The executives of CareFree Insurance,Inc.feel that "a minority of our employees perceive an authoritarian management style at CareFree." A random sample of 200 CareFree employees is selected to test this hypothesis.Eighty employees rate the management as authoritarian.Using $\alpha$ = 0.05,the observed value of chi-square is ___.

Question

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The observed value of chi-square is calculated using the formula $\chi^2 = \frac{(O - E)^2}{E}$, where O is the observed frequency and E is the expected frequency. Here, 80 out of 200 employees perceive the management as authoritarian, which is 40%. Under the null hypothesis, a minority (less than 50%) would perceive it as such, so E = 100. The chi-square value is calculated as $\chi^2 = \frac{(80 - 100)^2}{100} + \frac{(120 - 100)^2}{100} = 8.00$.

The Executives of CareFree Insurance,Inc α\alphaα = 005,the Observed Value of Chi-Square Is ___

The Executives of CareFree Insurance,Inc $\alpha$ = 005,the Observed Value of Chi-Square Is ___