Questions: In order to be accepted into a certain top university, applicants must score within the top 5% on the SAT exam. Given that the exam has a mean of 1000 and a standard deviation of 200, what is the lowest possible score a student needs to qualify for acceptance into the university? a. 1250 b. 1330 c. 1400 d. 1100

In order to be accepted into a certain top university, applicants must score within the top 5% on the SAT exam. Given that the exam has a mean of 1000 and a standard deviation of 200, what is the lowest possible score a student needs to qualify for acceptance into the university?
a. 1250
b. 1330
c. 1400
d. 1100

Transcript text: In order to be accepted into a certain top university, applicants must score within the top $5 \%$ on the SAT exam. Given that the exam has a mean of 1000 and a standard deviation of 200 , what is the lowest possible score a student needs to qualify for acceptance into the university? a. 1250 b. 1330 c. 1400 d. 1100

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the lowest possible SAT score that places a student in the top 5% of all test-takers. The SAT scores are normally distributed with a mean ($\mu$) of 1000 and a standard deviation ($\sigma$) of 200.

Step 2: Determine the Z-Score

To find the score corresponding to the top 5%, we first need to determine the Z-score that corresponds to the 95th percentile (since the top 5% is equivalent to the 95th percentile of the distribution).

Using a standard normal distribution table or a calculator, we find that the Z-score for the 95th percentile is approximately 1.645.

Step 3: Calculate the SAT Score

We use the Z-score formula to find the corresponding SAT score:

\[ Z = \frac{X - \mu}{\sigma} \]

Solving for $X$, we have:

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

Substituting the known values:

\[ X = 1.645 \cdot 200 + 1000 = 1329 \]

Since SAT scores are typically rounded to the nearest whole number, we round 1329 to 1330.