Atinder Assignment 3
Atinderpal Singh Saini
November 19, 2015
- For each of the following situations, state whether the parameter of interest is a mean or a proportion. It may be helpful to examine whether individual responses are numerical or categorical.
(1 pt) In a survey, one hundred college students are asked how many hours per week they spend on the Internet. # Mean, since the final answer would be a numerical value (number of hours).
(1 pt) In a survey, one hundred college students are asked: “What percentage of the time you spend on the Internet is part of your course work?” # Mean, since the final answer would be a numerical value as we can find the average of these percentages.
(1 pt) In a survey, one hundred college students are asked whether or not they cited information from Wikipedia in their papers. #Proportion, since the final answer would be a categorical value (Yes or No).
(1 pt) In a survey, one hundred college students are asked what percentage of their total weekly spending is on alcoholic beverages. # Mean, since the final answer would be a numerical value (percentage).
(1 pt) In a sample of one hundred recent college graduates, it is found that 85 percent expect to get a job within one year of their graduation date. # Proportion, since the final answer would be a categorical value, Yes or No, based on whether a student has got the job or not.
- (5 pt) In 2013, the Pew Research Foundation reported that “45% of U.S. adults report that they live with one or more chronic conditions”. However, this value was based on a sample, so it may not be a perfect estimate for the population parameter of interest on its own. The study reported a standard error of about 1.2%, and a normal model may reasonably be used in this setting. Create a 95% confidence interval for the proportion of U.S. adults who live with one or more chronic conditions. Also interpret the confidence interval in the context of the study.
Standar Error: 1.2%
95% Confidence Interval = point estimate (+ or -) Z * Standard Error
95% Confidence Interval = (0.45 (+ or -) 1.96 * 0.012) = ( 42.65% to 47.35% )
At a confidence interval of 95 % ,
42.65% to 47.35% of US Adults may report that they live with one or more chronic conditions
Since the point estimate reported is within the 95% confidence interval, the report is 95% of the times accurate.
- The nutrition label on a bag of potato chips says that a one ounce (28 gram) serving of potato chips has 130 calories and contains ten grams of fat, with three grams of saturated fat. A random sample of 35 bags yielded a sample mean of 136 calories with a standard deviation of 17 calories.
(4 pt) Write down the null and alternative hypotheses for a two-sided test of whether the nutrition label is lying. #Null Hypothesis : H0 :- One ounce (28 gram) serving of potato chips is equal to 130 calories # Ha :- One ounce (28 gram) serving of potato chips is not equal to 130 calories # where H0 is the null hypothesis and # Ha is the alternate hypothesis
(4 pt) Calculate the test statistic and find the p value. # Test Statistic is t = (Xbar - M0)/Standard Error # where Xbar <- Sample Mean and M0 <- Hypothesized Population Mean # Standard Error <- Standard Deviation / sqrt(Sample Size) # Therefore, t = ( 136 - 130 )/( 17 / sqrt(35) ) = 2.088 # Degree of freedom, df = (n-1) = 34, where n <- Sample Size # p value of the above found test statistic is 0.9816024
(2 pt) If you were the potato chip company would you rather have your alpha = 0.05 or 0.025 in this case? Why? # If p value is greater than significance level we cannot reject null hypothesis. # Current p value is greater than 0.025 and 0.05, thus, making the hypothesized mean correct and proving that the one ounce of potato chip has 130 calories based on hypothesized t test # Therefore, If I were the potato chip company, I would rather have my alpha = 0.05 in this case because my samples seem to be way more accurate
- Regression was originally used by Francis Galton to study the relationship between parents and children. He wondered if he could predict a man’s height based on the height of his father? This is the question we will explore in this problem. You can obtain data similar to that used by Galton as follows:
library(dplyr)##
## Attaching package: 'dplyr'
##
## The following objects are masked from 'package:stats':
##
## filter, lag
##
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(UsingR)## Loading required package: MASS
##
## Attaching package: 'MASS'
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## The following object is masked from 'package:dplyr':
##
## select
##
## Loading required package: HistData
## Loading required package: Hmisc
## Loading required package: grid
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
## Loading required package: ggplot2
##
## Attaching package: 'Hmisc'
##
## The following objects are masked from 'package:dplyr':
##
## combine, src, summarize
##
## The following objects are masked from 'package:base':
##
## format.pval, round.POSIXt, trunc.POSIXt, units
##
##
## Attaching package: 'UsingR'
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## The following object is masked from 'package:ggplot2':
##
## movies
##
## The following object is masked from 'package:survival':
##
## cancerlibrary(ggplot2)
height <- get("father.son")- (5 pt) Perform an exploratory analysis of the father and son heights. What does the relationship look like? Would a linear model be appropriate here?
