Question

Test whether mu 1 less than mu 2 at the alphaequals0.02 level of significance for the sample data shown in the accompanying table. Assume that the populations are normally distributed. Population 1 Population 2 n 32 25 x overbar 103.4 114.5 s 12.2 13.2

Answer 1

We have assumed variances are unequal

Step 1: Set up null and alternative hypotheses.

H₀: μ₁ = μ₂ (Note: H₀: μ₁ = μ₂ ⇔ H₀: μ₁ - μ₂ = 0)
H₁: μ₁ < μ₂ (Note: H₁: μ₁ < μ₂ ⇔ H₁: μ₁ - μ₂ < 0)

Step 2: Determine α (level of significance of hypothesis test).

α = 0.02 (Note: α = level of significance of hypothesis test = probability of making Type I error.)

Step 3: Calculate test statistic using x̅₁, x̅₂, σ₁, σ₂, n₁, and n₂;

x̅₁ for population 1: 103.4
s₁ (population 1 standard deviation) = 12.2
n₁ (sample size 1) = 32
x̅₂ for population 2: 114.5
s₂ (population 2 standard deviation) = 13.2
n₂ (sample size 2) = 25
(Note: From Step 1, we have H₀: μ₁ = μ₂ ⇔ H₀: μ₁ - μ₂ = 0; therefore, μ₁ - μ₂ = 0)

Test Statistic = -3.256147

Step 4: Determine P-value

Using the test statistic in Step 3 as a t-score, we will find the left-tailed area corresponding to this t-score.
(Note: We use the t-distribution since the population stanadard deviations (σ₁ and σ₂) are unknown.)

Left-tailed area corresponding to t-score of -3.26 = 0.001676
P-value = Left-Tailed Area = 0.001676

Step 5: Make Decision

In this case, p value (0.001676) < α (0.02)

If p-value is less than or equal to α then reject the null hypothesis.