X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
f | 5 | 8 | 15 | 14 | 10 | 6 | 2 |
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Let x be the random variable following Binomial distribution with parameter n = 6
x | f | fx |
0 | 5 | 0 |
1 | 8 | 8 |
2 | 15 | 30 |
3 | 14 | 42 |
4 | 10 | 40 |
5 | 6 | 30 |
6 | 2 | 12 |
N = Σf = 60 | Σfx = 162 |
N = 60 = total frequency
Then mean of observed data (x̄) = \(\frac{\sum fx}{\sum f}\)
= \(\frac{162}{60}\) = \(\frac{27}{10}\) = 2.7
Then np = \(\frac{27}{10}\)
or, 6 × p = \(\frac{27}{10}\)
or, p = \(\frac{27}{6 \times 10}\) = \(\frac{9}{20}\)
And
q = 1 – p = 1 – \(\frac{9}{20}\) = \(\frac{11}{20}\)
Then the probability mass function becomes
P(X=x) = p(x) = \(C(6, x)\left ( \frac{9}{20} \right )^x \left ( \frac{11}{20} \right )^{6-x}\)
The expected frequency is obtained by substituting the values of x = 0, 1, 2, 3, . . . . . .
x | \(p(x) = C(6, x)\left ( \frac{9}{20} \right )^x \left ( \frac{11}{20} \right )^{6-x}\) | Expected frequency = N × p(x) | |
0 | 0.277 | 1.662 | 2 |
1 | 0.1359 | 8.154 | 8 |
2 | 0.2708 | 16.248 | 16 |
3 | 0.3032 | 18.192 | 18 |
4 | 0.186 | 11.166 | 11 |
5 | 0.0608 | 3.654 | 4 |
6 | 0.0083 | 0.492 | 1 |
1 | 60 | 60 |
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