A biased coin is tossed four times and the number of heads noted. The experiment is performed 500

times in all and the results are summarised in the following table:

Number of heads 0 1 2 3 4

Frequency 12 50 151 200 87

Illustrate, with appropriate calculations how this set of data compares with a binomial distribution

with suitable *n* and *p* values.

To compare the observed data with a binomial distribution, we'll proceed with the following steps:

### 1. **Understand the Data and Parameters**:
   - The experiment involves tossing a biased coin 4 times.
   - The number of heads observed (0 to 4) is recorded over 500 trials.

### 2. **Observed Frequencies**:
   - Number of heads: 0, 1, 2, 3, 4
   - Frequencies: 12, 50, 151, 200, 87

### 3. **Determine Suitable \(n\) and \(p\)**:
   - Since the coin is tossed 4 times, \(n = 4\).
   - The probability of success (getting heads) is \(p\), which we'll estimate.

### 4. **Estimate the Probability \(p\)**:
   - We can estimate \(p\) using the average number of heads observed in the experiments. This is done by calculating the expected value (mean) of the binomial distribution.

\[
\text{Mean} = \sum (\text{Number of heads} \times \text{Frequency}) / \text{Total number of trials}
\]

\[
\text{Mean} = \sum (x \times f(x)) / 500
\]

Where \(x\) is the number of heads and \(f(x)\) is the corresponding frequency.

\[
\text{Mean of binomial distribution} = np
\]

Given that \(n = 4\), we can solve for \(p\).

### 5. **Calculate Expected Frequencies Using the Binomial Formula**:
   - The probability of getting \(x\) heads in 4 tosses is given by the binomial probability formula:

\[
P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}
\]

Where:
   - \(\binom{n}{x}\) is the binomial coefficient.
   - \(n = 4\) (number of trials).
   - \(p\) is the estimated probability of getting heads.

### 6. **Compare the Observed Frequencies with the Expected Frequencies**:
   - Calculate the expected frequencies for each number of heads using the binomial distribution with the estimated \(p\).
   - Compare these expected frequencies to the observed frequencies to see how well the binomial model fits the data.

Let's perform these calculations.### Results:

- **Mean Number of Heads (Observed)**: \(2.6\)
- **Estimated Probability of Heads (\(p\))**: \(0.65\)

### Expected Frequencies (Using the Binomial Distribution with \(n = 4\) and \(p = 0.65\)):

| Number of Heads | Observed Frequency | Expected Frequency |
|-----------------|--------------------|--------------------|
| 0               | 12                 | 7.50               |
| 1               | 50                 | 55.74              |
| 2               | 151                | 155.27             |
| 3               | 200                | 192.24             |
| 4               | 87                 | 89.25              |

### Comparison:
The expected frequencies, calculated using a binomial distribution with \(p = 0.65\), align reasonably well with the observed frequencies. This suggests that the data can be reasonably modeled by a binomial distribution with the parameters \(n = 4\) and \(p = 0.65\).

Would you like more details or have any questions?

### Further questions:
1. How would the expected frequencies change if \(p\) were different?
2. What is the chi-square goodness-of-fit test for this data?
3. How does the deviation between observed and expected frequencies affect the fit of the model?
4. Could a different distribution better model this data?
5. How would the results change if we increased the number of coin tosses?

### Tip:
When comparing observed data to a theoretical distribution, using a chi-square goodness-of-fit test can provide a quantitative measure of how well the data fits the model.

Illustrate how a biased coin toss experiment compares with a binomial distribution. Analyze 500 trials of tossing a coin 4 times, recording the frequency of heads. Explore calculations of expected frequencies, using suitable parameters (n = 4, p = 0.65) and compare them with observed frequencies.

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