Week 9 Lab4 Homework - Linear Regression
DATA 201 - Dylan Johnson
October 31, 2024
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsIn this lab we will go over
- Linear Regression
- Review Parametric Models
- Confidence Intervals
- Polynomial Regression
Please knit your final report as a PDF file. Submit both the PDF file and your RMD file in your Google drive homework folder.
Linear regression
We begin with a simulated example, to give you an orientation to the functions and their outputs, which you will use for subsequent exercises in the labe.
To do regression of \(y\) on \(x\) as a predictor, we can call the
lm function:
# These two commands are equivalent
fit <- lm(y ~ x, data=dat)
#fit <- lm(dat$y ~ dat$x)To get detailed information about the fit, such as coefficient estimates, \(t\)-statistics and \(p\)-values:
summary(fit)##
## Call:
## lm(formula = y ~ x, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.2041 -2.1343 0.6057 2.1197 9.7063
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3445 2.0164 1.163 0.26
## x -2.0669 0.1683 -12.279 3.48e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.341 on 18 degrees of freedom
## Multiple R-squared: 0.8933, Adjusted R-squared: 0.8874
## F-statistic: 150.8 on 1 and 18 DF, p-value: 3.483e-10The coefficients estimation can be accessed:
fit$coefficients## (Intercept) x
## 2.344487 -2.066860To get the estimated intercept term:
fit$coefficients[1]## (Intercept)
## 2.344487To get the estimated slope term:
fit$coefficients[2]## x
## -2.06686Exercise 1 a. What are the values of different parameters \(\beta_0\) and \(\beta_1\)?
# beta_0: 2.344487 and beta_1: 2.06686
# beta_0 is the Intercept (The b is slope intercept form) and beta_1 is the slope (The m is slope intercept form). The slope is negative indicating a downward regression.- Locate the R-squared and F statistics in your model output. Do they indicate that this linear model is a good fit? Explain your reasoning.
The Multiple R-squared value is 0.8933, the Adjusted R-squared is 0.8874, and the F-Statistic value is 150.8 with a p value of 3.483e-10. With the R-Squared values being high and close to one and the F-Statistic being high with a corresponding p value less than 0.05, they indicate that this linear model is a good fit.
e1b Answer:
Google Trends is a
Google web tool providing equally-spaced time series data on the search
volume. You may compare different topics to discover how peoples’
interests change over time. You already code in at least one of the two
most widely-used programming languages for data science (R
and Python). Which language is more popular over time?
The following picture shows how to retrieve the Google Trends data. You may download and analyze your topics of interests as well. The dataset we obtained contains the following variables:
- week: beginning date of the week (recent 5 years)
- python: trend of the search term Data science Python
- r: trend of the search term Data science r
Read the data. You may wish to view the dataset contents with additional code.
data_science <- read.csv("data_science.csv")
# convert string to date object
data_science$week <- as.Date(data_science$week, "%Y-%m-%d")
# create a numeric column representing the time
data_science$time <- as.numeric(data_science$week)
data_science$time <- data_science$time - data_science$time[1] + 1Exercise 2 The plot in the Google Trend page looks
somewhat linear. So we will try linear model first. Note that in the
lm function for the model \(y =
\beta_0 + \beta_1x + e\), you don’t need to add the intercept
term explicitly. Fitting the model with an intercept term is the default
when you pass the formula as y ~ x. If you would like to
fit a model without an intercept(\(y =
\beta_1x + e\), ), you need the formula
y ~ x - 1.
Run a regression of the r index on the predictor
time. Get the estimate of the slope.
# Insert your code here and save the estimated slope as
# `r.slope`. Then print the r.slope.
r.slope <- lm(r~time, data=data_science)
r.slope##
## Call:
## lm(formula = r ~ time, data = data_science)
##
## Coefficients:
## (Intercept) time
## -10.03418 0.03842Exercise 3. Let’s expand the analysis to include R and python
python.lm <- lm(python ~ time, data = data_science)
r.lm <- lm(r ~ time, data = data_science)Get detailed information about the fit for both models.
summary(r.lm)##
## Call:
## lm(formula = r ~ time, data = data_science)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.348 -7.076 -0.648 5.873 40.142
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.034179 1.116414 -8.988 <2e-16 ***
## time 0.038423 0.001065 36.089 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.019 on 258 degrees of freedom
## Multiple R-squared: 0.8347, Adjusted R-squared: 0.834
## F-statistic: 1302 on 1 and 258 DF, p-value: < 2.2e-16summary(python.lm)##
## Call:
## lm(formula = python ~ time, data = data_science)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.228 -8.692 -1.255 7.857 43.781
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.051606 1.350453 -9.665 <2e-16 ***
## time 0.038785 0.001288 30.116 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.91 on 258 degrees of freedom
## Multiple R-squared: 0.7785, Adjusted R-squared: 0.7777
## F-statistic: 907 on 1 and 258 DF, p-value: < 2.2e-16Read the output of summary(r.lm).
- Which of the following formula will you use to predict the search
index of
data science rat timet.
A. -10.034179 + 0.038423 t
B. 1.116414 + 0.001065 t
C. -10.034179 + 1.116414 t
D. 0.038423 + 0.001065 t
e3a Answer: Answer A because the F-Statistic is fairly large and the R-squared values
- Create two plots. One will contain a scatterplot of the
rindex as a function oftime, with the regression line drawn through the scatterplot. The second plot will contain thepythonindex as a function oftime, with the appropriate regression line drawn through the scatterplot. Present both plots side-by-side (1 row, 2 columns). Be sure to appropriately label each plot.
plot(data_science$time, data_science$r, main = "R Index as a Function of Time", xlab = "Time", ylab = "R Index", col = "blue", pch = 1)
abline(lm(r ~ time, data = data_science), col = "red", lwd = 2)
plot(data_science$time, data_science$python, main = "Python Index as a Function of Time", xlab = "Time", ylab = "Python Index", col = "green", pch = 1)
abline(lm(python ~ time, data = data_science), col = "orange", lwd = 2)
- Locate the p-value for the intercept and the slope for both the
rmodel and thepythonmodel, using the model output from (a). What are the null hypotheses in each case and what are your conclusions?
e3c Answer: For the R model the p-value is 0.038423 and the slope is 2.2e-16 and for the Python model the p-value is 0.038785 and the slope is 2.2e-16. In each case the null hypothesis is that there is no correlation between time as R or Python. Since the P values for both are less than 0.05 and the slopes are positive and roughly fit the shape of the scatter plots we will reject the null hypothesis and say that there is a correlation between time and R/Python more specifically the time spent using that programming language and your proficiency.
- Construct the confidence interval for the intercept and the slope
for each model, using the coefficient estimates and standard deviations.
(Please copy and paste the numbers you need from the output of
summaryfunction.) Will you accept or reject the null given the confidence interval you calculated?
summary_r <- summary(r.lm)
summary_python <- summary(python.lm)
intercept_r <- summary_r$coefficients["(Intercept)", "Estimate"]
slope_r <- summary_r$coefficients["time", "Estimate"]
se_intercept_r <- summary_r$coefficients["(Intercept)", "Std. Error"]
se_slope_r <- summary_r$coefficients["time", "Std. Error"]
intercept_python <- summary_python$coefficients["(Intercept)", "Estimate"]
slope_python <- summary_python$coefficients["time", "Estimate"]
se_intercept_python <- summary_python$coefficients["(Intercept)", "Std. Error"]
se_slope_python <- summary_python$coefficients["time", "Std. Error"]
confidence_level <- 0.95
t_critical <- qt(1 - (1 - confidence_level) / 2, df = summary_r$df[2])
ci_intercept_r <- c(intercept_r - t_critical * se_intercept_r, intercept_r + t_critical * se_intercept_r)
ci_slope_r <- c(slope_r - t_critical * se_slope_r, slope_r + t_critical * se_slope_r)
ci_intercept_python <- c(intercept_python - t_critical * se_intercept_python, intercept_python + t_critical * se_intercept_python)
ci_slope_python <- c(slope_python - t_critical * se_slope_python, slope_python + t_critical * se_slope_python)
cat("Confidence Intervals for model_r:\n")## Confidence Intervals for model_r:cat("Intercept:", ci_intercept_r, "\n")## Intercept: -12.23262 -7.835736cat("Slope:", ci_slope_r, "\n\n")## Slope: 0.03632639 0.04051944cat("Confidence Intervals for model_python:\n")## Confidence Intervals for model_python:cat("Intercept:", ci_intercept_python, "\n")## Intercept: -15.71092 -10.39229cat("Slope:", ci_slope_python, "\n\n")## Slope: 0.03624938 0.04132144Since the confidence intervals are not 0 we will reject the null hypothesis which means there is a significant relationship between time and python as well as time and r.
