a sample of 60 units was inspected to find the number of defective,which was grouped in 0,1,2,3,4 or 5 defectives .if X is the random variable defining the number of defective units, then the frequency table of the X is as follows

X012345Frequency517121583

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test at 1% level of significance whether this data can be considered to follow a binomial distribution.

“x=(0,1,2,3,4,5)”

observed frequencies:

“f=(5,17,12,15,8,3)”

relative frequencies are “r_x=f_x/60”

the sample mean is

“\mu=\sum xr_x=\frac{17+24+45+32+15}{60}=2.22”

binomial mean:

“\mu=np” , where “n=6” is the number of independent trials,

p is the success probability.

“p=2.22/6=0.37”

The chi-squared goodness-of-fit test:

the null hypothesis is that the distribution “Binom(6,0.37)” is an appropriate model for the number of successes.

test statistic is:

“\chi^2=\sum\frac{(f_x-E_x)^2}{E_x}”

where expected frequencies are “E_x=60p_x”

“p_x=C^x_np^x(1-p)^{n-x}”

“p_0=0.0625,p_1=0.2203,p_2=0.3234,p_3=0.2533”

“p_4=0.1116,p_5=0.0262”

“E_0=3.75,E_1=13.22,E_2=19.40,E_3=15.20”

“E_4=6.70,E_5=1.57”

“\chi^2=\frac{(5-3.75)^2}{3.75}+\frac{(17-13.22)^2}{13.22}+\frac{(12-19.4)^2}{19.4}+\frac{(15-15.2)^2}{15.2}+\frac{(8-6.7)^2}{6.7}+\frac{(3-1.57)^2}{1.57}=5.8775”

“df=n-2=4”

critical value:

“\chi^2_{crit}=13.277”

Since “\chi^2<\chi^2_{crit}” we accept the null hypothesis. Data can be considered to follow a binomial distribution.

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