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Tyler's art class is split into groups. His group has to stage a tableau vivant (French for "living picture") as part of a year-end project. Each member of the group suggests a painting to depict: Freedom from Want, A Sunday Afternoon on the Island of La Grande Jatte, The Sculpture Gallery, Alexander and Diogenes, and The Potato Eaters. To choose one of these paintings, they use a number generator to generate numbers from 1 to 5 and assign a number to each painting. First, they run 100 trials to check the validity of their model and record the outcomes shown in the table.

\begin{tabular}{|l|c|c|}
\hline Painting & Number & Times Result Occurred \\
\hline Freedom from Want & 1 & 19 \\
\hline A Sunday Afternoon on the Island of La Grande Jatte & 2 & 21 \\
\hline The Sculpture Gallery & 3 & 20 \\
\hline Alexander and Diogenes & 4 & 19 \\
\hline The Potato Eaters & 5 & 21 \\
\hline
\end{tabular}

Which statement is true based on the results in the table?

A. All of the outcomes are equally likely because their relative frequencies are similar.

B. All of the outcomes are equally likely because the relative frequencies add up to 100.

C. All of the outcomes are not equally likely because there is a variation in their relative frequencies.

D. All of the outcomes are not equally likely because the numbers 6 through 10 are not accounted for.

E. All of the outcomes are not equally likely because the relative frequency of generating a 1 is less than the relative frequency of generating a 5.



Answer :

GinnyAnswer
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.

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