QuestionFind the proportion of SAT verbal scores between 400 and 600, given a mean of 500 and a standard deviation of 100.

Studdy Solution

STEP 1

Assumptions1. SAT verbal scores follow a normal distribution. . The mean of the distribution is500.
3. The standard deviation of the distribution is100.
4. We are asked to find the proportion of scores that fall between400 and600.
5. We are using the $z$ distribution to find this proportion.

STEP 2

The $z$ score is a measure of how many standard deviations an element is from the mean. We can calculate it using the following formula $z = \frac{x - \mu}{\sigma}$ where $x$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

STEP 3

First, we calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100}$

STEP 4

Calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100} = -1$

STEP 5

Next, we calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100}$

STEP 6

Calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100} =1$

STEP 7

Now, we need to find the proportion of scores between $z_{400}$ and $z_{600}$ . In a standard normal distribution, the proportion of scores between -1 and1 (i.e., within one standard deviation from the mean) is approximately0.6826.
So, the proportion of scores that fall between400 and600 according to the $z$ distribution is0.6826.

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QuestionFind the proportion of SAT verbal scores between 400 and 600, given a mean of 500 and a standard deviation of 100.

Studdy Solution

STEP 1

Assumptions1. SAT verbal scores follow a normal distribution. . The mean of the distribution is500.
3. The standard deviation of the distribution is100.
4. We are asked to find the proportion of scores that fall between400 and600.
5. We are using the $z$ distribution to find this proportion.

STEP 2

The $z$ score is a measure of how many standard deviations an element is from the mean. We can calculate it using the following formula $z = \frac{x - \mu}{\sigma}$ where $x$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

STEP 3

First, we calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100}$

STEP 4

Calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100} = -1$

STEP 5

Next, we calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100}$

STEP 6

Calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100} =1$

STEP 7

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QuestionFind the proportion of SAT verbal scores between 400 and 600, given a mean of 500 and a standard deviation of 100.

Studdy Solution

STEP 1

Assumptions1. SAT verbal scores follow a normal distribution. . The mean of the distribution is500.3. The standard deviation of the distribution is100.4. We are asked to find the proportion of scores that fall between400 and600.5. We are using the zzz distribution to find this proportion.

STEP 2

The zzz score is a measure of how many standard deviations an element is from the mean. We can calculate it using the following formulaz=x−μσz = \frac{x - \mu}{\sigma}z=σx−μ​where xxx is the score, μ\muμ is the mean, and σ\sigmaσ is the standard deviation.

STEP 3

First, we calculate the zzz score for400.z400=400−500100z_{400} = \frac{400 -500}{100}z400​=100400−500​

STEP 4

Calculate the zzz score for400.z400=400−500100=−1z_{400} = \frac{400 -500}{100} = -1z400​=100400−500​=−1

STEP 5

Next, we calculate the zzz score for600.z600=600−500100z_{600} = \frac{600 -500}{100}z600​=100600−500​

STEP 6

Calculate the zzz score for600.z600=600−500100=1z_{600} = \frac{600 -500}{100} =1z600​=100600−500​=1

STEP 7

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions1. SAT verbal scores follow a normal distribution. . The mean of the distribution is500.
3. The standard deviation of the distribution is100.
4. We are asked to find the proportion of scores that fall between400 and600.
5. We are using the $z$ distribution to find this proportion.

The $z$ score is a measure of how many standard deviations an element is from the mean. We can calculate it using the following formula $z = \frac{x - \mu}{\sigma}$ where $x$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

First, we calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100}$

Calculate the $z$ score for400.
$z_{400} = \frac{400 -500}{100} = -1$

Next, we calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100}$

Calculate the $z$ score for600.
$z_{600} = \frac{600 -500}{100} =1$