Respuesta :
Answer:
And then we can conclude that the middle 75% values of her lap times are from a=126.835 to b=132.585
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) Solution to the problem
Let X the random variable that represent the times of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(129.71,2.5)[/tex]
Where [tex]\mu=129.71[/tex] and [tex]\sigma=2.5[/tex]
Since we want the middle 75% values then on tha tails we need 1-0.75 =0.25 or 25% of the data and since the distribution is symmetric we need 12.5 % of the values on each tail.
For this part we want to find a value a and b, such that we satisfy this condition:
[tex]P(X>b)=0.125[/tex] (a)
[tex]P(X<a)=0.125[/tex] (b)
We can use the z score formula in order to find the value a and b given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
As we can see on the figure attached the z value that satisfy the condition (b) with 0.125 of the area on the left and 0.875 of the area on the right it's z=-1.15. On this case P(Z<-1.15)=0.125 and P(z>-1.15)=0.875
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.125[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.125[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-1.15<\frac{a-129.71}{2.5}[/tex]
And if we solve for a we got
[tex]a=129.71 -1.15*2.5=126.835[/tex]
So the value of height that separates the bottom 12.5% of data from the top 87.5% is 126.835.
As we can see on the figure attached the z value that satisfy the condition (b) with 0.875 of the area on the left and 0.125 of the area on the right it's z=1.15. On this case P(Z<1.15)=0.875 and P(Z>1.15)=0.125
If we use condition (b) from previous we have this:
[tex]P(X>b)=P(\frac{X-\mu}{\sigma}>\frac{b-\mu}{\sigma})=0.125[/tex]
[tex]P(Z>\frac{b-\mu}{\sigma})=0.125[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]Z=1.15>\frac{b-129.71}{2.5}[/tex]
And if we solve for a we got
[tex]b=129.71 +1.15*2.5=132.585[/tex]
And then we can conclude that the middle 75% values of her lap times are from a=126.835 to b=132.585
