Question

The null and alternate hypotheses are:

H₀ : μ₁ = μ₂
H₁ : μ₁ ≠ μ₂

A random sample of 10 observations from one population revealed a sample mean of 23 and a sample standard deviation of 3.5. A random sample of 4 observations from another population revealed a sample mean of 27 and a sample standard deviation of 3.6.

At the 0.01 significance level, is there a difference between the population means?

1. State the decision rule. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

2. Compute the pooled estimate of the population variance. (Round your answer to 3 decimal places.)

3. Compute the test statistic. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

4. State your decision about the null hypothesis.

• Do not reject H₀.

• Reject H₀.

5. The  p-value is

• between 0.1 and 0.05

• less than 0.001

• between 0.02 and 0.05

• between 0.001 and 0.01

• between 0.1 and 0.2

Answer 1

Given that,

For population 1 : n1 = 10, x1-bar = 23 and s1 = 3.5

For population 2 : n2 = 4, x2-bar = 27 and s2 = 3.6

The null and alternative hypotheses are,

H0 : μ1 = μ2

H1 : μ1 ≠ μ2

a) Degrees of freedom = 10 + 4 - 2 = 12

t-critical values at significance level of 0.01 with 12 degrees of freedom are, tcrit = ± 3.055

Decision Rule : Reject H0, if t < -3.055 or t > 3.055

b) Pooled estimate of the population variance is,

=> Pooled variance = 12.428

c) Test statistic is,

=> Test statistic = t = -1.918

d) Since, test statistic = -1.918 > -3.055, we Do not reject H0.

e) p-value is lies between 0.1 and 0.05