91影视

The frequencies of the different leading digits from IRS tax files are recorded.

Assume that random sampling is conducted.

Let O denote the observed frequencies of the leading digits.

The observed frequencies are noted below:

\(\begin{aligned}{c}{O_1} = 55\\{O_2} = 25\\{O_3} = 17\;\;\\{O_4} = 24\end{aligned}\)

\({O_5} = 18\)

\(\begin{aligned}{c}{O_6} = 12\\{O_7} = 12\;\;\\{O_8} = 3\;\;\\{O_9} = 4\end{aligned}\)

The sum of all observed frequencies is computed below:

\(\begin{aligned}{c}n = 55 + 25 + ... + 4\\ = 170\end{aligned}\)

Let E denote the expected frequencies.

Let the expected proportion and expected frequencies of the i-th digit as given by Benford鈥檚 law.

As all the expected values are higher than 5, the requirements of the test are satisfied.

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

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

The alternative hypothesis is as follows:

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

The test is right-tailed.

If the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected.

The table below shows the necessary calculations:

The value of the test statistic is equal to:

\(\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \;\\ = 0.28667 + 0.809037 + ... + 1.866036\\ = 12.992\end{aligned}\)

Thus,\({\chi ^2} = 12.992\).

Let k be the number of digits, equal to 9.

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

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

The critical value of\({\chi ^2}\)at\(\alpha = 0.05\)with 8 degrees of freedom is equal to 15.507, taken from the chi-square table.

The p-value is,

\(\begin{aligned}{c}p - value = P\left( {{\chi ^2} > 12.992} \right)\\ = 0.112\end{aligned}\)

Since the test statistic value is less than the critical value and the p-value is greater than 0.05, the null hypothesis is failed to be rejected.

There is not enough evidence to conclude that the observed frequencies of the leading digits of the sizes of the electronic document files are not the same as the frequencies expected from Benford鈥檚 law.