Question

A random sample of n₁ = 49 measurements from a population with population standard deviation σ₁ = 5 had a sample mean of x₁ = 8. An independent random sample of n₂ = 64 measurements from a second population with population standard deviation σ₂ = 6 had a sample mean of x₂ = 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.

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

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

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

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

(b) State the hypotheses.

a. H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

b. H₀: μ₁ = μ₂; H₁: μ₁ > μ₂   

c. H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂

d. H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

(c) Compute x₁ − x₂.

x₁ − x₂ =

Compute the corresponding sample distribution value. (Test the difference μ₁ − μ₂. 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.

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

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

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

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

(f) Interpret the results.

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

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

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

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

Answer 1

I have answered the question below

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Answer:

H₀ :-  

H₁ :-  

Test Statistic :-

= (8 -11)/sqrt(25/49 + 36/64)

Z = -2.8966

Test Criteria :-

Reject null hypothesis if

Result :- Reject Null Hypothesis as 2.89655>2.58

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

Part b) H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

Part c) 11 - 13 = - 2

Z stat = -2.8966

Part d) P value = P ( Z < - 2.8966 ) = 0.0019

Part e)   Result :- Reject Null Hypothesis

Reject null hypothesis and conclude that data are statistically significant

Part f)

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