Let’s look at some histogram that show the relation between father and son’s height
ggplot(father.son,aes(x=father.son$sheight))+geom_histogram(binwidth=0.5)
cor(father.son,use="complete.obs")## fheight sheight
## fheight 1.0000000 0.5013383
## sheight 0.5013383 1.0000000# Medium positive correlation based on correlationBased on the above analysis, we can find that there is Medium positive correlation between Father and Son’s height based on correlation. A linear model would be appropriate here since it looks like a normal distribution.
- (5 pt) Use the lm function in R to fit a simple linear regression model to predict son’s height as a function of father’s height. Write down the model, ysheight = β0 + β1 x fheight
filling in estimated coefficient values and interpret the coefficient estimates.
Sons Height = 33.8866 + 0.5141 x Father’s Height, using the above equation of a linear regression model.
lin <- lm(sheight~fheight,data=father.son)
summary(lin)##
## Call:
## lm(formula = sheight ~ fheight, data = father.son)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.8772 -1.5144 -0.0079 1.6285 8.9685
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.88660 1.83235 18.49 <2e-16 ***
## fheight 0.51409 0.02705 19.01 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.437 on 1076 degrees of freedom
## Multiple R-squared: 0.2513, Adjusted R-squared: 0.2506
## F-statistic: 361.2 on 1 and 1076 DF, p-value: < 2.2e-16- (5 pt) Find the 95% confidence intervals for the estimates. You may find the confint() command useful.
confint(lin,level=0.95)## 2.5 % 97.5 %
## (Intercept) 30.2912126 37.4819961
## fheight 0.4610188 0.5671673The above command prints 95% Confidence Intervals. Kindly ignore the headers at the moment.
- (5 pt) Produce a visualization of the data and the least squares regression line.
plot(father.son$fheight,father.son$sheight)
fit <- lm(father.son$sheight~father.son$fheight)
abline(fit, col = 'red')
# The above plot shows the relation between the data and the red colored least squares regression line looks positive.
- (5 pt) Produce a visualization of the residuals versus the fitted values. (You can inspect the elements of the linear model object in R using names()). Discuss what you see. Do you have any concerns about the linear model?
resid(lin)## 1 2 3 4 5
## -7.549320518 -3.189432294 -3.937262188 -4.897126876 -1.035698698
## 6 7 8 9 10
## -2.043843444 -3.410828758 -3.165011904 -3.236827219 -4.334628227
## 11 12 13 14 15
## 1.022303623 -0.888502884 -0.939312002 -1.389889738 -1.848607024
## 16 17 18 19 20
## -2.591991494 -2.989964076 -2.052904112 -3.438394247 -2.717010607
## 21 22 23 24 25
## -3.605699113 -4.121340976 0.546305037 -0.312543404 -0.875422298
## 26 27 28 29 30
## -1.312839748 -1.192957249 -1.230181614 -1.812453255 -1.615901663
## 31 32 33 34 35
## -2.375460072 -2.204042616 -1.619202118 -3.046443067 -2.648375866
## 36 37 38 39 40
## -2.626129125 -2.805758011 -3.212520458 -4.390581934 1.251267433
## 41 42 43 44 45
## 0.423048888 -0.045610592 0.017480915 -0.497696232 -0.474796414
## 46 47 48 49 50
## 1.651610444 -0.093493379 -6.433844280 -1.501642356 -0.690917366
## 51 52 53 54 55
## 0.167196487 -0.296185711 -0.663097904 -0.437084264 0.035412008
## 56 57 58 59 60
## -0.270065897 -0.778626280 -0.794462663 -0.920354365 -1.725711657
## 61 62 63 64 65
## -1.230643968 -0.620670541 -1.364184860 -1.572385589 -1.470870305
## 66 67 68 69 70
## -1.513833405 -1.745407337 -2.069684088 -2.846581324 -3.329580662
## 71 72 73 74 75
## -2.946462676 -3.640587809 1.860225243 2.281611743 1.304137765
## 76 77 78 79 80
## 1.278954303 1.282886041 0.483236276 