Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. (You may find it useful to reference the appropriate table: z table or t table)
̐̄H₀: μ₁ - μ₂ ≥ 0
Hᐢ: μ₁ - μ₂ < 0
x̄₁ = 254
x̄₂ = 262
s₁ = 35
s₂ = 23
n₁ = 8
n₂ = 8
a-1. Calculate the value of the test statistic under the assumption that the population variances are equal. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
Test statistic
a-2. Find the p-value.
p-value < 0.01
0.01 ≤ p-value < 0.025
0.025 ≤ p-value < 0.05
0.05 ≤ p-value < 0.10
p-value ≥ 0.10
a-3. Do you reject the null hypothesis at the 1% level?
Yes, since the value of the p-value is less than the significance level.
No, since the value of the p-value is less than the significance level.
No, since the value of the p-value is greater than the significance level.
Yes, since the value of the p-value is greater than the significance level.
a-4. Interpret the results at α = 0.01.
We conclude that the population means differ.
We cannot conclude that the population means differ.
We conclude that population mean 1 is less than population mean 2.
We cannot conclude that population mean 1 is less than population mean 2.
b-1. Calculate the value of the test statistic under the assumption that the population variances are unknown and are not equal. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
Test statistic