MGMT2263
Worksheet #5
1) In a small private school, class sizes are kept to under 10 students. Three classes were chosen to see if there was any significant difference in the grades of the students. Seven students were sampled from each class. These were the results:
|
Class 1 |
Class 2 |
Class 3 |
|
98 |
82 |
64 |
|
86 |
92 |
74 |
|
76 |
90 |
76 |
|
66 |
99 |
86 |
|
30 |
74 |
67 |
|
76 |
61 |
80 |
|
85 |
59 |
82 |
No assumptions about the normality of the data were made. Is there any significant difference in the average grades of the three classes? Test at 5%. (chi-square = 0.3915; conclude no significant difference among classes)
2) In a study for a museum, people were asked to rate six different exhibits on a scale from 1 to 10, 10 being the best. Suppose that for a sample of 10 people, these were the results:
|
|
A |
B |
C |
D |
E |
F |
|
Person #1 |
8 |
3 |
2 |
3 |
3 |
7 |
|
Person #2 |
5 |
5 |
5 |
1 |
9 |
1 |
|
Person #3 |
8 |
10 |
8 |
7 |
10 |
8 |
|
Person #4 |
4 |
5 |
5 |
3 |
6 |
2 |
|
Person #5 |
10 |
9 |
9 |
7 |
10 |
6 |
|
Person #6 |
10 |
9 |
2 |
5 |
6 |
4 |
|
Person #7 |
5 |
6 |
8 |
8 |
2 |
3 |
|
Person #8 |
8 |
6 |
6 |
8 |
7 |
9 |
|
Person #9 |
10 |
10 |
10 |
10 |
10 |
10 |
|
Person
#10 |
6 |
6 |
5 |
7 |
8 |
6 |
Is there any significant difference in how people rank the 6 exhibits? Test at 5%. (chi-square = 6.5143; conclude no significant difference among exhibits)
3) In a steel plant, the hardness is supposed to be uniformly distributed between 65 and 75 on a certain scale. For a sample of 100 pieces, this was the distribution:
|
Hardness |
65 to under 67.5 |
67.5 to under 70 |
70 to under 72.5 |
72.5 to under 75 |
|
Frequency |
28 |
30 |
23 |
19 |
Is the distribution uniform? Test
at a 5% level of significance. (c2 = 2.96; conclude the distribution is
uniform)
4) A factory examines 7 widgets at a time for quality control purposes. It estimates that 24% of the widgets will be defective on average. For 500 samples, each consisting of 7 widgets, the factory had the following distribution of the number of defective widgets:
|
Value |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Observations |
85 |
159 |
144 |
76 |
32 |
4 |
0 |
0 |
Here is the appropriate binomial distribution:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
P(x) |
0.1465 |
0.3237 |
0.3067 |
0.1614 |
0.0510 |
0.0097 |
0.0010 |
0.0000 |
Is the factory estimate correct?
Test at a 5% level of significance (c2 = 4.7763; conclude the data follows a
binomial distribution with n=7 and p=0.24)
5) A city will install a traffic light at an intersection if it averages 3 vehicles per minute. In a sample of 500 minutes, the following distribution was observed:
|
Value |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
Observations |
26 |
84 |
120 |
109 |
74 |
51 |
21 |
12 |
1 |
2 |
Here is the appropriate Poisson distribution:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
P(x) |
0.0498 |
0.1494 |
0.2240 |
0.2240 |
0.1680 |
0.1008 |
0.0504 |
0.0216 |
0.0081 |
0.0027 |
Does the data follow a Poisson
distribution with a mean of 3? (c2 = 5.3892; conclude the data follows a
Poisson distribution with mean = 3)
6) In a survey at a mall during August, people were asked how much they spent that day. For a sample of 8 people these were the results:
|
50 |
75 |
100 |
120 |
140 |
150 |
240 |
1350 |
a)
Is the data normally distributed? Test at a 5% level of
significance. (test stat = 0.4109; data is not normally distributed)
b) Clearly
1350 is an outlier. If we remove this value, show why the remaining data is
normally distributed.
7) For a social service agency, this is the distribution of home status by gender:
|
|
Male |
Female |
|
Stable home |
3658 |
8099 |
|
Home with problems |
138 |
293 |
|
Transient home situation |
357 |
708 |
|
Homeless |
2450 |
3047 |
a) Does a person’s home situation depend on gender? Test at a 5% level of significance. (c2 = 300.8206; conclude home status depends on gender)
b) To what degree does home status depend on gender? (12.67%)
8) A non-profit organization wanted to see if the number of volunteer hours depended on a person’s work status. These were the results:
|
Contingency Table |
||||
|
|
Hours group |
|
||
|
Group |
< 10 |
10 to 19 |
20+ |
Grand Total |
|
Working |
10 |
4 |
0 |
14 |
|
Semi-retired |
13 |
23 |
2 |
38 |
|
Retired |
15 |
30 |
8 |
53 |
|
Grand Total |
38 |
57 |
10 |
105 |
a)
Which categories need
to be collapsed? (10 to 19 hours and 20+ hours)
b)
After collapsing
categories, test the hypothesis at 5%. (test stat = 9.019; conclude the number of
volunteer hours depends on a person’s work status)
c)
To what degree does
the number of hours depend on a person’s work status? (29.31%)