Question

A sociologist studying New York City ethnic groups wants to determine if there is a difference in income for immigrants from four different countries during their first year in the city. She obtained the data in the following table from a random sample of immigrants from these countries (incomes in thousands of dollars). Use a 0.05 level of significance to test the claim that there is no difference in the earnings of immigrants from the four different countries.

Country I	Country II	Country III	Country IV
12.5	8.5	20.8	17.3
9.1	17.5	16.8	8.6
10.8	19.1	22.8	14.8
8.1	10.4	5.8	21.5
16.8		19.1	19.4

(a) What is the level of significance?

(b) State the null and alternate hypotheses.

H₀: ?₁ = ?₂ = ?₃ = ?₄; H₁: Not all the means are equal.

H₀: ?₁ = ?₂ = ?₃ = ?₄; H₁: All four means are different.     

H₀: ?₁ = ?₂ = ?₃ = ?₄; H₁: Exactly two means are equal.

H₀: ?₁ = ?₂ = ?₃ = ?₄; H₁: Exactly three means are equal.

(c) Find SS_TOT, SS_BET, and SS_W and check that SS_TOT = SS_BET + SS_W. (Round your answers to three decimal places.)

(d) Find d.f._BET, d.f._W, MS_BET, and MS_W.

(e) Find the value of the sample F statistic.

(f) What are the degrees of freedom? numerator and denominator

(g) Find the P-value of the sample test statistic. 4 decimals

(h) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P value is less than or equal to the level of significance at ? = 0.05, we reject H₀.

Since the P value is greater than the level of significance at ? = 0.05, we do not reject H₀.     

Since the P value is greater than the level of significance at ? = 0.05, we reject H₀.

Since the P value is less than or equal to the level of significance at ? = 0.05, we do not reject H₀.

(i) Interpret your conclusion in the context of the application.

At the 5% level of significance there is insufficient evidence to conclude that the means are not all equal.

At the 5% level of significance there is sufficient evidence to conclude that the means are all equal.     

At the 5% level of significance there is insufficient evidence to conclude that the means are all equal.

At the 5% level of significance there is sufficient evidence to conclude that the means are not all equal.

PLEASE EXPLAIN STEPS TO HOW YOU REACHED THIS ANSWER

Answer 1

a)

level of significance=0.05

b)

H0: ?1 = ?2 = ?3 = ?4; H1: Not all the means are equal.

c )

applying one way ANOVA :

SStot =501.612

SSBET =96.172

SSW =405.440

d)

d.f.BET=3

d.f.W=16

MSBET =SSBET/dfBET =48.086

MSW =SSW/dfW=25.340

e)

value of the sample F statistic =MSBET/MSW =1.898 ~1.90

f)

degrees of freedom numerator =3

degrees of freedom denominator =16

g) P-value of the sample test statistic =0.1822 ( please try 0.1818 or 0.1821 if this comes wrong)

h)Since the P value is greater than the level of significance at ? = 0.05, we do not reject H0.

i)At the 5% level of significance there is insufficient evidence to conclude that the means are not all equal.