In: Statistics and Probability
1. A
university researcher is
interested in whether
recent recruitment efforts
have changed the
type of students
admitted to the
university. To test
this, she randomly
selects 50 freshmen
from the university
and records their
high school GPA.
The mean is 2.90
with a standard
deviation of 0.70.
The researcher also
knows that the mean
high school GPA of
all freshmen enrolled
at the university
five years ago was
2.75 with a
standard deviation of
0.36. The researcher
wants to know if
the high school GPA
of current freshmen
at the university
is different than
that of freshmen
from five years
ago.
(a) What are
the null and
alternative hypotheses in
this study (stated
mathematically)?
(b) Should the
researcher use a
one-tailed or a
two-tailed test?
(c) Compute the
appropriate test statistic
for testing the
hypothesis.
(d) Using α
= 0.05, what do
you conclude about
the high school GPA
of current freshman?
Be sure to include
a discussion of the
critical value in
your answer.
(e) What type
of error might the
researcher be making
in part (d)?
2. A researcher
believes that smoking
worsens a person’s
sense of smell. To
test this, he takes
a sample of 25
smokers and gives
them a test of
olfactory sensitivity. In
this test, higher
scores indicate greater
sensitivity. For his
sample, the mean
score on the test
is 15.1 with a
standard deviation of
1.2. The researcher
knows the mean
score in the
population is 15.5,
but the population
standard deviation is
unknown.
(a) What are
the null and
alternative hypotheses in
this study (stated
mathematically)?
(b) Should the
researcher use a
one-tailed or a
two-tailed test?
(c) Compute the
appropriate test statistic
for testing the
hypothesis.
(d) Using α
= 0.01, do you
conclude that smoking
affects a person’s
sense of smell? Be
sure to include a
discussion of the
critical value in
your answer.
(e) What type
of error might the
researcher be making
in part (d)?
1)
Ho : µ = 2.75
Ha : µ ╪ 2.75
b) two tailed
c)
Level of Significance , α =
0.050
population std dev , σ =
0.3600
Sample Size , n = 50
Sample Mean, x̅ = 2.9000
' ' '
Standard Error , SE = σ/√n = 0.3600 / √
50 = 0.0509
Z-test statistic= (x̅ - µ )/SE = (
2.900 - 2.75 ) /
0.0509 = 2.946
d)
critical z value, z* = ± 1.960
Decision: test stat > 196 Reject null hypothesis
Conclusion: There is enough evidence to conclude
that high school GPA
of current freshmen
at the university
is different than
that of freshmen
from five years ago.
e)
since, we reject the null hypothesis, so, there might be type I error