The following boxplot shows a side-by-side comparison over the level of efficiency Orange Juice and Ascorbic Acid have on guinea pigs’ tooth length overall.

Orange Juice seems to be overall more effective than Ascorbic Acid on Guinea Pigs’ tooth growth.

The following boxplot shows a side-by-side comparison over the level of efficiency Orange Juice and Ascorbic Acid have on guinea pigs’ tooth length by dosage amount.

Both Orange Juice and Ascorbic Acid have a higher effect on Guinea Pigs’ tooth growth as the dosage is increased.

The following boxplot shows a side-by-side comparison of the two above plots for easier observation.

Orange Juice tends to be more effective than Ascorbic Acid when administered in dosages of 0.5mg and 1.0mg. However, Ascorbic Acid appears to be just as effective as Orange Juice when both are administered in 2.0mg dosages.

Confidence Intervals (95%)

Guinea pigs who receive a supplement as ascorbic acid (any dosage)

VC<-filter(ToothGrowth,supp=="VC")
t.test(VC$len, conf.level = .95)
## 
##  One Sample t-test
## 
## data:  VC$len
## t = 11.24, df = 29, p-value = 4.363e-12
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  13.87675 20.04992
## sample estimates:
## mean of x 
##  16.96333

Guinea pigs who receive 2 mg of the supplement, delivered via orange juice

OJDose2 <-filter(ToothGrowth, supp=="OJ", dose == 2.0)
t.test(OJDose2$len, conf.level = .95)
## 
##  One Sample t-test
## 
## data:  OJDose2$len
## t = 31.038, df = 9, p-value = 1.833e-10
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  24.16069 27.95931
## sample estimates:
## mean of x 
##     26.06

T-tests used to compare mean growth between the following:

Between guinea pigs who receive ascorbic acid and those who receive orange juice, regardless of dosage (2-sided test)

\(H_{o} : \mu_{VC} = \mu_{OJ}\)

\(H_{a} : \mu_{VC} \not = \mu_{OJ}\)

OJ <- filter(ToothGrowth, supp =="OJ")
t.test(VC$len, OJ$len, alternative = c("two.sided"))
## 
##  Welch Two Sample t-test
## 
## data:  VC$len and OJ$len
## t = -1.9153, df = 55.309, p-value = 0.06063
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -7.5710156  0.1710156
## sample estimates:
## mean of x mean of y 
##  16.96333  20.66333

The t statistic result was t = -1.9153 ; with a p-value = 0.06063 Becuase the p-value is 0.06, we woud fail to reject the null hypothesis. It isn’t significant compared to the the standard of 0.5. There isn’t a significance between the compared population means.

Between guinea pigs who receive a 1 mg dose from orange juice and those who receive a 2 mg dose from orange juice (1-sided test)

\(H_{o} : \mu_{1mg} = \mu_{2mg}\)

\(H_{a} : \mu_{1mg} < \mu_{2mg}\)

OJDose1 <- filter(OJ, dose ==1.0)
t.test(OJDose1$len, OJDose2$len, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  OJDose1$len and OJDose2$len
## t = -2.2478, df = 15.842, p-value = 0.0196
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##        -Inf -0.7486236
## sample estimates:
## mean of x mean of y 
##     22.70     26.06

The t statistic result was t = -2.2478 ; with a p-value = 0.0196 We would indeed reject the null hypothesis since the p-value resulted in 0.0196. It is rather significant compared to the standard of 0.05. The compared population means are indeed difference on this observation.

Between guinea pigs who receive a 1 mg dose from ascorbic acid and those who receive a 2 mg dose from ascorbic acid (1-sided test)

\(H_{o} : \mu_{1mg} = \mu_{2mg}\)

\(H_{a} : \mu_{1mg} < \mu_{2mg}\)

VCDose1 <- filter(VC, dose==1.0)
VCDose2 <- filter(VC, dose==2.0)
t.test(VCDose1$len, VCDose2$len, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  VCDose1$len and VCDose2$len
## t = -5.4698, df = 13.6, p-value = 4.578e-05
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf -6.346525
## sample estimates:
## mean of x mean of y 
##     16.77     26.14

The t statistic result was t = -5.4698 ; with a p-value = 4.578e-05 Once again we would reject the null hypothesis; the result for the p-value was 4.578e-05. This is greatly significant compared to the standard p-value of 0.05. The compared polulation means are verey different again on this test.

Between guinea pigs who receive a 1 mg dose from ascorbic acid and those who receive a 1 mg dose from orange juice (2-sided test)

\(H_{o} : \mu_{1mgVC} = \mu_{1mgOJ}\)

\(H_{a} : \mu_{1mgVC} \not = \mu_{1mgOJ}\)

t.test(VCDose1$len, OJDose1$len, alternative = "two.sided")
## 
##  Welch Two Sample t-test
## 
## data:  VCDose1$len and OJDose1$len
## t = -4.0328, df = 15.358, p-value = 0.001038
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -9.057852 -2.802148
## sample estimates:
## mean of x mean of y 
##     16.77     22.70

The t statistic result was t = -4.0328 ; with a p-value = 0.001038 Because the p-value is 0.001038, we would reject the null hypothesis. The value is significantly compared to the o.05 standard. The two compared population menas are once again notably different.

Between guinea pigs who receive a 2 mg dose from ascorbic acid and those who receive a 2 mg dose from orange juice (2-sided test)

\(H_{o} : \mu_{2mgVC} = \mu_{2mgOJ}\)

\(H_{a} : \mu_{2mgVC} \not = \mu_{2mgOJ}\)

t.test(VCDose2$len, OJDose2$len, alternative = "two.sided")
## 
##  Welch Two Sample t-test
## 
## data:  VCDose2$len and OJDose2$len
## t = 0.046136, df = 14.04, p-value = 0.9639
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.63807  3.79807
## sample estimates:
## mean of x mean of y 
##     26.14     26.06

The t statistic result was t = 0.046136 ; with a p-value = 0.9639 On this scenario we would fail to reject the null hypothesis. The p-value was 0.9639. Greatly higher than 0.05 and highly significant when comparing these two population means.