12% of students at UOM are left-handed.  What is the probability that in a sample of 50 students, the sample proportion that are left-handed is less than 11%? Give your solution to 4 d.p.

This problem involves determining the probability that in a sample of 50 students, the sample proportion of left-handed students is less than 11%. To solve this, we can use the **normal approximation to the binomial distribution**, since the sample size is reasonably large.

The steps are as follows:

### 1. Define the problem:
- The population proportion (\( p \)) of left-handed students is 12%, or \( p = 0.12 \).
- The sample size is \( n = 50 \).
- We need to find the probability that the sample proportion \( \hat{p} \) is less than 11%, or \( \hat{p} < 0.11 \).

### 2. Mean and standard deviation of the sample proportion:
For a sample proportion, the mean (\( \mu_{\hat{p}} \)) and the standard deviation (\( \sigma_{\hat{p}} \)) are given by:

\[
\mu_{\hat{p}} = p = 0.12
\]
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.12(1 - 0.12)}{50}}
\]

### 3. Convert to a standard normal distribution:
We will convert \( \hat{p} = 0.11 \) into a z-score using the formula:

\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}
\]

This z-score tells us how many standard deviations away \( \hat{p} \) is from the mean, which we can then use to find the probability from the standard normal distribution table.

I'll calculate this now.The calculated z-score is approximately \( -0.218 \). 

Now, we will use this z-score to find the probability that the sample proportion is less than 11%. We can use the standard normal distribution table (or a normal distribution calculator) to find the cumulative probability corresponding to this z-score.

I'll calculate the probability next.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you can look up the cumulative probability for the z-score of \( -0.218 \) using a standard normal distribution table. The corresponding value will give you the probability that the sample proportion is less than 11%.

Would you like to continue with more questions or clarifications? Here are five related questions you can explore:

1. How is the normal approximation to the binomial distribution used in probability?
2. What is a z-score, and how is it applied in probability calculations?
3. What are the conditions for using a normal approximation for a binomial distribution?
4. How does the sample size affect the accuracy of the normal approximation?
5. What other distributions can be used for similar problems, and why?

**Tip:** When applying the normal approximation to a binomial distribution, ensure that \( np \) and \( n(1-p) \) are both greater than 5 to satisfy the conditions for using the normal model.

12% of students at UOM are left-handed. What is the probability that in a sample of 50 students, the sample proportion that are left-handed is less than 11%? Give your solution to 4 d.p.

Calculate the probability that less than 11% of students are left-handed in a sample of 50, using normal approximation to the binomial distribution. This example uses key statistical concepts like mean, standard deviation, and z-scores.

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