FizeauV

2021-08-19

Based on the Normal model N(100, 16) describing IQ scores, what percent of peoples

1. The distribution of IQ scores is normally distributed with a mean of 100 and a standard deviation of 16.

2. Transform the normal random variable,X, into the standard normal variable Z by using the following formula: $Z=\frac{X-\mu }{\sigma }$ .
a) $P\left(X>80\right)=P\left(\frac{X-100}{16}>\frac{80-100}{16}\right)=P\left(Z>-1.25\right)=1-P\left(Z\le -1.25\right)=1-0.1056=0.8944=89.44$
b) $P\left(X<90\right)=P\left(\frac{X-100}{16}<\frac{90-100}{16}\right)=P\left(Z\le -0.625\right)=0.2659=26.59$

c) $P\left(112

Jeffrey Jordon

The percentage needed is for the area shown here :

Let us suppose the number corresponding to the needed percentage is X, we get the following expression:

P(X>80)

Knowing that $Z=\frac{X-v}{\sigma }$

So, in the next steps , we will get the Z value from X and calculate its percentage as follows:

P(X >80)

$=P\left(\frac{X-v}{\sigma }>\frac{80-v}{\sigma }\right)$

$=P\left(Z>\frac{80-v}{\sigma }\right)$

$=P\left(Z>\frac{80-100}{16}\right)$

$=P\left(Z>-1.25\right)$

$=1-P\left(Z\le -1.25\right)$

=1-0.1056=0.8944

Therefore, 89.44% of people's IQs would be over 80

b)The percentage to be calculated is for the following colored area:

P(X<90)

$=P\left(Z<\frac{90-100}{16}\right)$

$=P\left(Z<-0.625\right)$

=0.266

Therefore, 26.6% of people's IQs would be under 80

c)The percentage to be calculated represents the data within the covered area:

P(112<X<132)

$=P\left(\frac{112-100}{16}

$=P\left(0.75

=0.2039

Therefore, 20.39% of people's IQs would be between 112 and 132

Jazz Frenia

Step 1. To find the percentage of people with IQ scores below 90, we can calculate the cumulative distribution function (CDF) at 90 using the Normal distribution formula:
$P\left(X\le 90\right)=\Phi \left(\frac{90-100}{\sqrt{16}}\right)$
Plugging in the values, we get:
$P\left(X\le 90\right)=\Phi \left(\frac{-10}{4}\right)$
Using a standard Normal distribution table or calculator, we find that $\Phi \left(-2.5\right)\approx 0.0062$. Therefore, the percentage of people with IQ scores below 90 is approximately 0.62%.
Step 2. To find the percentage of people with IQ scores above 120, we can calculate the complementary cumulative distribution function (CCDF) at 120:
$P\left(X>120\right)=1-\Phi \left(\frac{120-100}{\sqrt{16}}\right)$
Plugging in the values, we get:
$P\left(X>120\right)=1-\Phi \left(\frac{20}{4}\right)$
Using a standard Normal distribution table or calculator, we find that $\Phi \left(5\right)\approx 1$. Therefore, the percentage of people with IQ scores above 120 is approximately 0%.
Step 3. Finally, to find the percentage of people with IQ scores between 90 and 120, we can subtract the percentages from steps 1 and 2 from 100%:
$P\left(90
Plugging in the values, we get:
$P\left(90
$P\left(90
Therefore, the percentage of people with IQ scores between 90 and 120 is approximately 99.38%.
In summary, the percentages of people falling within certain IQ score ranges are approximately 0.62% below 90, 99.38% between 90 and 120, and 0% above 120.

fudzisako

- The percentage of people with IQ scores below 80 is approximately 10.56%.
- The percentage of people with IQ scores between 80 and 120 is approximately 78.88%.
- The percentage of people with IQ scores above 120 is approximately 10.56%.
Explanation:
Let's find the percentage of people with IQ scores below 80 and above 120.
To find the percentage below 80, we need to calculate the cumulative probability up to that point. We can use the cumulative distribution function (CDF) of the Normal distribution:
$P\left(X\le 80\right)=\Phi \left(\frac{80-100}{16}\right)$
Where $\Phi$ represents the CDF of the Normal distribution. Using this formula, we can find the cumulative probability:
$P\left(X\le 80\right)=\Phi \left(\frac{80-100}{16}\right)=\Phi \left(-1.25\right)=0.1056$
Therefore, approximately 10.56% of people have IQ scores below 80.
To find the percentage above 120, we can subtract the cumulative probability below 120 from 1:
$P\left(X>120\right)=1-P\left(X\le 120\right)=1-\Phi \left(\frac{120-100}{16}\right)$
Calculating this probability gives us:
$P\left(X>120\right)=1-\Phi \left(\frac{120-100}{16}\right)=1-\Phi \left(1.25\right)=1-0.8944=0.1056$
Therefore, approximately 10.56% of people have IQ scores above 120.
Now, let's calculate the percentage of people within the range of 80 to 120. To do this, we can subtract the cumulative probabilities below 80 and above 120 from 1:
$P\left(80120\right)$
Substituting the values we calculated earlier:
$P\left(80
Therefore, approximately 78.88% of people have IQ scores between 80 and 120.

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