Questions: Use the frequency distribution shown below to construct an expanded frequency distribution. High Temperatures ( ° F ) Class 18-28 29-39 40-50 51-61 62-72 73-83 84-94 Frequency, f 17 42 68 69 75 68 26 Complete the table below. High Temperatures ( ° F ) (Round to the nearest hundredth as needed.) Class Frequency, Midpoint Relative frequency Cumulative frequency 18-28 17 23 0.05 17 29-39 42 34 42

Solution Steps

To find the midpoint of each class, use the formula: $$\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$

For each class:

• $18-28$: $\frac{18 + 28}{2} = 23.0$
• $29-39$: $\frac{29 + 39}{2} = 34.0$
• $40-50$: $\frac{40 + 50}{2} = 45.0$
• $51-61$: $\frac{51 + 61}{2} = 56.0$
• $62-72$: $\frac{62 + 72}{2} = 67.0$
• $73-83$: $\frac{73 + 83}{2} = 78.0$
• $84-94$: $\frac{84 + 94}{2} = 89.0$

To find the relative frequency of each class, use the formula: $$\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Frequency}}$$

Given the total frequency is 365:

• $18-28$: $\frac{17}{365} \approx 0.0466$
• $29-39$: $\frac{42}{365} \approx 0.1151$
• $40-50$: $\frac{68}{365} \approx 0.1863$
• $51-61$: $\frac{69}{365} \approx 0.1890$
• $62-72$: $\frac{75}{365} \approx 0.2055$
• $73-83$: $\frac{68}{365} \approx 0.1863$
• $84-94$: $\frac{26}{365} \approx 0.0712$

To find the cumulative frequency of each class, sum the frequencies of all preceding classes including the current one:

• $18-28$: $17$
• $29-39$: $17 + 42 = 59$
• $40-50$: $17 + 42 + 68 = 127$
• $51-61$: $17 + 42 + 68 + 69 = 196$
• $62-72$: $17 + 42 + 68 + 69 + 75 = 271$
• $73-83$: $17 + 42 + 68 + 69 + 75 + 68 = 339$
• $84-94$: $17 + 42 + 68 + 69 + 75 + 68 + 26 = 365$

Final Answer