Confidence Intervals
Exercise 4.
summary(r.lm)##
## Call:
## lm(formula = r ~ time, data = data_science)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.348 -7.076 -0.648 5.873 40.142
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -10.034179 1.116414 -8.988 <2e-16 ***
## time 0.038423 0.001065 36.089 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.019 on 258 degrees of freedom
## Multiple R-squared: 0.8347, Adjusted R-squared: 0.834
## F-statistic: 1302 on 1 and 258 DF, p-value: < 2.2e-16- Get the coefficients of the linear models fit on
data science rsearch index using functioncoef. (This is equivalent to the dollar sign plus “coefficients”)
r_coef <- coef(r.lm)
print(r_coef)## (Intercept) time
## -10.03417888 0.03842291- Get the confidence intervals of the linear models fit on
data science rsearch index using functionconfint.
r_confint <- confint(r.lm)
print(r_confint)## 2.5 % 97.5 %
## (Intercept) -12.23262190 -7.83573586
## time 0.03632639 0.04051944- Get bootstrap confidence intervals of \(\beta_0\) and \(\beta_1\). Complete the following codes and print out your results. Compare the bootstrap CI with the parametric CI. What do you observe? How to make a conclusion on the null hypothesis based on your confidence interval?
bootOnce <- function() {
boot.index <- sample(1:nrow(data_science), replace = TRUE)
y.sample <- data_science$r[boot.index]
x.sample <- data_science$time[boot.index]
if (is.null(y.sample) || is.null(x.sample) || length(y.sample) == 0 || length(x.sample) == 0) {
stop("Error: y.sample or x.sample is NULL or has no values.")
}
fit.model <- lm(y.sample ~ x.sample)
coef <- coef(fit.model)
return(coef)
}
bootOnce()## (Intercept) x.sample
## -10.24290236 0.03866879Replicate the function bootOnce 1000 times to get 1000
bootstrapped statistics.
set.seed(123)
boot.stats <- replicate(1000, bootOnce())
boot.stats## [,1] [,2] [,3] [,4] [,5]
## (Intercept) -10.02108545 -11.29019263 -8.38941060 -11.1482317 -10.17283024
## x.sample 0.03827453 0.03953005 0.03646955 0.0403691 0.03807406
## [,6] [,7] [,8] [,9] [,10]
## (Intercept) -9.39104244 -8.39042543 -10.95282771 -10.37452567 -10.14999779
## x.sample 0.03774117 0.03705863 0.03930295 0.03939158 0.03879551
## [,11] [,12] [,13] [,14] [,15]
## (Intercept) -7.54252233 -10.69940813 -9.02989518 -9.08556479 -11.90514746
## x.sample 0.03603308 0.03818519 0.03727637 0.03835258 0.03978824
## [,16] [,17] [,18] [,19] [,20]
## (Intercept) -11.22663108 -9.00647917 -8.51185420 -8.77859247 -10.53731451
## x.sample 0.03972246 0.03716788 0.03655941 0.03650342 0.03887678
## [,21] [,22] [,23] [,24] [,25]
## (Intercept) -8.40302115 -12.02766263 -10.9582235 -7.12941788 -9.33212287
## x.sample 0.03641585 0.04050755 0.0400864 0.03588521 0.03730435
## [,26] [,27] [,28] [,29] [,30]
## (Intercept) -10.12224952 -9.05226010 -9.05152581 -9.5802980 -8.97917902
## x.sample 0.03746297 0.03756405 0.03787047 0.0380922 0.03728508
## [,31] [,32] [,33] [,34] [,35]
## (Intercept) -10.39980925 -13.67939482 -9.28647814 -10.83473308 -8.51096593
## x.sample 0.03940366 0.04373999 0.03950912 0.04047855 0.03706926
## [,36] [,37] [,38] [,39] [,40]
## (Intercept) -10.59837876 -12.28873035 -10.43429861 -10.37861519 -9.91967322
## x.sample 0.03788151 0.04059562 0.03791727 0.03897576 0.03869267
## [,41] [,42] [,43] [,44] [,45]
## (Intercept) -10.29839957 -10.22356452 -11.02449312 -10.9159024 -9.55112517
## x.sample 0.03802117 0.03794552 0.04012558 0.0392903 0.03739899
## [,46] [,47] [,48] [,49] [,50]
## (Intercept) -8.1889672 -9.66307945 -8.28000952 -11.04143031 -8.85000888
## x.sample 0.0382464 0.03838179 0.03568605 0.04003715 0.03639371
## [,51] [,52] [,53] [,54] [,55]
## (Intercept) -9.55476868 -10.54412796 -11.60531813 -11.21104068 -8.00960684
## x.sample 0.03749935 0.03892733 0.04098594 0.03943718 0.03657279
## [,56] [,57] [,58] [,59] [,60]
## (Intercept) -9.96972556 -10.08957551 -9.71054920 -11.49938771 -10.47691256
## x.sample 0.03750182 0.03847896 0.03911888 0.04078256 0.03784993
## [,61] [,62] [,63] [,64] [,65]
## (Intercept) -11.35337662 -9.63986893 -10.04241963 -8.01661737 -10.46101747
## x.sample 0.03925545 0.03795028 0.03842846 0.03620157 0.03918173
## [,66] [,67] [,68] [,69] [,70]
## (Intercept) -11.54248432 -10.56264251 -9.45734492 -9.75004475 -7.96510053
## x.sample 0.03962177 0.04026023 0.03820042 0.03922781 0.03655538
## [,71] [,72] [,73] [,74] [,75]
## (Intercept) -10.40435081 -9.63109963 -8.19178646 -10.45122010 -8.76565836
## x.sample 0.03778881 0.03895613 0.03609877 0.03916005 0.03682366
## [,76] [,77] [,78] [,79] [,80]
## (Intercept) -9.23147841 -9.25215176 -9.03135264 -9.01829939 -10.63910296
## x.sample 0.03770935 0.03777571 0.03777476 0.03720226 0.03860589
## [,81] [,82] [,83] [,84] [,85]
## (Intercept) -10.97682867 -9.52193073 -8.25942924 -9.44038438 -9.88120861
## x.sample 0.03951394 0.03748007 0.03723827 0.03847785 0.03764325
## [,86] [,87] [,88] [,89] [,90]
## (Intercept) -10.73876414 -10.80197922 -8.812584 -9.64352723 -9.52262899
## x.sample 0.03878724 0.03975191 0.036683 0.03776376 0.03736944
## [,91] [,92] [,93] [,94] [,95]
## (Intercept) -10.24841268 -9.95011406 -9.07087280 -10.02921883 -8.51868704
## x.sample 0.03901906 0.03825774 0.03696603 0.03881615 0.03704565
## [,96] [,97] [,98] [,99] [,100]
## (Intercept) -8.66876388 -10.53114305 -11.54004068 -9.45033274 -11.9115095
## x.sample 0.03673863 0.03931538 0.03988059 0.03770374 0.0405483
## [,101] [,102] [,103] [,104] [,105]
## (Intercept) -12.81041679 -8.77474037 -9.65859072 -11.21055287 -12.68854403
## x.sample 0.04093322 0.03650766 0.03737065 0.03791467 0.04200735
## [,106] [,107] [,108] [,109] [,110]
## (Intercept) -9.15907059 -9.37229926 -12.44891891 -10.91859385 -11.5337271
