Questions: WHERE APPROPRIATE, SHOW WORK, ROUND APPROPRIATELY USE PROPER UNITS. Use the following information to complete the tables and answer the questions. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. Researcher A: 3 4 11 15 16 17 22 44 37 16 Survival length (in months) Frequency Relative Frequency Cumulative Relative Frequency --------------------------------------------------------------------------------------------------------------------------------- 14 24 25 15 12 8 40 32 26 34 0.5-6.5 26 27 33 29 35 44 13 21 22 10 6.5-12.5 27 31 34 29 17 8 24 18 47 33 12.5-18.5 18.5-24.5 24.5-30.5 30.5 - 36.5 24 Pts ง 36.5-42.5 42.5 - 48.5

WHERE APPROPRIATE, SHOW WORK, ROUND APPROPRIATELY USE PROPER UNITS. Use the following information to complete the tables and answer the questions. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. Researcher A: 3 4 11 15 16 17 22 44 37 16 Survival length (in months) Frequency Relative Frequency Cumulative Relative Frequency --------------------------------------------------------------------------------------------------------------------------------- 14 24 25 15 12 8 40 32 26 34 0.5-6.5 26 27 33 29 35 44 13 21 22 10 6.5-12.5 27 31 34 29 17 8 24 18 47 33 12.5-18.5 18.5-24.5 24.5-30.5 30.5 - 36.5 24 Pts ง 36.5-42.5 42.5 - 48.5
Transcript text: WHERE APPROPRIATE, SHOW WORK, ROUND APPROPRIATELY & USE PROPER UNITS. Use the following information to complete the tables and answer the questions. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{2}{|l|}{ Researcher A: } & & & & & & & & \multirow[b]{2}{*}{16} & \begin{tabular}{l} Survival length \\ (in months) \end{tabular} & \multirow[t]{2}{*}{ Frequency } & \begin{tabular}{l} Relative \\ Frequency \end{tabular} & \begin{tabular}{l} Cumulative \\ Relative Frequency \end{tabular} \\ \hline 3 & 4 & 11 & 15 & 16 & 17 & 22 & 44 & 37 & & $0.5-6.5$ & & & \\ \hline 14 & 24 & 25 & 15 & 12 & 8 & 40 & 32 & 26 & 34 & $6.5-12.5$ & & & \\ \hline 26 & 27 & 33 & 29 & 35 & 44 & 13 & 21 & 22 & 10 & $12.5-18.5$ & & & \\ \hline 27 & 31 & 34 & 29 & 17 & 8 & 24 & 18 & 47 & 33 & $18.5-24.5$ & & & \\ \hline & & & & & & & & & & $24.5-30.5$ & & & \\ \hline & & & & & & & & & & 30.5 - 36.5 & & & \\ \hline 24 Pts & ง & & & & & & & & & $36.5-42.5$ & & & \\ \hline & & & & & & & & & & 42.5 - 48.5 & & & \\ \hline \end{tabular}
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Solution

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Solution Steps

To solve the problem, we need to calculate the frequency, relative frequency, and cumulative relative frequency for the given survival lengths. The frequency is the count of patients in each interval. The relative frequency is the frequency divided by the total number of patients. The cumulative relative frequency is the running total of the relative frequencies.

Step 1: Calculate Frequency for Each Interval

To determine the frequency for each interval, count the number of data points that fall within each specified range. The intervals and their corresponding frequencies are as follows:

  • Interval \( (0.5, 6.5) \): Frequency = 2
  • Interval \( (6.5, 12.5) \): Frequency = 5
  • Interval \( (12.5, 18.5) \): Frequency = 9
  • Interval \( (18.5, 24.5) \): Frequency = 5
  • Interval \( (24.5, 30.5) \): Frequency = 7
  • Interval \( (30.5, 36.5) \): Frequency = 7
  • Interval \( (36.5, 42.5) \): Frequency = 2
  • Interval \( (42.5, 48.5) \): Frequency = 3
Step 2: Calculate Relative Frequency for Each Interval

The relative frequency is calculated by dividing the frequency of each interval by the total number of patients, which is 40. The relative frequencies are:

  • Interval \( (0.5, 6.5) \): Relative Frequency = \( \frac{2}{40} = 0.05 \)
  • Interval \( (6.5, 12.5) \): Relative Frequency = \( \frac{5}{40} = 0.125 \)
  • Interval \( (12.5, 18.5) \): Relative Frequency = \( \frac{9}{40} = 0.225 \)
  • Interval \( (18.5, 24.5) \): Relative Frequency = \( \frac{5}{40} = 0.125 \)
  • Interval \( (24.5, 30.5) \): Relative Frequency = \( \frac{7}{40} = 0.175 \)
  • Interval \( (30.5, 36.5) \): Relative Frequency = \( \frac{7}{40} = 0.175 \)
  • Interval \( (36.5, 42.5) \): Relative Frequency = \( \frac{2}{40} = 0.05 \)
  • Interval \( (42.5, 48.5) \): Relative Frequency = \( \frac{3}{40} = 0.075 \)
Step 3: Calculate Cumulative Relative Frequency for Each Interval

The cumulative relative frequency is the running total of the relative frequencies. The cumulative relative frequencies are:

  • Interval \( (0.5, 6.5) \): Cumulative Relative Frequency = 0.05
  • Interval \( (6.5, 12.5) \): Cumulative Relative Frequency = 0.05 + 0.125 = 0.175
  • Interval \( (12.5, 18.5) \): Cumulative Relative Frequency = 0.175 + 0.225 = 0.400
  • Interval \( (18.5, 24.5) \): Cumulative Relative Frequency = 0.400 + 0.125 = 0.525
  • Interval \( (24.5, 30.5) \): Cumulative Relative Frequency = 0.525 + 0.175 = 0.700
  • Interval \( (30.5, 36.5) \): Cumulative Relative Frequency = 0.700 + 0.175 = 0.875
  • Interval \( (36.5, 42.5) \): Cumulative Relative Frequency = 0.875 + 0.05 = 0.925
  • Interval \( (42.5, 48.5) \): Cumulative Relative Frequency = 0.925 + 0.075 = 1.000

Final Answer

\[ \begin{align_} \text{Interval } (0.5, 6.5): & \text{ Cumulative Relative Frequency} = 0.05 \\ \text{Interval } (6.5, 12.5): & \text{ Cumulative Relative Frequency} = 0.175 \\ \text{Interval } (12.5, 18.5): & \text{ Cumulative Relative Frequency} = 0.400 \\ \text{Interval } (18.5, 24.5): & \text{ Cumulative Relative Frequency} = 0.525 \\ \text{Interval } (24.5, 30.5): & \text{ Cumulative Relative Frequency} = 0.700 \\ \text{Interval } (30.5, 36.5): & \text{ Cumulative Relative Frequency} = 0.875 \\ \text{Interval } (36.5, 42.5): & \text{ Cumulative Relative Frequency} = 0.925 \\ \text{Interval } (42.5, 48.5): & \text{ Cumulative Relative Frequency} = 1.000 \\ \end{align_} \]

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