Alright, Tyler's group conducted a simulation with 100 trials and recorded the following outcomes for each painting:
- "Freedom from Want" occurred 19 times.
- "A Sunday Afternoon on the Island of La Grande Jatte" occurred 21 times.
- "The Sculpture Gallery" occurred 20 times.
- "Alexander and Diogenes" occurred 19 times.
- "The Potato Eaters" occurred 21 times.
We will analyze these results to determine which statement is true:
1. Step 1: Calculate the relative frequencies (probabilities) for each painting:
- "Freedom from Want": [tex]\( \frac{19}{100} = 0.19 \)[/tex]
- "A Sunday Afternoon on the Island of La Grande Jatte": [tex]\( \frac{21}{100} = 0.21 \)[/tex]
- "The Sculpture Gallery": [tex]\( \frac{20}{100} = 0.20 \)[/tex]
- "Alexander and Diogenes": [tex]\( \frac{19}{100} = 0.19 \)[/tex]
- "The Potato Eaters": [tex]\( \frac{21}{100} = 0.21 \)[/tex]
2. Step 2: Examine the relative frequencies to see if they are exactly the same:
The relative frequencies are:
- 0.19
- 0.21
- 0.20
- 0.19
- 0.21
Clearly, these frequencies are not all exactly the same.
3. Step 3: Determine if the relative frequencies are similar (within a small tolerance):
To decide if the frequencies are roughly equal, we look at the range of frequencies and compare the maximum and minimum values:
- Maximum relative frequency = 0.21
- Minimum relative frequency = 0.19
Calculate the percentage difference between the max and min:
[tex]\[
\frac{0.21 - 0.19}{0.19} \approx 0.1053 \approx 10.53\%
\][/tex]
Given this difference (10.53%), it is greater than a small tolerance level like 5%.
4. Step 4: Evaluation of the presented statements:
- A. This suggests that all outcomes are equally likely because their relative frequencies are similar. However, the difference (10.53%) is not sufficiently small to consider them similar.
- B. This incorrectly asserts equal likelihood because all relative frequencies add up to 100%, which is always true for probabilities but does not imply equality of each event.
- C. This indicates outcomes are not equally likely due to variation in their relative frequencies. Given the 10.53% difference among frequencies, the outcomes indeed show variation.
- D. This incorrectly involves an irrelevant point about numbers 6 through 10, which are beyond our given scope (only numbers 1-5).
- E. This notes that relative frequency of generating a 1 is less than generating a 5, which is true, but it is a partial observation and doesn't address overall equal likelihood effectively.
Therefore, Statement C is the most accurate conclusion based on the experimental data:
C. All of the outcomes are not equally likely because there is a variation in their relative frequencies.
