What are the equations & answer to these 2 problems ?
1 answer:
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Let be the randomly selected lengths. Without loss of generality, suppose
where are independent random variables with the same uniform distribution on [0, 1].
By their mutual independence, we have
so that the joint density function is
where is the cube with vertices at (0, 0, 0) and (1, 1, 1).
Consider the plane
with . This plane passes through (0, 0, 0), (1, 0, 1), and (0, 1, 1), and thus splits up the cube into one tetrahedral region above the plane and the rest of the cube under it. (see attached plot)
The point (0, 0, 1) (the vertex of the cube above the plane) does not belong the region , since
. So the probability we want is the volume of the bottom "half" of the cube. We could integrate the joint density over this set, but integrating over the complement is simpler since it's a tetrahedron.
Then we have
Answer:
a. Graph the relative frequency distribution for these results. What type of graph is ideal?
we must use bar graph
[Graph is attached with the name graph 1]
b.
Toward the Frequency Relative Frequency
volatiles 17 0.566666667
solvent 2 0.066666667
left side 7 0.233333333
right side 4 0.133333333
[Graphs is added in the attachment with the name graph 2]
c.
17/30 = 0.567
d.
se=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.567(1-0.567)}{30}}=0.091
standard error shows the variation in sampling distribution.
e.
The most plausible values for the fraction of dodder seedlings that grow toward the volatiles is:
\hat{p}\pm 2\sqrt{\frac{p(1-p)}{n}}=0.0567\pm 2(0.091)\: \: or\: \: (0.385,0.749)
This does not included the fraction expected (0.25).