## x.sample 0.03763137 0.03742225 0.04074459 0.03904968 0.0409947
## [,111] [,112] [,113] [,114] [,115]
## (Intercept) -9.08743459 -11.22660006 -9.98159256 -11.63519190 -14.3634491
## x.sample 0.03871611 0.03919838 0.03862204 0.03960813 0.0427051
## [,116] [,117] [,118] [,119] [,120]
## (Intercept) -10.86648174 -10.93508234 -10.36371854 -8.67579047 -9.70258546
## x.sample 0.03899075 0.03873465 0.03846665 0.03646438 0.03903636
## [,121] [,122] [,123] [,124] [,125]
## (Intercept) -10.46015944 -9.59675129 -11.22080500 -10.23763158 -10.40132015
## x.sample 0.03913996 0.03853708 0.03875779 0.03852348 0.03874946
## [,126] [,127] [,128] [,129] [,130]
## (Intercept) -10.72561276 -9.26135980 -8.21410901 -9.61150363 -10.67987648
## x.sample 0.03811989 0.03688966 0.03732382 0.03721225 0.03889808
## [,131] [,132] [,133] [,134] [,135]
## (Intercept) -8.97607354 -11.67839168 -11.29489459 -10.51295820 -10.86378636
## x.sample 0.03675076 0.03946029 0.03904815 0.03829467 0.03946262
## [,136] [,137] [,138] [,139] [,140]
## (Intercept) -9.88365698 -8.63320131 -8.62660978 -11.73828599 -9.76732337
## x.sample 0.03774585 0.03753689 0.03699966 0.04066611 0.03864601
## [,141] [,142] [,143] [,144] [,145]
## (Intercept) -8.70477852 -13.2275574 -10.08391451 -8.04418897 -10.47463074
## x.sample 0.03731025 0.0416344 0.03772177 0.03603405 0.03868195
## [,146] [,147] [,148] [,149] [,150]
## (Intercept) -10.57146554 -10.03809765 -8.80670439 -11.20187916 -10.39396258
## x.sample 0.03928802 0.03833044 0.03763124 0.03928196 0.03802625
## [,151] [,152] [,153] [,154] [,155]
## (Intercept) -8.99281082 -9.21304761 -10.414492 -11.06048533 -10.28029392
## x.sample 0.03592365 0.03743215 0.040035 0.03905121 0.03765418
## [,156] [,157] [,158] [,159] [,160]
## (Intercept) -10.31208063 -10.31937854 -9.34147789 -9.26123805 -10.74262535
## x.sample 0.03854769 0.03907861 0.03895466 0.03805119 0.03810978
## [,161] [,162] [,163] [,164] [,165]
## (Intercept) -11.6959709 -10.46797336 -9.35948736 -11.25836152 -11.60695238
## x.sample 0.0399744 0.03877529 0.03878713 0.03899382 0.03997486
## [,166] [,167] [,168] [,169] [,170]
## (Intercept) -11.50631861 -10.08686291 -8.91262419 -9.43863651 -11.92633410
## x.sample 0.04025949 0.03797948 0.03844404 0.03789807 0.04014183
## [,171] [,172] [,173] [,174] [,175]
## (Intercept) -8.7111753 -10.83523459 -9.98175590 -9.07161336 -9.06346972
## x.sample 0.0366561 0.03910248 0.03773978 0.03736937 0.03678466
## [,176] [,177] [,178] [,179] [,180]
## (Intercept) -10.43443330 -11.28189203 -8.11782290 -11.82475606 -8.27634673
## x.sample 0.03920696 0.04010871 0.03646063 0.04001233 0.03710143
## [,181] [,182] [,183] [,184] [,185]
## (Intercept) -9.51733499 -8.71460954 -10.43974764 -10.7032796 -10.08556361
## x.sample 0.03801281 0.03728208 0.03861807 0.0396463 0.03744301
## [,186] [,187] [,188] [,189] [,190]
## (Intercept) -8.50603474 -8.87749296 -10.35009044 -12.33273084 -11.01722879
## x.sample 0.03763627 0.03778038 0.03864999 0.04009506 0.03987845
## [,191] [,192] [,193] [,194] [,195]
## (Intercept) -10.7448741 -11.81002899 -10.63835341 -9.37550699 -10.23512650
## x.sample 0.0396112 0.04116682 0.03832373 0.03824323 0.03880492
## [,196] [,197] [,198] [,199] [,200]
## (Intercept) -9.68572718 -9.82569630 -7.19062748 -10.55578223 -10.28337137
## x.sample 0.03847887 0.03857169 0.03569126 0.03783307 0.03776626
## [,201] [,202] [,203] [,204] [,205]
## (Intercept) -9.65242583 -8.94605879 -9.61268684 -10.60222198 -10.19632051
## x.sample 0.03811199 0.03870498 0.03820926 0.03896544 0.03766529
## [,206] [,207] [,208] [,209] [,210]
## (Intercept) -9.1480748 -9.82121210 -10.06428952 -11.02025562 -8.48025707
## x.sample 0.0371777 0.03785733 0.03765947 0.04007808 0.03736521
## [,211] [,212] [,213] [,214] [,215]
## (Intercept) -11.112027 -11.44189754 -10.01939197 -9.04769864 -11.28973687
## x.sample 0.040246 0.03997929 0.03881486 0.03761502 0.04005887
## [,216] [,217] [,218] [,219] [,220]
## (Intercept) -9.55432451 -9.37505106 -9.0581988 -9.08572536 -10.29793701
## x.sample 0.03685657 0.03930385 0.0371795 0.03750369 0.03856109
## [,221] [,222] [,223] [,224] [,225]
## (Intercept) -8.66698866 -9.23928953 -12.01457925 -10.43234762 -7.9512687
## x.sample 0.03605126 0.03820056 0.04061878 0.03888102 0.0361578
## [,226] [,227] [,228] [,229] [,230]
## (Intercept) -10.80551072 -10.65693823 -9.87904539 -8.28995733 -8.95149628
## x.sample 0.03961342 0.03945988 0.03814204 0.03492449 0.03723929
## [,231] [,232] [,233] [,234] [,235]
## (Intercept) -9.5860005 -12.532950 -10.29722280 -12.23879281 -8.43911013
## x.sample 0.0375625 0.039964 0.03833091 0.04164969 0.03594094
## [,236] [,237] [,238] [,239] [,240]
## (Intercept) -10.58666679 -9.59067900 -11.33970366 -10.38363967 -8.22959051
## x.sample 0.03903405 0.03836814 0.03998053 0.03914522 0.03684451
## [,241] [,242] [,243] [,244] [,245]
## (Intercept) -7.69600553 -10.10623784 -9.9382081 -8.77692921 -10.27745615
## x.sample 0.03530975 0.03724921 0.0381038 0.03868724 0.03957054
## [,246] [,247] [,248] [,249] [,250]
## (Intercept) -11.28322173 -9.10846452 -12.84035876 -8.64065917 -11.2946131
## x.sample 0.04013664 0.03792081 0.04050427 0.03661296 0.0390589
## [,251] [,252] [,253] [,254] [,255]
## (Intercept) -8.96856305 -9.22748975 -9.59744092 -9.0186461 -9.35027123
## x.sample 0.03773884 0.03757938 0.03851437 0.0371457 0.03810424
## [,256] [,257] [,258] [,259] [,260]
## (Intercept) -7.8754379 -8.77284195 -9.86244485 -9.88024084 -9.04530013
## x.sample 0.0370368 0.03784188 0.03760813 0.03869455 0.03708203
## [,261] [,262] [,263] [,264] [,265]
## (Intercept) -9.69623640 -10.96572488 -8.35990266 -9.31761778 -9.45241230
## x.sample 0.03855271 0.03849385 0.03711012 0.03798105 0.03830771
