Question

Answer IN R CODE please. Using the data below,

Create a scatterplot of y vs x (show this) and fit it a simple linear regression model using y as the response and plot the regression line (with the data). Show this as well. Test whether x is a significant predictor and create a 95% CI around the slope coefficient. What does the coefficient of determinations represent?

For x=20, create a CI for E(Y|X=20). Show this.

For x=150, can you use the model to estimate E(Y|X=150). Discuss.

Does the model appear to be linear with respect to x. Explain. Discuss, and if not, provide alternative model and repeat steps 1-6.

	y	x
1	311.8481	30.77326
2	440.9428	32.40036
3	41.6744	13.89724
4	417.7435	30.82836
5	177.3642	21.17247
6	639.0727	41.70052
7	179.9235	20.52949
8	19.64963	16.78782
9	1030.218	47.05621
10	211.6078	24.73312
11	468.797	33.30568
12	281.9641	27.20706
13	360.4149	28.98507
14	626.3254	33.98696
15	692.872	40.61913
16	840.8116	44.14024
17	71.51774	14.71966
18	97.75643	18.69047
19	251.0697	26.53534
20	81.51288	19.51529
21	270.3445	28.00065
22	1221.873	49.81578
23	110.3152	20.3347
24	595.4412	38.29436
25	126.2188	13.26268
26	11.15999	16.73084
27	230.5542	24.64804
28	77.3025	15.99319
29	1117.463	48.8532
30	122.5684	18.10108
31	932.665	44.75007
32	911.0599	44.23208
33	255.6625	24.33537
34	810.0097	41.18667
35	210.4745	20.06741
36	9.884425	11.10681
37	75.98362	11.67823
38	153.6595	20.20392
39	578.7254	38.05732
40	93.28379	12.89079
41	378.1102	27.82776
42	203.9408	25.8318
43	837.9018	43.87759
44	44.45671	11.49288
45	1145.79	48.94833
46	1073.485	47.3091
47	431.1394	30.53461
48	343.5504	28.65658
49	810.0665	41.25828

Please provide all relevant work in R code. The commands, the output and any interpretations/conclusions that are necessary.

Answer 1

now code :

### we have to do regression analysis : here y is dependent and x is independent :

y
x

### Create a scatterplot of y vs x (show this) and
## fit it a simple linear regression model using y as the
## response and plot the regression line (with the data)

## scatter plot :
plot(x,y , main = "scatter plot")

## fit regression model :
reg = lm(y~x)
reg

## estimated regression equation : yhat = -420.37 + ( 28.97*x)

summary(reg)

#### Q) Test whether x is a significant predictor and create
## a 95% CI around the slope coefficient.

## to test for coefficient of x : slope
## to test : Ho : β1 = 0 vs H1 : β1 ≠ 0

## test statistics : t = (b1 - β1) / se b1

b1 = 28.975
seb1 = 1.111

t = b1 / seb1
t

## p value = 2e-16

## Decision : we reject Ho if p value is less than alpha value using p value approach
## here p value is less than alpha value hence it is significant .

## Conclusion : slope is significant at given alpha level . that is x is significant .

#### Q1) 95 % confidence interval for slope :

## β1 = ( b1 ± t critical value * seb1 )

t_critical = abs(qt(0.05/2,47))
t_critical

lower_level = 28.975 - ( 2.011741*1.111 )
lower_level
upper_level = 28.975 + ( 2.011741*1.111 )
upper_level
## 95 % confidence interval for β1 = (26.739955 ,31.210044)

#### Q ) What does the coefficient of determinations represent?
## Answer : The coefficient of determination (denoted by R2) is a key output of regression
## analysis. It is interpreted as the proportion of the variance in the dependent variable
## that is predictable from the independent variable. it is lies in 0 to 1 .
  

#### Q ) For x=20, create a CI for E(Y|X=20). Show this. we can use direct command :

new.dat = data.frame(x=20)

predict(reg , newdata= new.dat ,interval = "confidence")

## lower limit = 126.2353 and upper limit = 192.0172

#### Q) For x=150, can you use the model to estimate E(Y|X=150). Discuss.

new.dat1 = data.frame(x=150)

predict(reg , newdata= new.dat1 ,interval = "confidence")

## Lower limit = 3653.956 , upper limit = 4197.698

#### Q) Does the model appear to be linear with respect to x. Explain. Discuss,

## to test : Ho : overall model is not linear vs H1 : overall model is singificant .

## test statistics = F = 680.5 and p value = 0

## Decision : we reject Ho if p value is less than alpha value using p value approach
## here p value is less than alpha value we reject Ho

## Conclusion ; there is enough evidence to conclude that overall model is significant .