Calculus-based Physics I, can someone explain this to me?
My apologies for the broadness of my question. I especially don't understand the notation being used here, but I know this is about data collection, specifically standard deviation and standard error. I mostly need help with the data collection of multiple variables, the formulae for standard deviation and standard error make no sense to me.
I could also use some examples.
Answers
2: For a sample of data [tex]x_1,x_2,\ldots,x_N[/tex], the mean of this sample denoted by [tex]\overline x[/tex] is the sum of the data divided by the number of data points,
[tex]\overline x = \dfrac{x_1+x_2+\cdots+x_N}N = \displaystyle\frac1N\sum_{i=1}^N x_i[/tex]
As an example, consider [tex]x_1=-1[/tex], [tex]x_2=1[/tex], and [tex]x_3=3[/tex]. Then
[tex]\overline x = \dfrac{-1+1+3}3 = 1[/tex]
3: Standard deviation is a measure of how dispersed a given data sample is relative to the mean. Consult the plot: for a normal distribution, approximately 68% of it lies within 1 standard deviation of the mean, approx. 95% within 2 standard deviations, and approx. 99.7% within 3 standard deviations.
For instance, if the data is pulled from a normally distributed population with mean 0 and standard deviation 1, if you were to randomly select any data from the population, then 68% of the time it will fall in the range (-1, 1); 95% of the time it will fall within (-2, 2); 99.7% of the time it fall within (-3, 3).
To compute the standard deviation for a sample, for each [tex]x_i[/tex] in [tex]x_1,x_2,\ldots,x_N[/tex], you
• take the difference between [tex]x_i[/tex] and the mean [tex]\overline x[/tex]
• square this difference
• sum all the squared differences
• divide the sum by N - 1 (for a sample) or N (for a population)
• take the square root
Here the standard deviation is denoted [tex]\sigma^x_{N-1}[/tex], which I would read as "the sample standard deviation of the data x" - sample because of the N - 1 subscript.
Continuing with the previous example, we'd have
[tex]\sigma^x_{N-1} = \displaystyle \sqrt{\frac{\left(-1-1\right)^2+\left(1-1\right)^2+\left(3-1\right)^2}{3-1}} = \sqrt4 = 2[/tex]
4: Not much more to say here, the standard error is basically a measure of how accurate a given estimate is about the population based on the sample data. It's analogous to uncertainty in measuring length with a ruler, for instance.
In our example,
[tex]\alpha^x = \dfrac2{\sqrt3}[/tex]
5: If x, y, and z are random variables, then I suppose ρ is meant to denote a function of these random variables (so that ρ itself is just another random variable). For instance, you could have ρ = x + 3y - 2z. Then [tex]\overline\rho[/tex] is the sample mean of ρ.
I'm not entirely sure about the notation [tex]x(\overline x,\sigma^x_{N-1},\alpha^x)[/tex], but I suspect it's just referring to sample x with mean [tex]\overline x[/tex] and standard deviation [tex]\sigma^x_{N-1}[/tex] with standard error [tex]\alpha^x[/tex].
∆ρ is just the differential of ρ, essentially capturing how ρ changes with respect to small changes in x, y, and z. The expression you see here follows from the chain rule for differentiation.
The formula you see for [tex]\sigma^\rho_{N-1}[/tex] is the sample standard deviation of ρ. Think of ∆ρ as a vector with 3 components. Then [tex]\sigma^\rho_{N-1}[/tex] is the magnitude of this vector.
Similarly, [tex]\alpha^\rho[/tex] is the standard error for ρ, and corresponds to the magnitude of the vector whose components are the standard errors of x, y, and z.