0.464206224 0.684499731
## 81 82 83 84 85
## 0.541204255 0.635911455 -0.167036297 0.447075292 -0.375922595
## 86 87 88 89 90
## -0.216290319 0.116000909 -0.540760234 -0.514137299 -0.965763441
## 91 92 93 94 95
## -0.408536148 -1.177500783 -0.045185671 -1.231809237 -1.256862315
## 96 97 98 99 100
## -1.283594057 -0.807166782 -1.300360878 -0.583115367 -1.592844342
## 101 102 103 104 105
## -1.694391204 -2.726917798 -1.696968809 -2.808427293 -3.721415968
## 106 107 108 109 110
## 2.481860301 2.991852489 2.191823951 2.316841460 1.663529726
## 111 112 113 114 115
## 1.509323610 1.810568992 1.750030102 1.182320365 1.412419651
## 116 117 118 119 120
## 0.593792097 0.609404304 0.130282046 0.484382391 0.873493002
## 121 122 123 124 125
## 0.234369977 -0.507444670 0.256542544 0.113457481 0.050642153
## 126 127 128 129 130
## -0.456481594 -0.714303903 -1.006836579 -0.642387828 -0.821788545
## 131 132 133 134 135
## -0.904210026 -1.227948506 -1.903205886 -3.391252138 2.614688577
## 136 137 138 139 140
## 2.476442442 2.256267373 2.481986131 2.739712775 1.972221011
## 141 142 143 144 145
## 1.206614827 1.720636825 0.898841785 0.733899166 1.800290795
## 146 147 148 149 150
## 0.561337916 0.719309646 0.393519322 0.726053610 0.182004139
## 151 152 153 154 155
## 0.015308736 0.133562580 -0.366483521 -0.228146526 -0.531668437
## 156 157 158 159 160
## -0.322913060 -0.986682154 -2.575892643 4.288392671 2.811879391
## 161 162 163 164 165
## 3.854350538 2.964102656 3.377198232 2.462113473 2.147711164
## 166 167 168 169 170
## 2.500735498 1.712579696 1.766511791 1.668353743 1.576386544
## 171 172 173 174 175
## 1.530073205 1.281262567 1.039887763 1.046845499 1.162054402
## 176 177 178 179 180
## 0.026946632 -0.515032586 -0.680480615 3.857967348 3.386384510
## 181 182 183 184 185
## 3.613236253 2.639723729 3.303625096 2.120430141 2.598257226
## 186 187 188 189 190
## 2.241349711 1.826296299 0.819806704 1.033570797 4.187133519
## 191 192 193 194 195
## 4.073441414 3.622528817 3.353553128 2.995469909 2.616144304
## 196 197 198 199 200
## 2.324543094 8.003255538 5.442858656 3.865042378 3.616255354
## 201 202 203 204 205
## 2.560981725 6.751380784 7.093602167 5.901547869 -4.818473013
## 206 207 208 209 210
## 5.339813720 3.591276323 3.711810584 -4.456682821 -2.036181060
## 211 212 213 214 215
## -7.357182705 3.280337809 -4.094563645 -7.402425332 -4.546391439
## 216 217 218 219 220
## -3.547572138 -2.996731758 -3.058385489 -4.114599078 -5.963614403
## 221 222 223 224 225
## -1.937790543 -2.947703222 -2.850429872 -4.008435512 -3.586611265
## 226 227 228 229 230
## -4.981214007 -0.455548844 -1.843159299 -1.066749004 -2.356374230
## 231 232 233 234 235
## -2.339795194 -2.037973366 -2.089951647 -2.198188552 -3.484180192
## 236 237 238 239 240
## -3.311773147 -4.661443598 -0.009016003 -0.379773844 -0.055167765
## 241 242 243 244 245
## -0.966452762 -0.676800413 -1.302263927 -1.650971009 -1.416779793
## 246 247 248 249 250
## -1.019166136 -2.330428929 -1.957194664 -1.842430538 -3.005425792
## 251 252 253 254 255
## -2.799590649 -2.524552853 -3.378735919 -4.134563822 0.902462882
## 256 257 258 259 260
## 0.600731530 0.536295550 0.953143203 -0.014966588 -0.237475177
## 261 262 263 264 265
## 0.265131054 -0.228149267 -5.391354890 -2.849681116 -1.850511525
## 266 267 268 269 270
## -3.589695641 -0.102142686 -1.043803584 -0.666873710 -0.613530122
## 271 272 273 274 275
## -0.782423016 -0.765505802 -0.981290464 -1.070444318 -0.568468247
## 276 277 278 279 280
## -1.610169354 -1.765660462 -1.642620503 -1.486344337 -2.148516404
## 281 282 283 284 285
## -2.177641887 -1.785823380 -2.458158060 -2.459333375 -3.180891716
## 286 287 288 289 290
## -2.358938099 -3.344303343 1.737870221 1.529971642 2.126584122
## 291 292 293 294 295
## 1.837056693 1.339263165 1.180360089 0.674466681 0.463427763
## 296 297 298 299 300
## -0.123244601 0.742131310 -0.293302886 -0.188462701 0.335081797
## 301 302 303 304 305
## 0.145177756 -0.264314883 -0.427049864 0.084303897 -0.927216569
## 306 307 308 309 310
## -0.791968391 -1.187685475 -0.234938274 -1.272797267 -1.642547450
## 311 312 313 314 315
## -0.822138191 -0.732354563 -1.497519308 -1.531592328 -2.319124873
## 316 317 318 319 320
## -1.420630068 -1.632832247 -1.879364631 -3.047175858 -2.731998400
## 321 322 323 324 325
## 3.238853634 2.267149069 2.325138632 1.685806236 1.877961690
## 326 327 328 329 330
## 1.616795594 0.998786670 1.879586855 1.013560311 1.266195138
## 331 332 333 334 335
## 0.417645158 0.461415500 0.255616549 0.157141033 0.998830954
## 336 337 338 339 340
## -0.265556588 0.105104884 -0.294442609 -0.151975195 -0.122545542
## 341 342 343 344 345
## 0.047138344 -0.187101821 -0.329330033 -1.375334272 -1.103788678
## 346 347 348 349 350
## -0.655542638 -1.121097939 -1.017182163 -1.979390319 2.955116189
## 351 352 353 354 355
## 2.662128458 1.985806920 1.903415364 2.163431715 1.717846883
## 356 357 358 359 360
## 2.148443709 0.821629353 1.132662390 1.446649298 1.690406308
## 361 362 363 364 365
## 1.158488113 0.988784179 1.243774578 0.607180922 0.101055811
## 366 367 368 369 370
## 0.542196241 0.177075898 0.517863104 -0.564554608 0.047002321
## 371 372 373 374 375
## -0.653364255 -0.963612058 -1.659336497 4.138422805 3.552991910
## 376 377 378 379 380
## 3.349257552 2.174165598 2.560942560 2.337009107 2.182338515
## 381 382 383 384 385
## 1.917710543 1.828863893 1.772559456 1.904477732 1.488699510
## 386 387 388 389 390
## 1.486941263 1.954373022 1.111390949 0.859528685 1.197521578
## 391 392 393 394 395
## 0.358195739 0.240338195 -0.148113758 4.182020693 3.565863898
## 396 397 398 399 400
## 3.250366427 2.324158514 2.389633338 2.537895513 2.374891897
## 401 402 403 404 405
## 2.617766937 1.520946125 0.864434305 0.729090171 4.699917114
## 406 407 408 409 410
## 4.672887051 3.795810853 3.707780237 3.487485465 2.732503400
## 411 412 413 414 415
## 1.666374075 1.963024008 5.064569422 5.484583810 3.387810643
## 416 417 418 419 420
## 3.637810778 2.024863332 4.629300349 8.151747800 -4.165330595
## 421 422 423 424 425
## -4.397042931 0.092183980 -8.271382039 -1.875112825 0.074072255
## 426 427 428 429 430
## -0.294812949 2.968543040 -1.649808898 -5.637762696 1.390831221
## 431 432 433 434 435
## 0.549083220 -4.497157752 -3.497778992 -4.525255176 -5.520827464
## 436 437 438 439 440
## -1.986475485 -2.387577959 -3.302808121 -3.025704983 -3.618434519
## 441 442 443 444 445
## -4.864695754 0.821873405 -1.462888129 -0.933447502 -2.631248222
## 446 447 448 449 450
## -2.581049069 -2.088575488 -2.392377869 -2.632835083 -2.985479647
## 451 452 453 454 455
## -3.746318585 -4.570049223 0.123878967 0.017222335 -0.636215736
## 456 457 458 459 460
## -0.343478636 -1.308177593 -0.996858657 -1.356835908 -1.482939709
## 461 462 463 464 465
## -1.045068074 -2.380873782 -2.035909345 -2.405678827 -2.577727027
## 466 467 468 469 470
## -2.282927149 -2.269134787 -2.804976822 -3.263977852 1.507905714
## 471 472 473 474 475
## 1.401090723 0.220064646 -0.064974348 0.629756131 -0.350102478
## 476 477 478 479 480
## -0.248751352 -0.711485837 -3.486091137 -4.676935637 -1.760437753
## 481 482 483 484 485
## -2.470469682 -0.068363565 -0.003059954 -0.667482049 -0.834641792
## 486 487 488 489 490
## -1.015693781 -0.740362645 -0.875387261 -1.605599758 -1.349426932
