Question

A random sample of

n_{1} = 49

measurements from a population with population standard deviation

σ_{1} = 3

had a sample mean of

x_{1} = 13.

An independent random sample of

n_{2} = 64

measurements from a second population with population standard deviation

σ_{2} = 4

had a sample mean of

x_{2} = 15.

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

(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 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 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.

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 sufficient 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

Given:

n_{1} = 49 σ1 = 3 1
= 13.

n_{2} = 64 σ2 = 4
2 = 15

Claim : The population means are different.

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

Answer - option A.

b) Hypothesis test :

*H*_{0}: μ_{1} = μ_{2}

_{H1}: μ_{1} ≠ μ_{2}

_{c)
1 -
2 = 13 - 15 = -2}

_{}

E) Since P-value < significance level 0.01, we reject the null hypothesis.

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

f) Interpretation :

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

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