The average time to run the 5K fun run is 24 minutes and the standard deviation is 2.7 minutes. 16 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X- N( b. What is the distribution of a? - N( c. What is the distribution ofz? Ex- N d. If one randomly selected runner is timed, find the probability that this runner's time will be between 23.1875 and 23.9875 minutes. e. For the 16 runners, find the probability that their average time is between 23.1875 and 23.9875 minutes. f. Find the probability that the randomly selected 16 person team will have a total time less than 372.8.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Part d , e and f
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The average time to run the 5K fun run is 24 minutes and the standard deviation is 2.7 minutes. 16 runners are
randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a
normal distribution.
a. What is the distribution of X? X- N
b. What is the distribution of ? - N(
c. What is the distribution of æ? x - N(
d. If one randomly selected runner is timed, find the probability that this runner's time will be between
23.1875 and 23.9875 minutes.
e. For the 16 runners, find the probability that their average time is between 23.1875 and 23.9875
minutes.
f. Find the probability that the randomly selected 16 person team will have a total time less than 372.8.
g. For part e) and f), is the assumption of normal necessary? O No Yes
h. The top 10% of all 16 person team relay races will compete in the championship round. These are the
10% lowest times. What is the longest total time that a relay team can have and still make it to the
championship round?
minutes
Hint:
Some Helpful Videos:
• Finding the Sampling Distribution [+]
• Finding a Probability Using the Central Limit Theorem [+]
• Finding Value Given a Probability Using the Central Limit Theorem [+]
• The Central Limit Theorem For Sums [+]
Hint
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(d)
Obtain the probability that this runner’s time will be between 23.1875 and 23.9875 minutes.
The probability that this runner’s time will be between 23.1875 and 23.9875 minutes is obtained below as follows:
The probability that this runner’s time will be between 23.1875 and 23.9875 minutes is 0.1979.
(e)
Obtain the probability that their average time is between 23.1875 and 23.9875 for the sample of 16 runners.
The probability that their average time is between 23.1875 and 23.9875 for the sample of 16 runners is obtained below as follows :
The required probability is,
The probability that their average time is between 23.1875 and 23.9875 for the sample of 16 runners is 0.3769.
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