91影视

The frequencies of the different leading digits of the amounts of 784 checks are recorded.

Assume that random sampling is conducted.

Let O denote the observed frequencies of the leading digits.

The observed frequencies are noted below:

\(\begin{aligned}{c}{O_1} = 0\\{O_2} = 15\;\;\\{O_3} = 0\;\;\\{O_4} = 76\end{aligned}\)

\({O_5} = 479\)

\(\begin{aligned}{c}{O_6} = 183\\{O_7} = 8\;\;\\{O_8} = 23\;\;\\{O_9} = 0\end{aligned}\)

The sum of all observed frequencies is computed below:

\(\begin{aligned}{c}n = 0 + 15 + ...... + 0\\ = 784\end{aligned}\)

Let E denote the expected frequencies.

Let the expected proportion and expected frequencies of the ith digit as given by Benford鈥檚 law.

Since the expected values are larger than 5, the requirements of the test are met.

The null hypothesis for conducting the given test is as follows:

The observed frequencies are the same as the frequencies expected from Benford鈥檚 law.

The alternative hypothesis is as follows:

The observed frequencies are not the same as the frequencies expected from Benford鈥檚 law.

The table below shows the necessary calculations:

The value of the test statistic is equal to:

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = 235.984 + 109.6146 + ....... + 36.064\\ = 3650.251\end{aligned}\]

Thus,\({\chi ^2} = 3650.251\).

Let k be the number of digits, which is 9.

The degrees of freedom for\({\chi ^2}\)is computed below:

\(\begin{aligned}{c}df = k - 1\\ = 9 - 1\\ = 8\end{aligned}\)

The chi-square table is used to obtain the critical value of\({\chi ^2}\)at\(\alpha = 0.01\)with 8 degrees of freedom is equal to 20.090.

The p-value is,

\(\begin{aligned}{c}p - value = P\left( {{\chi ^2} > 3650.251} \right)\\ = 0.000\end{aligned}\)

Since the test statistic value is greater than the critical value and the p-value is less than 0.01, the null hypothesis is rejected.

There is enough evidence to conclude thatthe observed frequencies are not the same as the frequencies expected from Benford鈥檚 law.

Since the observed frequencies differ from the expected frequencies, the check amounts are a result of fraud.