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Let's walk through the problem step by step. Here's the breakdown of each question and how to approach solving them:

### Part a:
**What proportion of the students scored at least 25 points on this test?**

Given that the scores follow a **Normal distribution** with a mean (\(\mu\)) of 22 and a standard deviation (\(\sigma\)) of 4, we are asked to find the proportion of students who scored at least 25 points.

1. **Step 1**: Convert the raw score of 25 to a **z-score** using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   where \(x = 25\), \(\mu = 22\), and \(\sigma = 4\).

2. **Step 2**: Use the z-score table (or normal distribution table) to find the probability that a score is less than 25, and subtract this from 1 to get the proportion of students who scored **at least** 25 points.

### Part b:
**What is the 31st percentile of the distribution of test scores?**

The percentile is the score below which a given percentage of scores fall. For the 31st percentile:

1. **Step 1**: Use the z-score corresponding to the 31st percentile (which is approximately -0.50 from the z-score table).
   
2. **Step 2**: Convert the z-score back to the original score using the formula:
   \[
   x = \mu + z\sigma
   \]
   where \(\mu = 22\), \(\sigma = 4\), and \(z \approx -0.50\).

### Part c:
**Find the values of \(a\) and \(b\) for the transformation of the test scores.**

The teacher wants to transform the scores to have a new mean of 80 and a new standard deviation of 10. The transformation is given by:
\[
\text{new score} = a + b(\text{old score})
\]

1. **Step 1**: Set up two equations based on the mean and standard deviation transformations:
   - For the mean: \(80 = a + b(22)\)
   - For the standard deviation: \(10 = b(4)\)
   
2. **Step 2**: Solve these equations to find \(a\) and \(b\).

### Part d:
**Would it be appropriate to use a Normal distribution to calculate the proportion of students who scored below 7 points on this assignment? Explain.**

Given that the maximum score on the assignment is 10 points, and the distribution of scores has a mean of 9.2 and a standard deviation of 2.1 points, the key question here is whether it's reasonable to apply the Normal distribution. 

1. **Step 1**: Check the distribution in relation to the possible scores. Since the maximum score is 10, we should consider whether a Normal distribution is suitable when the mean is close to the maximum score and the standard deviation might push some theoretical values outside the range of possible scores (negative values).

2. **Step 2**: Analyze whether the distribution might be skewed or whether the conditions for using a Normal distribution are violated (e.g., scores being bounded between 0 and 10).

---

Would you like to go into more detail for any of these parts or proceed with specific calculations?

Here are 5 follow-up questions to deepen your understanding:

1. How does the z-score relate to the percentile in a normal distribution?
2. What is the relationship between a linear transformation and the mean/standard deviation of a distribution?
3. Why might a normal distribution not be appropriate when dealing with a bounded set of scores?
4. How would you interpret a z-score of 0? What does it mean in the context of this problem?
5. How would changing the mean or standard deviation of a distribution affect its shape?

**Tip**: When converting between z-scores and raw scores, always keep track of whether you're using a cumulative probability (e.g., finding a percentile) or looking for a specific range (e.g., scores above or below a certain value).

The distribution of scores on a recent test closely followed a Normal distribution with a mean of 22 points and a standard deviation of 4 points.

Part a: What proportion of the students scored at least 25 points on this test?
Part b: What is the 31st percentile of the distribution of test scores?
Part c: The teacher wants to transform the test scores to have a mean of 80 and a standard deviation of 10. Find values of a and b for the transformation new score = a + b(old score).
Part d: Before the test, the teacher gave a review assignment with a maximum score of 10 points, a mean of 9.2 points, and a standard deviation of 2.1 points. Is it appropriate to use a Normal distribution to calculate the proportion of students who scored below 7 points?

Explore how to solve problems involving a Normal distribution of test scores with a mean of 22 and standard deviation of 4. Learn to find proportions using Z-scores, calculate percentiles, and transform test scores to a new scale. Suitable for high school students in Grades 11-12.

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