The probability of getting at least one defective disk in a lot of 50 is 0.2217.
What is probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.
Given that,
P (defective disc) = 0.005
P (non-defective) = 1 - 0.005 = 0.995
Now,
P(Getting one defective) = 1 - P(getting 50 non -defective)
= 1 - (0.995)^(50)
= 1- 0.7783
= 0.2217
Hence, the probability of getting at least one defective disk in a lot of 50 is 0.2217.
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Sorina has a mental age of 10 and an IQ of 125 as measured by the Stanford-Binet. Sorina's chronological age is
A) 6.
B) 8.
C) 9.
D) 10.
E) 12.5.
Sorina's chronological age is 8.
What is Intelligence Quotient?
The Intelligence Quotient (IQ) is an intelligence test that measures the intellectual abilities that a person has, that is to say, that it measures intelligence. To obtain a person's IQ, we divide the mental age by the chronological age and multiply the result by 100.
In this case, we have the IQ score, which is 125, and the mental age, which is 10, so we have to clear C (chronological age) to find the chronological age.
(mental age)/( chronological age) X 100 = IQ .
(10/C) X100 =125
10 X100 =125 X C
1000 = 125C
1000125 = C
8=C
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Sorina's chronological age is 8.
What is Intelligence Quotient?
The Intelligence Quotient (IQ) is an intelligence test that measures the intellectual abilities that a person has, that is to say, that it measures intelligence. To obtain a person's IQ, we divide the mental age by the chronological age and multiply the result by 100.
In this case, we have the IQ score, which is 125, and the mental age, which is 10, so we have to clear C (chronological age) to find the chronological age.
(mental age)/( chronological age) X 100 = IQ .
(10/C) X100 =125
10 X100 =125 X C
1000 = 125C
1000125 = C
8=C
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Type the correct answer in the box. Use numerals instead of words.
Acar enters a traffic circle at point R. and moves counterclockwise 296 feet before exiting the traffic circle at point P. To the nearest degre
measure of the central angle, ZPQR?
150 ft
296 ft
R
The measure of PQR is approximately
Reset
Next
The measure of ZPQR is approximately 113°. This can be solved usiing the concept of geometry.
What is Geometry?One of the first areas of mathematics is geometry, along with arithmetic. It is concerned with spatial characteristics like the separation, shape, size, and relative placement of objects. A geometer is a mathematician who specializes in geometry.
Therefore, to find the measure of ZPQR, we would follow the following steps:
Let x= measure of <PQR
Hence, the measure of <PQR
296 = x/360 x 3.14 x 2 x 150
Hence, x = 113.12 °
Approximately, the measure of ZPQR is 113 °
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How many fifths are there in 6 and 4/5?
Answer:
Step-by-step explanation:
0
The dot plot shows the ages of the 20 people currently in Ms. Jens classroom.
Which MEASURE OF VARIATION would be best to describe this distribution
A. median
B. interquartile range
C. range
D. mean
The correct measure of variation for best to describe this distribution is,
⇒ interquartile range
Option B is true.
What is interquartile range?For a data set has outliers and variability is often summarized by a statistic called the interquartile range, which is the difference between the first and third quartiles.
Given that;
The dot plot shows the ages of the 20 people currently in Ms. Jens classroom.
Hence, To describe this distribution we can use the interquartile range.
Thus, The correct measure of variation for best to describe this distribution is,
⇒ interquartile range
Option B is true.
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Packages move along a conveyor belt and are dumped into an ever growing pile. After t minutes, the pile grows at a rate of 2 + 8 t packages per hour. Define p(t) to equal the approximate number of packages in the pile after t minutes. If there are 13 packages in the pile when t = 0 minutes, how approximately how many packages are there in the pile after 30 minutes?
There are 15 packages in the pile after 30 minutes.
How to determine the number of packagesFrom the question, we have the following parameters that can be used in our computation:
Rate = 2 + 8t packages per hour
To find the number of packages after t minutes, we need to integrate the rate of growth of the pile over time.
We can use the formula:
p(t) = p(0) + ∫_0^t (2 + 8t) dt
where p(0) = 13 is the number of packages in the pile at t = 0.
