Question

Suppose that six coins are tossed simultaneously 640 times and the following frequency distribution in observed

Number of heads 0,1,2,3,4,5,6

Observed frequency 13,70,137,210,145,56,9

Test the null hypothesis that the coins are well balanced with Ω0.01

Answer 1

$\text{null hypothesis that the coins are well balanced}$

$\text {with Ω0.01}$

$\text{if all coins are correct, then the probability of }$

$\text{getting the heads for each of them is p = 0.5}$

$\text{then the random variable X is the number}$

$\text{of the rolled heads when throwing six coins - }$

$\text{ is binomial distribution with parameters}$

$\text{n = 6 and p = 0.5}$

$\text{we calculate the theoretical probabilities р(i)}\newline \text{each of the 7 possible values of the random variable X}$

$p_i =C^i_6p^i(1-p)^{6-i}$

$\text{place the data in the table}$

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} x_i & p_i & np_i&n_i&n_i-np_i &\frac{(n_i-np_i)^2}{np_i}\\ \hline 0 & 0.156& 10&13&3&0.9 \\ \hline 1 & 0.938& 60&70&10&1.67 \\ \hline 2 & 0.234& 160&137&-23&3.3 \\ \hline 3 & 0.313& 200&210&10&0.5 \\ \hline 4 & 0.234& 160&145&-15&1.41 \\ \hline 5 & 0.938& 60&56&-4&0.27\\ \hline 6 & 0.156& 10&9&-1&0.1 \\ \hline \Sigma& 1 & 640&640&-&8.15 \\ \hline \end{array}$

$\chi^2=8.15$

$\chi^2_{critical}=16.8\text{ ща }$

https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

$\chi^2<\chi^2_{critical}$

$\text{There is no reason to refute the hypothesis }$

$\text{about the correctness of the coins.}$