IQ is normally distributed with a mean of 100
and a standard deviation of 15.
Part A, we're asked to find the probability
that a person has an IQ greater than 90
and we're asked to give the answer as a decimal
to three decimal places,
as well as a percent to one decimal place.
We can indicate the probability
that a person has an IQ greater than 90 as the probability
that x is greater than 90.
To determine the probability, we will use the free online
normal distribution calculator linked below.
So going to the calculator,
we first select normal distribution at the top.
Next we enter the mean and standard deviation.
The mean is 100
and the standard deviation is 15.
Because we are determining probability,
we use one of the three options on the left.
We are trying to find the probability
that x is greater than 90
and therefore we select the second option
and enter the IQ score of 90 and then click calculate
and the probability is given below in red.
Also looking at the graph, notice how the area
below the normal curve to the right of the IQ score
of 90 is shaded.
This area represents the probability,
which to three decimal places is approximately 0.748.
Let's go ahead and record this.
0.748 as a percentage
is 74.8%.
Notice how we gave the probability as a decimal
to three decimal places, as well as a percent
to one decimal place.
For part B, you are asked to find the probability
that a person has an IQ of less than 120,
which we can indicate as the probability
that x is less than 120.
Going back to our calculator,
we now select the first option where we have the probability
of X less than and then we enter the IQ score of 120
and once again click calculate.
Looking at the graph, notice how we do have the area shaded
below the normal curve to the left of the IQ score of 120.
This area represents the probability
that x is less than 120.
To three decimal places, we have approximately 0.909.
Which is equal to 90.9%.
I need to be careful these next two questions are different.
In a sample of 1200 people,
how many people would have an IQ less than 105?
So we first will find the probability
that the IQ is less than 105, which is the probability
that x is less than 105.
But then to find the number of people out of 1200,
that have an IQ score of less than 105,
we need to multiply this probability by 1200.
Remember to find the percent of a number,
we convert the percent to a decimal and then multiply.
So we'll now find the probability as a decimal
and then multiply by 1200.
Going back to the calculator, we select the first option
and change the IQ score to 105, click calculate.
Once again the graph we have the area shaded to the left
at 105 below the normal curve
and the probability is given here in red.
To three decimal places, we have approximately 0.631.
So this is approximately equal to 0.631 times 1200.
Going to the calculator,
we have approximately 757 people.
In this problem, we round to the nearest person.
For Part D, in a sample of 600 people,
how many people would have an IQ greater than 135?
To answer the question we will find the probability
that x is greater than 135
and then multiply by 600 the sample size.
Going back to the calculator one last time.
We select the second option where we have the probability
that x is greater than and change the IQ score to 135
and click calculate.
Notice on the graph we have this small area shaded
to the right of the IQ score of 135
The probability rounded
to three decimal places would be approximately 0.010.
Notice how we have a nine in the third decimal place
and then an eight to the right,
so we round up the nine changes to 10, giving us 0.010.
And then we multiply by 600,
which gives us six people.
I hope you found this helpful.