Answer:
(a) E (X) = 61 and SD (X) = 9
(b) E (Z) = 0 and SD (Z) = 1
Step-by-step explanation:
The time of the finishers in the New York City 10 km run are normally distributed with a mean,<em>μ</em> = 61 minutes and a standard deviation, <em>σ</em> = 9 minutes.
(a)
The random variable <em>X</em> is defined as the finishing time for the finishers.
Then the expected value of <em>X</em> is:
<em>E </em>(<em>X</em>) = 61 minutes
The variance of the random variable <em>X</em> is:
<em>V</em> (<em>X</em>) = (9 minutes)²
Then the standard deviation of the random variable <em>X</em> is:
<em>SD</em> (<em>X</em>) = 9 minutes
(b)
The random variable <em>Z</em> is the standardized form of the random variable <em>X</em>.
It is defined as:
Compute the expected value of <em>Z</em> as follows:
![E(Z)=E[\frac{X-\mu}{\sigma}]\\=\frac{E(X)-\mu}{\sigma}\\=\frac{61-61}{9}\\=0](https://tex.z-dn.net/?f=E%28Z%29%3DE%5B%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%5D%5C%5C%3D%5Cfrac%7BE%28X%29-%5Cmu%7D%7B%5Csigma%7D%5C%5C%3D%5Cfrac%7B61-61%7D%7B9%7D%5C%5C%3D0)
The mean of <em>Z</em> is 0.
Compute the variance of <em>Z</em> as follows:
![V(Z)=V[\frac{X-\mu}{\sigma}]\\=\frac{V(X)+V(\mu)}{\sigma^{2}}\\=\frac{V(X)}{\sigma^{2}}\\=\frac{9^{2}}{9^{2}}\\=1](https://tex.z-dn.net/?f=V%28Z%29%3DV%5B%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%5D%5C%5C%3D%5Cfrac%7BV%28X%29%2BV%28%5Cmu%29%7D%7B%5Csigma%5E%7B2%7D%7D%5C%5C%3D%5Cfrac%7BV%28X%29%7D%7B%5Csigma%5E%7B2%7D%7D%5C%5C%3D%5Cfrac%7B9%5E%7B2%7D%7D%7B9%5E%7B2%7D%7D%5C%5C%3D1)
The variance of <em>Z</em> is 1.
So the standard deviation is 1.
Answer:
Option a: \frac{m^{5} }{162n} is the equivalent expression.
Explanation:
The expression is \frac{(3m^{-2} n)^{-3}}{6mn^{-2} } where m\neq 0, n\neq 0
Let us simplify the expression, to determine which expression is equivalent from the four options.
Multiplying the powers, we get,
\frac{3^{-3}m^{6} n^{-3}}{6mn^{-2} }
Cancelling the like terms, we have,
\frac{3^{-3}m^{5} n^{-1}}{6 }
This equation can also be written as,
\frac{m^{5}}{3^{3}6 n^{1} }
Multiplying the terms in denominator, we have,
\frac{m^{5} }{162n}
Thus, the expression which is equivalent to \frac{(3m^{-2} n)^{-3}}{6mn^{-2} } is \frac{m^{5} }{162n}
Hence, Option a is the correct answer. 23
Step-by-step explanation:
The least common denominator is 5y, and can be obtained by multiplying
-1/y by 5/5 to get -5/5y.
If we add the numbers using the least common denominator, we get:
-5/5y + 2/5y = -3/5y
Answer:
3m = -3
Step-by-step explanation:
You are given
2m - 6 = 8m,
hence,
2m - 8m = 6,
-6m = 6,
m = 6%2F%28-6%29,
m = -1.
Therefore, 3m = -3.
Pair 1: slope = (9 - 5)/(8+4) = 1/3
midpoint = ((-4+8)/2, (5+9)/2) = (2, 7)
perpendicular bisector passes through point (2, 7) with slope = -1/(1/3) = -3 giving the equation (y - 7)/(x - 2) = -3 or y - 7 = -3(x - 2) or y = -3x + 13 and y-intercept at y = 13.
Pair 2: slope = (6 - 4)/(-8-2) = -1/5
midpoint = ((2-8)/2, (4+6)/2) = (-3, 5)
perpendicular
bisector passes through point (-3, 5) with slope = -1/(-1/5) = 5 giving
the equation (y - 5)/(x + 3) = 5 or y - 5 = 5(x + 3) or y = 5x + 20
and y-intercept at y = 20.
Pair 3: slope = (2 - 4)/(7 - 5) = -1
midpoint = ((5+7)/2, (4+2)/2) = (6, 3)
perpendicular
bisector passes through point (6, 3) with slope = -1/(-1) = 1 giving
the equation (y - 3)/(x - 6) = 1 or y - 3 = (x - 6) or y = x - 3
and y-intercept at y = -3.
Pair 4: slope = (3 - 9)/(-4 - 2) = 1
midpoint = ((2-4)/2, (9+3)/2) = (-1, 6)
perpendicular
bisector passes through point (-1, 6) with slope = -1(1) = -1 giving
the equation (y - 6)/(x + 1) = -1 or y - 6 = -1(x + 1) or y = -x + 5
and y-intercept at y = 5.
Pair 5: slope = (-12 + 2)/(9 - 3) = -5/3
midpoint = ((3+9)/2, (-2-12)/2) = (6, -7)
perpendicular
bisector passes through point (6, -7) with slope = -1(-5/3) = 3/5 giving
the equation (y + 7)/(x - 6) = 3/5 or 5(y + 7) = 3(x - 6) or 5y = 3x - 53
and y-intercept at y = -10.6.
Pair 6: slope = (12 - 10)/(8 - 4) = 1/2
midpoint = ((4+8)/2, (10+12)/2) = (6, 11)
perpendicular
bisector passes through point (6, 11) with slope = -1(1/2) = -2 giving
the equation (y - 11)/(x - 6) = -2 or y - 11 = -2(x - 6) or y = -2x + 23
and y-intercept at y = 23.
Arrangement in order of y-intercepts from smallest to largest
a(3, -2) and b(9, -12)
a(5, 4) and b(7, 2)
a(2, 9) and b(-4, 3)
a(-4, 5) and b(8, 9)
a(2, 4) and b(-8, 6)
a(4, 10) and b(8, 12)