Evaluate
P(t) = 13 + 2t + 4t^2
For t = 30 minutes, which is equal to 0.5 hours, we have:
P(30) = 13 + 2 * 0.5 + 4 * 0.5^2
P(30) = 15
So, approximately there are 15 packages in the pile after 30 minutes.
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Halla el denominador que falta
2 10
_ = _
7 ?
Answer: Para encontrar el denominador que falta, podemos utilizar una regla de tres simple. En este caso, queremos encontrar el valor de la segunda fracción, por lo que podemos escribir:
2/7 = 10/x
Para despejar x, podemos multiplicar ambos lados de la ecuación por x y luego dividir por 2:
2/7 * x = 10
x = 10 * 7 / 2
x = 35
Por lo tanto, el denominador que falta en la segunda fracción es 35. La fracción completa es:
2/7 = 10/35
finding slope from 2 points color by number (1,-1) and (5,-2)
The required slope of the line is -1/4 which passes through points (1,-1) and (5,-2).
What is the slope of the line?The slope of the line is defined as the angle of the line. It is denoted by m
Slope m = (y₂ - y₁)/(x₂ -x₁ )
Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁,y₁) and Point 2 has coordinates (x₂, y₂)
Given,
Let
x₁ = 1 , y₁ = -1
x₂ = 5, y₂ = -2
∵ Slope m = (y₂ - y₁)/(x₂ -x₁ )
Substitute values in the formula, and we get
m = (-2 - (-1))/(5 - 1)
m = (-2 + 1)/(4)
m = -1/4
Thus, the required slope of the line is -1/4.
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Find a unit vector with positive first component that is orthogonal to the plane passing through the pointsP=(4,5,−7),Q=(13,14,2), andR=(13,14,8)
The unit vector with positive first component that is orthogonal to the plane passing through the points is 63i - 63j + 0k.
What is a unit vector?
A measurement that combines magnitude and direction is called a vector. A vector having magnitude 1 is referred to as a unit vector.
We are given three points P, Q and R.
The two vectors PQ and PR will lie on the plane through the three given points P(3, 0, 1), Q(4, -2, 1) & R(5, 3,-1).
The vector PQ = (9, 9, 9) and PR = (9, 9, 16) .
Therefore, the required vector is
N = PQ × PR
N = 63i - 63j + 0k
Hence, the unit vector with positive first component that is orthogonal to the plane passing through the points is 63i - 63j + 0k.
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Midwives Researchers Sally Tracy and associates undertook a cross-sectional study looking at the method of delivery and cost of delivery for first-time “low risk” mothers under three delivery scenarios: (1) Caseload midwifery (2) Standard hospital care (3) Private obstetric care The results of the study revealed that 58.5% of all births with midwifery were vaginal deliveries compared with 48.2% of standard hospital births and 30.8% of private obstetric care. In addition, the costs of delivery from midwifery was $3903.78 compared with $5279.23 for standard hospital care and $5413.69 for private obstetric care. (a) Why is this a cross-sectional observational study? (b) Name the explanatory variable in the study. (c) Name the two response variables in the study and determine whether each is qualitative or quantitative.
Note that:
a) This is a cross-sectional observational study because the data was collected from a single point in time and the observations were made without any manipulation of the variables by the researchers.
b) The explanatory variable in the study is the method of delivery (Caseload midwifery, Standard hospital care, Private obstetric care).
c) The two response variables in the study are the rate of vaginal deliveries (58.5%, 48.2%, 30.8%) and the cost of delivery ($3903.78, $5279.23, $5413.69). The rate of vaginal deliveries is a qualitative variable as it is a categorization of the type of delivery, while the cost of delivery is a quantitative variable as it is a measure of the amount of money spent on delivery.
What is a cross-sectional observational study?It is to be noted that a cross-sectional observational study is a type of study in which data is collected from a single point in time without any manipulation of variables by the researchers.
The data collected provides a snapshot of the variables at that point in time.
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Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) gives a good barometer of the overall stock market. On January 31, 2006, 9 of the 30 stocks making up the DJIA increased in price (The Wall Street Journal, February 1, 2006). On the basis of this fact, a financial analyst states we can therefore assume that 30% of the stocks traded on the New York Stock Exchange (NYSE) went up the same day.
