1. **State the problem:** We have 100 random numbers divided into 10 bins with observed frequencies: 6, 15, 14, 13, 5, 8, 11, 9, 11, 8. We want to test if these numbers are uniformly distributed in the interval [0,1] using the chi-square test for uniformity. 2. **Set up hypotheses:** - Null hypothesis $H_0$: The numbers are uniformly distributed. - Alternative hypothesis $H_1$: The numbers are not uniformly distributed. 3. **Calculate expected frequencies:** Since the distribution is uniform and there are 100 numbers divided equally into 10 bins, the expected frequency for each bin is $$E = \frac{100}{10} = 10.$$ 4. **Calculate the chi-square statistic:** Use the formula $$\chi^2 = \sum \frac{(O_i - E)^2}{E}$$ where $O_i$ are observed frequencies. Calculate each term: - Bin 1: $\frac{(6-10)^2}{10} = \frac{16}{10} = 1.6$ - Bin 2: $\frac{(15-10)^2}{10} = \frac{25}{10} = 2.5$ - Bin 3: $\frac{(14-10)^2}{10} = \frac{16}{10} = 1.6$ - Bin 4: $\frac{(13-10)^2}{10} = \frac{9}{10} = 0.9$ - Bin 5: $\frac{(5-10)^2}{10} = \frac{25}{10} = 2.5$ - Bin 6: $\frac{(8-10)^2}{10} = \frac{4}{10} = 0.4$ - Bin 7: $\frac{(11-10)^2}{10} = \frac{1}{10} = 0.1$ - Bin 8: $\frac{(9-10)^2}{10} = \frac{1}{10} = 0.1$ - Bin 9: $\frac{(11-10)^2}{10} = \frac{1}{10} = 0.1$ - Bin 10: $\frac{(8-10)^2}{10} = \frac{4}{10} = 0.4$ Sum these values: $$\chi^2 = 1.6 + 2.5 + 1.6 + 0.9 + 2.5 + 0.4 + 0.1 + 0.1 + 0.1 + 0.4 = 10.2.$$ 5. **Determine degrees of freedom:** $$df = k - 1 = 10 - 1 = 9,$$ where $k$ is the number of bins. 6. **Compare with critical value:** At significance level $\alpha = 0.05$, the critical chi-square value for 9 degrees of freedom is approximately 16.92. 7. **Conclusion:** Since $\chi^2 = 10.2 < 16.92$, we fail to reject the null hypothesis. **Final answer:** There is no sufficient evidence to conclude that the numbers are not uniformly distributed in [0,1].