We are tasked with finding the probability that 12 or more people out of 100 in a sample are left-handed, given that 10% of the population is left-handed. This involves using the normal approximation to the binomial distribution.

Step-by-Step Solution:

1. Define the problem parameters:

   • $n = 100$ (sample size)
   • $p = 0.10$ (probability of being left-handed)
   • $q = 1 - p = 0.90$ (probability of not being left-handed)

2. Find the mean ($\mu$) and standard deviation ($\sigma$) of the binomial distribution: $\mu = n \cdot p = 100 \cdot 0.10 = 10$ $\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{100 \cdot 0.10 \cdot 0.90} = \sqrt{9} = 3$

3. Set up the normal approximation: Since $n$ is large and $np > 5$ and $nq > 5$, the binomial distribution can be approximated by a normal distribution with: $N(\mu = 10, \sigma = 3)$

4. Apply the continuity correction: To find $P(X \geq 12)$, we adjust for the continuity correction: $P(X \geq 12) \approx P\left(Z \geq \frac{12 - 0.5 - \mu}{\sigma}\right)$ Substituting the values: $P(X \geq 12) \approx P\left(Z \geq \frac{12 - 0.5 - 10}{3}\right)$ $P(X \geq 12) \approx P\left(Z \geq \frac{1.5}{3}\right)$ $P(X \geq 12) \approx P(Z \geq 0.5)$

5. Use the standard normal table: The $Z$-score is 0.5. From the standard normal distribution table: $P(Z \geq 0.5) = 1 - P(Z \leq 0.5) = 1 - 0.6915 = 0.3085$

Final Answer:

The probability that 12 or more people in your sample are left-handed is approximately: P(X \geq 12) = 0.3085 \text{ (or 30.85%).}

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Do you have any questions or would you like me to elaborate on any step?

Related Questions:

1. What is the probability that fewer than 12 people are left-handed?
2. How does increasing the sample size affect the probability calculation?
3. What happens to the standard deviation if the percentage of left-handed people changes?
4. Why is the continuity correction important in this context?
5. Can you derive the same result without using the normal approximation?

Tip:

When using the normal approximation for a binomial distribution, always check that $np > 5$ and $nq > 5$ to ensure accuracy.