{r, echo=FALSE, warning=FALSE,include=FALSE} knitr::opts_chunk$set(echo = FALSE)
{r, echo=FALSE, warning=FALSE,include=FALSE} pacman::p_load(AER,tidyverse)
{r, echo=FALSE, warning=FALSE,include=FALSE} data(“STAR”) Lei Han–(Q1,Q2,Q9)
Carmen Li–(Q4,Q5,Q6，introduction)
Yihui Yang–(Q3,Q7,Q8)
INTRODUCTION
In this project, we try to figure out whether the math scores will have the relationship with the different class type and size. If they have correlation, we want to know if each class type have significantly influence with the math scores and each class type if they have big different or not. Moreover,how strong enough to claim their relation. So we are trying to provide evidences to prove the these questions. {r, echo=FALSE, warning=FALSE} #gender STAR %>% group_by(stark,gender) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(gender),y = Count,fill = gender)) + geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0)+ facet_wrap(~stark)+ labs(x= “Gender”, y= “Number of student”, title = “Student number in differnet class type”) {r, echo=FALSE, warning=FALSE} #ethnicity STAR %>% group_by(stark,ethnicity) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(ethnicity),y = Count,fill = ethnicity)) + geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0)+ facet_wrap(~stark)+ labs(x= “Ethnicity”, y= “Number of student”, title = “Student number in differnet class type”) {r, echo=FALSE, warning=FALSE} #birth STAR %>% #filter(stark==”small”) %>% group_by(stark,birth) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(birth),y = Count,fill = as.factor(birth))) + geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0,size = 2.5)+ facet_wrap(~stark)+ labs(x= “Birth Quarter”, y= “Number of student”, title = “Student number in differnet class type”) {r, echo=FALSE, warning=FALSE} # drop observations with NA value in math1 Math1 <- STAR %>% filter(!is.na(math1)) # check if there is any NA in stark sum(is.na(Math1$stark)) Math1$stark.small[Math1$stark==”small”] <- 1 Math1$stark.small[Math1$stark!="small"|is.na(Math1$stark)] <- 0 # generate stark.small to represent the child went to small type class Math1$stark.regular[Math1$stark=="regular"] <- 1 Math1$stark.regular[Math1$stark!="regular"|is.na(Math1$stark)] <- 0 # generate stark.regular to represent the child went to regular type class Math1$stark.regularaide[Math1$stark=="regular+aide"] <- 1 Math1$stark.regularaide[Math1$stark!="regular+aide"|is.na(Math1$stark)] <- 0 # generate stark.regularaide represent the child went to regular-with-aide type of class # stark.small = stark.regular = stark.regularaide = 0 represent no STAR class was attended
METHOD
_y__i_ = _β_₀ + _β_₁_X_₁ + _β_₂_X_₂ + _ϵ__i_
yi is math score in the 1st grade
x₁ is number of students assigned to the small class
x₂ is number of students assigned to the regular class
β₀ is the average of math score when the students in both small classes and regular class are zero
β₁ is the average change of math score when the student-number of small class have one-unit change
β₂ is the average change of math score when the student-number of regular class have one-unit change
ϵi is the error between the ture value and the fit value
RESULT
FITTING MODEL
_y__i_ = _β_₀ + _β_₁_X_₁ + _β_₂_X_₂ + _β_₃_X_₃ + _ϵ__i_
FITTING SUMMARY STATISTICS:
Codfficient Estimate Std.Error t_value Pr(>
————————– ———- ———– ——— ———-
Intercept 522.4761 0.9129 572.357 < 2e-16
Small 18.7197 1.4673 12.758 <2e-16
regular 9.2512 1.4302 6.469 1.06e-10
regular_with_aide 8.7684 1.4143 6.200 5.99e-10
Residual standard error: 42.58
Degree of freedom: 6596
Multiple R-squared: 0.0245
F-statistic: 55.22
P-value: <2.2e-16
