I need to compare the listed results to the theoretical probabilities and answer the questions below.Answer the following question: Which fits better to the theoretical probabilities?a) My personal observed results fit best to the theoretical probabilitiesb) The class overall observed results fit better to the theoretical probabilitiesc) Both sets of observed results fit equally well to the theoretical probabilities.d) Neither set of observed results fit the theoretical probabilities – they are both bad fits to the probabilities.I need help with figuring out what choice above is the best and need help writing an explanation to justify what I see in the results that caused me to select the answer above that I did. Thank youI'm team #2 in the table (see photos attached). **Table 4 - Results for the Whole Class**| Sum on the Two Dice | Summed Class Frequencies | Class Relative Frequency | Your Simulation Relative Frequency | Theoretical Probability (Relative Frequency) ||---------------------|--------------------------|--------------------------|-----------------------------------|---------------------------------------------|| 2 | 39 | 0.0325 | 0.01 | 0.028 || 3 | 53 | 0.0442 | 0.04 | 0.056 || 4 | 100 | 0.0833 | 0.08 | 0.083 || 5 | 151 | 0.1258 | 0.13 | 0.111 || 6 | 160 | 0.1333 | 0.09 | 0.139 || 7 | 195 | 0.1625 | 0.14 | 0.167 || 8 | 196 | 0.1633 | 0.21 | 0.139 || 9 | 128 | 0.1067 | 0.14 | 0.111 || 10 | 103 | 0.0853 | 0.14 | 0.083 || 11 | 50 | 0.0417 | 0.01 | 0.056 || 12 | 25 | 0.0208 | 0.01 | 0.028 |**Class Total Events**: 12,000**Description:**This table presents the results of a class experiment involving rolling two dice. The data shows the frequency and probability of each possible sum (from 2 to 12) that can result when adding the numbers on the two dice.- **Summed Class Frequencies**: The number of times each sum was rolled in the class experiment.- **Class Relative Frequency**: The proportion of each sum's occurrence relative to the total events (12,000).- **Your Simulation Relative Frequency**: The frequency data from an individual's simulation attempts.- **Theoretical Probability**: The expected probability of each **Table 3 – Comparison of Observed vs Theoretical Results**This table provides a comparison between the observed (empirical) relative frequency and the theoretical probability of the sums obtained from rolling two dice. It also calculates the difference between the observed and theoretical values.| Sum on two dice | Observed (Empirical) Relative Frequency | Theoretical Probability (Relative Frequency) | Difference Observed - Theoretical ||-----------------|-----------------------------------------|--------------------------------------------|----------------------------------|| 2 | 0.01 | 0.028 | -0.018 || 3 | 0.04 | 0.056 | -0.016 || 4 | 0.08 | 0.083 | -0.003 || 5 | 0.13 | 0.111 | 0.019 || 6 | 0.09 | 0.139 | -0.049 || 7 | 0.14 | 0.167 | -0.027 || 8 | 0.21 | 0.139 | 0.071 || 9 | 0.14 | 0.111 | 0.029 || 10 | 0.14 | 0.083 | 0.057 || 11 | 0.01 | 0.056 | -0.046 || 12 | 0.01 | 0.028 | -0.018 |**Explanation:**- The first column lists the possible sums that can be rolled with two dice, ranging from 2 to 12.- The second column shows the observed relative frequency, which represents how often each sum appeared in actual rolls.- The third column provides the theoretical probability for each sum, based on mathematical calculations of likelihood.- The fourth column calculates the difference between the observed frequencies and the theoretical probabilities, highlighting any discrepancies.This table helps in understanding how actual outcomes compare to theoretical expectations in a probability experiment involving rolling two dice.

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Description: I need to compare the listed results to the theoretical probabilities and answer the questions below.Answer the following question: Which fits better to the theoretical probabilities?a) My personal observed results fit best to the theoretical probabi

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I need to compare the listed results to the theoretical probabilities and answer the questions below.

Answer the following question: Which fits better to the theoretical probabilities?

a) My personal observed results fit best to the theoretical probabilities

b) The class overall observed results fit better to the theoretical probabilities

c) Both sets of observed results fit equally well to the theoretical probabilities.

d) Neither set of observed results fit the theoretical probabilities – they are both bad fits to the probabilities.

I need help with figuring out what choice above is the best and need help writing an explanation to justify what I see in the results that caused me to select the answer above that I did. Thank you

I'm team #2 in the table (see photos attached).

