This is a more theoretical question as i'm quite new to this... I was wondering if i'm using this: ln wage = b1 + ... + b3 experience + b4 experience ^2
If i'm using a squared X- variabele to know the effect I should use derivates etc but i'm wondering what my hypothesistests will be for the variables experience and experience ^2. Do I need 2 seperate hypotheses or just one and use in the alternative what i'm expecting experiences effect to be on wage over time?
fit2 <- lm(mpg ~ hp + drat + I(drat^2) , data = mtcars)
summary(fit2)
#>
#> Call:
#> lm(formula = mpg ~ hp + drat + I(drat^2), data = mtcars)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -5.1847 -2.4796 -0.3974 1.2020 7.5564
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.314340 23.589436 0.098 0.923
#> hp -0.052384 0.009573 -5.472 7.64e-06 ***
#> drat 9.499096 13.095200 0.725 0.474
#> I(drat^2) -0.658769 1.789196 -0.368 0.715
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.219 on 28 degrees of freedom
#> Multiple R-squared: 0.7424, Adjusted R-squared: 0.7148
#> F-statistic: 26.9 on 3 and 28 DF, p-value: 2.138e-08
fit <- lm(mpg ~ hp + drat, data = mtcars)
summary(fit)
#>
#> Call:
#> lm(formula = mpg ~ hp + drat, data = mtcars)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -5.0369 -2.3487 -0.6034 1.1897 7.7500
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 10.789861 5.077752 2.125 0.042238 *
#> hp -0.051787 0.009293 -5.573 5.17e-06 ***
#> drat 4.698158 1.191633 3.943 0.000467 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.17 on 29 degrees of freedom
#> Multiple R-squared: 0.7412, Adjusted R-squared: 0.7233
#> F-statistic: 41.52 on 2 and 29 DF, p-value: 3.081e-09
fit_l <- lm(log(mpg) ~ hp + drat + I(drat^2), data = mtcars)
summary(fit_l)
#>
#> Call:
#> lm(formula = log(mpg) ~ hp + drat + I(drat^2), data = mtcars)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.290377 -0.097397 0.001702 0.080792 0.305609
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.5194249 1.1299116 1.345 0.190
#> hp -0.0027235 0.0004585 -5.940 2.15e-06 ***
#> drat 0.8044142 0.6272477 1.282 0.210
#> I(drat^2) -0.0798903 0.0857008 -0.932 0.359
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.1542 on 28 degrees of freedom
#> Multiple R-squared: 0.7579, Adjusted R-squared: 0.732
#> F-statistic: 29.22 on 3 and 28 DF, p-value: 9.06e-09
The null hypothesis, H_0, is that the independent variables hp and drat have no association with mpg at a given level of confidence equal to 1 - \alpha where \alpha (such as 0.05) is the probability of observing a test statistic of at least as extreme purely by chance. For these models, we can reject H_1 for the hp variable, but not drat or drat^2.