Have you ever noticed that, when you tear a fingernail, it tends to tear to the...

Have you ever noticed that, when you tear a fingernail, it tends to tear to the side and not down into the finger? (Actually, the latter doesn’t bear too much thinking about.). Why might this be so? One possibility is that fingernails are tougher in one direction than another. A study of the toughness of human fingernails compared the toughness of nails along a transverse dimension (side to side) compared with a longitudinal direction, with 15 measurements of each (Farren et al., 2004). The toughness of fingernails along a transerve direction averaged 3.3 kJ/m2, with a standard deviation of 0.95, while the mean toughness along the longitudinal direction was 6.2 kJ/m2, with a standard deviation of 1.48 kJ/m2. a) Test for a significant difference in the toughness of these fingernails along two dimensions. b) As it turns out, all of the fingernails in this study came from the same volunteer. Discuss what the conclusion in part (a) means. What would be required to describe the fingernail toughness of all humans?

Expert Solution

Given that,
mean(x)=3.3
standard deviation , s.d1=0.95
number(n1)=15
y(mean)=6.2
standard deviation, s.d2 =1.48
number(n2)=15
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.145
since our test is two-tailed
reject Ho, if to < -2.145 OR if to > 2.145
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =3.3-6.2/sqrt((0.9025/15)+(2.1904/15))
to =-6.3865
| to | =6.3865
critical value
the value of |t α| with min (n1-1, n2-1) i.e 14 d.f is 2.145
we got |to| = 6.38647 & | t α | = 2.145
make decision
hence value of | to | > | t α| and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -6.3865 ) = 0
hence value of p0.05 > 0,here we reject Ho
ANSWERS
---------------
a.
null, Ho: u1 = u2
alternate, H1: u1 != u2
test statistic: -6.3865
critical value: -2.145 , 2.145
decision: reject Ho
p-value: 0
we have enough evidence to support the claim that significant difference in the toughness of these fingernails along two dimensions

b. yes,
all of the fingernails in this study came from the same volunteer.
One possibility is that fingernails are tougher in one direction than another.
A study of the toughness of human fingernails compared the toughness of nails along a transverse dimension (side to side) compared with a longitudinal direction,
with 15 measurements of each (Farren 2004)

orchestra answered 1 year ago

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Have you ever noticed that, when you tear a fingernail, it tends to tear to the...

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