GOF Examples
Uniform distribution
A die was rolled 54 times to see if the distribution
of the numbers is uniform. These were the results:
|
X |
1 |
2 |
3 |
4 |
5 |
6 |
|
O |
12 |
7 |
8 |
8 |
5 |
14 |
|
E |
|
|
|
|
|
|
Test at a 5% level of significance.
Binomial example
A factory receives TV panels in crates of 100. A
sample of 5 is taken to test that the percentage of defective panels does not
exceed 20%. In the course of a year, the factory received 500 shipments. In the
500 samples, the number of defective panels in each sample was as follows:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
|
O |
200 |
212 |
70 |
15 |
3 |
0 |
Does the data follow a binomial distribution with
n=5 and p=0.2? Test at a 10% level of significance.
Table of distribution:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
|
P(x) |
0.32768 |
0.4096 |
0.2048 |
0.0512 |
0.0064 |
0.00032 |
|
E
= 500*P(x) |
163.84 |
204.8 |
102.4 |
25.6 |
3.2 |
0.16 |
Need to combine classes of X = 3, 4, 5 since E <
5 for X = 4 and X = 5.
|
Class |
0 |
1 |
2 |
3+ |
|
O |
200 |
212 |
70 |
18 |
|
E |
163.84 |
204.8 |
102.4 |
28.96 |
Note that SE = 500.
Poisson example
Sheet metal is examined to determine the number of
blemishes per 1000 m of steel. In 400 samples, the number of
blemishes per 1000 m were as follows:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
O |
52 |
102 |
115 |
70 |
35 |
17 |
5 |
3 |
1 |
Does the data follow a Poisson distribution with µ =
2? Test at a 5% level of significance.
Table of distribution:
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
P(x) |
0.1353 |
0.2707 |
0.2707 |
0.1804 |
0.0902 |
0.0361 |
0.0120 |
0.0034 |
0.0009 |
|
E
= 400*P(x) |
54.12 |
108.28 |
108.28 |
72.16 |
36.08 |
14.44 |
4.8 |
- |
- |
We stop at X = 6 since its expected value is less
than 5.
E(0) + E(1) + E(2) + E(3) + E(4) + E(5) = 393.36.
Then E(6+) = 400 – 393.36 = 6.64. Need to combine
classes 6, 7, 8.
|
X |
0 |
1 |
2 |
3 |
4 |
5 |
6+ |
|
O |
52 |
102 |
115 |
70 |
35 |
17 |
9 |
|
E |
54.12 |
108.28 |
108.28 |
72.16 |
36.08 |
14.44 |
6.64 |
Normal distribution – no repeated values
A sample of sales were
taken from a dollar store:
|
5.62 |
14.69 |
23.74 |
31.71 |
39.21 |
|
7.19 |
17.94 |
31.52 |
35.31 |
39.91 |
Does the data follow a normal distribution? Test at
a 5% level of significance.
|
X |
Z |
F(z) |
S(z) |
S’(z) |
Max
D |
|
5.62 |
-1.49 |
0.0681 |
|
|
|
|
7.19 |
-1.37 |
0.0853 |
|
|
|
|
14.69 |
-0.78 |
0.2177 |
|
|
|
|
17.94 |
-0.53 |
0.2981 |
|
|
|
|
23.74 |
-0.07 |
0.4721 |
|
|
|
|
31.52 |
0.54 |
0.7054 |
|
|
|
|
31.71 |
0.55 |
0.7088 |
|
|
|
|
35.31 |
0.83 |
0.7967 |
|
|
|
|
39.21 |
1.14 |
0.8729 |
|
|
|
|
39.91 |
1.19 |
0.8830 |
|
|
|
Normal distribution – repeated values
A survey examined the number of times per six months
that people recycle:
|
2 |
3 |
3 |
3 |
4 |
|
4 |
7 |
8 |
20 |
20 |
Does the data follow a normal distribution? Test at
a 5% level of significance.
|
X |
Z |
F(z) |
S(z) |
S’(z) |
Max
D |
|
2 |
-0.78 |
0.2177 |
|
|
|
|
3 |
-0.64 |
0.2611 |
|
|
|
|
3 |
-0.64 |
0.2611 |
|
|
|
|
3 |
-0.64 |
0.2611 |
|
|
|
|
4 |
-0.49 |
0.3121 |
|
|
|
|
4 |
-0.49 |
0.3121 |
|
|
|
|
7 |
-0.06 |
0.4761 |
|
|
|
|
8 |
0.09 |
0.5359 |
|
|
|
|
20 |
1.83 |
0.9664 |
|
|
|
|
20 |
1.83 |
0.9664 |
|
|
|