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.

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.

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂    H₀: μ₁ = μ₂; H₁: μ₁ > μ₂H₀: 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 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