A sample of 76 stocks traded on the NYSE that day showed that 36 went up.
You are conducting a study to see if the proportion of stocks that went up is is significantly more than 0.3. You use a significance level of .
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =
What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =
The test static value for the stocks is obtained as 3.482.
What is proportion?
In general, the term "proportion" refers to a part, share, or amount that is compared to a whole. According to the definition of proportion, two ratios are in proportion when they are equal.
It is given that -
The number of stocks n = 76
X = Number of stocks that got increased in price = 36
p = Proportion of stocks that got increased in price.
The formula for p is p = X/n.
Substitute the values in the equation -
p = 76/36 = 2.11111
Now find the value for q -
q = 1 - p
= 1 - 2.11111
= -1.11111
P=0.3
Q = 1 - P
= 1 - 0.3
= 0.7
Now, the values for H0 and H1 are -
H0 : P = 0.3
H1 : P > 0.3
Under H0, the test statistic is given by -
Z = (p - P)/√(PQ/n)
Substitute the values in the equation -
Z = (2.11111 - 0.3)/√(0.3 × 0.7/76)
Z = 1.81111 /√0.002763
Z = 1.81111 /0.052
Z = 3.482
Therefore, test statistic is 3.482.
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what is the quotient?
Answer:
you are right
Step-by-step explanation:
The standard deviation of a sample was reported to be 4. The report indicated
that Σ ( x – average) ^2 = 544. What is the sample size?
Answer:
35
Step-by-step explanation:
The formula for the variance of a sample is:
Variance = Σ (x - average)^2 / (n - 1)
Where n is the sample size and x is each data point.
We are given the variance as 544 and the standard deviation as 4, and we know that:
Standard deviation = sqrt(Variance)
So,
Variance = Standard deviation^2
Variance = 4^2
Variance = 16
Now we have the variance and can find the sample size:
544 = 16 * (n - 1)
n - 1 = 544 / 16
n - 1 = 34
The sample size is n = 35.
Find three consecutive integers such that the sum of the first and twice the second is 56 minus three times the third.The smallest integer is __The middle integer is __The largest integer is __
The smallest integer is 10, the middle integer is 11, and the largest integer is 12.
What is the consecutive integers in algebra?
In algebra, consecutive integers are a sequence of numbers that follow each other in order, where each number is one more than the previous number.
Let's call the smallest integer "x". Then, the middle integer is "x + 1" and the largest integer is "x + 2".
Now, using the given information, we can set up an equation:
x + 2(x + 1) = 56 - 3(x + 2)
Expanding the equation and simplifying:
3x + 2 = 56 - 3x - 6
6x = 62
x = 10
So, the three consecutive integers are 10, 11, and 12.
Hence, The smallest integer is 10, the middle integer is 11, and the largest integer is 12.
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A particle that moves along a straight line has velocity
v(t)=t^2e^(-5t)
meters per second after t seconds. How many meters will it travel during the first t seconds (from time=0 to time=t)?
Use synthetic division to find the result when 4x³ + 7x² - 4x - 4 is divided by
x + 1. If there is a remainder, express the result in the form q(x) + r(x)
b(x)
Solving the following expression (4x4-7x3+7x2-12x+15)/(x-1) yields the result -7x2 + 4 + 19/ in the form q(x)+r(x)/b(x) (x-1).
what is expression ?It is permissible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, term, and mathematical operator Three components of a numerical method include numbers, variable, and processes (such as addition, subtraction, multiplication or division etc.) It is possible to oppose expressions and words. An expression, often known as an analytical form, is any mathematics statement that contains variables, values, and an arithmetic procedure between them. For instance, the word m in the balanced formula is separated from the terms 4m and 5 by that of the arithmetic symbol +, as is the term m in the expression 4m + 5.
given
Given expression is (4x⁴-7x³+7x²-12x+15)/(x-1)
then dividing this
=4x⁴/x-1-7x³/x-1+7x²/x-1-12x/x-1+15/x-1
=4x³-3x²+4x-8+7/x-1 in form of q(x)+r(x)/b(x)
= 4x⁴/x-1-7x³/x-1+7x²/x-1-12x/x-1+15/x-1
= -7x² + 4 + 19/(x-1)
Solving the following expression (4x4-7x3+7x2-12x+15)/(x-1) yields the result -7x2 + 4 + 19/ in the form q(x)+r(x)/b(x) (x-1).