## 491 492 493 494 495
## -0.864106255 -0.919440855 -0.822352004 -1.658792381 -1.976064268
## 496 497 498 499 500
## -1.658146023 -1.891878029 -2.229115654 -1.760182271 -2.455244350
## 501 502 503 504 505
## -3.012410151 -2.582018386 2.218731817 2.286712411 1.032497054
## 506 507 508 509 510
## 1.134678004 0.918782672 1.073320623 0.796418419 -0.170741632
## 511 512 513 514 515
## 0.287647361 -0.118998035 0.014531984 -0.341609600 -0.512497595
## 516 517 518 519 520
## -0.166179153 0.058683229 0.157404465 -0.070758990 -0.434102977
## 521 522 523 524 525
## -0.193019062 -0.485859072 -0.864082612 -1.463250029 -0.804839027
## 526 527 528 529 530
## -1.232652814 -1.401964163 -1.186450226 -1.064415076 -1.499661599
## 531 532 533 534 535
## -1.203505578 -2.081020787 -1.941188075 -2.456274391 -2.494140757
## 536 537 538 539 540
## 3.857451958 3.073832008 2.427127077 1.603469118 2.417727796
## 541 542 543 544 545
## 1.823501510 1.593041702 0.940236448 0.577220003 0.910136172
## 546 547 548 549 550
## 0.515233980 0.703753170 -0.233272042 0.196287715 0.100233396
## 551 552 553 554 555
## 0.476631073 -0.467795173 -0.484116104 -0.394556034 0.026199662
## 556 557 558 559 560
## 0.121192064 -0.615982756 -0.349044921 -0.111379734 -0.520127446
## 561 562 563 564 565
## -0.956518584 -0.900222426 -1.879284146 -1.571540721 3.390140321
## 566 567 568 569 570
## 2.714972594 2.529337171 2.331660578 2.185673182 2.205944965
## 571 572 573 574 575
## 1.617506217 1.841601958 0.994719840 1.925553736 1.311455510
## 576 577 578 579 580
## 0.206828320 1.042816425 0.505300846 0.523866950 0.632331719
## 581 582 583 584 585
## 0.122432249 0.605779235 0.420151602 -0.498952678 -0.565600690
## 586 587 588 589 590
## -0.006776941 -1.034715921 -0.712179528 4.139316630 3.906986317
## 591 592 593 594 595
## 3.293088672 2.426219526 2.224091815 2.731735177 2.764149030
## 596 597 598 599 600
## 1.733662782 2.318791395 2.003560969 1.544138142 1.888508198
## 601 602 603 604 605
## 1.384333399 1.614683392 1.222460389 1.958252376 0.723632483
## 606 607 608 609 610
## 0.246105987 -0.211022623 0.359560060 5.307896020 4.380018759
## 611 612 613 614 615
## 3.032073323 3.081352240 2.559210884 3.158959859 1.998324951
## 616 617 618 619 620
## 2.155955791 2.167794730 1.924072275 0.860063659 1.295236203
## 621 622 623 624 625
## 4.209837674 3.804104926 4.104828835 3.690333035 3.165760595
## 626 627 628 629 630
## 1.968800825 1.632180526 6.933304176 4.754464919 4.110180686
## 631 632 633 634 635
## 2.919973210 2.715892034 6.406283771 4.432464139 8.343688769
## 636 637 638 639 640
## -6.912291784 2.729159120 -2.864912195 -1.004644842 1.402358037
## 641 642 643 644 645
## 0.310794534 4.414340456 0.593138224 -6.197892353 -0.863888343
## 646 647 648 649 650
## -0.960672325 -3.003959566 -7.392165742 -2.912550245 -3.631370065
## 651 652 653 654 655
## -4.822340049 -2.113980471 -2.389505185 -2.863227011 -3.241792826
## 656 657 658 659 660
## -3.660770957 -3.557003416 0.347453597 -1.046381008 -1.210875925
## 661 662 663 664 665
## -1.199720032 -2.198016722 -1.753317097 -2.283677937 -2.250404110
## 666 667 668 669 670
## -3.154485655 -4.073990691 -3.496240977 1.147346305 -0.176037520
## 671 672 673 674 675
## -0.807078631 -0.557475725 -0.869483985 -0.886144096 -1.789389167
## 676 677 678 679 680
## -1.879991423 -2.216156355 -1.873581809 -2.065795216 -2.224427754
## 681 682 683 684 685
## -2.153025312 -2.756852481 -2.821668994 -2.891438726 -3.107287648
## 686 687 688 689 690
## 2.556559092 1.585146235 0.912853044 0.243258742 0.084565160
## 691 692 693 694 695
## -0.176879207 -0.582579831 0.728382928 -2.031688861 -5.673090644