In order for these statistics to make sense, each of x, y, and z must be samples of the same number of data. Say we take x as before [tex](x_1=-1,x_2=1,x_3=3)[/tex], along with [tex]y_1=0,y_2=4,y_3=-2[/tex] and [tex]z_1=-3,z_2=\frac12,z_3=10[/tex]. Suppose ρ = x + 3y - 2z. Then
• the sample means of y and z :
[tex]\overline y = \dfrac{0+4-2}3 = \dfrac23 \\\\ \overline z = \dfrac{-3+\frac12+10}3 = \dfrac52[/tex]
• the standard deviations of y and z :
[tex]\sigma^y_{N-1} = \sqrt{\dfrac{\left(0-\frac23\right)^2+\left(4-\frac23\right)^2+\left(2-\frac23\right)^2}{3-1}} = 2\sqrt{\dfrac73} \approx 3.06\\\\ \sigma^z_{N-1} = \sqrt{\dfrac{\left(-3-\frac52\right)^2+\left(\frac12-\frac52\right)^2+\left(10-\frac52\right)^2}{3-1}} = \dfrac{\sqrt{181}}2 \approx 6.73[/tex]
• the values of ρ :
[tex]\rho_1 = x_1+3y_1-2z_1 = -1+2\times0-2\times(-3) = 5 \\\\ \rho_2 = x_2+3y_2-2z_2 = 1+3\times4-2\times\dfrac12=12 \\\\ \rho_3 = x_3+3y_3-2z_3 = 3+3\times(-2)-2\times10 = -23[/tex]
• the sample mean of ρ :
[tex]\overline\rho = \dfrac{5+12-23}3 = -2[/tex]
• by the chain rule,
[tex]\Delta\rho = \Delta x+3\Delta y-2\Delta z[/tex]
so the standard deviation of ρ :
[tex]\sigma^\rho_{N-1} = \sqrt{\left(\sigma^x_{N-1}\right)^2 + \left(3\sigma^y_{N-1}\right)^2 + \left(-2\sigma^z_{N-1}\right)^2} \\\\\sigma^\rho_{N-1}= \sqrt{2^2 + 9\left(2\sqrt{\dfrac73}\right)^2 + 4\left(\dfrac{\sqrt{181}}2\right)^2} = \dfrac12\sqrt{\dfrac{703}3} \approx 7.65[/tex]
• the standard errors of y and z :
[tex]\alpha^y = \dfrac{2\sqrt{\frac73}}{\sqrt3} = \dfrac23\sqrt7 \approx 1.76 \\\\ \alpha^z = \dfrac{\frac{\sqrt{181}}2}{\sqrt3} = \dfrac12\sqrt{\dfrac{181}3} \approx 3.88[/tex]
• the standard error of ρ :
[tex]\alpha^\rho=\sqrt{\left(\alpha^x\right)^2+\left(3\alpha^y\right)^2+\left(-2\alpha^z\right)^2}\\\\\alpha^\rho=\sqrt{\left(\dfrac2{\sqrt3}\right)^2+9\left(\dfrac23\sqrt7\right)^2+4\left(\dfrac12\sqrt{\dfrac{703}3}\right)^2}=\sqrt{269}\approx16.40[/tex]
Related Questions
Question 4 A car of mass 820 kg has a maximum power or 30 kW and moves against a constant resistance of motion to 910 N. Calculate the maximum speed of the car in the following situation On ground level ?
[tex]p = fv \\ v = p \div f \\ 30 \div 910 = [/tex]
Answers
Explanation:
power = force × velocity
velocity=power/force
=(30×1000)/910
=32.97m/s
The correct answer is 32.97m/s.
How to calculate the maximum speed?By using the formula:
power = force × velocity
velocity = power / force
= (30×1000)/910
= 32.97m/s
The answer is 32.97m/s.
What is maximum speed physics?
Maximum speed is the highest rate of speed an athlete can attain. Acceleration refers to velocity, and because velocity has both a magnitude and direction associated with it, acceleration changes when athletes change the magnitude of their motion (how fast they are running), the direction of their motion, or both.
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The speed of light will be minimum while passing through
A) water
B) vaccum
C) air
D) glass
Answers
Answer:
The correct choice is option D. glass
Glass's optical density is very great. Which is why speed of light will be minimum when passing glass.
Answer:
glass
Explanation:
i think, sana makatulong
The type of force that will not cause a change in the speed of an object.
Answers
Explanation:
Balanced forces will cause no change in the speed of an object.
Which of the following is NOT an important phenomenon that commonly erodes and weathers exposed rock outcrops to form sedimentary material?
a) Lightning
b) Wind
c) Rain
d) Freezing/Thawing
Answers
Answer:
Explanation:
While each of these can cause erosion and weathering, lightning is probably the least important as it occurs less frequently and affects a much smaller surface area when it strikes.
Wind is not very effective by itself, but it can carry abrasives which work to degrade rock surfaces. It covers a very large area at once so the net effect can be moderate to large especially desert areas where plants are not readily available to disrupt the flow.
Rain covers huge areas and is quite common.
Freezing/Thawing cycles cover large areas and are quite common in the temperate and arctic latitudes and even in tropical altitudes.
Attached is a photo taken atop Half Dome in Yosemite National Park showing two of thousands of divots in the rock there caused by lightning strikes. The current in the lightning heats the stone causing water trapped in it to flash to steam. The increased pressure inside the stone can overwhelm the material strength and blow rock chunks over a fairly good sized area. This is a fairly rapid weathering and erosion when it occurs, but that is typically limited to a few dozen days per year and occurs mostly on high ground where lightning is more likely to strike earth.