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CAN SOMEONE HELP WITH THIS?✨
In the polynomial f(x) = 3x² - 2x + 5, f(0) is solved to get -2
From the equation of the tangent line
m= -2
b = 5
How to find the derivativeThe derivative of the polynomial f(x) = 3x² - 2x + 5 is solved to be
f'(x) = 2(3)x - 1(2)
f'(x) = 6x - 2
f(0) = -2
Equation of the tangent to the parabola f(x) = 3x² - 2x + 5
f'(x) = 6x - 2 at x = 0, f'(x) = -2 hence the slope, m is -2
Using the point slope formula for (0, 5)
(y - 5) = m (x - 0)
y - 5 = -2x - 0
y = -2x + 5
comparing with y = mx + b
m = -2 and b = 5
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who’s wrong? And why
Step-by-step explanation:
Ted is correct
When you multiply exponents, they actually add.
Note**
[tex]4x = {4x}^{1} [/tex]
Looking at pearl, when she did
[tex] {2x}^{2} \times 4 {x}^{1} [/tex]
It should have equaled
[tex]8 {x}^{3} [/tex]
Pearl's did not, but Ted's did
The deck of a bridge is suspended 205 feet above a river. If a pebble falls off the side of the bridge, the height, in feet, of the pebble above the water surface after t seconds is given by y = 205 − 16t2.
(a)
Find the average velocity (in ft/s) of the pebble for the time period beginning when t = 2 and lasting the following amount of time.
(i)
0.1 seconds
ft/s
(ii)
0.05 seconds
ft/s
(iii)
0.01 seconds
ft/s
(b)
Estimate the instantaneous velocity (in ft/s) of the pebble after 2 seconds.
ft/s
The required answers are
i) -65.6
ii) -64.8
iii) -64
iv) -64
VelocityThe pace at which an object's position changes in relation to a frame of reference and time is known as its velocity.
Since it is a vector quantity, the definition of velocity requires both magnitude (speed) and direction.
Its SI equivalent is the metre per second (ms-1). A body is considered to be accelerating if its velocity changes, either in magnitude or direction.
Now in the given question,
1. Average velocity
When t=2
i. [2, 2.1]
= y(2.1) - y(2) / 2.1 - 2
= 205 - 16(2.1)^2 - 205 + 16(2)^2 / 0.1
= 205 - 16*4.41 - 205 +16(4) / 0.1
= -70.56 +64 / 0.1
= -6.56 /0.1
= -65.6
ii. [2, 2.05]
= y(2.05) - y(4) / 4.05 - 4
= 205 - 16(2.05)^2 - 205 +16(2)^2 / 0.05
= 205 - 16*4.2025 - 205 + 16(4) / 0.05
= -3.24 / 0.05
= -64.8
iii. [2, 2.01]
= y(2.01) - y(2) / 2.01 - 2
= 205 - 16(2.01)^2 - 205 + 16(2)^2 / 0.01
= 205 - 16*4.0401 - 205 + 16(4) / 0.01
= -64.64 + 64 / 0.01
= -0.64 / 0.01
= -64
iv. Instantaneous velocity
y = 205 - 16t^2
dy/dx = -32t
Considering t = 2
Instantaneous velocity after 2 seconds = -32(2)
Instantaneous velocity after 2 seconds = -64
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How many 1/5 in cubes will it take to fill a prism that is 2 2/5 inches by 3 1/5 inches by 2 inches?
The number of 1/5 in a cube required to fill a prism which is 2 2/5 inches x 3 1/5 inches x 2 inches is, 2000 1/5 inch cubes
What is Volume?In mathematics, Space occupied by three dimensional objects is called Volume. Basically it is a space within a closed region of every three dimensional object.
Given that,
The dimensions of a prism 2 2/5 inches by 3 1/5 inches by 2 inches
The volume of the prism = 2 2/5 inches x 3 1/5 inches x 2 inches
= 16 inches³
To find the number of 1/5 inch cubes required to fill the prism,
we need to find the volume of one cube
and then divide the volume of the prism by the volume of one cube:
Volume of one cube = (1/5 inch)³
= (1/5)³ inch³
= 1/125 inch³
Volume of the prism / Volume of one cube = 16 / (1/125)
= 16 x 125
= 2000 cubes
So, it will take 2000, 1/5 inch cubes to fill the prism.
