1. **State the problem:** We want to test if the responses for compatibility levels between Microsoft and Linux operating systems differ significantly at a 5% significance level. 2. **Set hypotheses:** - Null hypothesis $H_0$: The distribution of compatibility responses is the same for Microsoft and Linux. - Alternative hypothesis $H_a$: The distributions differ. 3. **Data:** | Compatibility Level | Microsoft | Linux | |---------------------|-----------|-------| | High | 96 | 56 | | Moderate | 90 | 84 | | Low | 27 | 67 | 4. **Calculate totals:** - Microsoft total: $96 + 90 + 27 = 213$ - Linux total: $56 + 84 + 67 = 207$ - Grand total: $213 + 207 = 420$ 5. **Calculate expected counts under $H_0$:** For each cell, expected count = (row total * column total) / grand total. Column totals: - High: $96 + 56 = 152$ - Moderate: $90 + 84 = 174$ - Low: $27 + 67 = 94$ Expected counts for Microsoft: - High: $\frac{213 \times 152}{420} = 77.14$ - Moderate: $\frac{213 \times 174}{420} = 88.29$ - Low: $\frac{213 \times 94}{420} = 47.57$ Expected counts for Linux: - High: $\frac{207 \times 152}{420} = 74.86$ - Moderate: $\frac{207 \times 174}{420} = 85.71$ - Low: $\frac{207 \times 94}{420} = 46.43$ 6. **Compute Chi-square statistic:** $$\chi^2 = \sum \frac{(O - E)^2}{E}$$ Where $O$ is observed and $E$ is expected. Calculations: - Microsoft High: $\frac{(96 - 77.14)^2}{77.14} = 4.56$ - Microsoft Moderate: $\frac{(90 - 88.29)^2}{88.29} = 0.03$ - Microsoft Low: $\frac{(27 - 47.57)^2}{47.57} = 9.02$ - Linux High: $\frac{(56 - 74.86)^2}{74.86} = 4.56$ - Linux Moderate: $\frac{(84 - 85.71)^2}{85.71} = 0.03$ - Linux Low: $\frac{(67 - 46.43)^2}{46.43} = 9.02$ Sum: $$\chi^2 = 4.56 + 0.03 + 9.02 + 4.56 + 0.03 + 9.02 = 27.22$$ 7. **Degrees of freedom:** $df = (rows - 1)(columns - 1) = (2-1)(3-1) = 2$ 8. **Critical value at 5% significance and 2 df:** $\chi^2_{critical} = 5.991$ 9. **Decision:** Since $27.22 > 5.991$, we reject the null hypothesis. 10. **Conclusion:** There is sufficient evidence at the 5% significance level to conclude that the compatibility responses differ between Microsoft and Linux operating systems.