In a certain school district in a large metropolitan area, the SAT scores over that past five
years are normally distributed with a mean of 1489. Furthermore, is 1297 . What is the
score for this population?

(Enter answer rounded to the nearest whole number.)

Ask by Willis Kelley. in the United States

Mar 18,2025

Question

In a certain school district in a large metropolitan area, the SAT scores over that past five
years are normally distributed with a mean of 1489. Furthermore, is 1297 . What is the
score for this population?

(Enter answer rounded to the nearest whole number.)

Ask by Willis Kelley. in the United States

Mar 18,2025

UpStudy AI · Accepted Answer

To find the $$ P_{95} $$ score for the SAT scores in this population, we need to follow these steps:

1. **Understand the distribution**: The SAT scores are normally distributed with a mean ($$ \mu $$) of 1489. We also know that the first quartile ($$ Q_1 $$) is 1297.

2. **Find the standard deviation**: Since we have $$ Q_1 $$, we can use it to find the standard deviation ($$ \sigma $$). In a normal distribution, $$ Q_1 $$ corresponds to the 25th percentile. We can use the z-score for the 25th percentile to find $$ \sigma $$.

The z-score for the 25th percentile is approximately -0.674. The formula for $$ Q_1 $$ is:
   $$
   Q_1 = \mu + z \cdot \sigma
   $$
   Plugging in the values:
   $$
   1297 = 1489 + (-0.674) \cdot \sigma
   $$

3. **Solve for $$ \sigma $$**:
   Rearranging the equation gives:
   $$
   1297 - 1489 = -0.674 \cdot \sigma
   $$
   $$
   -192 = -0.674 \cdot \sigma
   $$
   $$
   \sigma = \frac{192}{0.674}
   $$

4. **Calculate $$ \sigma $$**:
   Now, let's calculate $$ \sigma $$.

5. **Find $$ P_{95} $$**: The z-score for the 95th percentile is approximately 1.645. We can use the formula:
   $$
   P_{95} = \mu + z \cdot \sigma
   $$

Now, let's perform the calculations.

First, we will calculate $$ \sigma $$ and then use it to find $$ P_{95} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{192}{0.674}$$
- step1: Convert the expressions:
  $$\frac{192}{\frac{337}{500}}$$
- step2: Multiply by the reciprocal:
  $$192\times \frac{500}{337}$$
- step3: Multiply:
  $$\frac{192\times 500}{337}$$
- step4: Multiply:
  $$\frac{96000}{337}$$
Calculate or simplify the expression $$ 1489 + 1.645 * (192/0.674) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1489+1.645\left(\frac{192}{0.674}\right)$$
- step1: Divide the numbers:
  $$1489+1.645\times \frac{96000}{337}$$
- step2: Multiply the numbers:
  $$1489+\frac{157920}{337}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{1489\times 337}{337}+\frac{157920}{337}$$
- step4: Transform the expression:
  $$\frac{1489\times 337+157920}{337}$$
- step5: Multiply the numbers:
  $$\frac{501793+157920}{337}$$
- step6: Add the numbers:
  $$\frac{659713}{337}$$
1. **Calculate the standard deviation ($$ \sigma $$)**:
   $$
   \sigma \approx 284.87
   $$

2. **Calculate the $$ P_{95} $$ score**:
   $$
   P_{95} = 1489 + 1.645 \cdot 284.87 \approx 1957.61
   $$

Now, rounding $$ P_{95} $$ to the nearest whole number gives us:
$$
P_{95} \approx 1958
$$

Thus, the $$ P_{95} $$ score for this population is **1958**.
The $$ P_{95} $$ score is approximately 1958.

In a certain school district in a large metropolitan area, the SAT scores over that past five
years are normally distributed with a mean of 1489. Furthermore, is 1297 . What is the
score for this population?

(Enter answer rounded to the nearest whole number.)

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Statistics Questions

In a certain school district in a large metropolitan area, the SAT scores over that past five years are normally distributed with a mean of 1489. Furthermore, is 1297 . What is the score for this population? (Enter answer rounded to the nearest whole number.)

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Statistics Questions

In a certain school district in a large metropolitan area, the SAT scores over that past five
years are normally distributed with a mean of 1489. Furthermore, is 1297 . What is the
score for this population?

(Enter answer rounded to the nearest whole number.)