Question

7. Suppose you are testing H0 : µ = 10 vs H1 : µ 6= 10. The sample is small (n = 5) and the data come from a normal population. The variance, σ 2 , is unknown. (a) Find the critical value(s) corresponding to α = 0.10. (b) You find that t = −1.78. Based on your critical value, what decision do you make regarding the null hypothesis (i.e. do you Reject H0 or Do Not Reject H0)?

Answer #1

3. Suppose you are testing H0 : = 10 vs H1 : 6= 10: The sample
is small (n = 5) and the data come from a normal population. The
variance, 2, is unknown. (a) Find the critical value(s)
corresponding to = 0:10. (b) You find that t = -1.78. Based on your
critical value, what decision do you make regarding the null
hypothesis (i.e. do you Reject H0 or Do Not Reject H0)?

1. For testing H0 : µ = 0 vs. Ha : µ > 0, H0 is rejected if X
>¯ 1.645, given n = 36 and σ = 6. What is the value of α, i.e.,
maximum probability of Type I error?
A. 0.90 B. 0.10 C. 0.05 D. 0.01
2. For testing H0 : µ = 0 vs. Ha : µ > 0, H0 is rejected if X
>¯ 1.645, given n = 36 and σ = 6. What...

Suppose that we wish to test H0: µ = 20 versus
H1: µ ≠ 20, where σ is known to equal 7. Also, suppose
that a sample of n = 49 measurements randomly selected
from the population has a mean of 18.
Calculate the value of the test statistic Z.
By comparing Z with a critical value, test
H0 versus H1 at α = 0.05.
Calculate the p-value for testing H0 versus
H1.
Use the p-value to test H0 versus...

H0: µ ≥ 205 versus
H1:µ < 205, x= 198,
σ= 15, n= 20, α= 0.05
test
statistic___________ p-value___________ Decision
(circle one) Reject
the
H0 Fail
to reject the H0
H0: µ = 26 versus
H1: µ<> 26,x= 22,
s= 10, n= 30, α= 0.01
test
statistic___________ p-value___________ Decision
(circle one) Reject
the
H0 Fail
to reject the H0
H0: µ ≥ 155 versus
H1:µ < 155, x= 145,
σ= 19, n= 25, α= 0.01
test
statistic___________ p-value___________ Decision
(circle one) Reject
the
H0 Fail
to reject the H0

Assuming that, in testing H0:
μ
=20 vs. H1
μ
≠20, you decide on the critical region X bar ≤ 15 and
X bar ≥ 25. Assume X is normally distributed, σ
2
= 25, and the following four random values
are observed: 9, 20, 15, 11.
a) Would you accept or reject H
0
?
b) What level of
α
is assumed here?
c) What probability value would you report?
d) What would be the appropriate critical region for...

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...

For the following hypotheses
H0 : µ ≤ µ0 vs Ha : µ > µ0
performed at the α significance level, the corresponding
confidence interval that would included all the µ0 values for which
one would fail to reject the null is
(a) 100(1 − α)% two-sided confidence interval
(b) 100(1 − α)% one-sided confidence interval with only upper
limit, i.e. (−∞, U)
(c) 100(1 − α)% one-sided confidence interval with only lower
limit, i.e. (L, ∞)

Suppose that you are testing the following hypotheses where the
variance is unknown: H0 : µ = 100 H0 : µ ≠ 100 The sample size is n
20. Find bounds on the P-value for the following values of the test
statistic. a. t0 = 2.75 b. t0 = 1.86 c. t0 = -2.05 d. t0 =
-1.86

You are testing H0: µ1 - µ 2 = 0 vs H1: µ1 - µ2 ≠ 0. In doing
so, you are choosing n1 observations from population 1, and n2
observations from population 2. Keeping the same total N = 100 = n1
+ n2, compute the power of the test to detect a difference of 0.6,
with a population standard deviation of 1.0 when: n1 = 0.1×n2, n1 =
0.2×n2, n1 = 0.3×n2, …, n1 = 1.0×n2, n1 =...

1. In testing a null hypothesis H0 versus an alternative Ha, H0
is ALWAYS rejected if
A. at least one sample observation falls in the non-rejection
region.
B. the test statistic value is less than the critical value.
C. p-value ≥ α where α is the level of significance. 1
D. p-value < α where α is the level of significance.
2. In testing a null hypothesis H0 : µ = 0 vs H0 : µ > 0,
suppose Z...

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