# Walden University Unit 6 Chi Square Goodness of Fit Test & Critical Value HW 1. Historically, the MBA program at Whatsamattu U. has about 40% of their stud

Walden University Unit 6 Chi Square Goodness of Fit Test & Critical Value HW 1. Historically, the MBA program at Whatsamattu U. has about 40% of their students choose a Leadership major, 30% choose a Finance major, 20% choose a Marketing major, and 10% choose no major. Does the most recent class of 200 MBA students fit that same pattern or has there been a shift in the choice of majors. Using the sample of 200 students (in the data file), conduct a Chi Square Goodness of Fit test to determine if the current distribution fits the historical pattern. Use a .05 significance level. 2. While job opportunities for men and women are considerably more balanced than they were 40 years ago, the career aspirations may still differ. Is there a difference in majors chosen by men and women? Using the sample of 200 MBA students (in the data file), conduct a Chi Square Test of Independence to determine if one’s choice of major is independent of their gender. Use a .05 significance level. ID

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

Gender

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

Finance

No Major

No Major

Finance

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

Finance

No Major

No Major

No Major

Finance

No Major

Finance

Finance

No Major

Finance

No Major

Finance

Finance

No Major

Finance

Finance

Employ

Unemployed

Full Time

Part Time

Full Time

Full Time

Unemployed

Full Time

Full Time

Part Time

Full Time

Part Time

Full Time

Full Time

Full Time

Part Time

Full Time

Full Time

Part Time

Full Time

Unemployed

Full Time

Part Time

Full Time

Full Time

Part Time

Full Time

Part Time

Unemployed

Part Time

Full Time

Full Time

Unemployed

Full Time

Full Time

Part Time

Part Time

Full Time

Unemployed

Full Time

Part Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Age

39

55

43

56

38

54

30

37

38

42

52

35

37

53

51

40

33

53

43

35

57

32

59

48

34

53

35

38

37

46

44

31

51

47

56

42

44

54

51

42

45

55

47

43

57

MBA_GPA

2.82

4

3.45

2.61

3.5

4

3

2.5

2.84

3.72

3.21

3.44

3.65

3.02

3.03

3.8

4

3.26

3.53

3.75

3.15

3.66

3.36

3.79

2.85

3.74

3.23

3.52

3.32

2.89

2.83

2.93

3.71

3.47

3.52

2.83

3.64

2.96

3.59

3.33

3.38

3.44

3.31

3.03

3.26

BS GPA

3

4

3.5

4

3.3

3.05

4

3.6

3.05

3.7

3.5

3.55

2.78

3.3

3.25

4

3.5

3.5

3.75

3.9

3.2

3.75

3.45

2.55

3.05

3.9

4

3.7

3.45

3.1

3.05

3.1

3.8

2.6

3.8

4

3.55

3.1

3.9

3.9

3.6

3.35

3.9

3.25

3.4

Hrs_Studying

10

15

3

4

5

5

6

6

6

6

6

6

6

6

6

6

6

7

6

7

6

8

8

8

8

8

2

2

2

2

1

1

1

4

4

4

6

6

6

6

6

6

7

7

7

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

1

1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Finance

No Major

Finance

Finance

Finance

No Major

Finance

No Major

Finance

Finance

Marketing

Marketing

Marketing

Leadership

Leadership

Marketing

Marketing

Marketing

Marketing

Marketing

No Major

No Major

No Major

No Major

Marketing

Leadership

Leadership

Leadership

Leadership

Leadership

No Major

Leadership

No Major

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

No Major

Marketing

Marketing

No Major

Finance

Finance

Finance

Finance

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Part Time

Full Time

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Part Time

Full Time

Part Time

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

