Solved on Sep 20, 2023

Find the percentage of people with IQ scores between $88$ and $116$ in a normal distribution with $\mu=100$ and $\sigma=15$ , rounded to the nearest tenth.

STEP 1

Assumptions1. IQ scores are normally distributed. The mean (average) IQ score is1003. The standard deviation of IQ scores is154. We are looking for the percentage of people with an IQ score between88 and116

STEP 2

First, we need to standardize the IQ scores to z-scores. The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating a z-score is $z = \frac{x - \mu}{\sigma}$ where- $x$ is the value we are standardizing- $\mu$ is the mean- $\sigma$ is the standard deviation

STEP 3

Now, plug in the given values for the lower limit of the IQ score range (88), the mean (100), and the standard deviation (15) to calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15}$

STEP 4

Calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15} = -0.8$

STEP 5

Now, plug in the given values for the upper limit of the IQ score range (116), the mean (100), and the standard deviation (15) to calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15}$

STEP 6

Calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15} =1.07$

STEP 7

We need to find the area under the normal distribution curve between these two z-scores. This area represents the percentage of people with an IQ score between88 and116. We can use the standard normal distribution table or a calculator with a normal distribution function to find these areas.

STEP 8

Find the area to the left of the upper z-score (1.07) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of116 or less.
$(Z \leq1.07) =0.8577$

STEP 9

Find the area to the left of the lower z-score (-.8) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of88 or less.
$(Z \leq -.8) =.2119$

STEP 10

Subtract the area to the left of the lower z-score from the area to the left of the upper z-score to find the area between the two z-scores. This gives us the percentage of people with an IQ score between88 and116.
$(-0.8 \leq Z \leq.07) =(Z \leq.07) -(Z \leq -0.8)$

STEP 11

Calculate the percentage of people with an IQ score between88 and116.
$(-0.8 \leq Z \leq.07) =0.8577 -0.2119 =0.6458$ To the nearest tenth,64.6% of people have an IQ score between88 and116.

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Find the percentage of people with IQ scores between $88$ and $116$ in a normal distribution with $\mu=100$ and $\sigma=15$ , rounded to the nearest tenth.

STEP 1

Assumptions1. IQ scores are normally distributed. The mean (average) IQ score is1003. The standard deviation of IQ scores is154. We are looking for the percentage of people with an IQ score between88 and116

STEP 2

STEP 3

Now, plug in the given values for the lower limit of the IQ score range (88), the mean (100), and the standard deviation (15) to calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15}$

STEP 4

Calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15} = -0.8$

STEP 5

Now, plug in the given values for the upper limit of the IQ score range (116), the mean (100), and the standard deviation (15) to calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15}$

STEP 6

Calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15} =1.07$

STEP 7

We need to find the area under the normal distribution curve between these two z-scores. This area represents the percentage of people with an IQ score between88 and116. We can use the standard normal distribution table or a calculator with a normal distribution function to find these areas.

STEP 8

Find the area to the left of the upper z-score (1.07) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of116 or less.
$(Z \leq1.07) =0.8577$

STEP 9

Find the area to the left of the lower z-score (-.8) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of88 or less.
$(Z \leq -.8) =.2119$

STEP 10

Subtract the area to the left of the lower z-score from the area to the left of the upper z-score to find the area between the two z-scores. This gives us the percentage of people with an IQ score between88 and116.
$(-0.8 \leq Z \leq.07) =(Z \leq.07) -(Z \leq -0.8)$

STEP 11

Calculate the percentage of people with an IQ score between88 and116.
$(-0.8 \leq Z \leq.07) =0.8577 -0.2119 =0.6458$ To the nearest tenth,64.6% of people have an IQ score between88 and116.

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Find the percentage of people with IQ scores between 888888 and 116116116 in a normal distribution with μ=100\mu=100μ=100 and σ=15\sigma=15σ=15, rounded to the nearest tenth.

STEP 1

Assumptions1. IQ scores are normally distributed. The mean (average) IQ score is1003. The standard deviation of IQ scores is154. We are looking for the percentage of people with an IQ score between88 and116

STEP 2

STEP 3

Now, plug in the given values for the lower limit of the IQ score range (88), the mean (100), and the standard deviation (15) to calculate the lower z-score.zlower=88−10015z_{lower} = \frac{88 -100}{15}zlower​=1588−100​

STEP 4

Calculate the lower z-score.zlower=88−10015=−0.8z_{lower} = \frac{88 -100}{15} = -0.8zlower​=1588−100​=−0.8

STEP 5

Now, plug in the given values for the upper limit of the IQ score range (116), the mean (100), and the standard deviation (15) to calculate the upper z-score.zupper=116−10015z_{upper} = \frac{116 -100}{15}zupper​=15116−100​

STEP 6

Calculate the upper z-score.zupper=116−10015=1.07z_{upper} = \frac{116 -100}{15} =1.07zupper​=15116−100​=1.07

STEP 7

We need to find the area under the normal distribution curve between these two z-scores. This area represents the percentage of people with an IQ score between88 and116. We can use the standard normal distribution table or a calculator with a normal distribution function to find these areas.

STEP 8

Find the area to the left of the upper z-score (1.07) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of116 or less.(Z≤1.07)=0.8577(Z \leq1.07) =0.8577(Z≤1.07)=0.8577

STEP 9

Find the area to the left of the lower z-score (-.8) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of88 or less.(Z≤−.8)=.2119(Z \leq -.8) =.2119(Z≤−.8)=.2119

STEP 10

STEP 11

Calculate the percentage of people with an IQ score between88 and116.(−0.8≤Z≤.07)=0.8577−0.2119=0.6458(-0.8 \leq Z \leq.07) =0.8577 -0.2119 =0.6458(−0.8≤Z≤.07)=0.8577−0.2119=0.6458To the nearest tenth,64.6% of people have an IQ score between88 and116.

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

Find the percentage of people with IQ scores between $88$ and $116$ in a normal distribution with $\mu=100$ and $\sigma=15$ , rounded to the nearest tenth.

Now, plug in the given values for the lower limit of the IQ score range (88), the mean (100), and the standard deviation (15) to calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15}$

Calculate the lower z-score.
$z_{lower} = \frac{88 -100}{15} = -0.8$

Now, plug in the given values for the upper limit of the IQ score range (116), the mean (100), and the standard deviation (15) to calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15}$

Calculate the upper z-score.
$z_{upper} = \frac{116 -100}{15} =1.07$

Find the area to the left of the upper z-score (1.07) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of116 or less.
$(Z \leq1.07) =0.8577$

Find the area to the left of the lower z-score (-.8) from the standard normal distribution table or using a calculator. This gives us the percentage of people with an IQ score of88 or less.
$(Z \leq -.8) =.2119$

Calculate the percentage of people with an IQ score between88 and116.
$(-0.8 \leq Z \leq.07) =0.8577 -0.2119 =0.6458$ To the nearest tenth,64.6% of people have an IQ score between88 and116.