## [,266] [,267] [,268] [,269] [,270]
## (Intercept) -10.15643885 -10.20525393 -9.95528795 -8.58493515 -9.29185061
## x.sample 0.03784719 0.03885069 0.03864326 0.03809711 0.03838706
## [,271] [,272] [,273] [,274] [,275]
## (Intercept) -9.76713376 -9.07482294 -11.43359587 -9.14866978 -7.77690023
## x.sample 0.03870165 0.03742597 0.03981862 0.03752127 0.03598963
## [,276] [,277] [,278] [,279] [,280]
## (Intercept) -10.02777714 -9.05688746 -8.17005792 -8.49116816 -10.27766703
## x.sample 0.03808091 0.03759218 0.03799833 0.03759342 0.03873496
## [,281] [,282] [,283] [,284] [,285]
## (Intercept) -8.95077767 -10.21388063 -11.64319150 -8.4028160 -10.31639695
## x.sample 0.03753221 0.03908214 0.03905751 0.0362804 0.04019808
## [,286] [,287] [,288] [,289] [,290]
## (Intercept) -9.98454007 -7.97076041 -11.81571714 -9.79911940 -10.63150919
## x.sample 0.03794497 0.03563034 0.04054022 0.03818271 0.03885449
## [,291] [,292] [,293] [,294] [,295]
## (Intercept) -10.52891133 -8.05180566 -9.52122335 -9.6311493 -10.95002214
## x.sample 0.03823742 0.03678627 0.03799287 0.0389684 0.04037781
## [,296] [,297] [,298] [,299] [,300]
## (Intercept) -9.93482124 -11.1116503 -8.65375595 -9.3602253 -10.08833419
## x.sample 0.03823793 0.0403845 0.03828536 0.0374725 0.03787018
## [,301] [,302] [,303] [,304] [,305]
## (Intercept) -9.35744813 -11.74563553 -9.3652721 -10.5611873 -9.65884020
## x.sample 0.03775424 0.03976137 0.0373441 0.0381117 0.03855448
## [,306] [,307] [,308] [,309] [,310]
## (Intercept) -10.40332345 -10.71959173 -9.40893140 -9.82124921 -9.15253607
## x.sample 0.03897498 0.03772416 0.03806305 0.03791537 0.03721417
## [,311] [,312] [,313] [,314] [,315]
## (Intercept) -11.4929413 -9.74128588 -11.20873951 -10.27192397 -10.04080774
## x.sample 0.0402759 0.03699373 0.03971901 0.03782939 0.03844792
## [,316] [,317] [,318] [,319] [,320]
## (Intercept) -10.34370771 -9.33649860 -9.96169164 -10.5418386 -10.81029544
## x.sample 0.03808314 0.03741953 0.03894867 0.0392403 0.03971357
## [,321] [,322] [,323] [,324] [,325]
## (Intercept) -7.65104316 -9.45902983 -8.61578693 -10.47672004 -10.19484137
## x.sample 0.03648428 0.03831942 0.03665268 0.03782628 0.03937934
## [,326] [,327] [,328] [,329] [,330]
## (Intercept) -12.91204173 -9.65113089 -10.40333206 -11.64214556 -8.45087238
## x.sample 0.04128255 0.03847144 0.03842321 0.03920917 0.03622707
## [,331] [,332] [,333] [,334] [,335]
## (Intercept) -9.91942612 -10.3060528 -10.46053821 -10.000417 -10.70872259
## x.sample 0.03713913 0.0390338 0.03867795 0.038122 0.03828913
## [,336] [,337] [,338] [,339] [,340]
## (Intercept) -10.77001120 -10.17417263 -9.48140627 -8.68054412 -11.11135099
## x.sample 0.03940178 0.03824428 0.03801445 0.03727173 0.03978096
## [,341] [,342] [,343] [,344] [,345]
## (Intercept) -9.14541155 -8.73806781 -9.10841121 -9.88153511 -9.78679031
## x.sample 0.03737911 0.03672852 0.03864812 0.03755174 0.03884751
## [,346] [,347] [,348] [,349] [,350]
## (Intercept) -10.41668120 -9.79451292 -11.397797 -9.26674233 -10.67566584
## x.sample 0.04007531 0.03774413 0.039838 0.03781755 0.03883244
## [,351] [,352] [,353] [,354] [,355]
## (Intercept) -9.46862155 -11.34997827 -11.13782370 -8.92018483 -11.89854724
## x.sample 0.03781059 0.04001072 0.03914967 0.03592963 0.04014518
## [,356] [,357] [,358] [,359] [,360]
## (Intercept) -9.91630090 -10.68083047 -8.92279451 -10.57150783 -8.37285366
## x.sample 0.03863178 0.03899501 0.03779705 0.03931221 0.03587715
## [,361] [,362] [,363] [,364] [,365]
## (Intercept) -10.91449748 -9.54574197 -13.02994472 -10.24969264 -11.1655758
## x.sample 0.03926871 0.03775752 0.04172335 0.03777065 0.0394131
## [,366] [,367] [,368] [,369] [,370]
## (Intercept) -10.68730454 -11.1318820 -9.33473599 -10.15061471 -10.54695153
## x.sample 0.03957578 0.0386215 0.03718373 0.03783199 0.03877732
## [,371] [,372] [,373] [,374] [,375]
## (Intercept) -10.46660731 -11.17750417 -9.8268736 -10.20468091 -9.12853606
## x.sample 0.03860974 0.03960928 0.0380997 0.03861743 0.03679655
## [,376] [,377] [,378] [,379] [,380]
## (Intercept) -10.93843289 -12.38773175 -9.90255268 -10.35079603 -10.57896269
## x.sample 0.03880284 0.04065144 0.03902357 0.03779628 0.03979491
## [,381] [,382] [,383] [,384] [,385]
## (Intercept) -11.14244949 -10.13203125 -10.79304595 -10.27866012 -11.89430461
## x.sample 0.04025321 0.04022374 0.03917986 0.03894804 0.04085942
## [,386] [,387] [,388] [,389] [,390]
## (Intercept) -11.68571434 -9.36203054 -8.98179036 -9.79146810 -10.2160741
## x.sample 0.03884908 0.03722484 0.03649956 0.03756326 0.0386312
## [,391] [,392] [,393] [,394] [,395]
## (Intercept) -9.29233082 -10.42238657 -10.60260390 -10.99810740 -12.5683668
## x.sample 0.03772586 0.03825361 0.03862099 0.04008688 0.0403459
## [,396] [,397] [,398] [,399] [,400]
## (Intercept) -10.39968780 -10.8503451 -10.12535043 -9.33274733 -11.05353985
## x.sample 0.03884441 0.0396581 0.03933214 0.03536145 0.03823189
## [,401] [,402] [,403] [,404] [,405]
## (Intercept) -9.56686507 -10.48367749 -10.04931378 -9.43705341 -9.76398859
## x.sample 0.03834456 0.03858071 0.03797313 0.03711998 0.03836356
## [,406] [,407] [,408] [,409] [,410]
## (Intercept) -9.62798988 -10.43182551 -10.60249261 -8.85540722 -9.78059731
## x.sample 0.03822204 0.03878773 0.03775592 0.03796518 0.03710004
## [,411] [,412] [,413] [,414] [,415]
## (Intercept) -10.0283041 -11.10876503 -8.32283629 -9.16922798 -10.25516699
## x.sample 0.0386923 0.04067242 0.03647169 0.03755501 0.03886941
## [,416] [,417] [,418] [,419] [,420]
## (Intercept) -10.88516871 -11.71240807 -8.27773458 -10.10964537 -8.46615377
## x.sample 0.03908868 0.03953858 0.03749486 0.03869619 0.03619088
## [,421] [,422] [,423] [,424] [,425]
## (Intercept) -11.64385740 -11.71026454 -11.97827467 -10.64370016 -8.87771754
## x.sample 0.03951182 0.04046802 0.04055037 0.03922637 0.03760899
## [,426] [,427] [,428] [,429] [,430]
## (Intercept) -7.431230 -13.94052617 -8.19170701 -10.09737820 -8.43020183
## x.sample 0.035777 0.04176583 0.03569963 0.03883811 0.03692192
## [,431] [,432] [,433] [,434] [,435]
## (Intercept) -8.91697257 -8.997011 -10.56224676 -10.61262597 -9.60577275
## x.sample 0.03639184 0.037308 0.03912265 0.03912073 0.03798526
## [,436] [,437] [,438] [,439] [,440]
## (Intercept) -8.53132032 -9.89173262 -9.62950843 -9.7980440 -11.29687193
## x.sample 0.03649663 0.03781898 0.03754842 0.0392223 0.04000028
## [,441] [,442] [,443] [,444] [,445]
## (Intercept) -9.53539918 -9.40213437 -9.96352591 -11.54190238 -7.13549322
## x.sample 0.03870838 0.03817683 0.03759827 0.03956167 0.03645167
## [,446] [,447] [,448] [,449] [,450]
## (Intercept) -10.53276493 -9.55495403 -8.51707465 -12.21201521 -10.02603145
## x.sample 0.03780608 0.03855039 0.03750526 0.04091401 0.03894641
## [,451] [,452] [,453] [,454] [,455]
## (Intercept) -9.09391436 -12.24679523 -8.85124550 -9.26313489 -9.50650251
## x.sample 0.03744637 0.04121995 0.03583319 0.03736668 0.03779831
## [,456] [,457] [,458] [,459] [,460]
## (Intercept) -11.38822967 -9.42661140 -12.45635090 -9.05658109 -8.78387104
## x.sample 0.03898729 0.03821715 0.04116803 0.03808103 0.03751767
## [,461] [,462] [,463] [,464] [,465]
## (Intercept) -11.013176 -9.35777621 -10.26692088 -10.42370264 -11.30153160
## x.sample 0.039808 0.03771802 0.03877022 0.03871482 0.03992795
## [,466] [,467] [,468] [,469] [,470]
## (Intercept) -10.3840820 -8.23381208 -10.33741358 -9.37291261 -9.55340507
## x.sample 0.0387252 0.03623594 0.03969548 0.03743895 0.03685438
## [,471] [,472] [,473] [,474] [,475]
## (Intercept) -10.30780719 -10.14309460 -12.47555697 -10.14129725 -9.33653817
## x.sample 0.03831982 0.03872355 0.03955822 0.03917501 0.03718146
## [,476] [,477] [,478] [,479] [,480]
## (Intercept) -10.2266768 -10.88418691 -9.7268591 -8.51992622 -11.25429318
## x.sample 0.0378184 0.03956836 0.0387558 0.03629624 0.03881255
## [,481] [,482] [,483] [,484] [,485]
## (Intercept) -10.03681583 -11.22163658 -11.92437891 -10.78699242 -12.15315392
## x.sample 0.03834286 0.03935247 0.04030157 0.03996148 0.04123564
## [,486] [,487] [,488] [,489] [,490]
## (Intercept) -8.30344788 -9.05254511 -11.52108102 -10.22262508 -9.44143785
## x.sample 0.03664301 0.03655528 0.03949223 0.03937339 0.03792715
## [,491] [,492] [,493] [,494] [,495]
## (Intercept) -11.05274803 -11.88089906 -9.5907736 -7.73853280 -8.45521705
## x.sample 0.04015853 0.03991594 0.0379465 0.03636533 0.03800849
## [,496] [,497] [,498] [,499] [,500]
## (Intercept) -6.79394109 -8.51630565 -10.61813397 -9.27304180 -9.18229152
## x.sample 0.03576686 0.03721745 0.03783971 0.03733388 0.03680209
## [,501] [,502] [,503] [,504] [,505]
## (Intercept) -8.72680769 -9.63689471 -9.78933746 -7.21638562 -9.75125096
## x.sample 0.03628286 0.03758297 0.03933132 0.03577997 0.03802011
## [,506] [,507] [,508] [,509] [,510]
## (Intercept) -11.04532215 -9.7936076 -10.43934301 -9.9854266 -9.68583066
## x.sample 0.03983591 0.0373031 0.03780225 0.0389279 0.03786379
## [,511] [,512] [,513] [,514] [,515]
## (Intercept) -11.61103684 -8.70514683 -8.66814751 -9.25650534 -9.18143402
## x.sample 0.03933964 0.03706052 0.03618709 0.03811993 0.03772389
## [,516] [,517] [,518] [,519] [,520]
## (Intercept) -12.43802878 -11.15908135 -8.33017785 -11.85659052 -8.89352252
## x.sample 0.04041524 0.03875451 0.03637847 0.03977082 0.03707149
## [,521] [,522] [,523] [,524] [,525]
## (Intercept) -10.86528319 -11.28435481 -11.46566734 -9.7414237 -10.73537586
## x.sample 0.03894485 0.03971064 0.03884697 0.0374677 0.03923873
## [,526] [,527] [,528] [,529] [,530]
## (Intercept) -9.63455783 -9.97405051 -9.43893592 -9.70735003 -9.59563364
## x.sample 0.03752832 0.03824975 0.03787246 0.03759757 0.03928976
## [,531] [,532] [,533] [,534] [,535]
## (Intercept) -10.27807016 -10.07411096 -10.06869753 -11.60561400 -10.41292623
## x.sample 0.03904625 0.03873645 0.03826882 0.04070181 0.03860844
## [,536] [,537] [,538] [,539] [,540]
## (Intercept) -8.22581011 -8.73329974 -9.74204051 -8.902277 -11.2923700
## x.sample 0.03691937 0.03798749 0.03761287 0.037063 0.0396806
## [,541] [,542] [,543] [,544] [,545]
## (Intercept) -8.59090026 -9.00357471 -10.42192375 -10.30784481 -12.01694450
## x.sample 0.03735767 0.03835146 0.04010163 0.03969228 0.03960084
## [,546] [,547] [,548] [,549] [,550]
## (Intercept) -9.35768885 -9.99548192 -9.89489925 -9.07395632 -10.18995367
## x.sample 0.03786688 0.03861585 0.03870892 0.03830024 0.03731948
## [,551] [,552] [,553] [,554] [,555]
## (Intercept) -10.56226514 -10.20669090 -10.86213151 -9.43030043 -9.67388483
## x.sample 0.03871398 0.03923185 0.03969419 0.03846055 0.03845166
## [,556] [,557] [,558] [,559] [,560]
## (Intercept) -10.49803962 -9.20857683 -8.85132623 -10.04130813 -9.57076302
## x.sample 0.03910275 0.03842854 0.03761823 0.03868214 0.03761795
## [,561] [,562] [,563] [,564] [,565]
## (Intercept) -9.72862539 -9.96524873 -11.41407359 -8.71487854 -9.64826282
## x.sample 0.03792845 0.03748163 0.04005067 0.03744278 0.03684436
## [,566] [,567] [,568] [,569] [,570]
## (Intercept) -9.4373079 -10.11817912 -9.53539194 -9.20921762 -12.2436238
## x.sample 0.0377538 0.03820053 0.03791152 0.03829853 0.0410206
## [,571] [,572] [,573] [,574] [,575]
## (Intercept) -9.73586255 -8.97173382 -10.26923604 -9.53914997 -8.49223515
## x.sample 0.03791085 0.03772334 0.03866161 0.03684985 0.03665715
## [,576] [,577] [,578] [,579] [,580]
## (Intercept) -9.86559091 -8.26642999 -8.62980952 -8.98486452 -8.68191320
## x.sample 0.03698607 0.03582941 