Question

A sample from a Normal distribution with an unknown mean µ and
known variance

σ = 45 was taken with n = 9 samples giving sample mean of ¯ y =
3.6.

(a) Construct a Hypothesis test with significance level α = 0.05 to
test whether the

mean is equal to 0 or it is greater than 0. What can you conclude
based on the

outcome of the sample?

(b) Calculate the power of this test if the true value of the mean
is µ = 3.

(c) Construct the Most Powerful Hypothesis test with significance
level α = 0.05 for

testing H _{0} : µ = 0 vs H _{a} : µ = 3.

(d) Is this test also Uniformly Most Powerful Hypothesis test for
testing H _{0} : µ = 0

vs H _{a} : µ > 0? Justify.

Answer #1

A sample data is drawn from a normal population with unknown
mean µ and known standard deviation σ = 5. We run the test µ = 125
vs µ < 125 on a sample of size n = 100 with x = 120. Will you be
able to reject H0 at the signiﬁcance level of 5%?
A. Yes
B. No
C. Undecidable, not enough information
D. None of the above

Suppose X1, · · · , Xn from a normal distribution N(µ, σ2 )
where µ is unknown but σ is known. Consider the following
hypothesis testing problem:
H0 : µ = µ0 vs. Ha : µ > µ0
Prove that the decision rule is that we reject H0 if
X¯ − µ0 σ/√ n > Z(1 − α),
where α is the significant level, and show that this is
equivalent to rejecting H0 if µ0 is less than the...

Problem 1. A random sample was taken from a normal population
with unknown mean µ and unknown variance σ2 resulting in
the following data. 10.66619, 10.71417, 12.12580, 12.09377,
12.04136, 12.17344, 11.89565, 12.82213, 10.13071, 12.35741,
12.48499, 11.79630, 11.32066, 13.53277, 11.20579, 11.39291,
12.39843 Perform a test of H0 : µ = 10 versus
H1 : µ ≠10 at the .05 level. Do the data support the
null hypothesis?

Let X1, X2, . . . , Xn be a random sample from the normal
distribution N(µ, 36). (a) Show that a uniformly most powerful
critical region for testing H0 : µ = 50 against H1 : µ < 50 is
given by C2 = {x : x ≤ c}. Find the values of c for α = 0.10.

Suppose we observe Y1,...Yn from a normal distribution with
unknown parameters such that ¯ Y = 24, s2 = 36, and n = 15.
(a) Find the rejection region of a level α = 0.05 test of H0 : µ
= 20 vs. H1 : µ 6= 20. Would this test reject with the given
data?
(b) Find the rejection region of a level α = 0.05 test of H0 : µ
≤ 20 vs. H1 : µ > 20....

To test H0: µ = 42.0 vs.
HA: µ ≠ 42.0, a sample of
n = 40
will be taken from a large population with σ=
9.90.
H0 will be rejected if the sample
mean is less than 40.3 or greater than
43.7.
Find and state the level of significance, α, to
three (3) places of decimal.

A random sample is selected from a normal population
with a mean of μ=50 and a standard deviation of σ=12. After a
treatment is administered to the individuals in the sample, the
sample mean is found to be M=55.
a. If the sample consists of n=16 scores, is the sample
mean sufficient to conclude that the treatment has a significant
effect? Use a two-tailed test with α =0.05. b. If the sample
consists of n=36 scores, is the sample mean...

Use µ = 2.5 and σ = 1 to generate many independent sets of ten
random numbers, each.
If I were to repeatedly carry out the one-sample t-test and
significance level α = 0.05 to test the hypothesis that the mean µ
used to generate those random numbers was µ = 2.5 for each new
vector of ten numbers, how often, on average, would you expect to
reject the null hypothesis? Explain how you come to your
conclusion.

A random
sample is selected from a normal population with a mean of µ = 30
and a standard deviation of σ= 8. After a treatment is administered
to the individuals in the sample, the sample mean is found to be x̅
=33.
Furthermore,
if the sample consists of n = 64 scores, is the sample
mean sufficient to conclude that the treatment has a significant
effect? Use a two-tailed test with α = .05.
4a. Which of
the following...

A sample of size 20 is drawn from a normal distribution with
unknown variance and unknown mean µ. The sample mean is ¯ x = 1.25
and the sample standard deviation is s = 0.25. The 99% lower
conﬁdence bound on µ is
A. 1.108 ≤ µ
B. 1.120 ≤ µ
C. µ ≤ 1.392
D. µ ≤ 1.380

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