From the table as above, we can know: $\hat\beta_0$ is 522.4761, a estimator of β₀: the average of math score when the students in both small classes and regular class are zero $\hat\beta_1$ is 18.7197, a estimator of β₁: the average change of math score when the student-number of small class have one-unit change $\hat\beta_2$ is 9.2512,a estimator of β₂: the average change of math score when the student-number of regular class have one-unit change For the 95% confident interval for the coeficient of class types $\hat\beta_1$ and $\hat\beta_2$, also, we can use theformula:
$$C.I.=[(\hat\beta_i-t_{1-\alpha/4} \hat{SE}_i ),(\hat\beta_i+t_{1-\alpha/4} \hat{SE}_i )]$$
{r, echo=FALSE, warning=FALSE,include=FALSE} model <- lm(math1~stark.small+stark.regular+stark.regularaide,data = Math1) confint(model,level=0.975)[2,] confint(model,level=0.975)[3,] $\hat\beta_1$ confident interval:(15.43010, 22.00925) $\hat\beta_2$ confident interval:(6.447590, 12.05475) For the $\hat\beta_1$ in different samplings for the dataset, we have 95% confident believe that the β₁ which the mean change of math score when the student-number of small class have one-unit change will exist between 15.43010 and 22.00925. For the $\hat\beta_2$in different samplings for the dataset, we have 95% confident believe that the β₂ which the average change of math score when the student-number of regular class have one-unit changewill exist between 6.447590 and 12.05475.
{r, echo=FALSE, warning=FALSE} model <- lm(math1~stark.small+stark.regular+stark.regularaide,data = Math1) plot(model)
math1 = 522.4761 + 18.7197_stark.small + 9.2512_stark.regular + 8.7684*stark.regularaide child does not attend STAR class has a average point of 522.48 So child went to small class has an average 18.7197 points higher than no-attend-STAR child child went to regular class has an average 9.2512 points higher than no-attend-STAR child child went to regular-with-aide class has an average 8.768 points higher than no-attend-STAR child
CONCLUSION
APEEDIX
library(AER) data(STAR) STAR %>% group_by(stark,gender) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(gender),y = Count,fill = gender)) +geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0)+facet_wrap(~stark)+labs(x= “Gender”, y= “Number of student”, title = “Student number in differnet class type”) STAR %>% group_by(stark,ethnicity) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(ethnicity),y = Count,fill = ethnicity)) +geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0)+ facet_wrap(~stark)+labs(x= “Ethnicity”, y= “Number of student”, title = “Student number in differnet class type”) STAR %>% #filter(stark==”small”) %>% group_by(stark,birth) %>% summarise(Count = n()) %>% ggplot(aes(x = as.factor(birth),y = Count,fill = as.factor(birth))) +geom_bar(stat = “identity”)+ geom_text(aes(label=Count),vjust = 0,size = 2.5)+facet_wrap(~stark)+labs(x= “Birth Quarter”, y= “Number of student”, title = “Student number in differnet class type”) Math1 <- STAR %>% filter(!is.na(math1)) sum(is.na(Math1stark)) < br > Math1stark.small[Math1$stark==”small”] <- 1 Math1stark.small[Math1stark!="small"|is.na(Math1stark)] < −0 < br > Math1stark.regular[Math1$stark==”regular”] <- 1 Math1stark.regular[Math1stark!="regular"|is.na(Math1stark)] < −0 < br > Math1stark.regularaide[Math1$stark==”regular+aide”] <- 1 Math1stark.regularaide[Math1stark!="regular+aide"|is.na(Math1$stark)] <- 0 model <- lm(math1~stark.small+stark.regular+stark.regularaide,data = Math1) confint(model,level=0.975)[2,] confint(model,level=0.975)[3,] model <- lm(math1~stark.small+stark.regular+stark.regularaide,data = Math1) plot(model)