Question

Independent random samples of the selling prices in 10 000 TL) of houses in three popular districts of four big cities were taken. The selling prices are as shown below. Assume that the populations from which the sample drawn are normally distributed. (use 2 digits after decimal point) DA DB District 145 52 54 46 SI SR 140 60 SD 37 49 1. Complete the following analysis of variance table. DY Sous Marae Between District Between 104.67 Citi Em 99.33 Total 1. At the 0.05 significance level, are the mean selling prices differ by districts? a) Null Hypothesis is O Hoc Mean selling prices for different districts are not equal O Hot Mean selling prices for different districts are equal b) Alternative Hypothesis is OH, Mean selling prices for different districts are equal for at least one pair OH, Mean selling prices for different districts are not equal for at least one pair F-Ratios d) Critical Value of the testato 596 IS e) Does it appear at that population means are different? O May be O Yes for at least one pair O No

Answer 1

Let $\bar{x_{gi}}$ denote mean of the i^th group/district i=1,2,3

Cg1
= 42 $\bar{x_{g2}}$ =53
193
= 52

denote mean of the j^th block/city j=1,2,3,4

Тbj

$\bar{x_{b1}}$=50.3333   $\bar{x_{b2}}$=51.6667 $\bar{x_{b3}}$ = 50   $\bar{x_{b4}}$=44  

denote the total mean

T

=49

T

Number of groups =k=3

Number of blocks= b=4

Let, Sum of squares due to groups: SSG = $\sum 4(\bar{x_{gi}}-\bar{x})^2$ = 296

Sum of squares due to blocks = SSB = $\sum 3(\bar{x_{bi}}-\bar{x})^2$ = 104.667

Total sum of squares =SST= $\sum (x_{ij}-\bar{x})^2$ =500

Error sum of squares = SST-SSG-SSB = 99.33

Question 1)

Source of variance	SS	DF	MSS=SS/DF	F ratio
Group	296	k-1=2	148	8.9398
Block	104.667	b-1=3	34.889	2.1074
Error	99.33	(k-1)(b-1)=6	16.555
Total	500	k*b-1=11

Question2)

a)

Null hypothesis is H0: Mean selling prices for different districts are equal.

b)

Alternative hypothesis is H1: Mean selling prices for different districts are not equal for at least one pair.

c) F-ratio is 8.9398

d) The critical value for the F-test at 5% level of significance is F_2,6;0.05. It is such that P[ F > F_2,6;0.05 ]=0.05

F_2,6;0.05 = 5.14

e) We H0 at 5% level of significance if F_cal > F_2,6;0.05

Hence, YES. It appears at 5% level of significance that population means are different for at least one pair.

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