36

58

46

53

59

49

34

46

46

33

56

39

51

55

38

33

34

31

37

46

31

47

54

52

43

44

34

59

45

30

32

32

40

48

51

30

31

35

33

35

31

38

46

45

59

58

46

3.04

2.98

2.8

3.75

3.64

3.65

3.18

3.44

3.06

3.51

3.33

2.81

3.64

3.05

2.85

3.56

2.92

3.35

3.46

3.59

3.11

3.65

3.17

2.97

3.77

3.21

3.17

3.65

2.94

3.53

3.65

3.61

3.7

2.91

3.09

3.77

3.79

3.59

3.38

4

2.97

3.44

3.64

3.48

2.76

3.73

2.91

4

3.1

3.05

3.75

3.65

3.8

3.3

4

3.15

3.75

3.4

3.05

3.8

3.4

3.25

3.6

3.1

3.5

3.35

3.75

3.2

3.7

3.5

3.1

3.9

3.2

3.15

3.65

3.1

3.7

3.6

3.7

3.9

3.1

3.25

3.95

3.8

3.6

3.5

3.5

3.1

3.65

3.55

3.4

3.1

3.8

3.05

7

7

7

3

3

3

3

3

3

10

2

2

8

7

3

7

5

7

10

8

6

8

7

5

8

6

6

10

5

8

7

8

8

5

6

9

8

7

8

8

8

8

8

8

8

8

8

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Finance

Finance

Finance

Finance

Finance

Finance

Finance

Finance

No Major

Marketing

Marketing

Leadership

Leadership

No Major

Leadership

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

No Major

Leadership

Leadership

Leadership

Leadership

Finance

No Major

No Major

Finance

Finance

Finance

Finance

Finance

Finance

Finance

Finance

Finance

Leadership

Leadership

Leadership

Finance

Finance

Finance

Finance

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Part Time

Full Time

Full Time

Full Time

Full Time

Part Time

Full Time

Full Time

Part Time

Full Time

Full Time

Unemployed

Full Time

Full Time

Part Time

Unemployed

Full Time

Part Time

Full Time

Full Time

Part Time

Full Time

Unemployed

Part Time

Full Time

Part Time

Full Time

Unemployed

Part Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Part Time

Full Time

35

53

31

50

38

50

48

53

53

30

32

42

56

46

49

32

36

42

37

31

31

42

39

47

28

28

52

35

38

44

38

52

53

53

31

47

51

37

46

48

54

48

36

39

28

45

31

3.78

3.5

3.13

3.14

3.24

3.56

3.16

3.53

3.7

3.3

4

3.5

3.39

3.65

2.78

3.44

3.88

2.84

3.53

3.22

3.56

3.2

3.56

3.41

3.56

3.34

2.56

3.76

3.55

3.88

3.31

3.09

3.82

3.01

3.66

3.64

3.59

3.49

3.13

3.83

3.04

3.91

3.56

3.96

3.46

3.22

3.27

3.95

3.4

3.15

3.25

3.3

3.5

3.25

3.55

3.15

3.35

3.6

3.4

3.4

3.8

3.7

3.6

3.95

3.95

3.6

3.3

3.8

3.25

3.3

3.6

3.7

3.6

3.6

3.8

3.45

3.9

3.45

3.15

4

3.2

3.85

3.7

3.65

3.55

3.2

3.9

3.15

4

3.7

4

3.4

3.15

3.2

9

7

6

6

6

7

6

7

6

6

7

7

7

8

8

7

9

9

7

6

8

6

6

7

8

7

7

8

7

8

7

6

9

6

8

8

7

7

6

8

6

10

8

9

7

6

6

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

0

1

1

1

1

1

1

1

0

1

1

1

1

0

1

1

1

0

1

1

1

0

1

Finance

Finance

Finance

Finance

Finance

Finance

Finance

Leadership

Leadership

Leadership

Leadership

No Major

No Major

No Major

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Marketing

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Full Time

Part Time

Full Time

Part Time

Unemployed

Full Time

Part Time

Full Time

Unemployed

Part Time

Unemployed

Part Time

Full Time

Unemployed

Full Time

Unemployed

Full Time

Full Time

Unemployed

Unemployed

Part Time

Full Time

Unemployed

Full Time

Part Time

Unemployed

Full Time

Part Time

Full Time

Part Time

Unemployed

Full Time

Part Time

Unemployed

Part Time

Full Time

Full Time

Part Time

Full Time

Unemployed

Full Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Part Time

47

35

52

52

55

52

46

31

33

45

50

33

37

33

46

55

30

51

35

40

29

52

27

51

56

