Question

Body fat and lean body mass can be estimated in living animals by measuring the total body electrical conductivity (TOBEC). This technique is useful when attempting to determine if "diet drugs" are working on laboratory rats over time

– the researchers need not sacrifice the animals to measure the amount of adipose body fat after periods of drug usage. However, the procedure requires that the animal be totally inside the measurement chamber, which is fairly small. Some, but not all the rats were so big that their tails had to be tucked under their bodies before putting them into the measurement chamber. This is causing concern among the researchers, because there is a possibility the tail position might alter the measurements. To see if the TOBEC measurement is altered by the tail position of the rats, an experiment is run on 16 rats ranging in weight from 210 to 505 g.

The rats were randomly assigned the order of measurement, with 8 rats measured in the tail-under position first, and 8 rats in the tail-extended position. The data for the lean mass measures are in the table below.

Test the hypothesis that there is no difference between the population mean TOBEC measurements of rats in the tail- under vs. the tail-extended positions. Conduct hypothesis test using =.05.

TOBEC Measures of Lean Mass (g)

Tail Under	Tail Extended
193.5	203.9
200.3	215.8
211.4	228.5
231.1	241.0
249.1	259.4
266.4	270.2
274.8	282.6
277.5	276.9
276.4	286.5
275.6	291.8
300.7	303.5
300.0	306.7
303.6	303.7
305.6	316.8
302.4	296.1
354.2	356.1

Answer 1

Given that,
mean(x)=270.1625
standard deviation , s.d1=43.6934
number(n1)=16
y(mean)=277.4688
standard deviation, s.d2 =39.9444
number(n2)=16
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.131
since our test is two-tailed
reject Ho, if to < -2.131 OR if to > 2.131
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =270.1625-277.4688/sqrt((1909.1132/16)+(1595.55509/16))
to =-0.4937
| to | =0.4937
critical value
the value of |t α| with min (n1-1, n2-1) i.e 15 d.f is 2.131
we got |to| = 0.49367 & | t α | = 2.131
make decision
hence value of |to | < | t α | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -0.4937 ) = 0.629
hence value of p0.05 < 0.629,here we do not reject Ho
ANSWERS
---------------
null, Ho: u1 = u2
alternate, H1: u1 != u2
test statistic: -0.4937
critical value: -2.131 , 2.131
decision: do not reject Ho
p-value: 0.629
we do not have enough evidence to support the claim that here is no difference between the population mean TOBEC measurements of rats in the tail- under vs. the tail-extended positions.