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