) The table below shows the number of vacation days taken by each respondent in
ID: 3363513 • Letter: #
Question
) The table below shows the number of vacation days taken by each respondent in the previous 90 days and their reported their level of happiness. For example, 9 people took no vacation days and reported being very happy. The respondents were taken from a random sample of n = 351 people.
Vacation days taken in last 90 days
0
1
2
3
4
5+
Very Happy
9 (9.76)
15 (14.05)
20 (21.47)
25 (25.37)
31 (31.23)
37 (35.13)
137
Fairly Happy
5 (12.25)
8 (17.64)
26 (26.95)
35 (31.85)
48 (39.2)
50 (44.1)
172
Not Very Happy
6 (1.5)
6 (2.15)
4 (3.29)
2 (3.89)
1 (4.79)
2 (5.38)
21
Not Happy
5 (1.5)
7 (2.15)
5 (3.29)
3 (3.89)
0 (4.79)
1 (5.38)
21
25
36
55
65
80
90
351
You can see that the chi-square test is not appropriate here because some expected cell counts are less than 5.
To do the hypothesis test to determine if there is an association between happiness and number of vacation days taken in the last 90 days, you will need to merge the data somehow to get large enough expected counts. You can merge categories down and/or across. Draw your new table with observed and expected counts below. There are numerous tables that can be constructed, but you will have to find one which gives a df of at least 3. If you combine too many so that your df = 1 or 2, then you have lost more data than is necessary.
Then do the test, using a significance level of 0.05.
A. Hypothesize:
H0:
Ha:
B. Prepare: = 0.05
Check the Central Limit Theory Conditions:
C. Compute to Compare: Use your calculator to find the test statistic and the p-value. Draw the distribution labeling the critical region(s) with ²*.
D. Interpret. Give your conclusion and statement.
Vacation days taken in last 90 days
0
1
2
3
4
5+
Very Happy
9 (9.76)
15 (14.05)
20 (21.47)
25 (25.37)
31 (31.23)
37 (35.13)
137
Fairly Happy
5 (12.25)
8 (17.64)
26 (26.95)
35 (31.85)
48 (39.2)
50 (44.1)
172
Not Very Happy
6 (1.5)
6 (2.15)
4 (3.29)
2 (3.89)
1 (4.79)
2 (5.38)
21
Not Happy
5 (1.5)
7 (2.15)
5 (3.29)
3 (3.89)
0 (4.79)
1 (5.38)
21
25
36
55
65
80
90
351
Explanation / Answer
Result:
To do the hypothesis test to determine if there is an association between happiness and number of vacation days taken in the last 90 days, you will need to merge the data somehow to get large enough expected counts. You can merge categories down and/or across. Draw your new table with observed and expected counts below. There are numerous tables that can be constructed, but you will have to find one which gives a df of at least 3. If you combine too many so that your df = 1 or 2, then you have lost more data than is necessary.
Then do the test, using a significance level of 0.05.
0-1
2
3
4+
Total
Very Happy
Observed
24
20
25
68
137
Expected
23.81
21.47
25.37
66.35
137.00
Fairly Happy
Observed
13
26
35
98
172
Expected
29.89
26.95
31.85
83.30
172.00
Not Very Happy, Not happy
Observed
24
9
5
4
42
Expected
7.30
6.58
7.78
20.34
42.00
Total
Observed
61
55
65
170
351
Expected
61.00
55.00
65.00
170.00
351.00
A. Hypothesize:
H0: there is no association between happiness and number of vacation days taken in the last 90 days
Ha: there is association between happiness and number of vacation days taken in the last 90 days
B. Prepare: = 0.05
Check the Central Limit Theory Conditions:
Critical chi square = 12.59 at 0.05 level.
C. Compute to Compare: Use your calculator to find the test statistic and the p-value. Draw the distribution labeling the critical region(s) with ²*.
Col 1
Col 2
Col 3
Col 4
Total
Row 1
Observed
24
20
25
68
137
Expected
23.81
21.47
25.37
66.35
137.00
(O - E)² / E
0.00
0.10
0.01
0.04
0.15
Row 2
Observed
13
26
35
98
172
Expected
29.89
26.95
31.85
83.30
172.00
(O - E)² / E
9.55
0.03
0.31
2.59
12.48
Row 3
Observed
24
9
5
4
42
Expected
7.30
6.58
7.78
20.34
42.00
(O - E)² / E
38.21
0.89
0.99
13.13
53.22
Total
Observed
61
55
65
170
351
Expected
61.00
55.00
65.00
170.00
351.00
(O - E)² / E
47.76
1.02
1.31
15.76
65.85
65.85
chi-square
6
df
2.89E-12
p-value
Calculated chi square = 65.85 > 12.59, the critical value.
Ho is rejected.
D. Interpret. Give your conclusion and statement.
We conclude that there is significant association between happiness and number of vacation days taken in the last 90 days.
0-1
2
3
4+
Total
Very Happy
Observed
24
20
25
68
137
Expected
23.81
21.47
25.37
66.35
137.00
Fairly Happy
Observed
13
26
35
98
172
Expected
29.89
26.95
31.85
83.30
172.00
Not Very Happy, Not happy
Observed
24
9
5
4
42
Expected
7.30
6.58
7.78
20.34
42.00
Total
Observed
61
55
65
170
351
Expected
61.00
55.00
65.00
170.00
351.00
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