Solved on Sep 20, 2023

Find the area under the normal curve to the right of an IQ score of $86$ , given a mean of $100$ and a standard deviation of $15$ . Round the answer to four decimal places.

STEP 1

Assumptions1. The IQ scores are normally distributed. . The mean of the IQ scores is100.
3. The standard deviation of the IQ scores is15.
4. We are looking for the area under the curve to the right of the score86.

STEP 2

First, we need to standardize the score86 to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. We can calculate it using the formula $z = \frac{X - \mu}{\sigma}$ where- X is the score we are interested in, - μ is the mean, and- σ is the standard deviation.

STEP 3

Now, plug in the given values for X, μ, and σ to calculate the z-score.
$z = \frac{86 -100}{15}$

STEP 4

Calculate the z-score.
$z = \frac{86 -100}{15} = -0.9333$

STEP 5

Now that we have the z-score, we need to find the area to the right of this z-score. Since the total area under the curve is1, the area to the right of the z-score is1 - the area to the left of the z-score. We can find the area to the left of the z-score using the standard normal distribution table or a calculator that can calculate normal probabilities.
$Area\, to\, the\, right =1 - Area\, to\, the\, left$

STEP 6

Using the standard normal distribution table or a calculator, we find that the area to the left of z = -0.9333 is approximately0.1753.

STEP 7

Now, plug in the value for the area to the left into the formula to find the area to the right.
$Area\, to\, the\, right =1 -0.1753$

STEP 8

Calculate the area to the right.
$Area\, to\, the\, right =1 -0.1753 =0.8247$ The area of the shaded region, to four decimal places, is0.8247.

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Find the area under the normal curve to the right of an IQ score of $86$ , given a mean of $100$ and a standard deviation of $15$ . Round the answer to four decimal places.

STEP 1

Assumptions1. The IQ scores are normally distributed. . The mean of the IQ scores is100.
3. The standard deviation of the IQ scores is15.
4. We are looking for the area under the curve to the right of the score86.

STEP 2

STEP 3

Now, plug in the given values for X, μ, and σ to calculate the z-score.
$z = \frac{86 -100}{15}$

STEP 4

Calculate the z-score.
$z = \frac{86 -100}{15} = -0.9333$

STEP 5

STEP 6

Using the standard normal distribution table or a calculator, we find that the area to the left of z = -0.9333 is approximately0.1753.

STEP 7

Now, plug in the value for the area to the left into the formula to find the area to the right.
$Area\, to\, the\, right =1 -0.1753$

STEP 8

Calculate the area to the right.
$Area\, to\, the\, right =1 -0.1753 =0.8247$ The area of the shaded region, to four decimal places, is0.8247.

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Find the area under the normal curve to the right of an IQ score of 868686, given a mean of 100100100 and a standard deviation of 151515. Round the answer to four decimal places.

STEP 1

Assumptions1. The IQ scores are normally distributed. . The mean of the IQ scores is100.3. The standard deviation of the IQ scores is15.4. We are looking for the area under the curve to the right of the score86.

STEP 2

STEP 3

Now, plug in the given values for X, μ, and σ to calculate the z-score.z=86−10015z = \frac{86 -100}{15}z=1586−100​

STEP 4

Calculate the z-score.z=86−10015=−0.9333z = \frac{86 -100}{15} = -0.9333z=1586−100​=−0.9333

STEP 5

STEP 6

Using the standard normal distribution table or a calculator, we find that the area to the left of z = -0.9333 is approximately0.1753.

STEP 7

Now, plug in the value for the area to the left into the formula to find the area to the right.Area to the right=1−0.1753Area\, to\, the\, right =1 -0.1753Areatotheright=1−0.1753

STEP 8

Calculate the area to the right.Area to the right=1−0.1753=0.8247Area\, to\, the\, right =1 -0.1753 =0.8247Areatotheright=1−0.1753=0.8247The area of the shaded region, to four decimal places, is0.8247.

Start learning now

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

Find the area under the normal curve to the right of an IQ score of $86$ , given a mean of $100$ and a standard deviation of $15$ . Round the answer to four decimal places.

Assumptions1. The IQ scores are normally distributed. . The mean of the IQ scores is100.
3. The standard deviation of the IQ scores is15.
4. We are looking for the area under the curve to the right of the score86.

Now, plug in the given values for X, μ, and σ to calculate the z-score.
$z = \frac{86 -100}{15}$

Calculate the z-score.
$z = \frac{86 -100}{15} = -0.9333$

Now, plug in the value for the area to the left into the formula to find the area to the right.
$Area\, to\, the\, right =1 -0.1753$

Calculate the area to the right.
$Area\, to\, the\, right =1 -0.1753 =0.8247$ The area of the shaded region, to four decimal places, is0.8247.