0.03706255 0.03735679 0.03649509
## [,581] [,582] [,583] [,584] [,585]
## (Intercept) -9.93693339 -8.85494787 -10.1655595 -10.38590366 -11.40578432
## x.sample 0.03882703 0.03704533 0.0380627 0.03865532 0.03901694
## [,586] [,587] [,588] [,589] [,590]
## (Intercept) -9.7787881 -9.80949647 -9.63291556 -10.58388386 -9.6147774
## x.sample 0.0385622 0.03815217 0.03747123 0.03907436 0.0380554
## [,591] [,592] [,593] [,594] [,595]
## (Intercept) -8.44614397 -9.92906692 -8.98870575 -11.96340047 -9.13085157
## x.sample 0.03634265 0.03902394 0.03710899 0.04041594 0.03683661
## [,596] [,597] [,598] [,599] [,600]
## (Intercept) -9.87894778 -8.19718097 -9.75877774 -9.0588197 -9.47248371
## x.sample 0.03777465 0.03623293 0.03892885 0.0382309 0.03666814
## [,601] [,602] [,603] [,604] [,605]
## (Intercept) -9.46285313 -11.55457891 -11.20463313 -9.40959497 -10.7071872
## x.sample 0.03851549 0.04012316 0.04013944 0.03723802 0.0394841
## [,606] [,607] [,608] [,609] [,610]
## (Intercept) -8.47244955 -11.01903606 -12.13538142 -7.77235090 -8.24737785
## x.sample 0.03664516 0.04065269 0.03979401 0.03577784 0.03699839
## [,611] [,612] [,613] [,614] [,615]
## (Intercept) -10.62928072 -9.37324777 -9.13764707 -9.65279888 -9.35887780
## x.sample 0.03931866 0.03798042 0.03732897 0.03831217 0.03750897
## [,616] [,617] [,618] [,619] [,620]
## (Intercept) -10.16261965 -8.8224562 -10.5701425 -10.67948945 -11.81942184
## x.sample 0.03785733 0.0372938 0.0387711 0.03850157 0.04002906
## [,621] [,622] [,623] [,624] [,625]
## (Intercept) -9.76165557 -12.64511485 -10.48633172 -10.23890314 -10.54659153
## x.sample 0.03825864 0.04033484 0.03820304 0.03895334 0.03819199
## [,626] [,627] [,628] [,629] [,630]
## (Intercept) -10.2630225 -11.83740714 -10.63504547 -9.57077268 -7.45618157
## x.sample 0.0382189 0.03996448 0.03873826 0.03778587 0.03692324
## [,631] [,632] [,633] [,634] [,635]
## (Intercept) -9.91082311 -8.48837889 -7.59187289 -10.37301897 -9.62598221
## x.sample 0.03822938 0.03669771 0.03576611 0.03848771 0.03770273
## [,636] [,637] [,638] [,639] [,640]
## (Intercept) -10.18025179 -11.49133025 -8.33053439 -8.69903344 -11.85745137
## x.sample 0.03803448 0.04047357 0.03633372 0.03704039 0.04003488
## [,641] [,642] [,643] [,644] [,645]
## (Intercept) -9.9734563 -9.14523520 -8.57862327 -10.2614656 -9.06102278
## x.sample 0.0387469 0.03702681 0.03692212 0.0384828 0.03784683
## [,646] [,647] [,648] [,649] [,650]
## (Intercept) -12.17936770 -11.08747827 -11.79265933 -11.641349 -9.98510996
## x.sample 0.04016185 0.03779539 0.03991385 0.040394 0.03874241
## [,651] [,652] [,653] [,654] [,655]
## (Intercept) -8.65761526 -9.73718430 -10.87271310 -8.38506038 -9.83396086
## x.sample 0.03738945 0.03825885 0.04014826 0.03655832 0.03763171
## [,656] [,657] [,658] [,659] [,660]
## (Intercept) -9.70286714 -9.80571360 -11.00708661 -10.36734846 -10.26659721
## x.sample 0.03719357 0.03855578 0.03976743 0.03811819 0.04024592
## [,661] [,662] [,663] [,664] [,665]
## (Intercept) -10.82916882 -9.84199235 -12.02916814 -12.21546060 -12.29458615
## x.sample 0.03826431 0.03738742 0.04072738 0.04066144 0.04086226
## [,666] [,667] [,668] [,669] [,670]
## (Intercept) -10.12375367 -7.34776919 -9.770366 -9.51586324 -9.00145401
## x.sample 0.03900021 0.03482534 0.037548 0.03749537 0.03650621
## [,671] [,672] [,673] [,674] [,675]
## (Intercept) -12.54335556 -8.22651932 -10.29310335 -10.48854320 -9.48138546
## x.sample 0.03989172 0.03702825 0.03910698 0.03899683 0.03784524
## [,676] [,677] [,678] [,679] [,680]
## (Intercept) -10.79487254 -10.72337540 -9.60362720 -11.42387236 -9.04215083
## x.sample 0.03787993 0.03851129 0.03811109 0.04043697 0.03768045
## [,681] [,682] [,683] [,684] [,685]
## (Intercept) -9.91466932 -9.63042391 -11.91840472 -8.43518024 -11.19339402
## x.sample 0.03812701 0.03872669 0.03985581 0.03661779 0.03895941
## [,686] [,687] [,688] [,689] [,690]
## (Intercept) -10.40560271 -8.96667664 -10.10680266 -8.83250587 -9.81636164
## x.sample 0.03899295 0.03595911 0.03802561 0.03681043 0.03812351
## [,691] [,692] [,693] [,694] [,695]
## (Intercept) -9.61908324 -9.66884840 -10.49939754 -10.97517127 -9.76754310
## x.sample 0.03870834 0.03804342 0.03918083 0.03976099 0.03841659
## [,696] [,697] [,698] [,699] [,700]
## (Intercept) -10.21554500 -9.16415228 -8.75794530 -9.9844441 -11.58689755
## x.sample 0.03866245 0.03724865 0.03736275 0.0391717 0.03994568
## [,701] [,702] [,703] [,704] [,705]
## (Intercept) -9.28401797 -9.47889848 -9.5601394 -9.61678624 -10.95827908
## x.sample 0.03761926 0.03736136 0.0371519 0.03835075 0.03850595
## [,706] [,707] [,708] [,709] [,710]
## (Intercept) -10.73752593 -8.76627955 -11.60607473 -10.57122554 -10.08308025
## x.sample 0.03898671 0.03696591 0.04002247 0.03931252 0.03821543
## [,711] [,712] [,713] [,714] [,715]
## (Intercept) -11.48710887 -12.02067721 -10.88426799 -9.10256709 -10.63264052
## x.sample 0.03940133 0.04089176 0.03913062 0.03695821 0.03857232
## [,716] [,717] [,718] [,719] [,720]
## (Intercept) -8.34750031 -10.76269650 -10.37010511 -11.8989870 -9.61533279
## x.sample 0.03708714 0.03895283 0.03880591 0.0402032 0.03771611
## [,721] [,722] [,723] [,724] [,725]
## (Intercept) -10.24968445 -8.18347912 -9.84752324 -9.42707612 -9.09790022
## x.sample 0.03859227 0.03682177 0.03883374 0.03760156 0.03654624
## [,726] [,727] [,728] [,729] [,730]
## (Intercept) -10.0824580 -10.22436399 -10.44148988 -10.11467650 -8.91917056
## x.sample 0.0381232 0.03857166 0.03928314 0.03792466 0.03837842
## [,731] [,732] [,733] [,734] [,735]
## (Intercept) -8.85403174 -9.63948732 -12.45418039 -9.21435877 -9.65248733
## x.sample 0.03815515 0.03771867 0.04072569 0.03741489 0.03746682
## [,736] [,737] [,738] [,739] [,740]
