Question

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
H_A: μ₁ − μ₂ < 0

x−1x−1 = 242	x−2x−2 = 262
s1 = 28	s2 = 29
n1 = 14	n2 = 14

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.)

a-2. Find the p-value.

• 0.05 p-value < 0.10
• p-value 0.10

• p-value < 0.01

• 0.01 p-value < 0.025
• 0.025 p-value < 0.05

a-3. Do you reject the null hypothesis at the 5% level?

• Yes, since the value of the p-value is greater 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 less than the significance level.

• No, since the value of the p-value is less than the significance level.

a-4. Interpret the results at αα = 0.05.

• We cannot conclude that the population means differ.

• We conclude that the population means differ.

• We cannot conclude that population mean 1 is less than population mean 2.

• We 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.)

b-2. Find the p-value.

• 0.05 p-value < 0.10
• p-value 0.10

• p-value < 0.01

• 0.01 p-value < 0.025
• 0.025 p-value < 0.05

b-3. Do you reject the null hypothesis at the 5% level?

• Yes, since the value of the p-value is greater than the significance 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.

b-4. Interpret the results at αα = 0.05.

• We cannot conclude that the population means differ.

• We conclude that the population means differ.

• We cannot conclude that population mean 1 is less than population mean 2.

• We conclude that population mean 1 is less than population mean 2.

Answer 1

a-1)

x1    =	242.000	x2      =	262.000
s1    =	28.000	s2      =	29.000
n1    =	14	n2     =	14
Pooled Variance Sp2=((n1-1)s21+(n2-1)*s22)/(n1+n2-2)=				812.5000

standard error se =Sp*√(1/n1+1/n2)=			10.7736
test stat t =(x1-x2-Δo)/Se=		-1.856

a-2)

0.025 p-value < 0.05

a-3)

Yes, since the value of the p-value is less than the significance level.

a-4)

We conclude that population mean 1 is less than population mean 2.

b-1)

standard error se=√(S21/n1+S22/n2)=			10.774
test stat t =(x1-x2-Δo)/Se=		-1.856

b-2)

0.025 p-value < 0.05

b-2)

Yes, since the value of the p-value is less than the significance level.

b-4)

We conclude that population mean 1 is less than population mean 2.