I need help answering these questions. Please help, this is for a lab (due tonight). Thank you!!Instructions: Compare the class 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. Write an explanation in 2-5 complete sentences to justify what you see in the results that caused you to select the answer above that you did.I'm team #2 in the table! (see photos attached for additional information). **Simulation Frequencies for Each Class Team**The table displays the frequencies of observed sums when rolling two dice for different class teams, each team having conducted 100 simulations.| Teams | Team 1 | Team 2 | Team 3 | Team 4 | Team 5 | Team 6 | Team 7 | Team 8 | Team 9 | Team 10 | Team 11 | Team 12 ||---------------|--------|--------|--------|--------|--------|--------|--------|--------|--------|---------|---------|---------|| Name 1 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- || Name 2 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- || Name 3 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- || Name 4 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- || Name 5 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- || Name 6 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |**Observed Sum on Two Dice Frequencies:**| Observed Sum | Team 1 | Team 2 | Team 3 | Team 4 | Team 5 | Team 6 | Team 7 | Team 8 | Team 9 | Team 10 | Team 11 | Team 12 ||--------------|--------|--------|--------|--------|--------|--------|--------|--------|--------|---------|---------|---------|| 2 | 2 | 5 | 1 | 2 | 0 | 0 | 4 | 2 | 7 | 6 | **Table 4 - Results for the Whole Class**This table presents the summarized results from a dice-rolling experiment conducted by a class, displaying various frequencies and probabilities associated with the sums rolled on two dice.| Sum on the Two Dice | Summed Class Frequencies | Class Relative Frequency | Your Stimulation 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**### Explanation:- **Summed Class Frequencies:** The number of times each sum appeared in the class's experiment.- **Class Relative Frequency:** The proportion of times each sum appeared, calculated by dividing the summed class frequencies by the total number of events (12,000).- **Your Stimulation Relative Frequency:** A comparison metric showing the relative frequency achieved by an individual or subgroup simulation of the same experiment.- **Theoretical Probability:** The expected relative frequency

Answered: Answer the following question: Which…

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I need help answering these questions. Please help, this is for a lab (due tonight). Thank you!!

Instructions: Compare the class 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.

Write an explanation in 2-5 complete sentences to justify what you see in the results that caused you to select the answer above that you did.

I'm team #2 in the table! (see photos attached for additional information).

**Simulation Frequencies for Each Class Team**

The table displays the frequencies of observed sums when rolling two dice for different class teams, each team having conducted 100 simulations.

| Teams | Team 1 | Team 2 | Team 3 | Team 4 | Team 5 | Team 6 | Team 7 | Team 8 | Team 9 | Team 10 | Team 11 | Team 12 |
|---------------|--------|--------|--------|--------|--------|--------|--------|--------|--------|---------|---------|---------|
| Name 1 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Name 2 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Name 3 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Name 4 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Name 5 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Name 6 | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |

**Observed Sum on Two Dice Frequencies:**

| Observed Sum | Team 1 | Team 2 | Team 3 | Team 4 | Team 5 | Team 6 | Team 7 | Team 8 | Team 9 | Team 10 | Team 11 | Team 12 |
|--------------|--------|--------|--------|--------|--------|--------|--------|--------|--------|---------|---------|---------|
| 2 | 2 | 5 | 1 | 2 | 0 | 0 | 4 | 2 | 7 | 6 |

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

This table presents the summarized results from a dice-rolling experiment conducted by a class, displaying various frequencies and probabilities associated with the sums rolled on two dice.

| Sum on the Two Dice | Summed Class Frequencies | Class Relative Frequency | Your Stimulation 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**

### Explanation:

- **Summed Class Frequencies:** The number of times each sum appeared in the class's experiment.
- **Class Relative Frequency:** The proportion of times each sum appeared, calculated by dividing the summed class frequencies by the total number of events (12,000).
- **Your Stimulation Relative Frequency:** A comparison metric showing the relative frequency achieved by an individual or subgroup simulation of the same experiment.
- **Theoretical Probability:** The expected relative frequency

Expert Solution

Step 1

To give answers to the asked questions, Chi-Squared test of Goodness fit can be used mostly with 100(1- $α$ )% assurance of its validity.

But for that test, actual observed and expected frequencies are required and this test can not be applied to relative frequencies as it has requirement of having frequencies must be greater than 5 , so It is mandatory to find frequencies from relative frequencies using N=1200.

Ho: Fit is good (No significant difference between observed and theoretical frequencies)

Vs.

H1: Fit is not good (always right tailed)

Using total frequency , N= 1200, Observed and theoretical frequencies can be obtained.

Test statistic for that test is,

$χ^{2} = \sum_{i - 1}^{n} \frac{{(O_{i} - T_{i})}^{2}}{T_{i}} ~ {χ^{2}}_{n - 1} w h e r e T_{i} i s t h e t h e o r i t i c a l f r e q u e n c i e s a n d O_{i} a r e O b s e r v e d c l a s s a n d p a r t i c u l a r f r e q u e n c i e s a n d n = 11 a n d t o t a l f r e q u e n c y = 1200$

Using formula for the data given , calculated values are,

${χ^{2}}_{c l a s s f r e q} = 18.34 < {χ^{2}}_{0.01, 10} = 23.209 \Rightarrow H o a c c e p t e d i . e f i t i s g o o d a n d {χ^{2}}_{p a r t i c u l a r f r e q} = 208.91 > {χ^{2}}_{0.01, 10} = 23.209 \Rightarrow H 0 r e j e c t e d i . e . f i t i s n o t g o o d .$

Larger is the Calculated value for the test statistics , more strong is the evidence to reject H0.

Hence, Fit is good for class frequencies and fit is not good for particular frequencies.

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Here, Mean absolute deviation can also be used to choose best fit among two.

Mean absolute deviation in statistics is defined as " Average absolute distance between observed and theoretical values" .Larger the value of MAD , more is the variability between actual and predicted values.

Given that , total number of frequencies = n=11.

$∴ M A D = \frac{1}{n} \sum_{i = 1}^{n} |A c t u a l - P r e d i c t e d| = \frac{1}{11} \sum_{i = 1}^{n} |A c t u a l - P r e d i c t e d| H e n c e, M A D (c l a s s f r e q u e n c i e s) = 0.00845 a n d M A D (P a r t i c u l a r Y o u r' s f r e q u e n c y) = 0.032182$

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