## 696 697 698 699 700
## 0.128901089 -0.848534235 -0.425037232 -0.026043450 -0.461419334
## 701 702 703 704 705
## -0.666059743 -0.583233298 -0.576804909 -1.025709317 -1.727829408
## 706 707 708 709 710
## -0.656146609 -1.173511313 -1.483330169 -1.102655255 -1.984596098
## 711 712 713 714 715
## -1.925230237 -2.075224673 -2.241378133 -2.105931208 -2.162257874
## 716 717 718 719 720
## -2.301354229 -2.591485584 -2.352828690 2.679176774 2.552197385
## 721 722 723 724 725
## 1.475345605 1.081885037 1.336060999 1.320918248 -0.163937907
## 726 727 728 729 730
## 0.226689115 0.155647222 0.615597172 0.841266298 0.306471871
## 731 732 733 734 735
## -0.177464407 0.347112755 0.422040392 -0.615294736 -0.348105883
## 736 737 738 739 740
## -0.791977664 -0.518978094 -0.636249443 -0.960427270 -0.107662718
## 741 742 743 744 745
## -1.446767273 -0.847153161 -0.853009530 -1.087654005 -1.037981022
## 746 747 748 749 750
## -1.341329846 -1.545713529 -2.050437241 -1.931290112 -3.112732942
## 751 752 753 754 755
## -2.649142210 -3.309949996 2.785279007 2.263076122 1.531179258
## 756 757 758 759 760
## 1.905201983 1.764683451 1.213794428 1.666860943 1.826353542
## 761 762 763 764 765
## 1.094292047 0.642080206 0.786003233 0.293665575 0.561125983
## 766 767 768 769 770
## 0.183822293 -0.296645354 0.459255598 0.257705551 -0.463951074
## 771 772 773 774 775
## -0.687939067 0.165302297 -0.485975879 -0.702747144 -0.516705767
## 776 777 778 779 780
## -0.798849741 -1.189519749 -1.284811307 -1.137687029 -2.215477427
## 781 782 783 784 785
## 3.981979061 3.257037431 2.441659396 1.865190889 2.184233319
## 786 787 788 789 790
## 2.438748847 1.523302204 1.744129527 1.792821336 0.977441966
## 791 792 793 794 795
## 0.761255989 0.987603262 0.845677064 0.751871029 1.311344445
## 796 797 798 799 800
## 1.121127694 0.939099389 -0.076258444 0.328537243 0.054671476
## 801 802 803 804 805
## 0.280665660 0.367567953 -0.843409666 -1.183476486 4.547311529
## 806 807 808 809 810
## 3.923177995 3.928424900 2.511007480 3.132594693 2.421288524
## 811 812 813 814 815
## 2.210515640 2.619886458 2.615086752 2.144988651 1.925027600
## 816 817 818 819 820
## 2.140899245 0.995929277 1.676481318 0.899878283 1.421671863
## 821 822 823 824 825
## 0.706562463 1.010632168 0.330221258 -0.676492952 5.512939661
## 826 827 828 829 830
## 3.582462901 3.565717639 3.207917457 2.921717005 2.891910573
## 831 832 833 834 835
## 1.970314406 2.583362960 2.434336198 1.514810192 1.805870562
## 836 837 838 839 840
## 1.103459451 4.316685112 4.858847022 3.737191773 2.971788177
## 841 842 843 844 845
## 3.110645457 2.544724923 2.029458434 6.796174721 5.605938444
## 846 847 848 849 850
## 4.054330034 3.889141502 3.209280692 5.693750433 5.221715677
## 851 852 853 854 855
## 8.968478813 -8.743284614 1.540916577 0.396943250 1.471849840
## 856 857 858 859 860
## 2.905875337 -1.061552721 -6.910148818 1.510890334 2.068747073
## 861 862 863 864 865
## -5.731984395 -2.572537507 -4.664720036 -2.422006258 -3.815460108
## 866 867 868 869 870
## -4.266914447 -0.520498395 -2.221230289 -2.771972252 -2.997079420
## 871 872 873 874 875
## -2.925450034 -3.579032884 -6.245864307 -0.992296736 -1.651270933
## 876 877 878 879 880
## -1.731973307 -1.784466334 -2.309649546 -2.035579488 -2.274169372
## 881 882 883 884 885
## -3.139890119 -3.757720851 -3.203750953 -4.590130726 0.172744471
## 886 887 888 889 890
## -0.916724729 -0.730549555 -0.887759564 -0.822638186 -1.364119639
## 891 892 893 894 895
## -1.144799831 -1.514609136 -2.067856275 -1.915276036 -1.980686728
## 896 897 898 899 900
## -1.697328359 -2.642619860 -2.485659262 -3.179389457 -3.698535260
## 