Question

A random sample of

n₁ = 49

measurements from a population with population standard deviation

σ₁ = 5

had a sample mean of

x₁ = 11.

An independent random sample of

n₂ = 64

measurements from a second population with population standard deviation

σ₂ = 6

had a sample mean of

x₂ = 14.

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 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.

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

(b) State the hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂    

H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂

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.

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

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

(f) Interpret the results.

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.    

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.

Answer 1

Answer)

A)

As the population s.d is known here we will use z test

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

B)

H0: μ1 = μ2; H1: μ1 ≠ μ2

C)

Test statistics is = (x1-x2)/(standard error

Standard error = √{(s1^2/n1)+(s2^2/n2)}

S1 = 5, s2 = 6

All the values are given in the question

After substitution

Test statistics is = -2.9

D)

From z table, p(z<-2.9) = 0.0019

But our test is two tailed so, p-value is = 2*0.0019 = 0.0038

E)

As p-value is less than 0.01

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.