35

46

39

31

52

35

32

44

43

38

54

30

38

45

48

43

34

54

36

45

55

45

3.43

3.85

3.89

3.37

3.32

3.54

3.8

3.74

3.6

2.6

3.8

2.67

3.95

3.56

3.79

3.93

3.79

3.71

3.05

3.22

3.85

3.82

3.23

3.56

3.53

3.62

3.8

3.47

3.64

3.03

3.17

3.22

3.92

3.82

3.26

3.8

3.2

3.46

3.67

4

3.66

3.96

3.75

3.83

3.55

3.36

3.21

3.45

3.95

3.9

3.45

3.3

3.55

3.9

3.85

3.45

3.55

3.3

3.45

4

3.75

3.75

4

3.85

3.85

3.35

3.2

3.95

3.95

3.95

3.65

3.65

4

3.95

3.35

3.65

3.15

3.25

3.2

4

3.95

3.55

3.85

3.2

3.35

3.75

3.4

3.85

4

3.85

3.85

3.2

3.35

3.25

7

9

8

7

6

7

8

8

7

7

6

7

9

8

8

9

8

8

6

6

9

9

9

7

7

9

9

6

7

5

6

6

10

9

7

8

6

6

8

7

8

10

8

8

6

6

6

187

188

189

190

191

192

193

194

195

196

197

198

199

200

1

0

1

1

1

1

1

1

1

1

1

1

1

1

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Leadership

Part Time

Part Time

Full Time

Full Time

Full Time

Full Time

Unemployed

Full Time

Unemployed

Unemployed

Unemployed

Unemployed

Unemployed

Full Time

34

54

36

24

34

45

33

22

27

33

36

34

55

33

2.97

3.99

3.07

3.65

3.67

3.06

3.98

3.93

3.41

3.43

3.7

3.76

3.9

3.23

3.15

4

3.15

3.65

3.85

3.35

3.7

4

3.3

3.5

3.65

3.75

3.9

3.3

5

10

6

7

8

6

8

10

6

7

7

8

8

6

Works FT

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

1

0

1

0

0

1

Variable descriptions

Gender = 0 (female), 1 (male)

Major = student’s major

Age = age of student in years

MBA_GPA = overall GPA in the MBA program

BS_GPA = overall GPA in the BS program

Hrs_Studying = average hours studied per week

Works FT = 0 (No), 1 (Yes)

0

0

0

1

1

1

0

1

1

0

1

0

1

0

1

1

0

1

1

1

0

1

0

1

1

1

0

0

0

1

1

1

1

1

0

1

1

)

1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

0

0

1

1

1

0

1

1

1

0

1

1

1

1

1

0

1

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

0

0

1

1

1

1

0

1

1

1

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

1

1

1

1

1

0

1

1

0

0

1

1

1

1

1

1

1

1

1

1

1

0

1

1

0

1

0

1

0

1

Chi-Square Goodness of Fit Test (Assuming Equal Expected)

Items

Pool

No Pool

Observed Expected

62

50.00

38

50.00

–

Chi Square

2.88

2.88

–

©2007 DrJimMirabella.com

Data

Level of Significance

Degrees of Freedom

0.05

1

Results

Critical Value

3.8415

Chi-Square Test Statistic

5.76

p -Value

0.0164

Reject the null hypothesis

This tests the null hypothesis that the distribution is equal across all categories.

It also tests if there is a difference in the frequencies of the categories / items.

Rejecting the null implies a difference in the categories / items.

Here we are testing if the sample fits the distribution of having 1/2 the homes with

The p-value of .0164 is less than .05, and so we reject the null hypothesis.

Thus we conclude that the sample does not fit the expected distribution.

There are significantly more homes with a pool.

Chi-Square Goodness of Fit Test (Assuming Unequal Expected)

Items

Brick

Stucco

Wood

Observed % Expected Expected

40

30.00%

30.00

35

30.00%

30.00

25

40.00%

40.00

–

Chi Square

3.33

0.83

5.63

–

©2007 DrJimMirabella.com

Data

Level of Significance

Degrees of Freedom

0.05

2

Results

Critical Value

5.9915

Chi-Square Test Statistic

9.79

p -Value

0.0075

Reject the null hypothesis

This tests the null hypothesis that the distribution is as expected.

In other words, it tests if the results fit the expected distribution.

Rejecting the null implies that the results do not fit the distribution.

