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 .  the teacher wants to transform the test scores so that they have an approximately normal distribution with a mean of 80 and a standard deviation of 10 points .  to do this she will use , new score=a+b(old score). Find the values of a and b that the teacher should use to transform the distribution of test scores

To transform the test scores from their current normal distribution to a new normal distribution, we use a linear transformation in the form:

\[
\text{new score} = a + b \times (\text{old score})
\]

### Given Information:

- Old mean \( \mu_{\text{old}} = 22 \)
- Old standard deviation \( \sigma_{\text{old}} = 4 \)
- New mean \( \mu_{\text{new}} = 80 \)
- New standard deviation \( \sigma_{\text{new}} = 10 \)

### Step 1: Solve for \( b \)

The standard deviation of the transformed scores will be \( b \times \sigma_{\text{old}} \). Since the teacher wants the new distribution to have a standard deviation of 10, we can set up the equation:

\[
b \times \sigma_{\text{old}} = \sigma_{\text{new}}
\]

Substitute the values:

\[
b \times 4 = 10
\]

Solve for \( b \):

\[
b = \frac{10}{4} = 2.5
\]

### Step 2: Solve for \( a \)

The mean of the transformed scores will be \( a + b \times \mu_{\text{old}} \). Since the teacher wants the new distribution to have a mean of 80, we can set up the equation:

\[
a + b \times \mu_{\text{old}} = \mu_{\text{new}}
\]

Substitute the values:

\[
a + 2.5 \times 22 = 80
\]

Simplify and solve for \( a \):

\[
a + 55 = 80
\]
\[
a = 80 - 55 = 25
\]

### Final Answer:

- \( a = 25 \)
- \( b = 2.5 \)

Thus, the transformation formula is:

\[
\text{new score} = 25 + 2.5 \times (\text{old score})
\]

Would you like a detailed explanation of how the transformation preserves the normal distribution? Here are some related questions:

1. How does scaling by \( b \) affect the variance of a distribution?
2. Can this transformation be applied to any normal distribution?
3. What happens if the old scores are not normally distributed?
4. How would the formula change if the desired new standard deviation were 15 instead of 10?
5. What is the inverse transformation to return to the old scores?

**Tip:** The linear transformation \( a + b(x) \) preserves the normal distribution's shape; it only changes the mean and standard deviation.

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. The teacher wants to transform the test scores so that they have an approximately normal distribution with a mean of 80 and a standard deviation of 10 points. To do this, she will use the formula: new score = a + b(old score). Find the values of a and b that the teacher should use to transform the distribution of test scores.

Learn how to apply a linear transformation to a normal distribution of test scores. This problem explains how to shift from a distribution with a mean of 22 and a standard deviation of 4 to a new distribution with a mean of 80 and a standard deviation of 10 using the formula new score = 25 + 2.5(old score). Suitable for Grades 10-12.

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