Answer:
x = 3 ± 2
Step-by-step explanation:
given
x² - 6x + 1 = 0 ( subtract 1 from both sides )
x² - 6x = - 1
To complete the square
add (half the coefficient of the x- term)² to both sides
x² - 6x + (- 3)² = - 1 + (- 3)²
x² - 6x + 9 = - 1 + 9
(x - 3)² = 8 ( take the square root of both sides )
x - 3 = ± = ± 2
( add 3 to both sides )
x = 3 ± 2
Answer:
a)
We want this probability:
And using the z score formula given by:
We got:
b) For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.75 of the area on the left and 0.25 of the area on the right it's z=0.674. On this case P(Z<0.674)=0.75 and P(z>0.674)=0.25
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 75% of data from the top 25% is 45.707.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:
Where and
And we want this probability:
And using the z score formula given by:
We got:
We want this probability:
And using the z score formula given by:
We got:
Part b
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.75 of the area on the left and 0.25 of the area on the right it's z=0.674. On this case P(Z<0.674)=0.75 and P(z>0.674)=0.25
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 75% of data from the top 25% is 45.707.