## (Intercept) -9.07372091 -10.3290940 -9.08643192 -9.95719290 -8.5315799
## x.sample 0.03677348 0.0390799 0.03747667 0.03831914 0.0365718
## [,741] [,742] [,743] [,744] [,745]
## (Intercept) -9.1612230 -10.21998203 -8.85818391 -12.31542070 -9.92688600
## x.sample 0.0355457 0.03950014 0.03691108 0.04075822 0.03814994
## [,746] [,747] [,748] [,749] [,750]
## (Intercept) -9.73976005 -10.32306300 -10.5814398 -9.08977127 -9.11423741
## x.sample 0.03775343 0.03945699 0.0395386 0.03758916 0.03815406
## [,751] [,752] [,753] [,754] [,755]
## (Intercept) -10.77289368 -8.75638432 -13.07107653 -12.412283 -10.81991068
## x.sample 0.03953666 0.03778033 0.04114252 0.040798 0.04107788
## [,756] [,757] [,758] [,759] [,760]
## (Intercept) -11.88066366 -11.48025685 -10.58114929 -10.65185890 -10.84407985
## x.sample 0.03998563 0.04010707 0.03945013 0.03881619 0.03995323
## [,761] [,762] [,763] [,764] [,765]
## (Intercept) -10.23791874 -9.96188194 -9.26652147 -9.84510771 -10.52469447
## x.sample 0.03801653 0.03820305 0.03689652 0.03788234 0.03885717
## [,766] [,767] [,768] [,769] [,770]
## (Intercept) -10.89507007 -9.57443880 -8.5225976 -9.50802149 -8.84835900
## x.sample 0.04035022 0.03845696 0.0362374 0.03762742 0.03713859
## [,771] [,772] [,773] [,774] [,775]
## (Intercept) -10.37627120 -10.30543651 -12.1384682 -10.69141532 -10.44013648
## x.sample 0.03822373 0.03848126 0.0396244 0.04028108 0.03798476
## [,776] [,777] [,778] [,779] [,780]
## (Intercept) -10.37885484 -11.88443251 -10.61099601 -11.55127713 -8.23658894
## x.sample 0.03897207 0.04034181 0.03865981 0.03890412 0.03576168
## [,781] [,782] [,783] [,784] [,785]
## (Intercept) -9.85682666 -11.60274912 -10.28698341 -8.36288316 -11.63494414
## x.sample 0.03806777 0.03954974 0.03897825 0.03749659 0.03968055
## [,786] [,787] [,788] [,789] [,790]
## (Intercept) -10.52255735 -8.26895458 -9.0265209 -10.84278581 -8.96661059
## x.sample 0.03825296 0.03775623 0.0380608 0.03978058 0.03711046
## [,791] [,792] [,793] [,794] [,795]
## (Intercept) -9.74293722 -11.16438282 -8.08597945 -11.68438560 -8.75120744
## x.sample 0.03749541 0.04058593 0.03622114 0.04014578 0.03800533
## [,796] [,797] [,798] [,799] [,800]
## (Intercept) -10.24635988 -10.65753628 -11.44126632 -9.49602482 -11.48662494
## x.sample 0.03943061 0.03900928 0.03955173 0.03730155 0.04011634
## [,801] [,802] [,803] [,804] [,805]
## (Intercept) -9.13244459 -12.53322624 -10.39857825 -10.95543941 -9.74048132
## x.sample 0.03734256 0.04024616 0.03807559 0.04031017 0.03965618
## [,806] [,807] [,808] [,809] [,810]
## (Intercept) -8.79935512 -9.95290279 -12.65893095 -9.79534720 -11.63091404
## x.sample 0.03643312 0.03863752 0.04025136 0.03799998 0.03928121
## [,811] [,812] [,813] [,814] [,815]
## (Intercept) -10.28845801 -11.53759993 -11.93047123 -10.25051389 -11.53083875
## x.sample 0.03820514 0.03966974 0.04045162 0.03800349 0.03964385
## [,816] [,817] [,818] [,819] [,820]
## (Intercept) -12.03526518 -8.72103398 -11.30587317 -9.54887225 -12.97646220
## x.sample 0.04065644 0.03667752 0.03959166 0.03815319 0.04169882
## [,821] [,822] [,823] [,824] [,825]
## (Intercept) -8.29063119 -9.66008148 -8.7692535 -9.29313369 -9.74810537
## x.sample 0.03713801 0.03773145 0.0378088 0.03764181 0.03796148
## [,826] [,827] [,828] [,829] [,830]
## (Intercept) -9.25905003 -11.17458929 -11.26231313 -9.63888864 -9.43363858
## x.sample 0.03733549 0.03899336 0.03932368 0.03811176 0.03790975
## [,831] [,832] [,833] [,834] [,835]
## (Intercept) -10.39041855 -7.27908943 -9.78154939 -9.85656799 -10.6590278
## x.sample 0.03848318 0.03550326 0.03847144 0.03875843 0.0398548
## [,836] [,837] [,838] [,839] [,840]
## (Intercept) -9.36268846 -10.36824971 -10.17133390 -10.51428841 -10.52482400
## x.sample 0.03764914 0.03827251 0.03811649 0.03932216 0.03780583
## [,841] [,842] [,843] [,844] [,845]
## (Intercept) -7.42003440 -8.99697613 -9.80194984 -9.53182601 -10.56001917
## x.sample 0.03532539 0.03732017 0.03731598 0.03792006 0.03886906
## [,846] [,847] [,848] [,849] [,850]
## (Intercept) -11.221673 -11.06744799 -8.86253231 -10.77089684 -11.75121541
## x.sample 0.039716 0.03917399 0.03684358 0.03920789 0.03976759
## [,851] [,852] [,853] [,854] [,855]
## (Intercept) -9.41136356 -9.3888488 -8.06660001 -9.41898063 -11.30539636
## x.sample 0.03768306 0.0370879 0.03619258 0.03820759 0.03991557
## [,856] [,857] [,858] [,859] [,860]
## (Intercept) -10.63473454 -10.75971843 -10.11734078 -9.31796619 -9.39413642
## x.sample 0.03888297 0.03923334 0.03746751 0.03755003 0.03834428
## [,861] [,862] [,863] [,864] [,865]
## (Intercept) -9.63089295 -11.92749613 -10.90979209 -10.95083354 -11.37689551
## x.sample 0.03824473 0.04074565 0.03939459 0.03933468 0.03952894
## [,866] [,867] [,868] [,869] [,870]
## (Intercept) -10.59584328 -9.3405544 -8.87381400 -8.22916350 -11.00208325
## x.sample 0.03922931 0.0373627 0.03735308 0.03639701 0.03978708
## [,871] [,872] [,873] [,874] [,875]
## (Intercept) -9.44821347 -10.50778927 -11.78031051 -8.07432893 -11.3526995
## x.sample 0.03797448 0.03763488 0.03917168 0.03687525 0.0400046
## [,876] [,877] [,878] [,879] [,880]
## (Intercept) -12.25156240 -10.47940917 -11.37243285 -8.98342631 -10.86311646
## x.sample 0.04073753 0.03885565 0.03992714 0.03623459 0.03850037
## [,881] [,882] [,883] [,884] [,885]
## (Intercept) -10.47460005 -9.10335686 -9.89878933 -10.29936532 -8.45082427
## x.sample 0.03885322 0.03707047 0.03802376 0.03909195 0.03626467
## [,886] [,887] [,888] [,889] [,890]
## (Intercept) -9.85965821 -9.41376493 -10.66212984 -8.501425 -9.85234494
## x.sample 0.03781615 0.03697933 0.03934696 0.036855 0.03873959
## [,891] [,892] [,893] [,894] [,895]
## (Intercept) -10.97367169 -11.40400373 -10.65623323 -10.04565824 -10.26154517
## x.sample 0.03994576 0.03944695 0.03861141 0.03826627 0.03885696
## [,896] [,897] [,898] [,899] [,900]
## (Intercept) -9.0419996 -11.32359819 -10.20589995 -9.28861130 -9.67641757
## x.sample 0.0362003 0.03935336 0.03782628 0.03903392 0.03873443
## [,901] [,902] [,903] [,904] [,905]
## (Intercept) -12.88124018 -8.79846950 -10.85200166 -9.16700005 -11.00242487
## x.sample 0.04071429 0.03691312 0.03928647 0.03735804 0.03973887
## [,906] [,907] [,908] [,909] [,910]
## (Intercept) -12.40990480 -9.57105538 -10.19637790 -10.8211930 -11.35885190
## x.sample 0.04013782 0.03808758 0.03851886 0.0391178 0.03990779
## [,911] [,912] [,913] [,914] [,915]
## (Intercept) -11.3157689 -9.43853108 -9.56809436 -9.66615949 -9.13243333
## x.sample 0.0399407 0.03708767 0.03737505 0.03838561 0.03679293
## [,916] [,917] [,918] [,919] [,920]
## (Intercept) -9.86726579 -10.35988933 -9.09108879 -10.32011255 -9.9975429
## x.sample 0.03748156 0.03878926 0.03616169 0.03981575 0.0384166
## [,921] [,922] [,923] [,924] [,925]
## (Intercept) -8.70980269 -11.5481947 -9.18688148 -8.93448237 -10.0460458
## x.sample 0.03800273 0.0400112 0.03832074 0.03643175 0.0387131
## [,926] [,927] [,928] [,929] [,930]
## (Intercept) -10.13723278 -11.25275215 -10.87331728 -7.74784352 -9.20888020
## x.sample 0.03795993 0.04053378 0.03965747 0.03678866 0.03778628
## [,931] [,932] [,933] [,934] [,935]
## (Intercept) -9.21122201 -9.65158344 -7.94567661 -8.62270546 -9.23210429
## x.sample 0.03704992 0.03828738 0.03678672 0.03720302 0.03666649
## [,936] [,937] [,938] [,939] [,940]
## (Intercept) -9.42338879 -9.41986869 -11.07677871 -11.37004163 -10.35122935
## x.sample 0.03630805 0.03833355 0.03972569 0.03929823 0.03924555
## [,941] [,942] [,943] [,944] [,945]
## (Intercept) -9.35201841 -7.97994109 -12.45404580 -8.95404640 -9.16990976
## x.sample 0.03876337 0.03503001 0.04009174 0.03797237 0.03701906
## [,946] [,947] [,948] [,949] [,950]
## (Intercept) -8.43262567 -10.43196006 -12.89639489 -11.05115099 -10.96272867
## x.sample 0.03664065 0.03996493 0.04243351 0.03935257 0.03919931
## [,951] [,952] [,953] [,954] [,955]
## (Intercept) -10.26658675 -9.04797952 -9.93209496 -10.21120384 -9.93458291
## x.sample 0.03811463 0.03910575 0.03737736 0.03875568 0.03716848
## [,956] [,957] [,958] [,959] [,960]
## (Intercept) -11.41280458 -9.5358692 -9.54694152 -13.35465578 -9.59919157
## x.sample 0.03981283 0.0385542 0.03860573 0.04106156 0.03873873
## [,961] [,962] [,963] [,964] [,965]
## (Intercept) -9.87976723 -11.1025934 -10.28759980 -7.6287752 -7.65043174
## x.sample 0.03778319 0.0387776 0.03917644 0.0359981 0.03649226
## [,966] [,967] [,968] [,969] [,970]
## (Intercept) -10.1237163 -9.27515327 -9.73789655 -10.79319233 -9.68285477
## x.sample 0.0375009 0.03741268 0.03822807 0.04009721 0.03860955
## [,971] [,972] [,973] [,974] [,975]
## (Intercept) -10.1570711 -11.19243348 -10.43894460 -11.42058912 -8.13035088
## x.sample 0.0389295 0.03990813 0.03855678 0.03991947 0.03699817
## [,976] [,977] [,978] [,979] [,980]
## (Intercept) -9.31792300 -11.79023656 -12.77505866 -9.76093470 -9.00560635
## x.sample 0.03851579 0.04022575 0.04067649 0.04002041 0.03662758
## [,981] [,982] [,983] [,984] [,985]
## (Intercept) -8.63342453 -12.33580556 -10.15677527 -10.55766669 -11.33462122
## x.sample 0.03650004 0.04006831 0.03747412 0.03912821 0.03912677
## [,986] [,987] [,988] [,989] [,990]
## (Intercept) -9.63986887 -10.22677238 -10.38547102 -9.63620040 -9.43015749
## x.sample 0.03793948 0.03754975 0.03912734 0.03717802 0.03750213
## [,991] [,992] [,993] [,994] [,995]
## (Intercept) -12.107626 -10.08156887 -10.494659 -10.91783851 -9.74137211
## x.sample 0.039768 0.03896809 0.038729 0.03981068 0.03874382
## [,996] [,997] [,998] [,999] [,1000]
## (Intercept) -11.58156310 -10.21864743 -9.91514275 -7.94619285 -12.58770007
## x.sample 0.04037803 0.03864032 0.03901666 0.03582816 0.04070542original_coef <- coef(r.lm)
parametric_ci <- confint(r.lm)
cat("Parametric Confidence Interval for Intercept (β0):", parametric_ci[1, ], "\n")## Parametric Confidence Interval for Intercept (β0): -12.23262 -7.835736cat("Parametric Confidence Interval for Slope (β1):", parametric_ci[2, ], "\n")## Parametric Confidence Interval for Slope (β1): 0.03632639 0.04051944ci_intercept_boot <- quantile(boot.stats[1, ], probs = c(0.025, 0.975))
ci_slope_boot <- quantile(boot.stats[2, ], probs = c(0.025, 0.975))
cat("Bootstrap Confidence Interval for Intercept (β0):", ci_intercept_boot, "\n")## Bootstrap Confidence Interval for Intercept (β0): -12.45405 -7.970619cat("Bootstrap Confidence Interval for Slope (β1):", ci_slope_boot, "\n")## Bootstrap Confidence Interval for Slope (β1): 0.03594066 0.04079954- Compare the bootstrap CI with the parametric CI. What do you observe? How would you make a conclusion on the null hypothesis based on your confidence interval?
e4d Answer: The Parametric and Bootstrap confidence intervals are very similar and both fit the model. Both CIs also don’t contain 0 so they both indicate that we reject the null hypothesis and that there is a significant relationship between time and r and that if time increases then time also increases.
BONUS/Extra Credit
The R-squared value of the linear model for the python
index indicates that there could be a better fit for the model. Test two
variations/transformations (e.g. polynomial, reciprocal, linear-log) to
see if you can produce a better fit. Show the variation, plot your
resulting model with the scatterplot, and reference the appropriate
output from the summary to support your accessment of whether or not
you’ve found a better model fit.
Add code blocks as needed. Write your narrative in between the code blocks.