What is the relationship between resistivity of a wire and length of a wire is?
Answers
[tex]\boxed{\sf R=\rho\dfrac{\ell}{A}}[/tex]
Or
[tex]\\ \rm\hookrightarrow \rho=R\dfrac{A}{l}[/tex]
Rho stands for resistivity of wireR stands for resistance of wire l stands for length of areaA stands for surface area of wireWhat is crowbar? Class 8
Answers
Answer:
Crowbar is a tool made of metal rods that are flattened at both ends, with one of them curved. There is also a crowbar that is curved at both ends.
hope it helped
Please help me find the magnitude and direction of the resultant force
Answers
value of g is independent of
Answers
Answer:
mass
Explanation:
The value of g (acceleration due to gravitation) is independent of the mass of the object.
A footballer kicks a ball at an angle of 45° with the horizontal. If the ball was in the air
for 10 s and lands 4000 m away determine its initial speed.
Answers
Answer:
HOPE THIS ANSWER WILL HELP YOU
60.96-mile distance in kilometers
Answers
Answer:
98.105610 Kilometers
Explanation:
Rounded to 8 digits
Answer:
98.11 km
Explanation:
60.96 mi (5280ft/mi)(12 in/ft)(2.54 cm/in)(1 m/ 100cm)(1 km/1000 m) = 98.10561024 km
however, reporting this precision to the nearest hundred millionth of a kilometer is silly as we only know the mile dimension to the nearest hundredth of a mile.
A hundredth of a mile is 5280/100 = 52.8 ft or just over 16 m
It would only be a little bit of a presumption to give our precision to the nearest 10 meters as I have done.
This also fits neatly into the significant digits rule. Both the original dimension and the converted dimension have four significant digits.
The length, breadth and height of a box is 50cm, 20cm and 10cm respectively then find surface area of its base and volume.
Answers
When calculating surface area of just the base, we ignore the height. Therefore, we just multiply the length and height values.
50cm x 20cm = 1000cm^2
Volume = 10,000cm^3
Volume of a box is calculated by multiplying length by breadth by height.
50cm x 20cm x 10cm = 10,000cm^3
Hope this helped!
I just hope this pic will help you
What is an electricity?
Answers
Explanation:
Electricity is the flow of electrical power or charge.
A scientist obtains images of a particular star, which appears as four separate stars.
Which phenomenon causes the star to appear this way?
O redshift
gravitational lensing
O principle of equivalence
O gravitational time dilation
Answers
Answer:
gravitational lensing
Explanation:
Answer:
Gravitational lensing
Explanation:
Driving home from school one day, you spot a ball rolling out into the street . (See the figure below) You brake for 1.50 s , slowing your 950-kg car from 16.0 m/s to 9.50 m/s.
a)What was the magnitude of the average force exerted on your car during braking?
b)What was the direction of the average force exerted on your car during braking?
c)How far did you travel while braking?
Answers
Answer:
a) 4116.67 N
b) negative (assuming car is traveling in positive direction)
c) 19.125 m
Explanation:
Part A)We can find the average force exerted on the car by using Newton's 2nd Law: F = ma.
We have the mass (950 kg) but we do not have the acceleration. Let's solve for this using kinematics.
We have the time during braking, the initial and final velocity, and we need to find the acceleration. To do this, we can use the following equation: v = v₀ + at.
v (final velocity) = 9.50 m/sv₀ (initial velocity) = 16.0 m/st (time) = 1.50 s a = ?Substitute these values into the equation.
v = v₀ + at9.50 m/s = 16.0 m/s + a (1.50 s) 9.50 = 16.0 + 1.50a -6.50 = 1.50a a = -4.333333... m/s²The car's acceleration during the braking period is -4.33 m/s², assuming the car is traveling in the positive direction.
Now, we can use this to plug into Newton's 2nd Law to find the force exerted on the car.
F = maF = (950 kg) (-4.33...) F = -4116.666667 NThe magnitude of the average force exerted on the car while braking is the absolute value, so it is 4116.67 N.
Part B)The direction of the average force is denoted in the negative sign. It is opposite the direction of the car's motion. Assuming that the car is traveling in the positive direction, the direction of the average force exerted on the car is in the negative direction since this force is slowing the car down while braking.
Part C)We can find the distance traveled during the braking period by using another kinematic equation.
We have the initial and final velocity, and the time traveled, so we can use this equation: Δx = v_avg · t.
Note that v_avg is the same thing as (v + v₀) / 2.Substitute the known values into the equation.
Δx = [(9.50 + 16.0)/2] · 1.50 Δx = 12.75 · 1.50Δx = 19.125 mThe car traveled 19.125 m while braking.
