Question

Consider the following hypothesis test.

H0: μ1−μ2 = 0
Ha: μ1−μ2 ≠ 0
The following results are for two independent samples taken from two populations.
Sample 1 Sample 2
N1 = 80 n2 = 70
X1 = 104 x1 = 106
δ1 = 8.4 δ2 = 7.6

Enter negative values as negative numbers.
a. What is the value of the test statistic?

Answer 1

The value of the test statistic (Z-score) for two independent samples with known population standard deviations is approximately -2.61.

Step-by-step explanation:

To calculate the value of the test statistic for comparing two independent sample means with known population standard deviations, we use the formula:

Z = (X₁ - X₂) - (μ₁ - μ₂) ÷ √((δ₁²/n₁) + (δ₂²/n₂))

Given:

• 
  
• n₁ = 80, n₂ = 70 (Sample sizes)
• 
  
• X₁ = 104, X₂ = 106 (Sample means)
• 
  
• δ₁ = 8.4, δ₂ = 7.6 (Population standard deviations)

Since the null hypothesis (H₀) states that μ₁ - μ₂ = 0, we can simplify the Z formula to:

Z = (104 - 106) ÷ √((8.4²/80) + (7.6²/70))

Performing the calculations gives:

Z = -2 ÷ √((70.56/80) + (57.76/70))

Z = -2 ÷ √(0.882 + 0.825)

Z = -2 ÷ √(1.707)

Z = -2 ÷ 1.3068

Z = -2.6136

The value of the test statistic (Z-score) is approximately -2.61.