Here we are testing if the sample fits the distribution of having 30% b

The p-value of .0075 is less than .05, and so we reject the null hypoth

Thus we conclude that the sample does not fit the expected distribut

There are significantly more homes than expected which are made of

Chi-Square Test of Independence

Row variable

Brick

Stucco

Wood

Total

Row variable

Brick

Stucco

Wood

0

0

Total

Data

Level of Significance

Number of Rows

Number of Columns

Degrees of Freedom

Observed Frequencies

Column variable

Pool

No Pool

30

10

18

17

14

11

62

38

0

Expected Frequencies

Column variable

Pool

No Pool

0

24.80

15.20

0.00

21.70

13.30

0.00

15.50

9.50

0.00

0.00

0.00

0.00

0.00

0.00

0.00

62

38

0

0

0

0

0

0.00

0.00

0.00

0.00

0.00

0

Total

40

35

25

0

0

100

Use the YELLOW cells to set up the Chi Square table.

The table can handle up to 5 rows and 5 columns of values.

If fewer rows or columns are needed, leave the excess blank.

The BLUE table computes the expected frequencies needed to comp

statistic. The only values that ultimately matter to you is in the RESUL

Total

0.00

0.00

0.00

0.00

0.00

0

40

35

25

0

0

100

0.05

3

2

2

Results

Critical Value

5.991465

Chi-Square Test Statistic

4.911472

p -Value

0.0858

Do not reject the null hypothesis

This tests the null hypothesis that the row variable and column variable are independent.

Rejecting the null implies that the two variables are related (one is dependent on the other).

Here we are testing if having a pool is independent of what

The p-value of .0858 is greater than .05, and so we do not r

There is insufficient evidence to conclude a relationship be

©2007 DrJimMirabella.com

CHAPTER SIX

CHI SQUARE TESTING

C

A

L being categorical in nature so we can analyze

Most hypothesis tests involve one variable

subgroups (e.g., scores for men vs. scores V

for women, with GENDER being the categorical

variable). In these parametric tests, the other

E variable must be a scale variable (i.e., interval

or ratio) since the tests involve computingRa mean. Yet there are many occasions in which

we have only categorical variables, and the technique is quite simple. Its applications

T

range from games of chance to analyzing surveys and polls.

,

Goodness of Fit

T

Let’s start with the simplest application in which we have a categorical variable in which

E

we expect all values to have equal occurrences.

When we flip a coin, we expect to get

50% heads and 50% tails. When we roll aRdie, we expect to roll a one 1/6 of the time, and

likewise for a two through six. With categorical

data, there is no averaging; we merely

R

count the frequencies and compute the percentages

(i.e., relative frequencies). So if you

E

flipped 12 heads and 8 tails, you would have 60% heads and 40% tails; the question is

N

whether that is significantly different from the 50/50 you expected to get. In evaluating if

C

it is a significant difference and if the coin flips do not fit with the distribution of a perfectly

E things – the percent breakdown of the sample

fair coin, we are essentially looking at two

Categorical Analysis

and the size of the sample. So getting 75% heads is not significant if it is based on only 4

coin flips, and yet 55% heads would be significant

if it were based on 10,000 flips. We see

1

newspapers report poll results and tell us 8

how a candidate has a lead, but that lead may be

insignificant if the sample is not large enough.

5

The proper way to word our test is to state9whether the results fit the expected distribution.

So if we flipped a coin 100 times and got T

59 heads and 41 tails, does that outcome fit with

the distribution of a fair coin?

S

Ho: The coin flips fit the distribution of a fair coin.

Ha: The coin flips do not fit the distribution of a fair coin.

Copyright 2011, Savant Learning SystemsTM

Introduction to Statistics by Jim Mirabella

6-1

Chapter Six: Chi Square Testing

C

A

L

The math behind this analysis is actually quite simple. The statistics is called CHI SQUARE and

V between the observed and expected values,

the computation involves computing the differences

E

squaring those differences, and dividing by the expected

value. We had 59 heads but out of 100

coin flips we would have expected 50. The difference

R between the observed 59 and the expected

50 is 9, and 9 squared is 81, and if we divide that

T 81 by the expected value of 50 we get 1.62,

which is the value shown in the Chi Square column

, for Heads. We do the same for Tails and also

get 1.62. Adding them up we get 3.24, our Chi Square test statistic. Statistics tables tell us the

magic number above which the Chi Square statistic is considered to be significant. If our observed

values exactly matched the expected values (i.e.,Twe flipped 50 heads and 50 tails), the computed

E as it gets. So the larger this value, the more

Chi Square statistic would be zero, which is as small

that the results veer from the expected results. In R

this case we have a p-value of .0719. This means

that if you took a perfectly fair coin and flipped it R

100 times, there is a 7.19% probability of getting

a 59/41 split. Since we set a significance level of E

5%, that means that if these results could happen

by chance more than 5% of the time, we don’t draw conclusions about the coin being unfair (i.e.,

N

it could have come from a fair coin). We would not reject the null hypothesis in this case.