You shake a bottle of soda and take off the cap. If the soda shoots out of the
bottle with a velocity of 17 m/s, how high will it go?
A. 16.9 m
B. 14.7 m
C. 10.6 m
D. 12.2 m
Answers
Answer: B) 14.7 meters
============================================================
Explanation:
The bottle cap is a projectile, so we'll use the aptly named projectile formula
That formula (the meters version of it anyway) is approximately
h = -4.9t^2 + v*t + s
where,
t = elapsed time in secondsh = height at time tv = starting velocitys = starting heightWe'll assume that s = 0, though realistically it's probably going to be a bit larger than this (since the person is holding it above the ground). For the sake of simplicity, we'll stick with s = 0.
We're told that the initial velocity is 17 m/s, which means v = 17.
All of that means the formula mentioned earlier updates into this:
h = -4.9t^2 + 17t
If we were to graph this, or apply the -b/(2a) formula, then you should find that the vertex occurs when t = 1.73469 approximately.
Plug this into the equation we found to get...
h = -4.9t^2 + 17t
h = -4.9(1.73469)^2 + 17(1.73469)
h = 14.7449
h = 14.7
The bottle cap will reach a peak height of approximately 14.7 meters and does so at around the 1.7 second mark. The total flight time is approximately 2*1.7 = 3.4 seconds. This all assumes that the starting height and ending height are both 0 meters off the ground.
The soda will go as high as 14.7 meters. Therefore, option (B) is correct.
What is the conservation of energy principle?We can solve this problem using the conservation of energy principle, which states that the initial energy of a system is equal to the final energy. At the start, the soda in the bottle has some potential energy due to its position and kinetic energy due to its motion. When the soda shoots out of the bottle, it converts all of its initial energy to potential energy as it reaches its maximum height.
Using the conservation of energy, we can write:
Initial energy = Final energy
[tex]0.5 * m * v^2 = m * g * h[/tex]
where m is the mass of the soda, v is its velocity, g is the acceleration due to gravity ([tex]9.8 m/s^2[/tex]), and h is the height it reaches.
Solving for h, we get:
h = [tex](v^2)/(2g) = (17^2)/(29.8)[/tex]= 14.7 m
Therefore, the soda will go as high as 14.7 meters. Option B is correct.
Learn more about the conservation of energy principle, here:
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(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is 9.79m/s2 is moved to a location where the acceleration due to gravity is 9.82m/s2. What is its new period
Answers
I assume this is the motion of the simple pendulum
T = 2π × [tex]\sqrt{\frac{L}{g } }[/tex]
=> [tex]\frac{T1}{T2 }[/tex] = [tex]\sqrt{\frac{g2}{g1} }[/tex]
Given T1= 3s g1= 9.79 g2= 9.82
=> T2 = 3.00459 s
50 POINTS!
Motion Diagrams: Observe each motion and describe how to object is moving. If the motion diagrams show more than one kind, highlight and label each kind. Please thoroughly explain. Thank you so much!
Answers
No:1
The object is moving with constant or uniform acceleration and in average speed
No:-2The object is de accelerating
No:-3The object deaccelerated and came to rest so fast.
No:-4The object moves slowly first then accelerated.
No:-5The object accelerated at first so fast then move with constant acceleration then again accelerated .
Which feature of a heating curve indicates a change of state
Answers
Answer:
The diagonal or the inclined lines shows the changes in terms of temperature, and the horizontal lines shows the changing of phases.
Explanation:
hope it is useful
Which Scientist discovered
atomic Theory?
Answers
Answer:
John dalton discovered
Name another way that energy is made
Answers
Answer:
The three major categories of energy for electricity generation are fossil fuels (coal, natural gas, and petroleum), nuclear energy, and renewable energy sources. Most electricity is generated with steam turbines using fossil fuels, nuclear, biomass, geothermal, and solar thermal energy.
Explanation:
Solar
Wind
Geothermal
Hydrogen
TidaWave
Hydroelectric
Biomass
A car speeds up from 0.0 m/s to 14.0 m/s
in 3.25 s.
The acceleration of the car is:
A. 14.0 m/s2
B. 4.31 m/s2
C. 45.5 m/s2
D. 10.8 m/s2
Answers
Explanation:
Its speed is zero at the beginning of the 3.25 s and 14 m/s at the end.
I'm thinking you probably wanted to find the car's acceleration.
Acceleration =(change in speed) / (time for the change).
Acceleration =(14m/s )/ (3.25s) .
Acceleration = 4.31m/s² .
I hope it will help you so.
Have a great day.
Be safe and be happy.
