Question

From previous studies, it has been generally believed that Northern Hemisphere icebergs have a mean depth of 270 meters. An environmentalist has suggested that global warming has caused icebergs to have greater depth. A team of scientists visiting the Northern Hemisphere observed a random sample of 41 icebergs. The depth of the base of the iceberg below the surface was carefully measured for each. The sample mean and standard deviation were calculated to be 276 meters and 20 meters respectively.

a)What is/are the parameters of interest relevant to this hypothesis test? Choose all parameters that you use to set up the null and alternative hypotheses, as well as those referenced in the assumptions and derivation of the relevant test statistic.

A. The mean depth (in m) of the 41 icebergs in the study.
B. The mean depth (in m) of all Northern Hemisphere icebergs.
C. 270 meters
D. 41
E. None of the above

b)Compute the test statistic (Please round your answer to three decimal places)

c)Which of the following ranges the P-value must lie in? [You will need the t-table to answer this question.]

A. <0.005
B. 0.05-0.01
C. 0.01-0.025
D. 0.025-0.05
E. 0.05-0.10
F. >0.10

Part d) Based on the PP-value that was obtained, you would (Select all that apply):

A. believe the null hypothesis is true.
B. fail to reject the null hypothesis at all.
C. reject the null hypothesis at α=0.05α=0.05 level of significance
D. neither reject nor accept the null hypothesis.
E. reject the null hypothesis at α=0.1α=0.1 level of significance
F. None of the above

Answer 1

Given that,
population mean(u)=270
sample mean, x =276
standard deviation, s =20
number (n)=41
null, Ho: μ=270
alternate, H1: μ!=270
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.021
since our test is two-tailed
reject Ho, if to < -2.021 OR if to > 2.021
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =276-270/(20/sqrt(41))
to =1.9209
| to | =1.9209
critical value
the value of |t α| with n-1 = 40 d.f is 2.021
we got |to| =1.9209 & | t α | =2.021
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 != 1.9209 ) = 0.0619
hence value of p0.05 < 0.0619,here we do not reject Ho
ANSWERS
---------------
a.
null, Ho: μ=270
alternate, H1: μ!=270
option:c
The mean depth (in m) of all Northern Hemisphere icebergs is 270m
b.
test statistic: 1.9209
critical value: -2.021 , 2.021
decision: do not reject Ho
c.
p-value: 0.0619
option:E
d.
i.
we do not have enough evidence to support the claim that Northern Hemisphere icebergs have a mean depth of 270 meters.

ii.
level of significance 0.10
Given that,
population mean(u)=270
sample mean, x =276
standard deviation, s =20
number (n)=41
null, Ho: μ=270
alternate, H1: μ!=270
level of significance, α = 0.1
from standard normal table, two tailed t α/2 =1.684
since our test is two-tailed
reject Ho, if to < -1.684 OR if to > 1.684
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =276-270/(20/sqrt(41))
to =1.9209
| to | =1.9209
critical value
the value of |t α| with n-1 = 40 d.f is 1.684
we got |to| =1.9209 & | t α | =1.684
make decision
hence value of | to | > | t α| and here we reject Ho
p-value :two tailed ( double the one tail ) - Ha : ( p != 1.9209 ) = 0.0619
hence value of p0.1 > 0.0619,here we reject Ho
ANSWERS
---------------
null, Ho: μ=270
alternate, H1: μ!=270
test statistic: 1.9209
critical value: -1.684 , 1.684
decision: reject Ho
p-value: 0.0619
we have enough evidence to support the claim that Northern Hemisphere icebergs have a mean depth of 270 meters.