R Project #1: Hypothesis Testing With t-Test

 QUESTION:

Consider the gain in weight of 19 female rats between 28 and 84 days after birth. 12 were fed on high protein diet and 7 on a low protein diet. Using the data given below, test the hypothesis that there is no difference in weight gain between female rats raised on a high-protein diet versus those raised on a low-protein diet. Use a significance level of 𝛼 = 0.05 and assume equal variances. 

Weight Gain Measured in Grams:

High protein: 134,146,104,119,124,161,107,83,113,129,97,12 

Low protein: 70,118,101,85,107,132,94 

SOLUTION:

The Null Hypothesis here is that there is no weight gain difference between the 2 groups of rats. It will be shown that there is no reason to reject the Null Hypothesis. This will be done using the R code shown below. 

#This will be a test involving two population means

#mu1 is population mean-of-weight-gain of high protein rats

#mu2 is population mean-of-weight-gain of low protein rats

# H0: mu1 - mu2 = 0 i.e. no difference in weight gain (THIS IS THE NULL HYPOTHESIS)

#Ha: mu1 - mu2 <>0  therefore, 2-tailed test (THIS IS THE ALTERNATE HYPOTHESIS)

 

#create vectors of weight gain in rats

high_protein<-c(134,146,104,119,124,161,107,83,113,129,97,12)

low_protein<-c(70,118,101,85,107,132,94)

 

#Because each vector has a low number of data points, i.e. n<30,

#a t-test will be done to test the null hypothesis H0.

 

#FIRST:determine if high_protein and low_protein are

#normal or approximately normal by plotting their

#stem and leaf plots

stem(high_protein)

The decimal point is 2 digit(s) to the right of the |

   0 | 1

  0 | 8

  1 | 00112233

  1 | 56

 

stem(low_protein)

The decimal point is 1 digit(s) to the right of the |

    6 | 0

   8 | 54

  10 | 178

  12 | 2

#SECOND: perform a second test to determine if

#high_protein and low_protein are normal or approximately normal

#by plotting their histograms

hist(high_protein,main="high protein",xlab="weight gain")

hist(low_protein,main="low protein",xlab="weight gain")

#both vectors have mounds in both stem and leaf plots and

#in their histograms, therefore, presume normal populations

#Therefore, t-test is permissible.

 

#dof = degrees of freedom

dof = length(high_protein) + length(low_protein)-2

dof  

[1] 17

#perform  t-test for the 2 independent samples

t.test(high_protein,low_protein,alternative="two.sided",conf.level=0.95,var.equal=TRUE)

               Two Sample t-test

data:  high_protein and low_protein

t = 0.62634, df = 17, p-value = 0.5394

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

 -23.09271  42.59271

sample estimates:

mean of x mean of y

   110.75    101.00

#The 95% confidence interval includes  i.e. the interval's range is -23.09271 to 42.59271.

#The p-value is 0.5394, which is much greater than alpha=0.05.

#Therefore, there is insufficient evidence to reject H0, i.e. there is no difference in

#weight gain between the 2 groups of female rats.