Questions: Use the frequency distribution shown below to construct an expanded frequency distribution. High Temperatures (°F) Class 20-30 31-41 42-52 53-63 64-74 75-85 86-96 Frequency, f 18 42 67 68 78 67 25 Complete the table below. High Temperatures (°F) (Round to the nearest hundredth as needed.) Class Frequency, f Midpoint Relative frequency Cumulative frequency 20-30 18 31-41 42 42-52 67 53-63 68 64-74 78 75-85 67 86-96 25

Use the frequency distribution shown below to construct an expanded frequency distribution. High Temperatures (°F) Class 20-30 31-41 42-52 53-63 64-74 75-85 86-96 Frequency, f 18 42 67 68 78 67 25 Complete the table below. High Temperatures (°F) (Round to the nearest hundredth as needed.) Class Frequency, f Midpoint Relative frequency Cumulative frequency 20-30 18 31-41 42 42-52 67 53-63 68 64-74 78 75-85 67 86-96 25
Transcript text: Use the frequency distribution shown below to construct an expanded frequency distribution. High Temperatures ( ${ }^{\circ} \mathrm{F}$ ) \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline Class & $20-30$ & $31-41$ & $42-52$ & $53-63$ & $64-74$ & $75-85$ & $86-96$ \\ \hline Frequency, f & 18 & 42 & 67 & 68 & 78 & 67 & 25 \\ \hline \end{tabular} Complete the table below. High Temperatures ( ${ }^{\circ} \mathrm{F}$ ) (Round to the nearest hundredth as needed.) \begin{tabular}{|l|c|c|c|l|} \hline Class & Frequency, $\mathbf{f}$ & Midpoint & \begin{tabular}{l} Relative \\ frequency \end{tabular} & \begin{tabular}{l} Cumulative \\ frequency \end{tabular} \\ \hline $20-30$ & 18 & $\square$ & $\square$ & $\square$ \\ $31-41$ & 42 & $\square$ & $\square$ & $\square$ \\ $42-52$ & 67 & $\square$ & $\square$ & $\square$ \\ $53-63$ & 68 & $\square$ & $\square$ & $\square$ \\ $64-74$ & 78 & $\square$ & $\square$ & $\square$ \\ $75-85$ & 67 & $\square$ & $\square$ & $\square$ \\ $86-96$ & 25 & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

To construct the expanded frequency distribution, we need to calculate the midpoint, relative frequency, and cumulative frequency for each class. The midpoint is the average of the lower and upper bounds of each class. The relative frequency is the frequency of the class divided by the total frequency. The cumulative frequency is the sum of the frequencies of the current class and all previous classes.

Step 1: Calculate Midpoints

The midpoint for each class is calculated as the average of the lower and upper bounds of the class. The midpoints are:

  • \( 20-30 \): \( \frac{20 + 30}{2} = 25.00 \)
  • \( 31-41 \): \( \frac{31 + 41}{2} = 36.00 \)
  • \( 42-52 \): \( \frac{42 + 52}{2} = 47.00 \)
  • \( 53-63 \): \( \frac{53 + 63}{2} = 58.00 \)
  • \( 64-74 \): \( \frac{64 + 74}{2} = 69.00 \)
  • \( 75-85 \): \( \frac{75 + 85}{2} = 80.00 \)
  • \( 86-96 \): \( \frac{86 + 96}{2} = 91.00 \)
Step 2: Calculate Relative Frequencies

The relative frequency for each class is calculated as the frequency of the class divided by the total frequency. The total frequency is \( 365 \). The relative frequencies are:

  • \( 20-30 \): \( \frac{18}{365} \approx 0.0493 \)
  • \( 31-41 \): \( \frac{42}{365} \approx 0.1151 \)
  • \( 42-52 \): \( \frac{67}{365} \approx 0.1836 \)
  • \( 53-63 \): \( \frac{68}{365} \approx 0.1863 \)
  • \( 64-74 \): \( \frac{78}{365} \approx 0.2137 \)
  • \( 75-85 \): \( \frac{67}{365} \approx 0.1836 \)
  • \( 86-96 \): \( \frac{25}{365} \approx 0.0685 \)
Step 3: Calculate Cumulative Frequencies

The cumulative frequency for each class is the sum of the frequencies of the current class and all previous classes. The cumulative frequencies are:

  • \( 20-30 \): \( 18 \)
  • \( 31-41 \): \( 18 + 42 = 60 \)
  • \( 42-52 \): \( 60 + 67 = 127 \)
  • \( 53-63 \): \( 127 + 68 = 195 \)
  • \( 64-74 \): \( 195 + 78 = 273 \)
  • \( 75-85 \): \( 273 + 67 = 340 \)
  • \( 86-96 \): \( 340 + 25 = 365 \)

Final Answer

\[ \boxed{ \begin{array}{|l|c|c|c|c|} \hline \text{Class} & \text{Frequency, } \mathbf{f} & \text{Midpoint} & \text{Relative frequency} & \text{Cumulative frequency} \\ \hline 20-30 & 18 & 25.00 & \frac{18}{365} & 18 \\ 31-41 & 42 & 36.00 & \frac{42}{365} & 60 \\ 42-52 & 67 & 47.00 & \frac{67}{365} & 127 \\ 53-63 & 68 & 58.00 & \frac{68}{365} & 195 \\ 64-74 & 78 & 69.00 & \frac{78}{365} & 273 \\ 75-85 & 67 & 80.00 & \frac{67}{365} & 340 \\ 86-96 & 25 & 91.00 & \frac{25}{365} & 365 \\ \hline \end{array} } \]

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