Question

2. Using a computer program, a data set with 20 observations on and was generated (GENERATEDATA.csv). Again, use the same instructions as above (in #1) to upload and import this dataset into an RStudio Cloud work session. (1 point) )(3 points) Make a scatter plot and describe the relationship between Y and X e) (4 points) Using R commands we've used before, calculate the means and standard deviations for both Y and X (3) (2 points) Using R, calculate the correlation coefficient value between Y and X (4) (5 points) Using the values found above, use the formulas given in the Lecture to manually calculate (show your work) the slope and Y-intercept for the least squares regression line. Write out the equation for the least squares regression line (s)(3 points) Refer to the OpenIntro.org Lab (Linear Regression) to perform the Im) command in R computing the linear regression line with R. Write out the regression line equation from the Output from R. (6) (2 points) As shown in the Lab, add the regression line to your scatter plot using the abline() command. (1 point) What percent of the variability in Y is explained by X? Give the appropriate statistic. (8) (4 points) As shown in the Lab, plot the residuals versus the x values; use the abline() command to draw the dashed line at y-0. Give a short paragraph commenting on this residual plot graph () (3 points) Make a histogram of the residuals and comment on what this shows you.

here is the data

Y X

34.38 22.06

30.38 19.88

26.13 18.83

31.85 22.09

26.77 17.19

29.00 20.72

28.92 18.10

26.30 18.01

29.49 18.69

31.36 18.05

27.07 17.75

31.17 19.96

27.74 17.87

30.01 20.20

29.61 20.65

31.78 20.32

32.93 21.37

30.29 17.31

28.57 23.50

29.80 22.02

this is q1

Answer 1

reg<-read.csv ("D:\\Dipalil\reg.csv") head (reg) # Yx # 1 34.38 22.06 # # 2 30.38 19.88 ## 3 26.13 18.83 ## 4 31.85 22.09 ## 5 26.77 17.19 ## 6 29.00 20.72 plot (reg$x, reg$Y)

reg$Y 28 30 32 34 17 ο ο 18 Η Ο ο 19 Ο 20 ο reg$X Ο Ο 21 22 23

library (dplyr) ## Attaching package: 'dplyr' ## The following objects are masked from 'package:stats': # filter, lag ## The following objects are masked from 'package:base': intersect, set diff, setequal, union library (tidyr) summary<-reg $x summarise (mean X=mean (X, na.rm=T), Sd X=sd (X, na.rm=T), mean Y=mean (Y, na.rm=T), Sd Y=sd (Y, na.rm=T)) summary ## ## 1 mean_x Sd_x mean_Y Sd_Y 19.73 1.860555 29.6775 2.214581

cor (reg$x, reg$Y) ## [1] 0.5226194 4 & 5 fit_ln<-lm (Y-X, data=reg) summary (fit_im) IEE EI Call: Im (formula = Y - X, data = reg) SIE SIE EI ## Residuals: Min 10 Median ## -3.4527 -1.3224 0.1483 30 1.4459 Max 3.2531 SIE SIE SIE SIE Coefficients: Estimate Std. Error t value Pr(tl) (Intercept) 17.4042 4.7391 3.672 0.00174 ** ## x 0.6221 0.2392 2.601 0.01807 * ## ---

## Signif. codes: 01***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1'' 1 ## Residual standard error: 1.94 on 18 degrees of freedom ## Multiple R-squared: 0.2731, Adjusted R-squared: 0.2327 ## F-statistic: 6.764 on 1 and 18 DF, p-value: 0.01807 Y=0.6221*x+17.4042 library (ggplot2) ggplot (reg, aes (X, Y)) + stat_summary (fun.data=mean_cl_normal) + geom smooth (method='lm') +labs (title = "Scatter Plot abline") +theme (plot.title = element text (hjust = 0.5))

Scatter Plot abline 32.5 30.0- 27.5- 20 *

27% of variability in Y is explained by X Multiple R-squared: 0.2731 res<-resid(fit_lm) res FIE I TE 1 2 4 3.25309246 0.60919050 -2.98764301 0.70443056 -1.32745916 -1.29334269 10 11 12 0.25646322 -2.30755109 0.45944586 2.72756638 -1.37581462 1.34942544 14 15 16 ## -0.78046222 0.04013024 -0.63979826 1.73548264 2.23231615 2.11789324 19 20 ## -3.45267872 -1.32068691 17 18 TE FEIE E plot (reg$x, res, ylab="Residuals", xlab="X", main="Residual Plot with abline") abline(0, 0)

Residual Plot with abline 3 2 o 1 o 0 Residuals 0 O -1 0 -2 0 -3 E R 18 19 20 21 22 23 A "good" residual plot will look as if the points (residuals) are randomly scattered. That is, there will be no patterns to the residuals. Does that appear to be the case here?

The uses of the Residual Plot 1. To check to see if a linear relationship exists between the explanatory and response variable. If the residuals form a non-linear pattern, the relationship between the explanatory and response variable will be non-linear. 2. Identify possible outliers. 3. To check to see if the variation in the residuals remains constant over the entire range of the explanatory variable. hist (res)

Histogram of res 5 4 3 Frequency 2 1 0 SEO res The overall pattern of the residuals should be similar to the bell-shaped pattern observed when plotting a histogram of normally distributed data.