Question

A random sample of

*n*_{1} = 49

measurements from a population with population standard deviation

*σ*_{1} = 5

had a sample mean of

*x*_{1} = 8.

An independent random sample of

*n*_{2} = 64

measurements from a second population with population standard deviation

*σ*_{2} = 6

had a sample mean of

*x*_{2} = 11.

Test the claim that the population means are different. Use level of significance 0.01.(a) Check Requirements: What distribution does the sample test statistic follow? Explain.

The standard normal. We assume that both population
distributions are approximately normal with unknown standard
deviations.The Student's *t*. We assume that both population
distributions are approximately normal with unknown standard
deviations. The standard normal. We assume
that both population distributions are approximately normal with
known standard deviations.The Student's *t*. We assume that
both population distributions are approximately normal with known
standard deviations.

(b) State the hypotheses.

*H*_{0}: *μ*_{1} =
*μ*_{2}; *H*_{1}:
*μ*_{1} <
*μ*_{2}*H*_{0}:
*μ*_{1} ≠ *μ*_{2};
*H*_{1}: *μ*_{1} =
*μ*_{2} *H*_{0}:
*μ*_{1} = *μ*_{2};
*H*_{1}: *μ*_{1} >
*μ*_{2}*H*_{0}:
*μ*_{1} = *μ*_{2};
*H*_{1}: *μ*_{1} ≠
*μ*_{2}

(c) Compute

*x*_{1} − *x*_{2}.

*x*_{1} − *x*_{2} =

Compute the corresponding sample distribution value. (Test the
difference *μ*_{1} − *μ*_{2}. Round
your answer to two decimal places.)

(d) Find the *P*-value of the sample test statistic. (Round
your answer to four decimal places.)

(e) Conclude the test.

At the *α* = 0.01 level, we reject the null hypothesis
and conclude the data are not statistically significant.At the
*α* = 0.01 level, we reject the null hypothesis and conclude
the data are statistically significant. At
the *α* = 0.01 level, we fail to reject the null hypothesis
and conclude the data are not statistically significant.At the
*α* = 0.01 level, we fail to reject the null hypothesis and
conclude the data are statistically significant.

(f) Interpret the results.

Reject the null hypothesis, there is sufficient evidence that there is a difference between the population means.Fail to reject the null hypothesis, there is insufficient evidence that there is a difference between the population means. Reject the null hypothesis, there is insufficient evidence that there is a difference between the population means.Fail to reject the null hypothesis, there is sufficient evidence that there is a difference between the population means.

Answer #1

a)

The standard normal. We assume that both population distributions are approximately normal with known standard deviations

b)

H0: μ1 = μ2; H1: μ1 ≠ μ2

c)

x1 -x2 = 8 - 11 = -3

Pooled Variance

sp = sqrt(s1^2/n1 + s2^2/n2)

sp = sqrt(25/49 + 36/64)

sp = 1.0357

Test statistic,

z = (x1bar - x2bar)/sp

z = (8 - 11)/1.0357

z = -2.90

d)

P-value Approach

P-value = 0.0037

e)

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

f)

Reject the null hypothesis, there is sufficient evidence that there
is a difference between the population means

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An independent random sample of
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σ2 = 4
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x2 = 15.
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level of significance 0.01.
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deviation
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(a) Check Requirements: What distribution does the sample test
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