C

E

Had there been 60

heads and 40 tails, the

p-value would drop to

1

.0455, and there is less

8

than a 5% chance of

5

getting such a result

by random chance.

9

We would then reject

T

the null hypothesis

S

and conclude that the

results are not from the distribution of a fair coin (i.e., the coin is not fair).

Be careful not to jump to conclusions merely on the percentage of heads or tails. As stated earlier,

it can be deceiving and depends on the sample size too.

Copyright 2011, Savant Learning SystemsTM

Introduction to Statistics by Jim Mirabella

6-2

Chapter Six: Chi Square Testing

Here you see 75% heads,

and it is not significant

since it is based on only 4

coin flips.

And here you see 51%

heads which is significant

because it is based on

10,000 flips. The closer

the results are to equality,

the larger the sample you

need to reject the null, and

vice versa. So for a small

sample to have significant

results, the deviation from

equality must be large.

With only 10 coin flips,

we got significant results

because of the 90% heads

in the sample.

Here you can see how this

test can be applied to a

6-sided die to determine if

the rolls follow a uniform

distribution in which all

six sides have an equal

likelihood of occurrence.

Copyright 2011, Savant Learning SystemsTM

C

A

L

V

E

R

T

,

T

E

R

R

E

N

C

E

1

8

5

9

T

S

Introduction to Statistics by Jim Mirabella

6-3

Chapter Six: Chi Square Testing

So Chi Square can do something simple like determining if a coin is fair or if dice are loaded by

comparing the results of a test sample to an expected distribution. And likewise you can take

polling results and compare the vote distribution of two candidates to see if there is a definitive

winner. Truly powerful and yet simple.

Now what if we don’t expect an even spread of the values? Maybe we expect one value to occur

more or less than the others (which is typically the case). We can still use the Chi Square Goodness

of Fit test, but we just need to load the expected values manually. Let’s start with another game

of chance – Roulette. On a roulette wheel there are 38 numbers (1 – 36, 0 and 00). Numbers 1

through 36 are evenly split between red and black, while the 0 and 00 are both green. So the wheel

has 18 red, 18 black and 2 green. When peopleCbet on red or black, the payoff is even money

(which is fine if the green numbers weren’t there,A

but they are what gives the house its edge). 95%

of the time, red or black wins and the house pays off, but 5% of the time, green wins and the house

L

pretty much cleans the table. The casinos have computers that monitor every game in the house,

and they essentially conduct Chi Square testing toVdetermine if a pattern doesn’t fit.

E

R

T

,

T

E

R

R

E

N where we assume Unequal Expected values.

Here we use the Chi Square Goodness of Fit test

The 200 spins show 100 red, 80 black and 20 green,

C which probably doesn’t feel all that odd. A

perfectly fair roulette wheel should have 47.37%Ered and black, and 5.26% green (18/38 red and

black, 2/38 green). According to these results, the p-value is .0039 which means that this should

occur .39% of the time with a fair game – mighty suspicious. The house should get green only

1

about 5% of the time and here they are getting green about twice as often. It is subtle enough to

8 see how with a small data set you can expose

get by the non-statistician in the crowd, but you can

corruption. It is doubtful that any major casino5 cheats like this, but they use these analytical

techniques to determine if any gamblers are cheating

9 or if their games are paying off too much, as

it is a business and they wish to make money. T

This Goodness of Fit technique is the method S

used to test if a sample comes from a Normal

distribution; essentially we expect to see a certain percentage within 1 standard deviation, within

2 standard deviations, and within 3 standard deviations, and the frequencies are compared to those

expected results. .

Copyright 2011, Savant Learning SystemsTM

Introduction to Statistics by Jim Mirabella

6-4

Chapter Six: Chi Square Testing

Crosstabulations

We’ve seen how Chi Square testing can be used to test for the goodness of fit of a single categorical

variable. Its other common use is to test for the relationship of two categorical variables.

With so many employee surveys and customer surveys conducted regularly, the temptation is to

analyze them by merely looking at questions individually, but that tells us less than you might

think. On employee surveys that use …

Purchase answer to see full

attachment

## Leave a Reply

Want to join the discussion?Feel free to contribute!