Question

We are given that n=15, the sample mean Ῡ=2.5, the sample
standard deviation s=1.5 and random variable Y distributed Normal
with mean µ and variance σ^{2}, where both µ and
σ^{2} are unknown and we are being concentrated on testing
the following set of hypothesis about the mean parameter of the
population of interest.

We are to test:

H_{0} : µ ≥ 3.0 versus H_{1} : µ < 3.0.

Compute the following:

**a)** P- value of the test

**b)** Probability of making Type II
error and the power of this test at µ= 2.0

Answer #1

a) Test for single mean:

Test Statistic:

The p-value is p = 0.1088

b) P(type II erro) and Power test calculations:

We are given that n=15, the sample mean Ῡ=2.5, the sample
standard deviation s=1.5 and random variable Y distributed Normal
with mean µ and variance σ2, where both µ and
σ2 are unknown and we are being concentrated on testing
the following set of hypothesis about the mean parameter of the
population of interest.
We are to test:
H0 : µ ≥ 3.0 versus H1 : µ < 3.0.
Compute the following:
a) P- value of the test
b) ...

We are given that n=15, the sample mean Ῡ=2.5, the sample
standard deviation s=1.5 and random variable Y distributed Normal
with mean µ and variance σ2, where both µ and
σ2 are unknown and we are being concentrated on testing
the following set of hypothesis about the mean parameter of the
population of interest.
We are to test:
H0 : µ ≥ 3.0 versus H1 : µ < 3.0.
Compute the following:
a) P- value of the test
b) ...

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
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We are given that P = population proportion of persons 20-30
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H0: p≤0.67 versus H1: p>0.67
The observed sample proportion is pˆp^ = 61/79 =
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compute the following:
a) P- value of the test
b) Probability of making Type II error and the
power of this test at p= 0.87

Suppose the values
1.39 2.22 2.38 1.60 1.50
are a random sample from a distribution assumed to be Normal but
for which the mean and variance are unknown.
(a) Give a 95% confidence interval for the mean.
(b) Test the hypotheses H0: μ = 2.5 versus H1: μ ̸= 2.5 at the
0.05 significance level.
(c) Give the p-values for testing the following hypotheses: i.
H0: μ=2.5versusH1: μ̸=2.5 ii. H0: μ≤2.5versusH1: μ>2.5 iii. H0:
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Which would result in the highest probability of a Type II
error?
µ = 42; n = 100
µ = 42; n = 10
µ = 41; n = 100
µ = 41; n = 10
µ = 40.9; n = 15
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Suppose that we will take a random sample of size n
from a population having mean µ and standard deviation σ.
For each of the following situations, find the mean, variance, and
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(a) µ = 20, σ = 2, n = 41
(Round your answers of "σ" and "σ2" to 4 decimal
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µ
σ2
σ
(b) µ = 502, σ = .7, n = 132
(Round your answers of...

Suppose we have a random sample of size 50 from a N(μ,σ2) PDF.
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Calculate the test statistic from your sample mean. Then
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A sample is drawn from a population and we estimate that the
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