A study was conducted on students from a particular high school over the last 8 years. The following information was found regarding standardized tests used for college admitance. Scores on the SAT test are normally distributed with a mean of 1108 and a standard deviation of 206 . Scores on the ACT test are normally distributed with a mean of 21.3 and a standard deviation of 4.8 . It is assumed that the two tests measure the same aptitude, but use different scales. If a student gets an SAT score that is the 35-percentile, find the actual SAT score. SAT score = 1029 Round answer to a whole number. What would be the equivalent ACT score for this student? ACT score $$ = $$ Round answer to 1 decimal place. If a student gets an SAT score of 1396, find the equivalent ACT score. ACT score =

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UpStudy AI · Accepted Answer

**Step 1. Find the $$35^\text{th}$$ percentile SAT score.**

The SAT scores are normally distributed with
$$
\mu_{\text{SAT}} = 1108 \quad \text{and} \quad \sigma_{\text{SAT}} = 206.
$$
Let $$z$$ be the standardized score such that
$$
P(Z < z) = 0.35.
$$
Looking up the standard normal table (or using an appropriate calculator), we find
$$
z \approx -0.385.
$$

Now, the corresponding SAT score is given by
$$
\text{SAT score} = \mu_{\text{SAT}} + z \cdot \sigma_{\text{SAT}}.
$$
Substitute the known values:
$$
\text{SAT score} \approx 1108 + (-0.385)(206) \approx 1108 - 79.4 \approx 1028.6.
$$
Rounded to a whole number, the SAT score is
$$
1029.
$$

---

**Step 2. Find the equivalent ACT score for the $$35^\text{th}$$ percentile (SAT score of $$1029$$).**

The ACT scores are normally distributed with
$$
\mu_{\text{ACT}} = 21.3 \quad \text{and} \quad \sigma_{\text{ACT}} = 4.8.
$$
Because the aptitude is measured on different scales but with the same relative performance, we use the same $$z$$-score:
$$
\text{ACT score} = \mu_{\text{ACT}} + z \cdot \sigma_{\text{ACT}}.
$$
Substitute $$z \approx -0.385$$:
$$
\text{ACT score} \approx 21.3 + (-0.385)(4.8) \approx 21.3 - 1.848 \approx 19.452.
$$
Rounded to one decimal place, the equivalent ACT score is
$$
19.5.
$$

---

**Step 3. Find the equivalent ACT score for an SAT score of $$1396$$.**

First, compute the $$z$$-score for an SAT score of $$1396$$:
$$
z = \frac{\text{SAT score} - \mu_{\text{SAT}}}{\sigma_{\text{SAT}}} = \frac{1396 - 1108}{206} = \frac{288}{206} \approx 1.398.
$$

Now, use this $$z$$-score to find the equivalent ACT score:
$$
\text{ACT score} = \mu_{\text{ACT}} + z \cdot \sigma_{\text{ACT}} \approx 21.3 + 1.398 \times 4.8.
$$
Calculate:
$$
1.398 \times 4.8 \approx 6.7104,
$$
so
$$
\text{ACT score} \approx 21.3 + 6.7104 \approx 28.0104.
$$
Rounded to one decimal place (if required), the ACT score is
$$
28.0.
$$

---

**Final Answers:**

- SAT score corresponding to the $$35^\text{th}$$ percentile: $$1029$$.
- Equivalent ACT score: $$19.5$$.
- For an SAT score of $$1396$$, the equivalent ACT score: $$28.0$$.
- SAT score at the 35th percentile: 1029 - Equivalent ACT score: 19.5 - For an SAT score of 1396, the equivalent ACT score is 28.0

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