# Problem 1: 
`Teaching Styles`<-c(3.03, 3.15, 3.18, 3.03, 2.88, 2.70, 3.03, 3.10, 2.95, 3.13)
`Learning Styles`<-c(2.64, 2.62, 2.63, 2.75, 2.53, 2.64, 2.64, 2.66, 2.71, 2.70)
`Average Performance`<-c(1.60, 1.80, 3.0, 2.40, 1.20, 1.50, 1.77, 3.30, 2.40, 1.90)
Z<-cbind(`Teaching Styles`, `Learning Styles`,`Average Performance` )
Z<-data.frame(Z)
Z
##    Teaching.Styles Learning.Styles Average.Performance
## 1             3.03            2.64                1.60
## 2             3.15            2.62                1.80
## 3             3.18            2.63                3.00
## 4             3.03            2.75                2.40
## 5             2.88            2.53                1.20
## 6             2.70            2.64                1.50
## 7             3.03            2.64                1.77
## 8             3.10            2.66                3.30
## 9             2.95            2.71                2.40
## 10            3.13            2.70                1.90
model <- lm(`Average Performance` ~ `Teaching Styles` + `Learning Styles`, data = Z)
summary(model)
## 
## Call:
## lm(formula = `Average Performance` ~ `Teaching Styles` + `Learning Styles`, 
##     data = Z)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.6221 -0.4064 -0.1026  0.2072  1.0036 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)
## (Intercept)        -15.101      9.000  -1.678    0.137
## `Teaching Styles`    2.162      1.382   1.564    0.162
## `Learning Styles`    4.021      3.344   1.203    0.268
## 
## Residual standard error: 0.5871 on 7 degrees of freedom
## Multiple R-squared:  0.4097, Adjusted R-squared:  0.2411 
## F-statistic: 2.429 on 2 and 7 DF,  p-value: 0.158

It can be seen that p-value of the F-statistic is 0.158, which is not significant. This means that none of the predictor variables (Learning Styles and Teaching Styles) is significantly related to the outcome variable which is the “Average Performance”.

summary(model)$coefficient
##                     Estimate Std. Error   t value  Pr(>|t|)
## (Intercept)       -15.100978   9.000353 -1.677821 0.1372807
## `Teaching Styles`   2.161701   1.382365  1.563770 0.1618474
## `Learning Styles`   4.021102   3.343708  1.202588 0.2682306