Question

The quantity of dissolved oxygen is a measure of water pollution in lakes, rivers, and streams. Water samples were taken at four different locations in a river in an effort to determine if water pollution varied from location to location. Location I was 500 meters above an industrial plant water discharge point and near the shore. Location II was 200 meters above the discharge point and in midstream. Location III was 50 meters downstream from the discharge point and near the shore. Location IV was 200 meters downstream from the discharge point and in midstream. The following table shows the results. Lower dissolved oxygen readings mean more pollution. Because of the difficulty in getting midstream samples, ecology students collecting the data had fewer of these samples. Use a 10% level of significance. Do we reject or not reject the claim that the quantity of dissolved oxygen does not vary from one location to another?

Location I	Location II	Location III	Location IV
7.2	6.2	4.9	4.7
6.8	7.1	5.2	5.6
7.4	7.3	4.9	6.9
6.9	7.9	5.2
6.5		4.8

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: ?₁ = ?₂ = ?₃ = ?₄; H₁: Exactly three means are equal.   H_o: ?₁ = ?₂ = ?₃ = ?₄; H₁: All four means are different.    H_o: ?₁ = ?₂ = ?₃ = ?₄; H₁: Not all the means are equal.   H_o: ?₁ = ?₂ = ?₃ = ?₄; H₁: Exactly two means are equal.

(b) Find SS_TOT, SS_BET, and SS_W and check that SS_TOT = SS_BET + SS_W. (Use 3 decimal places.)

SSTOT	=
SSBET	=
SSW	=

Find d.f._BET, d.f._W, MS_BET, and MS_W. (Use 3 decimal places for MS_BET, and MS_W.)

dfBET	=
dfW	=
MSBET	=
MSW	=

Find the value of the sample F statistic. (Use 3 decimal places.)


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

(c) Find the P-value of the sample test statistic. (Use 4 decimal places.)

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

Since the P value is greater than the level of significance at ? = 0.1, we do not reject H₀. Since the P value is less than or equal to the level of significance at ? = 0.1, we reject H₀.    Since the P value is greater than the level of significance at ? = 0.1, we reject H₀. Since the P value is less than or equal to the level of significance at ? = 0.1, we do not reject H₀.

(e) Interpret your conclusion in the context of the application.
At the 10% level of significance there is insufficient evidence to conclude that the means are not all equal. At the 10% level of significance there is sufficient evidence to conclude that the means are not all equal.    At the 10% level of significance there is insufficient evidence to conclude that the means are all equal . .At the 10% level of significance there is sufficient evidence to conclude that the means are all equal.

Answer 1

a)Ho: 1 = 2 = 3 = 4; H1: Not all the means are equal.

b)

SStot=18.729

SSbet=14.163

SSW =4.566

dfBET =3

dfW =13

MSBET =4.721

MSW=0.351

value of the sample F statistic =13.441

(numerator df =3

denominator df=13

c) P-value of the sample test statistic =0.0003

d)

Since the Pvalue is less than or equal to the level of significance at ? = 0.1, we reject H0.

e)

At the 10% level of significance there is sufficient evidence to conclude that the means are not all equal.