901 902 903 904 905
## -4.187737492 1.975566233 1.138536012 0.039401362 0.476744056
## 906 907 908 909 910
## -0.358221327 0.233489628 0.326412660 -0.385024862 -5.470119405
## 911 912 913 914 915
## -0.943585721 -3.858235770 1.721985741 0.411202107 -0.570615118
## 916 917 918 919 920
## -0.584183699 -0.618681331 -1.021739968 0.031520090 -1.290869634
## 921 922 923 924 925
## -1.464087582 -1.339519526 -0.922204781 -1.427366675 -1.125437078
## 926 927 928 929 930
## -1.689962615 -1.566518148 -1.994245576 -2.162774545 -1.965427571
## 931 932 933 934 935
## -1.932382933 -2.225195673 -2.552031012 -2.562924943 1.567945618
## 936 937 938 939 940
## 1.705210854 1.478785397 1.632961910 0.689264831 1.447444579
## 941 942 943 944 945
## 0.386428459 0.627950374 0.422352636 0.435988172 0.030293202
## 946 947 948 949 950
## 0.403164614 0.131251766 0.216064953 0.141792061 -0.575767023
## 951 952 953 954 955
## -0.894873844 -1.086366203 -0.876240723 -0.332976163 -0.722298637
## 956 957 958 959 960
## -0.940250308 -0.954278596 -1.291021065 -1.371546810 -1.005424910
## 961 962 963 964 965
## -1.193308727 -1.825408209 -1.966702622 -2.205402332 -2.280651354
## 966 967 968 969 970
## -2.668852466 -2.801679762 3.350174082 1.944998698 1.834501349
## 971 972 973 974 975
## 1.240292894 1.318071750 1.018360789 1.293638770 0.673796039
## 976 977 978 979 980
## 0.978309682 0.547953795 0.711440073 0.545759462 0.662990964
## 981 982 983 984 985
## 0.273431755 0.180373017 0.208792946 -0.016661794 0.164318768
## 986 987 988 989 990
## 0.179359096 0.199053067 0.037893666 -0.341811880 -1.120156569
## 991 992 993 994 995
## -0.837729072 -1.109617521 -0.933549710 -1.109290052 -0.735894733
## 996 997 998 999 1000
## -1.633203894 3.028308450 2.181361796 3.050111495 2.889867743
## 1001 1002 1003 1004 1005
## 2.195399495 1.535078644 2.191354836 1.104142120 1.014290782
## 1006 1007 1008 1009 1010
## 1.997966028 1.384245909 1.004036442 0.598681741 1.378897978
## 1011 1012 1013 1014 1015
## 1.086436763 0.119568553 0.163910582 0.322355679 0.554619095
## 1016 1017 1018 1019 1020
## -0.414866878 0.171365277 -0.236351870 -0.520593564 -1.634873942
## 1021 1022 1023 1024 1025
## 3.436372382 3.506197296 3.668358411 2.939776083 2.466993091
## 1026 1027 1028 1029 1030
## 2.385146036 2.790695191 2.584203554 1.583647664 1.912701017
## 1031 1032 1033 1034 1035
## 1.586134683 1.580070675 1.332671761 1.946137415 1.254843310
## 1036 1037 1038 1039 1040
## 1.309027961 0.602998826 0.414716046 -0.364875489 -1.278200379
## 1041 1042 1043 1044 1045
## 4.136000253 3.682608593 3.774236726 3.211133930 2.378770554
## 1046 1047 1048 1049 1050
## 2.131856040 2.588984847 2.049679070 1.836506945 1.220304149
## 1051 1052 1053 1054 1055
## 0.962337404 4.331060985 4.028872564 4.195720592 3.619775234
## 1056 1057 1058 1059 1060
## 2.313494450 2.811195962 2.270234930 1.997450895 5.865147140
## 1061 1062 1063 1064 1065
## 4.680763131 3.452156878 2.926232447 1.284000836 4.123250188
## 1066 1067 1068 1069 1070
## 7.400658027 -7.523332626 5.628732981 -4.251839278 -7.593037101
## 1071 1072 1073 1074 1075
## -3.658603782 -6.671128941 -8.877150660 2.423122015 -2.290051307
## 1076 1077 1078
## -1.483926919 -0.950717935 -3.015475796ggplot(lin, aes(x=residuals(lin))) + geom_density() + labs(x='Residuals',y='Density',title='Density Plot for Residuals')
# The plot seems like a nearly normal distributed. This means we can create a linear model which will be fairly accurate
- (5 pt) Using the model you fit in part (b) predict the height was 5 males whose father are 50, 55, 70, 75, and 90 inches respectively. You may find the predict() function helpful.
