1. **State the problem:** We are given a frequency distribution of high temperatures with class intervals and frequencies. We need to construct an expanded frequency distribution table including midpoints, relative frequencies, and cumulative frequencies. 2. **Given data:** Class intervals: 15-25, 26-36, 37-47, 48-58, 59-69, 70-80, 81-91 Frequencies ($f$): 17, 45, 67, 68, 78, 67, 23 3. **Formulas and rules:** - Midpoint of a class = $\frac{\text{Lower limit} + \text{Upper limit}}{2}$ - Relative frequency = $\frac{f}{\text{Total frequency}}$ - Cumulative frequency = sum of frequencies up to that class 4. **Calculate total frequency:** $$\text{Total frequency} = 17 + 45 + 67 + 68 + 78 + 67 + 23 = 365$$ 5. **Calculate midpoints:** - For 15-25: $\frac{15 + 25}{2} = 20$ - For 26-36: $\frac{26 + 36}{2} = 31$ - For 37-47: $\frac{37 + 47}{2} = 42$ - For 48-58: $\frac{48 + 58}{2} = 53$ - For 59-69: $\frac{59 + 69}{2} = 64$ - For 70-80: $\frac{70 + 80}{2} = 75$ - For 81-91: $\frac{81 + 91}{2} = 86$ 6. **Calculate relative frequencies (rounded to nearest hundredth):** - For 15-25: $\frac{17}{365} \approx 0.05$ - For 26-36: $\frac{45}{365} \approx 0.12$ - For 37-47: $\frac{67}{365} \approx 0.18$ - For 48-58: $\frac{68}{365} \approx 0.19$ - For 59-69: $\frac{78}{365} \approx 0.21$ - For 70-80: $\frac{67}{365} \approx 0.18$ - For 81-91: $\frac{23}{365} \approx 0.06$ 7. **Calculate cumulative frequencies:** - For 15-25: 17 - For 26-36: $17 + 45 = 62$ - For 37-47: $62 + 67 = 129$ - For 48-58: $129 + 68 = 197$ - For 59-69: $197 + 78 = 275$ - For 70-80: $275 + 67 = 342$ - For 81-91: $342 + 23 = 365$ 8. **Final expanded frequency distribution table:** | Class | Frequency ($f$) | Midpoint | Relative Frequency | Cumulative Frequency | |--------|-----------------|----------|--------------------|----------------------| | 15-25 | 17 | 20 | 0.05 | 17 | | 26-36 | 45 | 31 | 0.12 | 62 | | 37-47 | 67 | 42 | 0.18 | 129 | | 48-58 | 68 | 53 | 0.19 | 197 | | 59-69 | 78 | 64 | 0.21 | 275 | | 70-80 | 67 | 75 | 0.18 | 342 | | 81-91 | 23 | 86 | 0.06 | 365 |