1. **State the problem:** We want to test if the average maximum heart rates are equal across the four workouts using a significance level $\alpha = 0.01$. This is a one-way ANOVA test. 2. **Set up hypotheses:** - Null hypothesis $H_0$: $\mu_1 = \mu_2 = \mu_3 = \mu_4$ (all group means are equal) - Alternative hypothesis $H_a$: At least one group mean is different 3. **Calculate group means:** Workout #1 mean: $\frac{159 + 170 + 152 + 169 + 172}{5} = \frac{822}{5} = 164.4$ Workout #2 mean: $\frac{199 + 179 + 162 + 180 + 177}{5} = \frac{897}{5} = 179.4$ Workout #3 mean: $\frac{185 + 169 + 157 + 176 + 168}{5} = \frac{855}{5} = 171$ Workout #4 mean: $\frac{190 + 184 + 177 + 186 + 162}{5} = \frac{899}{5} = 179.8$ 4. **Calculate overall mean:** $$\bar{x} = \frac{822 + 897 + 855 + 899}{20} = \frac{3473}{20} = 173.65$$ 5. **Calculate Sum of Squares Between (SSB):** $$SSB = 5[(164.4 - 173.65)^2 + (179.4 - 173.65)^2 + (171 - 173.65)^2 + (179.8 - 173.65)^2]$$ $$= 5[( -9.25)^2 + 5.75^2 + (-2.65)^2 + 6.15^2]$$ $$= 5[85.56 + 33.06 + 7.02 + 37.82] = 5 \times 163.46 = 817.3$$ 6. **Calculate Sum of Squares Within (SSW):** Calculate variance within each group and sum: Workout #1: $\sum (x_i - 164.4)^2 = (159-164.4)^2 + (170-164.4)^2 + (152-164.4)^2 + (169-164.4)^2 + (172-164.4)^2 = 29.16 + 31.36 + 153.76 + 21.16 + 57.76 = 293.2$ Workout #2: $\sum (x_i - 179.4)^2 = 380.8$ Workout #3: $\sum (x_i - 171)^2 = 292$ Workout #4: $\sum (x_i - 179.8)^2 = 334.8$ Total SSW = 293.2 + 380.8 + 292 + 334.8 = 1300.8 7. **Calculate degrees of freedom:** Between groups: $df_1 = k - 1 = 4 - 1 = 3$ Within groups: $df_2 = N - k = 20 - 4 = 16$ 8. **Calculate Mean Squares:** $$MSB = \frac{SSB}{df_1} = \frac{817.3}{3} = 272.43$$ $$MSW = \frac{SSW}{df_2} = \frac{1300.8}{16} = 81.3$$ 9. **Calculate F-statistic:** $$F = \frac{MSB}{MSW} = \frac{272.43}{81.3} \approx 3.35$$ 10. **Decision rule:** At $\alpha = 0.01$ and degrees of freedom $(3,16)$, the critical F-value from F-distribution tables is approximately 4.43. 11. **Conclusion:** Since $F = 3.35 < 4.43$, we fail to reject the null hypothesis. **Interpretation:** There is not enough evidence at the 0.01 significance level to conclude that the average maximum heart rates differ among the four workouts.