[6.03] What is the value of the x variable in the solution to the following system of equations?
1 answer:
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Answer:
a)
b)
c) The value of height that separates the bottom 75% of data from the top 25% is 9.9745.
Step-by-step explanation:
1) 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".
2) 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
3) Part b
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
4) Part c
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=0.6745. On this case P(Z<0.6745)=0.75 and P(z>0.6745)=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 9.9745.
1) (23^-2)^3 (5
3^2)^2 / (3^-2)(5
2)^2 = 2
2) (3^3) (4^0)^2 (32)^-3 (2^2) = 1/2
3) (3^74^7) (2
5)^-3 (5)^2 / (12^7) (5^-1) (2^-4) = 2
4) (2.3)^-1 (2^0) / (2.3)^-1 = 1
<u>Step-by-step explanation</u>:
Step 1 :
(23^-2)^3 (5
3^2)^2 / (3^-2)(5
2)^2
⇒ (2^3) (3^-6) (5^2) (3^4) / (3^-2) (10^2)
⇒ (2.2^2.5^2) (3^-6.3^4) / (3^-2) (10^2)
⇒ (2)(10^2) (3^-2) / (3^-2) (10^2)
⇒ 2
Step 2 :
(3^3) (4^0)^2 (32)^-3 (2^2)
Any number with power 'zero' is 1.
⇒ (3^3) (1^2) (3^-3) (2^-3) (2^2)
⇒ (2^(-3+2))
⇒ 2^-1 = 1/2
Step 3 :
(3^74^7) (2
5)^-3 (5)^2 / (12^7) (5^-1) (2^-4)
⇒ (12^7) (2^-3) (5^-3) (5^2) / (12^7) (5^-1) (2^-4)
⇒ (12^7) (2^-3) (5^-3) (5^2) / (12^7) (5^-1) (2^-4)
⇒ (2^-3) (5^(-3+2)) / (5^-1) (2^-4)
⇒ (2^-3) (5^-1) / (5^-1) (2^-4)
⇒ 1 / (2^-1)
⇒ 2
Step 4 :
(2.3)^-1 (2^0) / (2.3)^-1
⇒ (2^0)
⇒ Any number with power 'zero' is 1
⇒ 1