Let's go step by step through the given data to derive the relative frequency table, an appropriate graph, a bar graph, and a histogram.
### Step 1: Understanding the categorical data
We have the following data for books and movies across different genres:
| | Comedy | Adventure | Biography | Romance | Thriller |
|--------|--------|-----------|-----------|---------|----------|
| Book | 15 | 25 | 31 | 46 | 78 |
| Movie | 54 | 62 | 15 | 41 | 52 |
### Step 2: Calculate Total Numbers
First, we sum up the total numbers for both books and movies to figure out the relative frequencies.
Total number of books:
[tex]\[ 15 + 25 + 31 + 46 + 78 = 195 \][/tex]
Total number of movies:
[tex]\[ 54 + 62 + 15 + 41 + 52 = 224 \][/tex]
### Step 3: Compute Relative Frequencies
Next, we compute the relative frequencies for each category within books and movies.
#### For Books:
1. Comedy:
[tex]\[
\frac{15}{195} \approx 0.0769 \text{ (rounded to four decimal places)}
\][/tex]
2. Adventure:
[tex]\[
\frac{25}{195} \approx 0.1282
\][/tex]
3. Biography:
[tex]\[
\frac{31}{195} \approx 0.1590
\][/tex]
4. Romance:
[tex]\[
\frac{46}{195} \approx 0.2359
\][/tex]
5. Thriller:
[tex]\[
\frac{78}{195} \approx 0.4000
\][/tex]
#### For Movies:
1. Comedy:
[tex]\[
\frac{54}{224} \approx 0.2411
\][/tex]
2. Adventure:
[tex]\[
\frac{62}{224} \approx 0.2768
\][/tex]
3. Biography:
[tex]\[
\frac{15}{224} \approx 0.0670
\][/tex]
4. Romance:
[tex]\[
\frac{41}{224} \approx 0.1830
\][/tex]
5. Thriller:
[tex]\[
\frac{52}{224} \approx 0.2321
\][/tex]
### Step 4: Construct the Relative Frequency Table
We now can form the relative frequency table using the computed values:
| | Comedy | Adventure | Biography | Romance | Thriller |
|--------------|---------|-----------|-----------|---------|----------|
| Books | 0.0769 | 0.1282 | 0.1590 | 0.2359 | 0.4000 |
| Movies | 0.2411 | 0.2768 | 0.0670 | 0.1830 | 0.2321 |
### Step 5: Create a Graph (Bar Graph)
To visualize the data, a bar graph can be an excellent way to compare the relative frequencies across the different genres for books and movies.
1. X-axis: Represents the genres (Comedy, Adventure, Biography, Romance, Thriller).
2. Y-axis: Represents the relative frequencies.
3. Bars: Two bars for each genre, one for books and one for movies.
### Step 6: Create a Histogram
A histogram is typically used for continuous data and for this reason, it might be more illustrative to use a grouped bar chart instead. However, if you intended to compare the distribution relative to each category (genre) between books and movies, separate histograms could be used but are less common in this scenario.
Now you are set with all the necessary information and calculations, and the visual presentation can be implemented using a suitable software tool to produce the bar graph and histograms.