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Find the volume of a sphere with radius 13. Use 3.14 for pi, round your answer to the nearest hundredth if necessary, and do not include units.
the volume of a sphere:
V=[tex]\frac{4\pi r^3}{3} =\frac{4*3.14*13^3}{3}[/tex]≈9198.106
EXTRACURRICULARS The table shows the results of a survey about what students plan to do after school. How can you determine the probability that a student selected at random will be doing homework or participating in a sport after school? Justify your reasoning.
As a result, there is a 52% chance that a pupil chosen at random will be doing homework or playing an activity after school.
What is a Probability example?By reducing the number of positive outcomes by the overall number of potential outcomes, probability—the chance that a will occur—is determined. The most basic illustration is a coin toss. There are just two results when users flip a coin: either it shows up heads or tails.
We need to discover the sum of the percentage of students who plan to do homework and the percentage of students who plan to engage in a sport in order to calculate the likelihood that a student chosen at random will be doing homework or playing a sport after school.
Therefore, the likelihood that a pupil chosen at random will be working on homework or playing an activity after school is:
24% + 28% = 52%
This is because the two events are not mutually exclusive, and a student may plan to do both activities after school.
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A 30-foot ladder reaches the wall at 20 feet above the ground. Find the measure of the angle that ladder makes with the ground and round to the nearest tenth of a degree
The measure of the angle that ladder makes with the ground 41.8°.
What is the measure of the angle that the ladder makes with the ground?Given the data in the question;
This forms a right triangle.
Angle that ladder makes with the ground θ = ?Opposite of angle θ = 20ftHypotenuse = 30ftTo find the angle, we use trigonometric ratio.
Sine = Opposite / Hypotenuse
sinθ = 20ft / 30ft
sinθ = 2/3
θ = sin⁻¹( 2/3 )
θ = 41.8°
Therefore, the measure of the angle is 41.8°.
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3
Select the correct answer.
Which expression is a prime polynomial?
A. X^3 - 1
B.X^2 + 1
C.X^3 + 1
D.X^2 - 1
Answer: The correct answer is D, X^2 - 1.
A prime polynomial is a polynomial that has no real or complex roots (i.e., no solutions), except for 0. X^2 - 1 has two roots, -1 and 1, both of which are not 0. Hence, it is a prime polynomial.
Step-by-step explanation:
Betty is listening to a playlist of 16 songs. She randomly shuffles the tracks on the playlist after each song. What is the probability to the nearest tenth of a percent that the same song is played twice in a row?
Answer: 6.3%
Step-by-step explanation:
The probability that the same song is played twice = Number of favourite outcomes / Total outcomes.
= 1 / 6
= 0.0625
in per cent = 6.25%
Probability to the nearest tenth of % = 6.3%
.: Probability - Probability is a number that reflects the chance or likelihood that a particular event will occur.
Probability can be expressed as proportional that ranging from 0 to 1.
They can also be expressed as percentages ranging from 0% to 100%.
1. Yes-AA
2. Yes- SAS
3. Yes- SSS
4. No- Not Similar
The triangles are similar to the AA property.
What are similar triangles?If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable.
Similar figures are items that feature more than two figures that are the same shape but different sizes.
Given triangles ANF and HES
for ΔANF,
∠A = 29°, ∠N = 106°, and ∠F = 45°(from angle sum property)
for ΔHES
∠H = 45°, ∠S = 29° and ∠E = 106°(from angle sum property)
so ∠A = ∠S
∠N = ∠E
and ∠F = ∠H
so ΔANF ≈ ΔHES
by AA similarity.
Hence option A is correct.
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makayla is taking a math course and is working with the perimeter of rectangles. she knows the perimeter and length of her rectangle but wants to solve for the width. rearrange the following equation for w, where p is the perimeter, l is the length, and w is the width of the rectangle.
The equation for w(width of rectangle) is: W = (P - 2L)/2
The correct answer is an option (c)
We know that the formula for the perimeter of the rectangle is:
P = 2L + 2W
where P is the perimeter of the rectangle,
L is the length of the rectangle,
and W is the width of the rectangle.
