Question

Fit a Binomial distribution for the following data and test at 5% level of significance that Binomial distribution is a good fit.

Answer 1

Let x be the number of defective items is a Binomial variate with the parameter n = 5, and P is obtained as below: 

Let f be the number of samples, then from the frequency distribution:

5p = 600/200, p = 3/5 = 0.6, q=0.4

Then the p.m.f is:-

p(x) = ⁿC_xp^x q^n-x ; x = 0,1,2 ….. n

p(x) = ⁵Cx 0.6^x 0.45^{5 – x} ; x = 0,1,2, 5

Theoretical frequency : T_x = p(x) N

T₀ = ⁵C₀ 0.6⁰ 0.4^5-0 × 200 = 2.048

Using recurrence relation :

The fitted observed and theoretical frequency distribution (Approx) is:

H₀: Binomial distribution is good fit (0_i = E_i) 

H₁ Binomial distribution is not good fit (0_i ≠ E_i) {upper tail text K₂} Under H₀, the χ²-test statistic is:-

since P is estimated/calculated from the data and so (n – 1 – 1) = (n-2) d.f

χ² cal =0.1206

At ∝ = 5% for (n- 2) = 5-2 = 3 d.f the upper tail critical value K²= 7.81

Here χ² cal lies in AR (Acceptance region).

Therefore H₀ is accepted.

Conclusion: B.D is a good fit.