Problem Description:
We are given a dataset with variables y_FIT, x1_SC, x2_ECI, and x3_SSO. The goal is to explore relationships and test hypotheses regarding these variables.
y_FIT<-c(55, 70, 58, 74, 86, 98, 96, 70, 40, 67, 41, 41, 47, 45, 92, 50, 98,
42, 64, 54)
x1_SC<-c(1, 2, 2, 3, 3, 3, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2)
x2_ECI<-c(85, 67, 53, 52, 76, 64, 75, 75, 68, 74, 40, 66, 65, 70, 70, 60, 77,
65, 65, 50)
x3_SSO<-c(52, 51, 50, 56, 55, 61, 65, 56, 45, 52, 45, 48, 50, 49, 61, 52, 63,
49, 55, 55)
mydata<-data.frame(y_FIT,x1_SC,x2_ECI,x3_SSO)
Hypothesis One
Objective: Assess the correlation between x3_SSO and y_FIT.
cor.test(x3_SSO,y_FIT)
##
## Pearson's product-moment correlation
##
## data: x3_SSO and y_FIT
| I reject the null hypothesis that fitness does not increase as the # of socially supportive others decreases because the observed correlation coefficient r = 0.919 has p-value (.0000000108) less than .05. The null hypothesis is rejected and hence the conclusion that fitness increase as the # of socially supportive others increases |
## t = 9.8761, df = 18, p-value = 1.082e-08
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.8026266 0.9678203
## sample estimates:
## cor
## 0.9188066
# Correlation Test correlation_result<- cor.test(mydata$x3_SSO, mydata$y_FIT) # Results print("Pearson's product-moment correlation") print(correlation_result)
Observation: A significant correlation exists between x3_SSO and y_FIT (cor = 0.92, p-value < 0.001).
Hypothesis Two
Objective: Compare means of y_FIT between two classes based on x1_SC.
LoClass<- ifelse(x1_SC<=1,1,0)
LoClass<- ordered(LoClass, levels = c(0,1), labels = c('Mid&Upper', 'Low'))
table(LoClass)
## LoClass
## Mid&Upper Low
## 10 10
t.test(y_FIT ~ LoClass, options(digits = 5), alternative = "two.sided")
##
## Welch Two Sample t-test
##
## data: y_FIT by LoClass
## t = 3.5, df = 18, p-value = 0.0026
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 10.158 40.642
## sample estimates:
## mean in group Mid&Upper mean in group Low
## 77.1 51.7
Observation: There is a significant difference in means between 'Mid&Upper' (mean = 77.1) and 'Low' (mean = 51.7) classes (p-value = 0.0026).
Hypothesis Three
Objective: Explore the linear relationship between y_FIT and x3_SSO using linear regression.
fit1 = lm(y_FIT ~ x3_SSO, data = mydata)
summary(fit1)
##
## Call:
## lm(formula = y_FIT ~ x3_SSO, data = mydata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.39 -5.48 -0.72 4.98 16.61
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -113.640 18.123 -6.27 6.5e-06 ***
## x3_SSO 3.328 0.337 9.88 1.1e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.3 on 18 degrees of freedom
## Multiple R-squared: 0.844, Adjusted R-squared: 0.836
## F-statistic: 97.5 on 1 and 18 DF, p-value: 1.08e-08
Observation: The linear regression model indicates a significant relationship between y_FIT and x3_SSO (p-value< 0.001, R-squared = 0.844).
Standardized Coefficient
Objective: Calculate the standardized coefficient between standardized y_FIT and x3_SSO.
beta<-lm(scale(y_FIT) ~scale(x3_SSO), data =mydata)
beta
##
## Call:
## lm(formula = scale(y_FIT) ~ scale(x3_SSO), data = mydata)
##
## Coefficients:
## (Intercept) scale(x3_SSO)
## -3.11e-16 9.19e-01
Observation: The standardized coefficient for x3_SSO is 0.919, indicating a strong standardized relationship.
This structured approach provides a clear presentation of the hypotheses, methods, and results, making it easier for readers to understand and interpret the analysis.