**Table 4 - Results for the Whole Class**

| Sum on the Two Dice | Summed Class Frequencies | Class Relative Frequency | Your Simulation Relative Frequency | Theoretical Probability (Relative Frequency) |
|---------------------|--------------------------|--------------------------|-----------------------------------|---------------------------------------------|
| 2 | 39 | 0.0325 | 0.01 | 0.028 |
| 3 | 53 | 0.0442 | 0.04 | 0.056 |
| 4 | 100 | 0.0833 | 0.08 | 0.083 |
| 5 | 151 | 0.1258 | 0.13 | 0.111 |
| 6 | 160 | 0.1333 | 0.09 | 0.139 |
| 7 | 195 | 0.1625 | 0.14 | 0.167 |
| 8 | 196 | 0.1633 | 0.21 | 0.139 |
| 9 | 128 | 0.1067 | 0.14 | 0.111 |
| 10 | 103 | 0.0853 | 0.14 | 0.083 |
| 11 | 50 | 0.0417 | 0.01 | 0.056 |
| 12 | 25 | 0.0208 | 0.01 | 0.028 |

**Class Total Events**: 12,000

**Description:**

This table presents the results of a class experiment involving rolling two dice. The data shows the frequency and probability of each possible sum (from 2 to 12) that can result when adding the numbers on the two dice.

- **Summed Class Frequencies**: The number of times each sum was rolled in the class experiment.
- **Class Relative Frequency**: The proportion of each sum's occurrence relative to the total events (12,000).
- **Your Simulation Relative Frequency**: The frequency data from an individual's simulation attempts.
- **Theoretical Probability**: The expected probability of each

**Table 3 – Comparison of Observed vs Theoretical Results**

This table provides a comparison between the observed (empirical) relative frequency and the theoretical probability of the sums obtained from rolling two dice. It also calculates the difference between the observed and theoretical values.

| Sum on two dice | Observed (Empirical) Relative Frequency | Theoretical Probability (Relative Frequency) | Difference Observed - Theoretical |
|-----------------|-----------------------------------------|--------------------------------------------|----------------------------------|
| 2 | 0.01 | 0.028 | -0.018 |
| 3 | 0.04 | 0.056 | -0.016 |
| 4 | 0.08 | 0.083 | -0.003 |
| 5 | 0.13 | 0.111 | 0.019 |
| 6 | 0.09 | 0.139 | -0.049 |
| 7 | 0.14 | 0.167 | -0.027 |
| 8 | 0.21 | 0.139 | 0.071 |
| 9 | 0.14 | 0.111 | 0.029 |
| 10 | 0.14 | 0.083 | 0.057 |
| 11 | 0.01 | 0.056 | -0.046 |
| 12 | 0.01 | 0.028 | -0.018 |

**Explanation:**

- The first column lists the possible sums that can be rolled with two dice, ranging from 2 to 12.
- The second column shows the observed relative frequency, which represents how often each sum appeared in actual rolls.
- The third column provides the theoretical probability for each sum, based on mathematical calculations of likelihood.
- The fourth column calculates the difference between the observed frequencies and the theoretical probabilities, highlighting any discrepancies.

This table helps in understanding how actual outcomes compare to theoretical expectations in a probability experiment involving rolling two dice.

Expert Solution

Step 1

We will use the chi-square test to determine the goodness of the fit.

a) Comparison of Personal Observed vs Theoretical Results

Null Hypothesis: There is Relationship between theoretical result and observed result

Alternate Hypothesis: There is no relationship

Sum of 2 die	Observed(O)	Expected(E)	(O-E)^2/E
2	39	33.6	0.8679
3	53	67.2	3.0006
4	100	99.6	0.0016
5	151	133.2	2.3787
6	160	166.8	0.2772
7	195	200.4	0.1455
8	196	166.8	5.118
9	128	133.2	0.2030
10	103	99.6	0.1161
11	50	67.2	4.4024
12	25	33.6	2.2012

Test Statistics:

$\begin{array}{rcl} C h i ‐ s q u a r e s t a t i s t i c s & = & \sum_{} \frac{{(O - E)}^{2}}{E} \\ = & 18.7614 \end{array}$

Critical value= $C h i ‐ s q u a r e w i t h d e g r e e o f f r e e d o m 10 a n d s i g n i f i c a n c e l e v e l o f 0.05 = 18.31$

As test statistics is more than critical value we reject the null hypothesis

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