We need to rearrange this equation for w.
Consider equation,
P = 2L + 2W
P - 2L = 2L + 2W - 2L ......(subtract 2L from each side of equation)
P - 2L = 2W
(P - 2L)/2 = 2W/2 .....(divide each side by 2)
W = (P - 2L)/2
Therefore, the correct answer is W equals the quantity P minus 2 times L all over 2
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The complete question is:
Makayla is taking a math course and is working with the perimeter of rectangles. She knows the perimeter and length of her rectangle but wants to solve for the width. Rearrange the following equation for W, where P is the perimeter, L is the length, and W is the width of the rectangle. P = 2L + 2W.
a) W = 2P − 2L
b) W = P/ 2-2 times L
c) W equals the quantity P minus 2 times L all over 2
d) W = 2P + 2L
Answer:
c) W equals the quantity P minus 2 times L all over 2
Step-by-step explanation:i just took the test
I need help please!
What did Nixon's visit to China prove to the US?
A. If you can deal with one communist nation, others will likely join in
B. The Soviet Union was close to falling apart and communism would leave Eastern Europe
C. Most communist nations actually practice capitalism and have free elections
D. The communist world was not a single entity, and had many different positions
Nixon's visit to China was intended to advance their relationship and create new trade opportunities.
Why was Nixon's visit to China?The President stated on live television that he would visit the PRC in 1972 on July 15, 1971. The American public was able to see photos of mainland China for the first time in more than 20 years during the week-long tour, which took place from February 21 to 28, 1972.
Nixon intended to protect the global populace, which meant he wanted to improve relations amongst the world's superpowers. Nixon's visit to China was intended to advance their relationship and create new trade opportunities.
A significant step in normalizing diplomatic ties with China, Nixon's visit was a big success. The following year, American tourists began to travel there, and American business organizations established a booming trade with China. The United States and China achieved complete diplomatic ties by the year 1979.
Therefore, the correct answer is option B. The Soviet Union was close to falling apart and communism would leave Eastern Europe.
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Let Y1,Y2,....,Y5 be a random sample of size 5 from a normal population with mean 0 and variance 1 and let Y bar=(1/5)sumation from i=1 to 5 Yi. Let Y6 be another independent observation from the same population. What is the distribution ofa) W= sumation from i=1 to 5 of Yi^2 ? Why ?b) U= sumation from i=1 to 5 of ( Yi- Y bar)^2? Why?c) sumation from i=1 to 5 of ( Yi- Y bar)^2 + Y6^2 ? Why?
a) From the given information, [tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
be a random sample of size 5 from a normal population with mean 0 and variance 1. Hence,
[tex]W=(\sum_{i=1}^5 Y_{i}^2)\sim X_{5}^2[/tex]
b) From the given information, the random variable is [tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
and [tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
representing the standard normal variables. Hence,
.[tex]U=(\sum_{i=1}^5 (Y_{i} - \bar{Y})^2) \sim X_{5}^2[/tex]
c) From the given information,[tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
representing the standard normal variables and moreover, [tex]Y_{6}[/tex]
is independent standard normal variable. Hence, [tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2\sim X_{5}^2[/tex]
and
.[tex]Y_{6}^2\sim X_{1}^2[/tex]
What is Chi-square distribution?
Chi-square distribution with k degrees of freedom is defined as the sum of the k standard normal variables. This is also defined as the square of the deviations taken from the mean. Chi-square distribution is used for testing goodness of fit and independence in contingency tables.
• If [tex]z_{1}, z_{2},...........z_{k}[/tex]
are independent standard normal variables, then
[tex]\sum_{i=1}^k z_{i}^2\sim X_{k}^2[/tex]
• If [tex]X\sim X_{m}^2[/tex]
and [tex]Y\sim X_{n}^2[/tex]
, then
[tex](X+Y)\sim X_{m+n}^2[/tex]
Step: 1
(a)
The distribution of [tex]\sum_{i=1}^5 Y_{i}^2[/tex]
is obtained below:
From the given information, [tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
be a random sample of size 5 from a normal population with mean 0 and variance 1. Hence,
[tex]W=(\sum_{i=1}^5 Y_{i}^2)\sim X_{5}^2[/tex]
The random variable, [tex]W=\sum_{i=1}^5 Y_{i}^2[/tex]
representing the sum of squares of standard normal variables and hence, the distribution of [tex]W=\sum_{i=1}^5 Y_{i}^2[/tex]
is identified as chi-square distribution.
