Find the z-score boundaries that separate a normal distribution as described in each of the following. a. The middle 20% from th
1 answer:
Answer:
a) The boundaries are
b) The boundaries are .
c) The boundaries are .
d) The boundaries are .
Step-by-step explanation:
Z-score:
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.
a. The middle 20% from the 80% in the tails.
The middle 20% is between the 50 - (20/2) = 40th percentile and the 50 + (20/2) = 60th percentile:
40th percentile: Z has a pvalue of 0.4, so Z = -0.253.
60th percentile: Z has a pvalue of 0.6, so Z = 0.253.
The boundaries are .
b. The middle 50% from the 50% in the tails.
The middle 50% is between the 50 - (50/2) = 25th percentile and the 50 + (50/2) = 75th percentile:
25th percentile: Z has a pvalue of 0.25, so Z = -0.675.
75th percentile: Z has a pvalue of 0.75, so Z = 0.675.
The boundaries are .
c. The middle 95% from the 5% in the tails.
The middle 95% is between the 50 - (95/2) = 2.5th percentile and the 50 + (95/2) = 97.5th percentile:
2.5th percentile: Z has a pvalue of 0.025, so Z = -1.96.
97.5th percentile: Z has a pvalue of 0.975, so Z = 1.96.
The boundaries are .
d. The middle 99% from the 1% in the tails.
The middle 99% is between the 50 - (99/2) = 0.5th percentile and the 50 + (99/2) = 99.5th percentile:
0.5th percentile: Z has a pvalue of 0.005, so Z = -2.575.
99.5th percentile: Z has a pvalue of 0.995, so Z = 2.575.
The boundaries are .
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Answer:
(a) Please see the first figure attached
(b) The coordinates of the centroid are
(c) due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of and
, the centroid will lie
Step-by-step explanation:
Hi, let us first solve part (a). Since for any given values of and
we will obtain two linear functions:
and
with we can assure that our region is going to be a triangle. To see this, please take a look at the plot I generated using Wolfram. In this case, I have used two specific values for m and n but keeping the condition
.
Now, for part (b) let me start remembering what the centroid is: the centroid of a triangle is the point where the three medians of the triangle meet. And a median of a triangle is a line segment from one vertex to the midpoint on the opposite side of the triangle (see the second figure where the medians are depicted in red and the centroid of the triangle, G is depicted in blue). For a given triangle , the coordinates of its centroid
are given by:
and
Now let's apply this to our problem. Take a look at the first figure. The vertex A has clearly coordinates for any value of
and
since the two lines have their intersection with y-axis in this point.
To obtain the coordinates of and
, let's use the given functions and the fact that the coordinate x is limited to 1. Then, we have:
For A:
then, when
, substituting in the formula
For B and doing the same as for A:
then, when
, substituting in the formula
Thus, the coordinates of the vertices of the triangle are: ,
and
and the coordinates of the centroid are:
and
.
Summarizing: the coordinates of the centroid of the region are
Now, for part (c), due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of and
, the centroid will lie. Other important points of the triangle, like the orthocentre and circumcentre, can lie outside in obtuse triangles. In right triangles, the orthocentre always lies at the right-angled vertex.
