Answer to Question #241865 in Statistics and Probability for Md. Jahangir Alam

Question

The grades in a statistics course for a particular semester were as follows:

x= 1, 2, 3, 4, 5, 6, 7, 8

f= 136, 60, 34, 12, 55, 1, 3, 1

Test the hypothesis, at the 0.05 level of significance, does the distribution of grades

follow poisson distribution.

Answer 1

Let X denote the grades in a statistics course.

Hypothesis tested is,

H₀: X ~ Poisson distribution against H₁: X does not follow a Poisson distribution

We first compute the parameter $\lambda=(\displaystyle\sum{(f*x)})/n=716/302=2.37(2 decimal places).$

We then compute respective probabilities for $x=1,2,3,4,5,6,7,8$ using the Poisson distribution given by;

$P$_x $=P(X=x)=(e^{-\lambda}*\lambda^x)/x!$

Therefore,

P₁=P(X=1)=e^-2.37*(2.37)=0.22

P₂=P(X=2)=e^-2.37*(2.37²)/2!=0.26

P₃=P(X=3)=e^-2.37*(2.37³)/3!=0.21

P₄=P(X=4)=e^-2.37*(2.37⁴)/4!=0.12

P₅=P(X=5)=e^-2.37*(2.37⁵)/5!=0.06

P₆=P(X=6)=e^-2.37*(2.37⁶)/6!=0.02

P₇=P(X=7)=e^-2.37*(2.37⁷)/7!=0.008

P₈=P(X=8)=e^-2.37*(2.37⁸)/8!=0.002

Next is to determine the expected frequencies, E_X=n*P_X

E₁=n*p₁=302*0.22=66.87

E₂=n*p₂=302*0.26=79.28

E₃=n*p₃=302*0.21=62.65

E₄=n*p₄=302*0.12=37.13

E₅=n*p₅=302*0.06=17.61

E₆=n*p₆=302*0.02=6.96

E₇=n*p₇=302*0.008=2.36

E₈=n*p₈=302*0.02=0.70

Since the expected frequencies for x=7 and x=8 are less than 5, they are both combined with the expected frequency for x=6 to get a total expected frequency of 10.01.

With these values, we can now find the chi square test statistic defined by,

$\chi$²(calculated)=$\displaystyle\sum_{i=1}^{6}(O_{i}-E_{i})^2/E_i$

=(136-66.87)²/66.87+(60-79.28)²/79.28+(34-62.65)²/62.65+(12-37.13)²/37.13+(55-17.62)²/17.62+(5-10.01)²/10.01=188.17(2 decimal places).

$\chi$²  table value with k-1= 6-1=5degrees of freedom at $\alpha$ =0.05=11.0705.

$\chi^2$(calculated)=188.17 is greater than the table value of 11.0705 hence, we reject the null hypothesis that X follows a Poisson distribution and conclude that there is no sufficient evidence to show that grades in a statistics class follow a Poisson distribution.