Respuesta :
Answer:
His jump was of 272.45 inches
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 263, \sigma = 14[/tex]
75th percentile
X when Z has a pvalue of 0.75. So X when Z = 0.675
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.675 = \frac{X - 263}{14}[/tex]
[tex]X - 263 = 14*0.675[/tex]
[tex]X = 272.45[/tex]
His jump was of 272.45 inches
Answer:
[tex]z=0.674<\frac{a-263}{14}[/tex]
And if we solve for a we got
[tex]a=263 +0.674*14=272.436[/tex]
So the value of height that separates the bottom 75% of data from the top 25% is 272.436.
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".
Solution to the problem
Let X the random variable that represent the distances of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(263,14)[/tex]
Where [tex]\mu=263[/tex] and [tex]\sigma=14[/tex]
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.25[/tex] (a)
[tex]P(X<a)=0.75[/tex] (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:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.75[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.75[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=0.674<\frac{a-263}{14}[/tex]
And if we solve for a we got
[tex]a=263 +0.674*14=272.436[/tex]
So the value of height that separates the bottom 75% of data from the top 25% is 272.436.