Hint:
Use chi-square distribution to find the distribution of
.[tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
Step: 2
(b)
From the given information, the random variable is [tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
and [tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
representing the standard normal variables. Hence,
.[tex]U=(\sum_{i=1}^5 (Y_{i} - \bar{Y})^2) \sim X_{5}^2[/tex]
The random variable [tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
representing the sum of squares of deviations taken from the mean for 5 standard normal variables and hence, the distribution of
[tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
is chi-square distribution with 5 degrees of freedom.
Hint:
Use [tex](X+Y)\sim X_{m+n}^2[/tex]
to find the distribution of
[tex]\sum_{i=1}^5 (Y_{i} - \bar{Y})^2+Y_{6}^2[/tex]
Step: 3
(c)
From the given information,[tex]Y_{1}, Y_{2},...........Y_{5}[/tex]
representing the standard normal variables and moreover, [tex]Y_{6}[/tex]
is independent standard normal variable. Hence, [tex]U=\sum_{i=1}^5 (Y_{i} - \bar{Y})^2\sim X_{5}^2[/tex]
and
.[tex]Y_{6}^2\sim X_{1}^2[/tex]
Therefore,
[tex]\sum_{i=1}^5 (Y_{i} - \bar{Y})^2+Y_{6}^2\sim X_{5+1}^2[/tex]
[tex]\sim X_{6}^2[/tex]
The distribution of [tex]\sum_{i=1}^5 (Y_{i} - \bar{Y})^2+Y_{6}^2[/tex]
is identified as chi-square distribution with degrees of freedom is obtained by adding the degrees of freedom of[tex]\sum_{i=1}^5 (Y_{i} - \bar{Y})^2[/tex]
and the degrees of freedom of [tex]Y_{6}^2[/tex].
The distribution of [tex]W=\sum_{i=1}^5 Y_{i}^2[/tex]
is chi-square distribution with 5 degrees of freedom.
To learn more about the chi-square distribution from the given link
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Find the exact lengths of the sides of quadrilateral ABCD with vertices A(4, 5), B(-3, 3), C(-6,-13) and D(6,-2).
As a result, the answers to the above quadrilateral issue are A.5, B.2, 3, C.-1/5, and D.2/3.
what is quadrilateral ?In terms of the geometry, a quadrilateral is really a four-sided polygon with four corners and four corners. Its origins originate from the Latin terms tetra and latus (meaning "side"). A rectangle is a this double form with four sides, 4 vertices, with four corners. Two main types of convex and convex exist. Convex bedbugs or bed bug also have a number of subclasses, such as trapezoids, parallelograms, quarters, rhombuses, and squares. Corresponding points come in many irregular configurations, such as quadrilaterals, trapezoids, rectangles, kites, square, and rhombuses.
given
The vertices of the quadrilateral ABCD are A(-4, -5), B(-3, 0), C(0, 2), and D. (5, 1).
We need to determine whether ABCD is a trapezoidal shape or not.
We are known that a trapezoid has one set of parallel opposed sides, making it a quadrilateral. Moreover, the slopes of two parallel lines are equal.
To start, the slopes of sides AB, BC, CD, and DA will be calculated.
It has
The incline for
AB =0+5/ -3+4
5. In the first step.
Step 2: The slope in BC is
=> 2-0 / 0+3
=> 2/3.
Step 3: The CD's gradient is
1-2/5-0 is equivalent to -1/5.
The gradient of DA is equal to in step four.
=>-5.1/-4.4/2.3
=>2/3.
As a result, the answers to the above quadrilateral issue are A.5, B.2, 3, C.-1/5, and D.2/3.
To know more about quadrilateral visit:
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Solve for x
4|3x+5|-4=64
Answer:
x=4
Step-by-step explanation:
12x+20-4=64
12x+16=64
12x=64-64
12x=48
x=4