ggplot(father.son, aes(x=fheight, y=sheight)) +
geom_point(shape=1)+labs(x='Father Height',y='Son Height',title='Scatterplot between Father height and son height') + geom_smooth(method='lm')
new.df <- data.frame(fheight=c(50,55,70,75,90))
predict(lin,newdata=new.df)## 1 2 3 4 5
## 59.59126 62.16172 69.87312 72.44358 80.15498- (5 pt) What do the estimates of the slope and height mean? Are the results statistically significant? Are they practically significant?
- An investigator is interested in understanding the relationship, if any, between the analytical skills of young gifted children and the father’s IQ, the mother’s IQ, and hours of educational TV. The data are here: library(openintro) data(gifted)
- (5 pt) Run two regressions: one with the child’s analytical skills test score (“score”) and the father’s IQ (“fatheriq”) and the child’s score and the mother’s IQ score (“motheriq”).
library(openintro) ## Please visit openintro.org for free statistics materials
##
## Attaching package: 'openintro'
##
## The following object is masked from 'package:MASS':
##
## mammals
##
## The following object is masked from 'package:datasets':
##
## carsdata(gifted)
glimpse(gifted)## Observations: 36
## Variables: 8
## $ score (int) 159, 164, 154, 157, 156, 150, 155, 161, 163, 162, 154...
## $ fatheriq (int) 115, 117, 115, 113, 110, 113, 118, 117, 111, 122, 111...
## $ motheriq (int) 117, 113, 118, 131, 109, 109, 119, 120, 128, 120, 117...
## $ speak (int) 18, 20, 20, 12, 17, 13, 19, 18, 22, 18, 19, 20, 20, 2...
## $ count (int) 26, 37, 32, 24, 34, 28, 24, 32, 28, 27, 32, 33, 35, 2...
## $ read (dbl) 1.9, 2.5, 2.2, 1.7, 2.2, 1.9, 1.8, 2.3, 2.1, 2.1, 2.2...
## $ edutv (dbl) 3.00, 1.75, 2.75, 2.75, 2.25, 1.25, 2.00, 2.25, 1.00,...
## $ cartoons (dbl) 2.00, 3.25, 2.50, 2.25, 2.50, 3.75, 3.00, 2.50, 4.00,...score - Score in test of analytical skills. fatheriq - Father’s IQ. motheriq - Mother’s IQ. speak - Age in months when the child first said ‘mummy’ or ‘daddy’. count - Age in months when the child first counted to 10 successfully. read - Average number of hours per week the child’s mother or father reads to the child. edutv - Average number of hours per week the child watched an educational program on TV during the past three months. cartoons - Average number of hours per week the child watched cartoons on TV during the past three months.
linfather <- lm(score~fatheriq,data=gifted)
linfather##
## Call:
## lm(formula = score ~ fatheriq, data = gifted)
##
## Coefficients:
## (Intercept) fatheriq
## 130.4294 0.2501linmother <- lm(score~motheriq,data=gifted)
linmother##
## Call:
## lm(formula = score ~ motheriq, data = gifted)
##
## Coefficients:
## (Intercept) motheriq
## 111.0930 0.4066- (5 pt) What are the estimates of the slopes for father and mother’s IQ score with their 95% confidence intervals? (Note, estiamtes and confidence intervals are usually reported: Estimate (95% CI: CIlower, CIupper)
Father slope is 0.2501 (CIlower: , CIupper)
Mother slope is 0.4066 (CIlower: , CIupper)
(5 pt) How are these interpreted?
(5 pt) What conclusions can you draw about the association between the child’s score and the mother and father’s IQ? # There is less positive correlation between father’s IQ and child’s score and there is a little higher, yet less positive correlation between mother’s IQ and child’s score.