In a particular year, the mean score on the ACT test was 22.4 and the standard deviation was 4.6. The mean score on the SAT mathematics test was 487 and the standard deviation was 115. The distributions of both scores were approximately bell-shaped. Round the answers to at least two decimal places.

Question

(a) Find the -score for an ACT score of 17.

Answer 1

To calculate the z-score for an ACT score of 17, you can use the formula for the z-score:

\[ z = \frac{(X - \mu)}{\sigma} \]

where:

• \(X\) is the score,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

For the ACT:

• Mean (\(\mu\)) = 22.4
• Standard deviation (\(\sigma\)) = 4.6
• Score (\(X\)) = 17

Now, substituting the values into the formula:

\[ z = \frac{(17 - 22.4)}{4.6} \]

Calculating the numerator:

\[ 17 - 22.4 = -5.4 \]

Now, calculating the z-score:

\[ z = \frac{-5.4}{4.6} \approx -1.173913043478261 \]

Rounding to two decimal places, the z-score is:

\[ z \approx -1.17 \]

So the z-score for an ACT score of 17 is \(-1.17\).

Answer 2

(b) Find the -score for a SAT score of 529.

Answer 3

To calculate the z-score for an SAT score of 529, we can use the same formula as before:

\[ z = \frac{(X - \mu)}{\sigma} \]

where:

• \(X\) is the score,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

For the SAT:

• Mean (\(\mu\)) = 487
• Standard deviation (\(\sigma\)) = 115
• Score (\(X\)) = 529

Now, substituting the values into the formula:

\[ z = \frac{(529 - 487)}{115} \]

Calculating the numerator:

\[ 529 - 487 = 42 \]

Now, calculating the z-score:

\[ z = \frac{42}{115} \approx 0.3652173913043478 \]

Rounding to two decimal places, the z-score is:

\[ z \approx 0.37 \]

So the z-score for an SAT score of 529 is approximately \(0.37\).

Answer 4

c) Jose's ACT score had a -score of 0.99. What was his ACT score?

Answer 5

To find Jose's ACT score using the z-score formula, we can rearrange the formula to solve for \(X\). The formula is:

\[ z = \frac{(X - \mu)}{\sigma} \]

Rearranging the formula gives us:

\[ X = z \cdot \sigma + \mu \]

Given:

• \(z = 0.99\)
• Mean (\(\mu\)) = 22.4
• Standard deviation (\(\sigma\)) = 4.6

Now, substituting the values into the equation:

\[ X = 0.99 \cdot 4.6 + 22.4 \]

Calculating \(0.99 \cdot 4.6\):

\[ 0.99 \cdot 4.6 = 4.554 \]

Now, adding this to the mean:

\[ X = 4.554 + 22.4 = 26.954 \]

Rounding to two decimal places, Jose's ACT score is:

\[ X \approx 26.95 \]

Thus, Jose's ACT score was approximately \(26.95\).

Answer 6

(d) Emma's SAT score had a -score of -1.6. What was her SAT score?

Answer 7

To find Emma's SAT score using the z-score formula, we can rearrange the formula as follows:

\[ z = \frac{(X - \mu)}{\sigma} \]

Rearranging gives us:

\[ X = z \cdot \sigma + \mu \]

Given:

• \(z = -1.6\)
• Mean (\(\mu\)) = 487
• Standard deviation (\(\sigma\)) = 115

Now, substituting the values into the equation:

\[ X = -1.6 \cdot 115 + 487 \]

Calculating \(-1.6 \cdot 115\):

\[ -1.6 \cdot 115 = -184 \]

Now, adding this result to the mean:

\[ X = -184 + 487 = 303 \]

Thus, Emma's SAT score is:

\[ X = 303 \]

So, Emma's SAT score was 303.