1. **State the problem:** We want to test if there is a difference between the mean lifetimes of light bulbs in Infocus and Proxima projectors using a two-tailed test at the 0.05 significance level. 2. **Hypotheses:** - Null hypothesis $H_0$: $\mu_1 = \mu_2$ (means are equal) - Alternative hypothesis $H_1$: $\mu_1 \neq \mu_2$ (means differ) 3. **Type of test statistic:** Since population variances are assumed equal and samples are small, use the two-sample pooled t-test. 4. **Calculate sample means:** - Infocus: $\bar{x}_1 = \frac{650 + 1096 + 997 + 1057 + 796 + 1088 + 634 + 1134}{8} = \frac{7452}{8} = 931.5$ - Proxima: $\bar{x}_2 = \frac{833 + 955 + 821 + 804 + 903 + 923 + 639 + 907}{8} = \frac{6785}{8} = 848.125$ 5. **Calculate sample variances:** - Infocus variance $s_1^2 = \frac{\sum (x_i - \bar{x}_1)^2}{n_1 - 1} = 44122.071$ - Proxima variance $s_2^2 = \frac{\sum (x_i - \bar{x}_2)^2}{n_2 - 1} = 10289.839$ 6. **Calculate pooled variance:** $$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} = \frac{7 \times 44122.071 + 7 \times 10289.839}{14} = \frac{308854.497 + 72028.873}{14} = \frac{380883.37}{14} = 27206.0$$ 7. **Calculate test statistic:** $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} = \frac{931.5 - 848.125}{\sqrt{27206.0 \left(\frac{1}{8} + \frac{1}{8}\right)}} = \frac{83.375}{\sqrt{27206.0 \times 0.25}} = \frac{83.375}{\sqrt{6801.5}} = \frac{83.375}{82.47} = 1.011$$ 8. **Degrees of freedom:** $df = n_1 + n_2 - 2 = 14$ 9. **Critical values for two-tailed test at $\alpha=0.05$:** From t-distribution table, $t_{0.025,14} = \pm 2.145$ 10. **Conclusion:** Since $|t| = 1.011 < 2.145$, we fail to reject $H_0$. There is not enough evidence to conclude a difference in mean lifetimes. **Final answer:** No, we cannot conclude that the mean lifetimes differ between Infocus and Proxima projectors at the 0.05 significance level.