To determine whether a coin is unfair based on the number of heads obtained after tossing the coin 1000 times, we are given the condition:

$\left| \frac{h - 500}{50} \right| \geq 1.645$

Where:

• $h$ is the number of heads obtained.
• 500 is the expected number of heads (since 1000 tosses would give 500 heads in a fair coin).
• 50 is the standard deviation in this case (since $\sigma = \sqrt{1000 \times 0.5 \times 0.5} = 50$).

Step-by-step approach:

1. Remove the absolute value: The inequality can be split into two cases:

   $\frac{h - 500}{50} \geq 1.645 \quad \text{or} \quad \frac{h - 500}{50} \leq -1.645$

2. Solve each case:

   For the first case: $\frac{h - 500}{50} \geq 1.645$ Multiply both sides by 50: $h - 500 \geq 1.645 \times 50$ $h - 500 \geq 82.25$ Add 500 to both sides: $h \geq 582.25$ Since $h$ must be an integer, round it up: $h \geq 583$

   For the second case: $\frac{h - 500}{50} \leq -1.645$ Multiply both sides by 50: $h - 500 \leq -1.645 \times 50$ $h - 500 \leq -82.25$ Add 500 to both sides: $h \leq 417.75$ Round it down: $h \leq 417$

Conclusion:

The number of outcomes that result in heads must either be greater than or equal to 583, or less than or equal to 417, for the coin to be considered unfair. In other words:

• If $h \geq 583$ or $h \leq 417$, the coin is unfair.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. What is the meaning of the 1.645 value in this context?
2. How would the inequality change if the number of tosses was 10,000 instead of 1,000?
3. Can we interpret this result as a hypothesis test?
4. What is the significance of the number 50 in this problem?
5. How would we calculate the probability of getting 583 heads or more if the coin is fair?

Tip: In hypothesis testing, a value like 1.645 typically corresponds to a confidence level, here likely 90%, which helps to determine the threshold for fairness.