For a set of 70 coin flips, we record 21 heads and 49 tails.

If p is the true probability of heads, what is our best estimate for p here?
What is our best estimate for the standard deviation?

Given the data from the 70 coin flips, we can use the following steps to estimate the probability \( p \) of flipping heads and calculate the standard deviation:

### 1. **Best Estimate for \( p \) (Probability of Heads):**

The best estimate for \( p \), the probability of getting heads, is the ratio of the number of heads to the total number of flips. This is simply the sample proportion:

\[
p = \frac{\text{Number of Heads}}{\text{Total Number of Flips}} = \frac{21}{70} = 0.3
\]

So, the best estimate for \( p \) is **0.3**.

### 2. **Best Estimate for the Standard Deviation:**

The standard deviation of a binomial distribution is given by the formula:

\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]

Where:
- \( n \) is the number of trials (70 coin flips),
- \( p \) is the estimated probability of heads (0.3).

Substituting the values:

\[
\sigma = \sqrt{70 \cdot 0.3 \cdot (1 - 0.3)} = \sqrt{70 \cdot 0.3 \cdot 0.7} = \sqrt{14.7} \approx 3.83
\]

Thus, the best estimate for the standard deviation is **3.83**.

### Summary:
- **Best estimate for \( p \):** 0.3
- **Best estimate for the standard deviation:** 3.83

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Would you like further details or have any questions?

Here are five questions to expand on this topic:
1. How do we calculate the standard error of the proportion for this data?
2. What would change if we observed 35 heads and 35 tails?
3. Can we use confidence intervals to estimate \( p \) more accurately?
4. How would increasing the number of flips affect the standard deviation?
5. What is the difference between binomial and normal approximation for such problems?

**Tip:** Always remember that the standard deviation measures the variability of outcomes around the mean in a binomial distribution!

For a set of 70 coin flips, we record 21 heads and 49 tails.
If p is the true probability of heads, what is our best estimate for p here?
What is our best estimate for the standard deviation?

Learn how to estimate the probability of heads and calculate the standard deviation from a set of 70 coin flips with 21 heads and 49 tails. This problem uses key probability concepts and binomial distribution formulas. Suitable for students in Grades 9-12.

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