Interpretation: on average, an increase in 1 unit of IQ while holding everthing else constant will result in an increase of $8.303 in wage. The intercept tells us that if a worker has 0IQ, then his/her wage will be $116.992 monthly. It is meaningful to understand the coeffiecient. It presents a wage gap between a worker with xIQ and yQ will be |x-y|*8.303 on average. As for my intuitive, someone with a higher IQ likely to work more effectively, think logically and process at higher rate. That ability surely reflect on wage to be higher, live up to their quality.
so the predicted change in wage for a ncrease in 2 standard deviation in IQ is $249.964.
If a person with IQ=90, then increase 2sd in IQ would make their IQ=90 +2sd. Same happen to person with IQ=110. Because this is a linear model, the predicted change in wage would stay the same as calculated, an increase of $249.964 respond to a 2sd IQ boost no matter what their IQ is. We conclude that the predicted change in wage level does not differ for an individual with IQ=90 and an individual with IQ=110.
Question 1d
R-squared closer to 0 , the less proportion of sample variance in y explained by independent variable. In this case, it is 0.09554 means that IQ explain only 9.554% of variation in wage. Meaning that while IQ does explain the differences in wage, it does not explain most of the variation consider other factors. The F-statistics 98.55 tells us how strong the overall relationship between IQ and wage. It is very high, implies that IQ does help explain it. The p-value <2.2e^-16, which is very very small, indicates that probability of observing result which we set the null hypothesis is true (that IQ has no effect on wage) is pretty much 0. Together, IQ and wage is statistically significant and highly unlikely to be due to chance
Question 1e
Yes it does. The F-statistics 98.55 prove that IQ has an effect on wage. In this case, the coefficient of IQ is positive, meaning higher IQ on avergae cause higher wage.
We see that t-calc is greater than t-crit (1.651>1.647) so we reject the null hypothesis
Conclusion
The t-calc is greater than t-crit at 5% significant level so we reject null hypothesis that, controlling for IQ and father’s education, mother’s education has no effect on wage in favour of the alternative hypothesis that mother’s education has a positive effect on wage. We conclude that there is a sufficient evidence that mother’s education has a positive effect on wage once we control for IQ and father’s education.
We see that Fcalc >Fcrit (110.228>3.008) so we reject the null hypothesis that, controlling for IQ, parent education has no effect on wage in favour of the alternative hypothesis that parent education has an effect on wage.
Conclusion
The F-test here is used for joint hypothesis test, instead of doing each model one by one. Meaning we test whether both parent (father and mother) education has no effect on wage once we control for IQ. In this case, there is a sufficient evidence that parent education has effect on wage. Therefore father and mother education are jointly significant and help explain variation in wage.
We see that Fcalc>Fcrit (32.605>2.617) so we reject null hypothesis that IQ, mother’s education and father’s education has no effect on wage in favour of the alternative hypothesis that at least one of them has effect on wage.
Conclusion
The F-test of all three predictors at the 5% significance level provides sufficient evidence that IQ, mother’s education, and father’s education are jointly significant in explaining the variation in wage. This means that, when considered together, these variables contribute meaningfully to the model, and at least one of them has a statistically significant effect on wage. Therefore, we reject the null hypothesis that none of the predictors has effect on wage.
We see that Fcalc<Fcrit (0.108<3.008) so we fail to reject null hypothesis
Conclusion
The F-test result in the Fcalc being less than critical value, so we fail to reject null hypothesis at 5% significant level. Meaning we do not have enough evidence to say that the null hypothesis is false. There is insufficient evidence to claim that an extra year of mother’s education has a different effect from father’s education, or that it does not have exactly twice the effect on wage as an extra IQ point. Therefore, the data does not against the assumption that mother’s education has the same effect as father’s education and twice the effect of IQ on wage.
We see that tcalc<tcrit (0.373<1.647) so we fail to reject null hypothesis
Conclusion
The t-statistic is less than the critical value at 5% significant level so we fail to reject the null hypothesis. We say that we do not have enough evidence to state that the null hypothesis is false. Therefore, there is an insufficient evidence that an extra year of mother’s education has different effect on wage as an extra year of father’s education. The data does not contradict the null hypothesis.
We doing a two-tailed test, and see that |tcalc|<tcrit (|-1.468|<1.963) so we fail to reject null hypothesis
Conclusion
The t-statistic is less than the critical value at 5% significant so we fail to reject the null hypothesis. Meaning that we do not have enough evidence to say that null hypothesis is false. Therefore, there is insufficient evidence that living in the city has an effect on wage and that the data does not against the null hypothesis that living in city has no effect on wage.
Question 4d
mean(wage2$IQ, na.rm=TRUE)
[1] 101.2824
mean(wage2$educ, na.rm=TRUE)
[1] 13.46845
predicted wage for individual living in city with average IQ and average education