103 164.2586 165.9620
377 167.7060 167.0064 168.4055
378 165.2311 164.3875 166.0746
379 164.2652 163.3541 165.1764
380 168.5511 167.8874 169.2147
381 161.7903 160.6876 162.8930
382 169.2151 168.5735 169.8567
383 162.5750 161.5353 163.6147
384 167.1023 166.3725 167.8322
385 161.3677 160.2304 162.5051
386 171.3279 170.7145 171.9412
387 162.4543 161.4050 163.5035
388 166.0762 165.2872 166.8651
389 166.3780 165.6073 167.1487
390 167.1023 166.3725 167.8322
391 162.9976 161.9910 164.0041
392 168.0682 167.3850 168.7513
393 162.8768 161.8609 163.8928
394 164.4463 163.5482 165.3444
395 164.9896 164.1296 165.8496
396 167.6456 166.9432 168.3480
397 171.6900 171.0746 172.3055
398 162.9976 161.9910 164.0041
399 165.1103 164.2586 165.9620
400 167.1023 166.3725 167.8322
401 164.1445 163.2245 165.0644
402 169.8791 169.2536 170.5046
403 166.2572 165.4793 167.0352
404 169.3962 168.7596 170.0328
405 166.0762 165.2872 166.8651
406 168.5511 167.8874 169.2147
407 172.8370 172.2018 173.4721
408 165.5932 164.7737 166.4128
409 167.7060 167.0064 168.4055
410 169.5773 168.9453 170.2093
411 165.8951 165.0948 166.6953
412 170.1206 169.4993 170.7418
413 168.6718 168.0126 169.3310
414 166.0158 165.2231 166.8085
415 170.1809 169.5606 170.8012
416 164.5067 163.6129 165.4005
417 166.6798 165.9266 167.4329
418 166.6194 165.8628 167.3760
419 165.6536 164.8379 166.4693
420 171.5693 170.9548 172.1838
421 172.1126 171.4922 172.7330
422 165.8951 165.0948 166.6953
423 166.4987 165.7351 167.2623
424 164.6274 163.7422 165.5126
425 169.2151 168.5735 169.8567
426 167.9474 167.2590 168.6359
427 162.5147 161.4702 163.5591
428 174.7686 174.0601 175.4772
429 163.7219 162.7707 164.6732
430 165.4121 164.5807 166.2436
431 160.8848 159.7073 162.0623
432 169.2755 168.6356 169.9153
433 167.5853 166.8799 168.2906
434 162.0317 160.9486 163.1149
435 167.2231 166.4996 167.9466
436 169.2151 168.5735 169.8567
437 164.0238 163.0950 164.9526
438 168.3096 167.6366 168.9827
439 166.0762 165.2872 166.8651
440 158.3495 156.9539 159.7451
441 169.3358 168.6977 169.9740
442 167.4645 166.7533 168.1758
443 166.9816 166.2453 167.7180
444 165.9554 165.1590 166.7519
445 165.8347 165.0306 166.6388
446 165.8347 165.0306 166.6388
447 165.5932 164.7737 166.4128
448 171.1468 170.5337 171.7599
449 164.0238 163.0950 164.9526
450 167.0420 166.3089 167.7751
451 165.1103 164.2586 165.9620
452 166.5591 165.7990 167.3191
453 171.2675 170.6543 171.8807
454 166.6798 165.9266 167.4329
455 163.0579 162.0561 164.0598
456 171.1468 170.5337 171.7599
457 172.6559 172.0251 173.2866
458 167.7664 167.0696 168.4631
459 165.0500 164.1941 165.9058
460 161.3677 160.2304 162.5051
461 165.3518 164.5163 166.1873
462 163.8427 162.9005 164.7849
463 170.8449 170.2312 171.4587
464 163.4201 162.4461 164.3941
465 165.7743 164.9664 166.5822
466 175.3723 174.6323 176.1123
467 164.2049 163.2893 165.1204
468 171.6297 171.0147 172.2446
469 163.2390 162.2512 164.2269
470 171.2675 170.6543 171.8807
471 165.1103 164.2586 165.9620
472 166.0158 165.2231 166.8085
473 162.8165 161.7958 163.8371
474 182.5556 181.2803 183.8309
475 163.1183 162.1211 164.1155
476 165.5329 164.7094 166.3564
477 170.0602 169.4380 170.6824
478 167.5249 166.8166 168.2332
479 172.4144 171.7888 173.0400
480 170.6035 169.9881 171.2189
481 163.1787 162.1861 164.1712
482 175.6137 174.8602 176.3672
483 165.7743 164.9664 166.5822
484 162.0317 160.9486 163.1149
485 163.1183 162.1211 164.1155
486 166.1969 165.4153 166.9785
487 165.3518 164.5163 166.1873
488 170.5431 169.9272 171.1590
489 165.1707 164.3231 166.0183
490 161.6092 160.4917 162.7267
491 161.9714 160.8834 163.0594
492 165.8951 165.0948 166.6953
493 161.8506 160.7528 162.9484
494 163.7219 162.7707 164.6732
495 163.8427 162.9005 164.7849
496 165.2914 164.4519 166.1309
497 168.4907 167.8248 169.1567
498 169.2151 168.5735 169.8567
499 166.3176 165.5433 167.0919
500 165.6536 164.8379 166.4693
501 170.7846 170.1705 171.3987
502 168.6115 167.9501 169.2729
503 171.2675 170.6543 171.8807
504 165.7743 164.9664 166.5822
505 161.3074 160.1650 162.4497
506 170.4828 169.8663 171.0993
507 166.4987 165.7351 167.2623
___________________________________________________________________________________________

a) equation of fitted regression line is,

hgt = 105.83246 +  0.60364 * sho.gi

b)

slope = 0.60364

intercept = 105.83246

Interpretation of slope: If the height increased by 1 ft, we predict the sho.gi increase by approximately 0.60364 cm

Interpretation of intercept- If the height is 0 ft, then we predict the sho.gi is 105.83246 cm

c)

R2 = 0.4432

R2 = 0.4432 indicates that adequacy of the fitted model is satisfactory.

d) At sholder girth of 100 cm

hgt = 105.83246 +  0.60364 * sho.gi

hgt = 105.83246 +  0.60364 * 100

hgt = 166.1965
hgt = 166 ft ________________rounded to whole number.

For remaining question see output of r code.

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