1)
x<- c(3.9,7.8,1.1,6.6,3.4,2.5,6,9.1,4.6,2)
y <- c(17,32,6,25,23,18,30,33,25,11)
model <- lm (y ~x)
summary(model)
Call: lm(formula = y ~ x) Residuals: Min 1Q Median 3Q Max -4.681 -2.754 -1.116 3.215 5.088 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.2220 2.5324 2.852 0.021419 * x 3.1442 0.4764 6.600 0.000169 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 3.742 on 8 degrees of freedom Multiple R-squared: 0.8449, Adjusted R-squared: 0.8255 F-statistic: 43.56 on 1 and 8 DF, p-value: 0.0001693
y^= 7.222 + 3.1442 x
2)
Using Excel
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.919158799 | |||||
R Square | 0.844852898 | |||||
Adjusted R Square | 0.825459511 | |||||
Standard Error | 3.741928104 | |||||
Observations | 10 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1 | 609.9837925 | 609.9837925 | 43.56396678 | 0.000169334 | |
Residual | 8 | 112.0162075 | 14.00202593 | |||
Total | 9 | 722 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 99.0% | Upper 99.0% | |
Intercept | 7.222042139 | 2.532438664 | 2.851813251 | 0.021419261 | -1.275270473 | 15.71935475 |
x | 3.144246353 | 0.476379272 | 6.600300507 | 0.000169334 | 1.545809378 | 4.742683329 |
99% confidence interval for slope
1.545809 | 4.742683 |
3)
p-value for one-tailed test = 0.00016/2 = 0.00008
p-value < alpha
we reject the null hypothesis
4)
p-value = 0.00017 < alpha
hence we reject the null hypothesis
5)
R^2 = 0.8449
hence 84.49 % of variation is explained by this model
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