A coin is tossed 500 times. Tail & head turns up 250 times. Is coin unbiased.

In order to determine whether the coin is unbiased, we can conduct a **hypothesis test** for a **binomial distribution**. Specifically, we want to test if the coin is fair (i.e., the probability \( p \) of getting a head or a tail is 0.5).

### Null Hypothesis (\( H_0 \)):
- The coin is unbiased, meaning \( p = 0.5 \).

### Alternative Hypothesis (\( H_1 \)):
- The coin is biased, meaning \( p \neq 0.5 \).

#### Step 1: Test Statistic
Since we are conducting a large number of trials (500 coin tosses), we can use the **normal approximation** to the binomial distribution. According to this approximation, the number of heads follows a normal distribution:

\[
X \sim N(np, np(1 - p))
\]

Where:
- \( n = 500 \) (number of trials)
- \( p = 0.5 \) (the hypothesized probability of heads)

Thus, the mean and variance are:
\[
\mu = np = 500 \times 0.5 = 250
\]
\[
\sigma^2 = np(1 - p) = 500 \times 0.5 \times 0.5 = 125
\]
\[
\sigma = \sqrt{125} \approx 11.18
\]

#### Step 2: Z-Score Calculation
We calculate the **Z-score** to determine how many standard deviations the observed number of heads (250) is from the mean:

\[
Z = \frac{X - \mu}{\sigma}
\]
\[
Z = \frac{250 - 250}{11.18} = 0
\]

#### Step 3: P-Value
A Z-score of 0 corresponds to a p-value of 1. This means that the observed result is exactly what we would expect under the null hypothesis.

#### Step 4: Conclusion
Given that the Z-score is 0 and the p-value is very high, we **fail to reject the null hypothesis**. Therefore, there is no significant evidence to suggest that the coin is biased. 

### Final Answer:
The coin appears to be **unbiased** based on the given data.

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Would you like more details or have any questions?

Here are five related questions:
1. How do you interpret p-values in hypothesis testing?
2. What is the difference between a Type I and Type II error?
3. How does the normal approximation work for binomial distributions?
4. What is the Central Limit Theorem, and how is it related to this test?
5. How would the test change if the number of tosses was much smaller, say 10?

**Tip**: Always ensure that the number of trials \( n \) is large enough when using the normal approximation for binomial distributions!

Determine if a coin is fair after 500 tosses with 250 heads and 250 tails using hypothesis testing for binomial distribution. Learn key statistical concepts such as Z-score, normal approximation, and p-value. Suitable for undergraduate students studying statistics.

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