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A federal agency studied the records pertaining to the number of job-related injuries at an underground coal mine. The value for the past 100 months were as follows:
Injuries per Month 0 1 2 3 4 5 6
Frequency of Occurrence 35 40 13 6 4 1 1

i) Apply the Chi-Square test to these data to test the hypothesis that the underlying distribution is Poisson.

ii) Apply the Chi-Square test to these data to test the hypothesis that the underlying distribution is Poisson with mean 1.0. Use level of significance $\alpha$= 0.05 and $\chi ^2$= 5.99, $\chi ^2 _{0.05,3}$= 7.81

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i) Define hypothesis

$H_0:$ data fits to Poisson distribution

$H_a:$ data does not fits to Poisson distribution

For Poisson distribution,

$P_{i}=\frac{e^{-\alpha} \alpha^x}{x !}$

$\alpha=\overline{x}=\frac{\sum f_{i} m_{i}}{n}$

$\therefore \alpha = \frac{0 \times 35 + 1 \times 40 + 2 \times 13 + 3 \times 6 + 4 \times 4 + 5 \times 1 + 6 \times 1}{100}$

$\therefore \alpha = 1.11$

$X_i$ $O_i$ $P_i$ $E_i = h * P_i$ $\frac{\left(0_{i}-E_{i}\right)^{2}}{E_{i}}$
0 35 0.3296 32.96 0.126
1 40 0.3658 36.58 0.320
2 13 0.2030 20.30 2.625
3 6 $\quad$ | 0.0751 7.51 $\quad$ | $\quad$ |
4 4 $\quad$ 12 0.0209 2.09 $\quad$ 10.16 0.333
5 1 $\quad$ | 0.0046 0.46 $\quad$ | $\quad$ |
6 1 $\quad$ | 0.0010 0.10 $\quad$ | $\quad$ |
- $\sum = 100$ $\sum = 1.0000$ $\sum = 100$ $\chi_0^2 = 3.400$

To find $\alpha$:

We need to estimate value of $\alpha$

$\therefore \quad S=1$

$\therefore \quad d \cdot f=k-s-1=4-1-1=2$

$\because x_{0.05, 2}^{2}=5.99$

$\chi_{0}^{2}=3.404\lt\chi^{2}_{0.05, 2} = 5.99$

$\therefore H_0$ is accepted.

$\therefore$ given data fits to Poisson distribution.

ii) Define hypothesis:

$H_0:$ data fits to Poisson distribution

$H_a:$ data does not fits to Poisson distribution

$\because p_{i}=\frac{e^{-\alpha} \alpha^{x}}{x !} \quad \alpha_{i}=1$

$X_i$ $O_i$ $P_i$ $E_i = h * P_i$ $\frac{\left(0_{i}-E_{i}\right)^{2}}{E_{i}}$
0 35 0.3279 36.79 0.087
1 40 0.3679 36.79 0.280
2 13 0.1839 18.39 1.580
3 6 $\quad$ | 0.0613 6.13 $\quad$ | $\quad$ |
4 4 $\quad$ 12 0.0153 1.53 $\quad$ 10.16 1.963
5 1 $\quad$ | 0.0031 0.31 $\quad$ | $\quad$ |
6 1 $\quad$ | 0.0006 0.06 $\quad$ | $\quad$ |
- $\sum = 100$ $\sum = 1.0$ $\sum = 100$ $\chi_0^2 = 3.91$

To find $\alpha$:

We do not estimate value of $\alpha$

$\therefore s = 0$

$\therefore d.f = k - s - 1 = 4 - 0 - 1 = 3$

$\because \chi_{0.05, 3}^2 = 7.81$

$\because \chi_{0}^{2} = 3.91 \lt \chi_{0.05, 3}^{2}=7.81$

$\therefore H_0$ is accepted.

$\therefore$ given data fits to Poisson distribution.