Hope I have helped
What is the amount of force exerted by gas particles?
Answers
Answer:
Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
in the same viewing window. Compare the magnitudes of f (x) and g(x) when x is close to 0. Use the comparison to write a short paragraph explaining why
Answers
Answer:
Due to inertia of restttttttrestrestrestrest
Who among the old Greek Philosophers/Scientists was the first to systematically study the
Stars?
Answers
Previous Answer:
Aristotle
Change:
Aristarchus
Answer:
Aristarchus of Samos was the first Greek philosopher to believe the solar system was organized around the Sun, rather than the Earth. The heliocentric was very unpopular during Aristarchus' lifetime, although it would inspire astronomers centuries later. Ptolemy was an astronomer and mathematician.
9.1 x 105 – 7.2 x 105
=
Answers
Answer:
1.9 x 10⁵
Explanation:
I ASSUME your question is supposed to read
9.1 x 10⁵ – 7.2 x 10⁵ =
9.1 - 7.2 = 1.9 and the powers are the same so just tag them along
its the same as
910000 - 720000 = 190000
1.9 x 105 could be another possibility if the question is actually posed correctly.
A plane leaves Houston and flies 400 km north to Dallas. The pilot realizes he has forgotten his
golf clubs and returns to Houston to pick them up on the way back to Houston, the plane runs
out of gas and is forced to land in Huntsville, 100 km North of Houston. The trip from Houston to
Dallas took 120 min and the trip from Dallas to Huntsville took 70 min.
A. What is the total distance (in meters) covered by the airplane?
B. What is the plane's total displacement (in meters)?
C What is the average velocity for: 190 MM
i Houston to Dallas
ii. Dallas to Huntsville
Answers
(A) The total distance covered by the plane is 500,000 m
(B) The plane's total displacement is 300,000 m
(C) The average velocity of the plane is 100 m/s
The given parameters:
initial displacement of the plane = 400 km Dallas
final displacement of the train = 100 km Huntsville
the time taken for the initial displacement = 120 min
final time for the second trip = 70 min
A sketch of the plane's trip is as follows:
Dallas
400 km ↑ ↓
↑ ↓ Huntsville 100 km
↑
↑
Houston
(A) The total distance covered by the plane:
The total distance is the total path travelled by the plane.
Total distance = 400 km + 100 km = 500 km, = 500,000 m
(B) The plane's total displacement:
The total displacement is the change in the position of the plane.
The displacement = 400 km - 100 km = 300 km = 300,000 m
(C) The average velocity of the plane
[tex]Average \ velocity = \frac{change \ in \ displacement }{change \ in \ time} \\\\Average \ velocity = \frac{400 \ km - 100 \ km}{120 \ \min - \ 70 \ \min} \\\\Average \ velocity =\frac{300 \ km}{2\ hr - 1.167 \ hr} \\\\Average \ velocity = \frac{300 \ km}{0.833 \ hr} \\\\Average \ velocity = 360.14 \ km/hr\\\\Average \ velocity = \frac{360.14 \ km/h}{3.6 \times \frac{km/hr}{m/s} } = 100 \ m/s[/tex]
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What is the relationship between the energy of the system, the energy of the surroundings, and the energy of the universe
Answers
Answer: see below:
The universe experiences the opposite energy change as the surroundings to keep the total energy of the system constant. The energy of the system, surroundings, and universe must be equivalent. The energy of the universe must remain 0, so the energy of the system and surroundings cannot change. I reaaallly hope this helped, luv ya, have a nice day!! <333
A light bulb uses 40.0 W of power when connected to a 110 V (DC) outlet. If the same bulb is connected to a European 220 V outlet, how much power will the bulb use?
Answers
If the bulb's resistance doesn't change, the power it uses is proportional to the square of the voltage. So doubling the voltage quadruples the power.
On 220 V, the bulb uses (4 • 40w)= 160 W.
It glows very brightly.
For a very short time.
Because it also gets very hot, and burns out.
After a few seconds, it uses zero watts.
Answer:
160
Explanation:
Acellus
A positively charged glass rod is bought close to a suspended metal needle. What
can we say about the charge on the needle given that the needle is
a) attracted ?
b) repelled ?
Answers
Answer:
attracted
Explanation:
opposite charges attract each other when the rub against each other
Answer:
This depends because the electrostatic force obeys the principle that states that force between both of the particles does not get affected by the charges of the other thus if the needle is getting attracted it possess negative charges( the opposite charge) .And if they repel it means they have the same charges ( positive charges).
what happens to the value of g when we move from equator to the pole.the weight of an object is greater at poles than at equator .
why?
Answers
Hope you understand
