Data Cont.

As mentioned above, we will focus on CP and Height variables of Pidgey species after evolution. Specifically, - hp_new: Post-evolution Hit Points. - cp_new: Post-evolution Combat Power. The type of these numeric variables is continuous.

pokemon <- read_csv("C:/Users/JustinTSD/Desktop/Intro To Stats Assignment 4/pokemon.csv")
Pokemon_sub = pokemon %>% dplyr:: select(name, species, cp_new, hp_new)
Pidgey=Pokemon_sub%>%filter(species=="Pidgey")
Pidgey

Descriptive Statistics and Visualisation

plot(hp_new~cp_new, data=Pidgey,xlab="CP",ylab="HP",main="CP & HP of Pidgey after evolution")
PidgeyHPModel<-lm(hp_new~cp_new,data=Pidgey)
abline(PidgeyHPModel,col="red")

Decsriptive Statistics Cont.

CP_New=Pidgey %>% summarise(Min = min(cp_new,na.rm = TRUE),Q1 = quantile(cp_new,probs = .25,na.rm = TRUE),
                            Median = median(cp_new, na.rm = TRUE),
                            Q3 = quantile(cp_new,probs = .75,na.rm = TRUE),
                            Max = max(cp_new,na.rm = TRUE),Mean = mean(cp_new, na.rm = TRUE),
                            SD = sd(cp_new, na.rm = TRUE),n = n(),Missing = sum(is.na(cp_new))) 
HP_New=Pidgey %>% summarise(Min = min(hp_new,na.rm = TRUE),Q1 = quantile(hp_new,probs = .25,na.rm = TRUE),
                            Median = median(hp_new, na.rm = TRUE),
                            Q3 = quantile(hp_new,probs = .75,na.rm = TRUE),
                            Max = max(hp_new,na.rm = TRUE),Mean = mean(hp_new, na.rm = TRUE),
                            SD = sd(hp_new, na.rm = TRUE),n = n(),Missing = sum(is.na(hp_new)))
Factors=c("CP_New","HP_New")
Pidgey_DescriptiveStat=rbind(CP_New,HP_New)
PidgeyStat=data.frame(Factors,Pidgey_DescriptiveStat)
PidgeyStat

Hypothesis Testing

• We will investigate if higher CP will lead to be higher HP of Pidgey and if there is a relationship between CP and HP by performing an appropriate hypothesis testing which is Correlation and Simple Linear Regression with a predictor variable, x, is cp_new and dependent variable, y, is hP_new.

PidgeyHPModel%>%summary()

## 
## Call:
## lm(formula = hp_new ~ cp_new, data = Pidgey)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.686  -2.652   0.487   3.446   5.775 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 24.334296   1.346442   18.07   <2e-16 ***
## cp_new       0.090086   0.003197   28.18   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.149 on 37 degrees of freedom
## Multiple R-squared:  0.9555, Adjusted R-squared:  0.9543 
## F-statistic: 793.9 on 1 and 37 DF,  p-value: < 2.2e-16

Hypthesis Testing Cont.

\[R^2 = \frac{bLxy}{Lyy}=0.955\] - A total of 95.5% of the variability in a Pidgey’s HP can be explained by a linear relationship with Pidgey’s CP. The F-test for linear regression: H0: The data do not fit the linear regression model HA: The data fit the linear regression model F-statistic is 793.9 on df1=1 and df2=37 with p-value<.001 < 0.05 level of significance, we reject H0. There was statistically significant evidence that the data fit a linear regression model. - The intercept is a = 24.334. It shows that when CP of Pidgey is 0, Pidgey’s HP is 24.334. We test the statistical significance of a: \[H_0:\alpha=0\] \[H_A:\alpha\neq0\] t statistic = 18.07, p<0.001. The intercept is statistically significant at level of 0.05.

PidgeyHPModel%>%confint()%>%round(3)

##              2.5 % 97.5 %
## (Intercept) 21.606 27.062
## cp_new       0.084  0.097

95% CI for a to be [21.606,27.062], so we reject H0.

Hypthesis Testing Cont.

• The slope is b = 0.09, when CP increases one unit, Pidgey’s HP will increase 0.09. We will test the statistical significance of b: \[H_0:\beta=0\] \[H_A:\beta\neq0\] t statistic = 28.18, p<0.001<0.05. So we reject H0. There was statistically significant evidence that CP was positive related to HP.

Hypthesis Testing Cont.

cor(Pidgey$hp_new,Pidgey$cp_new)

## [1] 0.9774826

library(Hmisc)
bivariate=as.matrix(dplyr::select(Pidgey,hp_new,cp_new))
rcorr(bivariate,type="pearson")

##        hp_new cp_new
## hp_new   1.00   0.98
## cp_new   0.98   1.00
## 
## n= 39 
## 
## 
## P
##        hp_new cp_new
## hp_new         0    
## cp_new  0

The correlation between HP & CP to be r = 0.98 and p-value=0.

Hypthesis Testing Cont.

A hypothesis test for r: \[H_0:r=0\] \[H_A:r\neq0\] \[t=r\sqrt\frac{(n-2)}{(1-r^2)}=29.956\] A two-tailed p-value:

2*pt(q=29.956,df=39-2,lower.tail = FALSE)

## [1] 1.515035e-27

p-value<0.001<0.05 level of significance, we reject H0. There was a statistically significant positive correlation between HP and CP of Pidgey species after evolution. We test H0 using a confidence interval approach. r is converted to a z-score: \[r=z=\frac{1}{2}ln(\frac{1+r}{1-r})=\frac{1}{2}ln(\frac{1+0.98}{1-0.98})=2.298\] —

References

• Kain, E 2016, “‘Pokémon Go’ Is The Biggest Mobile Game In US History - And It’s About To Top Snapchat”, Forbes, 13 July, viewed 27 May 2017, https://www.forbes.com/sites/erikkain/2016/07/13/pokemon-go-is-the-biggest-mobile-game-in-us-history-and-its-about-to-top-snapchat/#1f8f864f5d5c.

• OpenIntro, “Pokémon Go Evolutions (Pokemon)”, electronic data set, OpenIntro, viewed 25 May 2017, https://www.openintro.org/stat/data/?data=pokemon.