Math / Data & Statistics

QuestionAn aptitude test is designed to measure leadership abilities of the test subjects. Suppose that the scores on the test are normally distributed with a mean of 570 and a standard deviation of 120. The individuals who exceed 700 on this test are considered to be potential leaders. What proportion of the population are considered to be potential leaders? Round your answer to at least four decimal places. $\square$

Studdy Solution

STEP 1

What is this asking? What percentage of people taking a leadership test score above 700, if the average score is 570 and scores typically vary by 120 points? Watch out! Don't forget to convert the score of 700 into a z-score before looking up the proportion in the z-table!

STEP 2

1. Calculate the z-score
2. Find the corresponding proportion
3. Calculate the final proportion

STEP 3

Alright, let's kick things off by figuring out how far 700 is from the average score, in terms of standard deviations.
This is what a z-score tells us!

STEP 4

The formula for the z-score is $z = \frac{x - \mu}{\sigma}$ , where $x$ is the raw score (700 in our case), $\mu$ is the population mean (570), and $\sigma$ is the standard deviation (120).

STEP 5

Plugging in our values, we get $z = \frac{700 - 570}{120} = \frac{130}{120} \approx 1.0833$ .
So, a score of 700 is 1.0833 standard deviations above the mean.
Pretty impressive!

STEP 6

Now, we need to find the proportion of people who have a z-score less than our calculated z-score of 1.0833.
We can look this up in a z-table or use a calculator.

STEP 7

Looking up 1.08 in the z-table gives us a proportion of approximately 0.8599.
Since our z-score is a little bigger (1.0833), the actual proportion will be slightly higher.
Let's say approximately 0.8608.

STEP 8

Remember, the z-table tells us the proportion of people below a certain z-score.
We want the proportion above our z-score, since we're looking for potential leaders (those who scored higher than 700).

STEP 9

Since the total proportion is 1 (or 100%), we can subtract the proportion we found from 1 to get the proportion above the z-score: $1 - 0.8608 = 0.1392$ .

STEP 10

Approximately 0.1392, or 13.92%, of the population are considered potential leaders based on this test.

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Studdy Solution

STEP 1

What is this asking? What percentage of people taking a leadership test score above 700, if the average score is 570 and scores typically vary by 120 points? Watch out! Don't forget to convert the score of 700 into a z-score before looking up the proportion in the z-table!

STEP 2

1. Calculate the z-score
2. Find the corresponding proportion
3. Calculate the final proportion

STEP 3

Alright, let's kick things off by figuring out how far 700 is from the average score, in terms of standard deviations.
This is what a z-score tells us!

STEP 4

The formula for the z-score is $z = \frac{x - \mu}{\sigma}$ , where $x$ is the raw score (700 in our case), $\mu$ is the population mean (570), and $\sigma$ is the standard deviation (120).

STEP 5

Plugging in our values, we get $z = \frac{700 - 570}{120} = \frac{130}{120} \approx 1.0833$ .
So, a score of 700 is 1.0833 standard deviations above the mean.
Pretty impressive!

STEP 6

Now, we need to find the proportion of people who have a z-score less than our calculated z-score of 1.0833.
We can look this up in a z-table or use a calculator.

STEP 7

Looking up 1.08 in the z-table gives us a proportion of approximately 0.8599.
Since our z-score is a little bigger (1.0833), the actual proportion will be slightly higher.
Let's say approximately 0.8608.

STEP 8

Remember, the z-table tells us the proportion of people below a certain z-score.
We want the proportion above our z-score, since we're looking for potential leaders (those who scored higher than 700).

STEP 9

Since the total proportion is 1 (or 100%), we can subtract the proportion we found from 1 to get the proportion above the z-score: $1 - 0.8608 = 0.1392$ .

STEP 10

Approximately 0.1392, or 13.92%, of the population are considered potential leaders based on this test.

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Studdy Solution

STEP 1

What is this asking? What percentage of people taking a leadership test score above 700, if the average score is 570 and scores typically vary by 120 points? Watch out! Don't forget to convert the score of 700 into a *z*-score before looking up the proportion in the *z*-table!

STEP 2

1. Calculate the *z*-score2. Find the corresponding proportion3. Calculate the final proportion

STEP 3

Alright, let's **kick things off** by figuring out how far 700 is from the average score, in terms of standard deviations.This is what a *z*-score tells us!

STEP 4

The formula for the *z*-score is z=x−μσz = \frac{x - \mu}{\sigma}z=σx−μ​, where xxx is the **raw score** (700 in our case), μ\muμ is the **population mean** (570), and σ\sigmaσ is the **standard deviation** (120).

STEP 5

Plugging in our values, we get z=700−570120=130120≈1.0833z = \frac{700 - 570}{120} = \frac{130}{120} \approx 1.0833z=120700−570​=120130​≈1.0833.So, a score of 700 is **1.0833 standard deviations** above the mean.Pretty impressive!

STEP 6

Now, we need to find the proportion of people who have a *z*-score *less than* our calculated *z*-score of 1.0833.We can look this up in a *z*-table or use a calculator.

STEP 7

Looking up 1.08 in the *z*-table gives us a proportion of approximately 0.8599.Since our *z*-score is a little bigger (1.0833), the actual proportion will be slightly higher.Let's say approximately **0.8608**.

STEP 8

Remember, the *z*-table tells us the proportion of people *below* a certain *z*-score.We want the proportion *above* our *z*-score, since we're looking for potential leaders (those who scored *higher* than 700).

STEP 9

Since the total proportion is 1 (or 100%), we can subtract the proportion we found from 1 to get the proportion above the *z*-score: 1−0.8608=0.13921 - 0.8608 = 0.13921−0.8608=0.1392.

STEP 10

Approximately **0.1392**, or **13.92%**, of the population are considered potential leaders based on this test.

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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.

What is this asking? What percentage of people taking a leadership test score above 700, if the average score is 570 and scores typically vary by 120 points? Watch out! Don't forget to convert the score of 700 into a z-score before looking up the proportion in the z-table!

1. Calculate the z-score
2. Find the corresponding proportion
3. Calculate the final proportion

Alright, let's kick things off by figuring out how far 700 is from the average score, in terms of standard deviations.
This is what a z-score tells us!

The formula for the z-score is $z = \frac{x - \mu}{\sigma}$ , where $x$ is the raw score (700 in our case), $\mu$ is the population mean (570), and $\sigma$ is the standard deviation (120).

Plugging in our values, we get $z = \frac{700 - 570}{120} = \frac{130}{120} \approx 1.0833$ .
So, a score of 700 is 1.0833 standard deviations above the mean.
Pretty impressive!

Now, we need to find the proportion of people who have a z-score less than our calculated z-score of 1.0833.
We can look this up in a z-table or use a calculator.

Looking up 1.08 in the z-table gives us a proportion of approximately 0.8599.
Since our z-score is a little bigger (1.0833), the actual proportion will be slightly higher.
Let's say approximately 0.8608.

Remember, the z-table tells us the proportion of people below a certain z-score.
We want the proportion above our z-score, since we're looking for potential leaders (those who scored higher than 700).

Since the total proportion is 1 (or 100%), we can subtract the proportion we found from 1 to get the proportion above the z-score: $1 - 0.8608 = 0.1392$ .

Approximately 0.1392, or 13.92%, of the population are considered potential leaders based on this test.