Question

A study presents data on widths and lengths of the native butter clam. Here is a small random sample of dimensions for four of these clams that we will use to test whether, in general, the clams are longer than they are wide.

Width	Length
3.2 4.4 4.8 5.8	4.1 5.6 6.6 7.0

(a) Use software to carry out a paired t test to see if the mean of differences, width minus length, is negative for the larger population of clams from which the sample was taken: first report the t statistic using software. (Round your answer to two decimal places.)
t =  

(b) Next report the P-value using software. (Round your answer to three decimal places.)


(c) Is there evidence at the 5% level that the population mean is negative?

Yes

No     

(d) To see how choice of test procedure can play a role in conclusions, carry out a two-sample t test to see if the difference between population means is negative. (This is not the appropriate procedure, because the samples of widths and lengths are not independent, but paired together.) Report the t statistic for this test using software. (Round your answer to two decimal places.)
t =  

(e) Report the P-value using software. (Round your answer to three decimal places.)


(f) Would the two-sample test provide evidence at the 5% level that the population mean is negative?

Yes

No     

Answer 1

R-code :

Paired t-test

> w<-c(3.2,4.4,4.8,5.8)
> h<-c(4.1,5.6,6.6,7)
> d<-w-h
> d
[1] -0.9 -1.2 -1.8 -1.2
> mean(d)
[1] -1.275
> sd(d)
[1] 0.3774917

Function for doing lower tailed t-test :

t_lt<-function(mu0,xbar,sigma,n,alpha)
{
z_t<-(xbar-mu0)*sqrt(n)/(sigma)
z_c<-qt(alpha,df=n-1)
pvalue <- pt(z_t,df=n-1)
conc<-ifelse(pvalue>alpha,"Fail to reject H0","Reject H0")
cint<-c("- \U221E",xbar+abs(z_c)*(sigma/sqrt(n)))
op<-c(z_t,z_c,conc,pvalue,cint)
op
}

> t_lt(0,-1.275,0.377,4,0.05)
[1] "-6.76392572944297" "-2.35336343480183" "Reject H0" "0.00330129528999329"
[5] "- ∞" "-0.831390992539856"

Two-sample unequal variance t-test :

> mean(w)
[1] 4.55
> mean(h)
[1] 5.825
> sd(w)
[1] 1.075484
> sd(h)
[1] 1.291962

Function for computing t-test :

twosamuneq <- function(xbar1,xbar2,s1,s2,n1,n2,alpha)
{
t1<-s1^2/n1
t2<-s2^2/n2
nr<-(t1+t2)^2
t3<-(t1^2)/(n1-1)
t4<-(t2^2)/(n2-1)
dr<-t3+t4
d<-nr/dr
s<-(t1+t2)^0.5
z_t<-(xbar1-xbar2)/s
z_c<-qt(1-(0.5*alpha),df=d)
pvalue <- 2*(1-pt(abs(z_t),df=d))
conc<-ifelse(pvalue>alpha,"Fail to reject H0","Reject H0")
cint<-c(xbar1-xbar2-z_c*s,xbar1-xbar2+z_c*s)
moe<-z_c*s
op<-c(z_t,z_c,conc,pvalue,cint,d,s,moe)
op
}

p-value :

> 1-pt(-1.517,df=6)
[1] 0.9099711