Question

The null and alternate hypotheses are:

H₀ : μ₁ = μ₂
H₁ : μ₁ ≠ μ₂

A random sample of 11 observations from one population revealed a sample mean of 23 and a sample standard deviation of 4.6. A random sample of 8 observations from another population revealed a sample mean of 28 and a sample standard deviation of 3.6.

At the 0.05 significance level, is there a difference between the population means?

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

4. State your decision about the null hypothesis.

• Reject H₀.

• Do not reject H₀.

5. The  p-value is

• between 0.05 and 0.02

• less than 0.001

• between 0.05 and 0.1

• between 0.001 and 0.01

• between 0.1 and 0.2

Answer 1

n 1 = 11

n 2=8

s1 = 4.6

s2 = 3.6

a) decision rule : >2.11,we reject null hypothesis

b) pooled estimate =

c)

Null and alternative hypothesis is

Vs

Level of significance = 0.05

Before doing this test we have to check population variances are equal or not.

Null and alternative hypothesis is

Vs

Test statistic is

F = Larger variance / Smaller variance = 21.16 / 12.96 = 1.6327

Degrees of freedoms

Degrees of freedom for numerator = n1 - 1 = 11 - 1 = 10

Degrees of freedom for denominator = n2 - 1 = 8 - 1 = 7

Critical value = 3.637                ( using f-table )

F test statistic < critical value    we fail to reject null hypothesis.

Conclusion: Population variances are equal.

So we have to use pooled variance.

Test statistic

Formula

d.f = n1 + n2 – 2 = 11+8-2=17

critical value =2.11

d)

critical value < ,Reject Ho

Reject H₀.

e)  

p-value = 0.0206 ( using t table )

The p-value is between 0.05 and 0.02

p-value , Reject H0

conclusion : There is a difference between the population means