Question

Use the following information to answer the next seven exercises: 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; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34

Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29

Complete the tables using the data provided:

Short answer

The tables for Researcher A and Researcher B can be prepared as follows:

Step-by-step solution

Given Information

We have data in the given question about the average (mean) length of time in months that AIDS patients live once they begin treatment. The data is given below as:

Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34

Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29

Calculation

The frequency of a grouped distribution is the number of values that lie inside the given class interval. For example, for the given data, the frequency for the class interval 0.5-6.5 for Researcher A is 2 because there are two observations in this interval. For the rest class intervals, the frequencies are determined in the same way.

The relative frequency is the ratio of a specific class's frequency to the sum of all frequencies. The following is the formula for computing relative frequencies:

$\mathrm{Relative}\mathrm{Frequency}=\frac{\text{Frequency}}{\text{Sum of frequencies}}$

For example, for Researcher A, the relative frequency for the first class interval 0.5-6.5 is:

$\mathrm{Relative}\mathrm{Frequency}=\frac{2}{40}=0.05$

Other class intervals can be found in the same way.

The previous relative frequencies are added together to form the cumulative relative frequency. Add all of the previous relative frequencies to the current row's relative frequency to get the cumulative relative frequency. For the Researcher A data, the cumulative relative frequencies for the class intervals 0.5-6.5 and 6.5-12.5 are:

$$\begin{gathered} \mathrm{Cumulative}\mathrm{Frequency}\mathrm{of}(0.5-6.5)=0.05 \\ \mathrm{Cumulative}\mathrm{Frequency}\mathrm{of}(6.5-12.5)=0.05+0.175=0.225 \end{gathered}$$

For rest classes, the cumulative relative frequency can be found in the same way.

Hence, the table can be prepared as follows:

Final answer

As a result, with the help of the calculations as explained above, the tables for Researcher A and Researcher B can be prepared as shown: