Direct computation:
Parameterize the top part of the circle [tex]x^2+y^2=1[/tex] by
[tex]\vec r(t)=(x(t),y(t))=(\cos t,\sin t)[/tex]
with [tex]0\le t\le\pi[/tex], and the line segment by
[tex]\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)[/tex]
with [tex]0\le t\le1[/tex]. Then
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)[/tex]
[tex]=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt[/tex]
[tex]=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}[/tex]
Using the fundamental theorem of calculus:
The integral can be written as
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}[/tex]
If there happens to be a scalar function [tex]f[/tex] such that [tex]\vec F=\nabla f[/tex], then [tex]\vec F[/tex] is conservative and the integral is path-independent, so we only need to worry about the value of [tex]f[/tex] at the path's endpoints.
This requires
[tex]\dfrac{\partial f}{\partial x}=\sin x\implies f(x,y)=-\cos x+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=\cos y=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\sin y+C[/tex]
So we have
[tex]f(x,y)=-\cos x+\sin y+C[/tex]
which means [tex]\vec F[/tex] is indeed conservative. By the fundamental theorem, we have
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=f(2,-\pi)-f(1,0)=-\cos2-(-\cos1)=\boxed{\cos1-\cos2}[/tex]
The line integral of (sin x dx + cos y dy) over curve C is calculated using parameterization and direct computation for the circle part and the line segment part, or alternative methods such as polar coordinates or contour integration.
The question involves evaluating the line integral of the function (sin x dx + cos y dy) along a given curve C, which in this case consists of two parts: the upper semicircle of x2 + y2 = 1 and a line segment from (−1, 0) to (2, −π). To evaluate this, we use two methods: direct computation and using Green's theorem when applicable. For the semicircle, we parameterize it by setting x = cos(θ), y = sin(θ), where θ ranges from 0 to π. The line segment can be parameterized by a straight line equation derived from the two points. The integral along the curve is computed separately for each piece, and the results are added together to find the integral over the entire curve. To evaluate the line integral in an alternate method, one might use tricks such as transforming to polar coordinates or employing complex analysis techniques such as contour integration, depending on the nature of the integral.
What is the value of X?
Answer:
x = 55°
Step-by-step explanation:
The sum of the angles in a triangle is 180°, so you have ...
x + 50° +75° = 180°
x = 55° . . . . . . . . . . . . subtract 125°
A roulette wheel has the numbers 1 through 36, 0, and 00. A bet on three numbers pays 11 to 1 (that is, if you bet $1 and one of the three numbers you bet comes up, you get back your $1 plus another $11). How much do you expect to win with a $1 bet on three numbers?
Answer:
You are expected to lose $0.05 (or win -$0.05)
Step-by-step explanation:
Since the roulette wheel has the numbers 1 through 36, 0, and 00, there are 38 possible outcomes.
In this bet, you are allowed to pick 3 out of the 38 numbers. Thus, your chances of winning (P(W)) and losing (P(L)) are:
[tex]P(W)=\frac{3}{38}\\P(L) = 1 - P(W)\\P(L) = \frac{35}{38}\\[/tex]
The expected value of the bet is given by the sum of the product of each outcome pay by its probability. Winning the bet means winning $11 while losing the bet means losing $1. The expected value is:
[tex]EV = (11*\frac{3}{38}) -(1*\frac{35}{38})\\EV = -\$0.0525[/tex]
Therefore, with a $1 bet, you are expected to lose roughly $0.05
Find the work done by F= (x^2+y)i + (y^2+x)j +(ze^z)k over the following path from (4,0,0) to (4,0,4)
a. The line segment x=4 y=0 z=[0;4]
Find a scalar potential function f for F, such that
F=∇f.
The work done by F over the line segment is?
b. The helix r(t)= (4cost)i +(4sint)j +(2t/pi)k t=[0;2pi]
Find df/dt for F?
The Work done by F over the Helix?
c. The x axis from (4,0,0) to (0,0,0) followed by the line z=x, y=0 from (0,0,0) to (4,0,4)
What is the integral to comput the work done by F along the x-axis from followed by the line z=x?
What is the work done by F over the 2 curves
[tex]\vec F(x,y,z)=(x^2+y)\,\vec\imath+(y^2+x)\,\vec\jmath+ze^z\,\vec k[/tex]
We want to find [tex]f(x,y,z)[/tex] such that [tex]\nabla f=\vec F[/tex]. This means
[tex]\dfrac{\partial f}{\partial x}=x^2+y[/tex]
[tex]\dfrac{\partial f}{\partial y}=y^2+x[/tex]
[tex]\dfrac{\partial f}{\partial z}=ze^z[/tex]
Integrating both sides of the latter equation with respect to [tex]z[/tex] tells us
[tex]f(x,y,z)=e^z(z-1)+g(x,y)[/tex]
and differentiating with respect to [tex]x[/tex] gives
[tex]x^2+y=\dfrac{\partial g}{\partial x}[/tex]
Integrating both sides with respect to [tex]x[/tex] gives
[tex]g(x,y)=\dfrac{x^3}3+xy+h(y)[/tex]
Then
[tex]f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+h(y)[/tex]
and differentiating both sides with respect to [tex]y[/tex] gives
[tex]y^2+x=x+\dfrac{\mathrm dh}{\mathrm dy}\implies\dfrac{\mathrm dh}{\mathrm dy}=y^2\implies h(y)=\dfrac{y^3}3+C[/tex]
So the scalar potential function is
[tex]\boxed{f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+\dfrac{y^3}3+C}[/tex]
By the fundamental theorem of calculus, the work done by [tex]\vec F[/tex] along any path depends only on the endpoints of that path. In particular, the work done over the line segment (call it [tex]L[/tex]) in part (a) is
[tex]\displaystyle\int_L\vec F\cdot\mathrm d\vec r=f(4,0,4)-f(4,0,0)=\boxed{1+3e^4}[/tex]
and [tex]\vec F[/tex] does the same amount of work over both of the other paths.
In part (b), I don't know what is meant by "df/dt for F"...
In part (c), you're asked to find the work over the 2 parts (call them [tex]L_1[/tex] and [tex]L_2[/tex]) of the given path. Using the fundamental theorem makes this trivial:
[tex]\displaystyle\int_{L_1}\vec F\cdot\mathrm d\vec r=f(0,0,0)-f(4,0,0)=-\frac{64}3[/tex]
[tex]\displaystyle\int_{L_2}\vec F\cdot\mathrm d\vec r=f(4,0,4)-f(0,0,0)=\frac{67}3+3e^4[/tex]
I need help with this challenge question pls someone help asap
Answer:
Step-by-step explanation:
You are designing a 1000 cm^3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of aluminum used up by the can will therefore beA = 8r^2 + 2pi rhWhat is the ratio now of h to r for the most economical can?
Answer:
h /r = 2.55
Step-by-step explanation:
Area of a can:
Total area of the can = area of (top + bottom) + lateral area
lateral area 2πrh without waste
area of base (considering that you use 2r square) is 4r²
area of bottom ( for same reason ) 4r²
Then Total area = 8r² + 2πrh
Now can volume is 1000 = πr²h h = 1000/πr²
And A(r) = 8r² + 2πr(1000)/πr²
A(r) = 8r² + 2000/r
Taking derivatives both sides
A´(r) = 16 r - 2000/r²
If A´(r) = 0 16 r - 2000/r² = 0
(16r³ - 2000)/ r² = 0 16r³ - 2000 = 0
r³ = 125
r = 5 cm and h = ( 1000)/ πr² h = 1000/ 3.14* 25
h = 12,74 cm
ratio h /r = 12.74/5 h /r = 2.55
To find the ratio of h to r for the most economical can, we need to minimize the amount of aluminum used. The ratio is 0.
Explanation:To find the ratio of h to r for the most economical can, we need to minimize the amount of aluminum used. The total amount of aluminum used is given by the equation A = 8r^2 + 2πrh. To minimize A, we can take the derivative of A with respect to h and set it equal to 0. Solving this equation will give us the value of h in terms of r, and we can then find the ratio of h to r.
Taking the derivative of A with respect to h, we get dA/dh = 2πr. Setting this equal to 0 and solving for h, we find that h = 0. To find the ratio of h to r, we divide both sides of the equation by r, giving us h/r = 0.
Therefore, the ratio of h to r for the most economical can is 0.
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Which of the following are true about the correlation coefficient r? Select one or more:
If the correlation coefficient is +1, then the slope of the regression line is also +1.
The correlation coefficient is always greater than 0.
If the correlation coefficient is close to 0, that means there is a strong linear relationship between the two variables.
The correlation coefficient will change if we change the units of measure.
The correlation coefficient is always between -1 and +1.
If the correlation coefficient is positive, the slope of the regression line will also be positive.
Answer:
Which is the output of the formula =AND(12>6;6>3;3>9)?
A.
TRUE
B.
FALSE
C.
12
D.
9
Step-by-step explanation:
Answer:
A correlation coefficient close to zero
Step-by-step explanation:
This makes linear regression unreasonable because the correlation coefficient shows how related the data points are, -1 and 1 being very strong and 0 being uncorrelated. So the line of best fit would not make sense because the data points are random in relation to one another.
Step-by-step explanation:
A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 418 gram setting. It is believed that the machine is underfilling the bags. A 9 bag sample had a mean of 411 grams with a standard deviation of 20 . A level of significance of 0.025 will be used. Assume the population distribution is approximately normal. Is there sufficient evidence to support the claim that the bags are underfilled?
Answer: There is sufficient evidence to support the claim that the bags are under-filled.
Step-by-step explanation:
Since we have given that
[tex]H_0:\mu=418\\\\H_a:\mu<418[/tex]
Sample mean = 411
Standard deviation = 20
n = 9
So, the test statistic value is given by
[tex]z=\dfrac{\bar{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}\\\\\\z=\dfrac{411-418}{\dfrac{20}{\sqrt{9}}}\\\\\\z=\dfrac{-7}{\dfrac{20}{3}}\\\\\\z=-1.05[/tex]
At 0.025 level of significance,
critical value z = -2.306
since -2.306<-1.05
so, we will reject the null hypothesis.
Yes, there is sufficient evidence to support the claim that the bags are underfilled.
Evaluate each of the following line integrals.
(a)
integral.gif
C
x dy − y dx, c(t) = (cos(t), sin(t)), 0 ≤ t ≤ 2π
(b)
integral.gif
C
x dy + y dx, c(t) = (2 cos(πt), 2 sin(πt)), 2 ≤ t ≤ 4
Answer:
a) 2π
b) 0
Step-by-step explanation:
Recall that for a parametrized differentiable curve C = (x(t), y(t)) with the parameter t varying on some interval [a, b]
[tex]\large \int_{C}[P(x,y)dx+Q(x,y)dy]=\int_{a}^{b}[P(x(t),y(t))x'(t)+Q(x(t),y(t))y'(t)]dt[/tex]
Where P, Q are scalar functions
a)
C(t) = (x(t), y(t)) = (cos(t), sin(t)), 0 ≤ t ≤ 2π
P(x,y) = -y ==> P(x(t),y(t)) = -y(t) = -sin(t)
Q(x,y) = x ==> Q(x(t),y(t)) = x(t) = cos(t)
x'(t) = -sin(t)
y'(t) = cos(t)
[tex]\large \int_{C}[-ydx+xdy]=\int_{0}^{2\pi}[(-sin(t))(-sin(t))+cos(t)cos(t)]dt=\\\\\int_{0}^{2\pi}[sin^2(t)+cos^2(t)]dt=\int_{0}^{2\pi}dt=2\pi[/tex]
b)
C(t) = (x(t), y(t)) = (2cos(πt), 2sin(πt)), 2 ≤ t ≤ 4
P(x,y) = y ==> P(x(t),y(t)) = y(t) = 2sin(πt)
Q(x,y) = x ==> Q(x(t),y(t)) = x(t) = 2cos(πt)
x'(t) = -2πsin(πt)
y'(t) = 2πcos(πt)
[tex]\large \int_{C}[ydx+xdy]=\int_{2}^{4}[(2sin(\pi t))(-2\pi sin(\pi t))+2cos(\pi t)2\pi cos(\pi t)]dt=\\\\4\int_{2}^{4}[cos^2(\pi t)-sin^2(\pi t)]dt=4\int_{2}^{4}cos(2\pi t)dt=\\\\4\left[\frac{sin(2\pi t)}{2\pi}\right]_2^4=\frac{2}{\pi}(sin(8 \pi)-sin(4\pi))=0[/tex]
A survey of 1010 college seniors working towards an undergraduate degree was conducted. Each student was asked, "Are you planning or not planning to pursue a graduate degree?" Of the 1010 surveyed, 658 stated that they were planning to pursue a graduate degree. Construct and interpret a 98% confidence interval for the proportion of college seniors who are planning to pursue a graduate degree. Round to the nearest thousandth.
Answer:
The 98% confidence interval would be given (0.616;0.686).
We are confident at 98% that the true proportion of people that they were planning to pursue a graduate degree is between (0.616;0.686).
Step-by-step explanation:
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Description in words of the parameter p
[tex]p[/tex] represent the real population proportion of people that they were planning to pursue a graduate degree
[tex]\hat p[/tex] represent the estimated proportion of people that they were planning to pursue a graduate degree
n=1010 is the sample size required
[tex]z_{\alpha/2}[/tex] represent the critical value for the margin of error
The population proportion have the following distribution
[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]
Numerical estimate for p
In order to estimate a proportion we use this formula:
[tex]\hat p =\frac{X}{n}[/tex] where X represent the number of people with a characteristic and n the total sample size selected.
[tex]\hat p=\frac{658}{1010}=0.651[/tex] represent the estimated proportion of people that they were planning to pursue a graduate degree
Confidence interval
The confidence interval for a proportion is given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 98% confidence interval the value of [tex]\alpha=1-0.98=0.02[/tex] and [tex]\alpha/2=0.01[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=2.33[/tex]
And replacing into the confidence interval formula we got:
[tex]0.651 - 2.33 \sqrt{\frac{0.651(1-0.651)}{1010}}=0.616[/tex]
[tex]0.651 + 2.33 \sqrt{\frac{0.651(1-0.651)}{1010}}=0.686[/tex]
And the 98% confidence interval would be given (0.616;0.686).
We are confident at 98% that the true proportion of people that they were planning to pursue a graduate degree is between (0.616;0.686).
A sample of 30 distance scores measured in yards has a mean of 7, a variance of 16, and a standard deviation of 4. You want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. What are the new mean, median, variance, and standard deviation?
Answer:
21, 144, 12
Step-by-step explanation:
Given that a sample of 30 distance scores measured in yards has a mean of 7, a variance of 16, and a standard deviation of 4.
Let X be the distance in yard.
i.e. each entry of x is multiplied by 3.
New mean variance std devition would be
E(3x) = [tex]3E(x) = 21[/tex]
Var (3x) = [tex]3^2 Var(x) = 9(16) =144[/tex]
Std dev (3x) = [tex]\sqrt{144 } =12[/tex]
Thus we find mean and std devition get multiplied by 3, variance is multiplied by 9
The new mean after converting the distances from yards to feet is 21, the median remains unchanged at 7, the new variance is 144, and the new standard deviation is 12.
To find the new mean, we multiply the original mean by the conversion factor. Since there are 3 feet in a yard, we have:
New Mean = Original Mean × Conversion Factor
= 7 yards × 3 feet/yard
= 21 feet
The median is a measure of central tendency that represents the middle value of a data set. When each data point is multiplied by a constant factor, the median is also multiplied by that factor. However, since the median is the middle value of an ordered list of data, and we are not changing the order or adding or removing any data points, the median in terms of yards and feet is the same numerical value, even though the units are different:
New Median = [tex]Original Median Ã[/tex]— Conversion Factor
= 7 yards (since the median is the same numerical value)
For the variance, when each data point is multiplied by a constant, the variance is multiplied by the square of that constant:
New Variance = [tex]Original Variance × (Conversion Factor)^2[/tex]
= [tex]16 yards^2 × (3 feet/yard)^2[/tex]
= [tex]16 yards^2 × 9 (feet^2/yard^2)[/tex]
= [tex]144 feet^2[/tex]
Finally, the standard deviation is the square root of the variance, so to find the new standard deviation, we take the square root of the new variance:
New Standard Deviation = [tex]√New Variance[/tex]
= [tex]√144 feet^2[/tex]
= 12 feet
Two teams play a series of games, the first team to win 4 games is the winner overall. Suppose that one of the teams is stronger and has probability 0.6 to win each game, independent of any other games. What is the probability that the stronger team wins the series in exactly i games. Do it for i = 4,5,6,7. Compare the probability that the stronger team wins with the probability that it would win a 2 out of 3 series.
Answer:
Step-by-step explanation:
Figuring it out rn
the height of woman ages 20-29 is normally distributed , with a mean of 64.4 inches. assuming the standard diviation = 2.3 inches. are you more likely to randomly select 1 woman with a height less than 64.9 inches or are you more likely to select a sample of 22 woman with a mean height less than 64.9 inches.(USING STANDARD NORMAL TABLE)A. what is the probability of randomly selecting 1 woman with a height less than 64.9 inches?B. what is the probability of selecting a sample of 22 woman with a mean height less then 64.9 inches?C. are you more likely to randomly select 1 woman with a height less than 64.9 inches or are you more likely to select a sample of 22 woman with a mean height of 64.9 inches?
We use the Z-score to determine the probability of selecting a single woman with a height less than 64.9 inches, and the concept of a sampling distribution (standard error) for a sample of 22 women. Generally, the larger the sample size, the closer the sample mean gets to the population mean, making it usually more probable to select one woman with a height of less than 64.9 inches than for a sample of 22 women.
Explanation:This question is about probability and statistics, and specifically about the Normal Distribution and its applications. We know that the height of women aged 20-29 is normally distributed with a mean of 64.4 inches and a standard deviation of 2.3 inches.
A. To find the probability of randomly selecting one woman with a height less than 64.9 inches, we need to convert 64.9 into a z-score using the formula Z = (X - μ) / σ. Here, X represents the given height (64.9 inches), μ is the mean (64.4 inches), and σ is the standard deviation (2.3 inches). The resulting Z score is then looked up in the standard normal table to determine the probability.B. The likelihood of selecting a sample of 22 women with a mean height less than 64.9 inches would involve using the concept of a sampling distribution. Here, the standard deviation is divided by the square root of the sample size (σ/√n) to calculate the standard error. This would then be used in a similar fashion as in part A, by calculating a z-score to find the probability.C. We can't definitively answer this without exact values but generally, the larger the sample size the closer the sample mean gets to the population mean due to the law of large numbers. Therefore, it would usually be more likely to select one woman with a height less than 64.9 inches than to select a sample of 22 women with a mean height less than 64.9 inches.Learn more about Normal Distribution / Probability here:https://brainly.com/question/30653447
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Calculate probabilities for selecting 1 woman and a sample of 22 women with heights less than 64.9 inches; Compare which selection is more likely.
Given:
Mean (μ) = 64.4 inches
Standard deviation (σ) = 2.3 inches
Probability for 1 Woman:Calculate the z-score:
z = (x - μ) / σ
= (64.9 - 64.4) / 2.3
= 0.5 / 2.3
= 0.2174
Now, we look up the probability corresponding to a z-score of 0.2174 in the standard normal table.
The probability for z = 0.2174 is approximately 0.5869.
So, the probability of randomly selecting 1 woman with a height less than 64.9 inches is approximately 0.5869.
Probability for 22 Women:Calculate standard error:
SE = σ / √n
= 2.3 / √(22)
≈ 0.489
Find the z-score for the sample mean:
z = ([tex]^-_x[/tex] - μ) / σ
= (64.9 - 64.4) / 0.489
= 0.5 / 0.489
≈ 1.021
Now, we look up the probability corresponding to a z-score of 1.021 in the standard normal table.
The probability for z = 1.021 is approximately 0.8451.
So, the probability of selecting a sample of 22 women with a mean height of less than 64.9 inches is approximately 0.8451.
Comparing the probabilities:→ Probability of randomly selecting 1 woman with a height less than 64.9 inches: 0.5869
→ Probability of selecting a sample of 22 women with a mean height less than 64.9 inches: 0.8451
Since 0.8451 is greater than 0.5869, it is more likely to select a sample of 22 women with a mean height of less than 64.9 inches.
Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up. Construct a 95% confidence interval for the population proportion who claim they always buckle up. What is the error bound?
Answer:
The 95% confidence interval would be given (0.761;0.839). The error bound is [tex]Me=\pm 0.0392[/tex]
Step-by-step explanation:
1) Data given and notation
n=400 represent the random sample taken
X=320 represent the people drivers claimed they always buckle up
[tex]\hat p=\frac{320}{400}=0.8[/tex] estimated proportion of people drivers claimed they always buckle up
[tex]\alpha=0.05[/tex] represent the significance level (no given, but is assumed)
Confidence =95% or 0.95
p= population proportion of people drivers claimed they always buckle up
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
2) Calculating the interval for the proportion
The confidence interval would be given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.96[/tex]
And replacing into the confidence interval formula we got:
[tex]0.8 - 1.96 \sqrt{\frac{0.8(1-0.8)}{400}}=0.761[/tex]
[tex]0.8 + 1.96 \sqrt{\frac{0.8(1-0.8)}{400}}=0.839[/tex]
And the 95% confidence interval would be given (0.761;0.839). The error bound is [tex]Me=\pm 0.0392[/tex]
We are confident that about 76.1% to 83.9% of people drivers that they always buckle up at 95% of confidence
The time in seconds that it takes for a sled to slide down a hillside inclined at an angle θ is given by the formula below, where d is the length of the slope in feet. Find the time it takes to slide down a 2000 ft slope inclined at 30°. (Round your answer to one decimal place.) t = d 16 sin θ
Answer: It takes 15.8 seconds to slide down a 2000 ft slope.
Step-by-step explanation:
Since we have given that
[tex]t=\sqrt{\dfrac{d}{16\sin\theta}}[/tex]
where, t is the time taken,
d is the length to slide down a slope
sin θ is the angle at which it inclined.
So, we have d = 2000 ft
θ = 30°
So, the time taken is given by
[tex]t=\sqrt{\dfrac{2000}{16\sin 30^\circ}}\\\\t=\sqrt{\dfrac{2000}{8}}\\\\t=\sqrt{250}\\\\t=15.81\ seconds[/tex]
Hence, it takes 15.8 seconds to slide down a 2000 ft slope.
The time it takes to slide down a 2000 ft slope inclined at 30° is 250 seconds, calculated using the equation t = d / (16 sin θ).
Explanation:We're given the equation for calculating time of a sled sliding down a slope as t = d / (16 sin θ). Here, d is the length of the slope, and θ is the incline of the hill slope.
To calculate the time it takes to slide down a 2000 ft slope inclined at 30°, we substitute the respective values into the formula: t = 2000 / (16 sin 30). The sin value for 30° is 0.5, so the formula simplifies to t = 2000 / (16 * 0.5), which yields t = 2000 / 8. By performing that division, we see that t = 250 seconds. Therefore, it takes 250 seconds to slide down a 2000 ft slope inclined at 30°.
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Pedro thinks that he has a special relationship with the number 3. In particular, Pedro thinks that he would roll a 3 with a fair 6-sided die more often than you'd expect by chance alone. Suppose p is the true proportion of the time Pedro will roll a 3.
(a) State the null and alternative hypotheses for testing Pedro's claim. (Type the symbol "p" for the population proportion, whichever symbols you need of "<", ">", "=", "not =" and express any values as a fraction e.g. p = 1/3)
H0 = _______
Ha = _______
(b) Now suppose Pedro makes n = 30 rolls, and a 3 comes up 6 times out of the 30 rolls. Determine the P-value of the test:
P-value =________
Answer:
p value = 0.3122
Step-by-step explanation:
Given that Pedro thinks that he has a special relationship with the number 3. In particular,
Normally for a die to show 3, probability p = [tex]\frac{1}{6} =0.1667[/tex]
Or proportion p = 0.1667
Pedro claims that this probability is more than 0.1667
[tex]H_0 = P =0.1667______H_a = P>0.1667_____[/tex]
where P is the sample proportion.
b) n=30 and [tex]P = 6/30 =0.20[/tex]
Mean difference = [tex]0.2-0.1667=0.0333[/tex]
Std error for proportion = [tex]\sqrt{\frac{p(1-p)}{n} } \\=0.0681[/tex]
Test statistic Z = p difference/std error = 0.4894
p value = 0.3122
p >0.05
So not significant difference between the two proportions.
The null hypothesis is that Pedro's claim is not true, while the alternative hypothesis is that Pedro's claim is true. The P-value is approximately 0.082.
Explanation:
(a) The null hypothesis states that Pedro's claim is not true, so the null hypothesis is H0: p = 1/6. The alternative hypothesis states that Pedro's claim is true and he rolls a 3 more often than expected, so the alternative hypothesis is Ha: p > 1/6.
(b) To determine the P-value, we can use the binomial distribution. The probability of rolling a 3 with a fair 6-sided die is 1/6. Using the binomial distribution, we can calculate the probability of rolling a 3 six times out of 30 rolls. This gives a P-value of approximately 0.082, which is the probability of observing a result as extreme as the one obtained or more extreme, assuming the null hypothesis is true.
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Express the null hypothesis and the alternative hypothesis in symbolic form. The owner of a football team claims that the average attendance at games is over 63,500 , and he is therefore justified in moving the team to a city with a larger stadium.
Answer:
[tex]H_{0}: \mu \leq 63500\\H_A: \mu > 63500[/tex]
Step-by-step explanation:
We are given the following in the question:
The owner of a football team claims that the average attendance at games is over 63,500.
He wants to justify that the team needs to be moved to a larger stadium outside the city.
If the attendance is larger than 63,500 the team would be moved to a larger stadium and if it is less than or equal to 63,500 that it would not.
Thus, the null and alternate hypothesis will be designed as:
[tex]H_{0}: \mu \leq 63500\\H_A: \mu > 63500[/tex]
The null hypothesis says that the average attenders is equal to or less than 63,500 and alternate supports the claim that the attenders average is greater than 63,500.
Write an equation of the line passing through the point (3, -1) and parallel to the line y=2/3x - 5. Show work
Answer:
The answer is: y = 2/3x - 3
Step-by-step explanation:
Given point: (3, -1)
Given equation: y = 2/3x - 5, which is in the form y = mx + b where m is the slope and b is the y intercept.
Parallel lines have the same slope. Use the point slope form of the equation with the point (3, -1) and substitute:
y - y1 = m(x - x1)
y - (-1) = 2/3(x - 3)
y + 1 = 2/3x - 6/3
y + 1 = 2/3x - 2
y = 2/3x - 3
Proof:
f(3) = 2/3(3) - 3
= 6/3 - 3
= 2 - 3
= -1, giving the point (3, -1)
Hope this helps! Have an Awesome Day!! :-)
You were given the following joint probability function for
Y1 = { 0, if the child survived, 1, if not,
and
Y2 = { 0 if no belt used, 1 if adult belt used, and 2 if car seat belt used
Notice that Y1 is the number of fatalities per child and, since children's car seats usually utilize two belts, Y2 is the number of seat belts in use at the time of accident.
Given:
Y1
y2 0 1 total
0 0.38 0.17 0.55
1 0.14 0.02 0.16
2 0.24 0.05 0.29
Total 0.76 0.24 1
Are Y1 and Y2 independent? Why or why not?
Answer: No, Y1 and Y2 are not independent
Step-by-step explanation:
Because they don't satisfy this condition:
FsubscriptY1Y2(Y1,Y2) = FsubscriptY1(y1) × FsubscriptY2(y2)
... for all given values of Y1 and Y2
This is the condition for independence.
How do we know that Y1 and Y2 don't satisfy this condition?
We use the information in the Joint Probability Distribution Table.
Let's see if the condition stands when Y1 is zero and Y2 is zero
FsubscriptY1Y2(0,0) = 0.38
FsubscriptY1(0) × FsubscriptY2(0) = 0.76×0.55 = 0.418
We can see that 0.38 is not equal to 0.418
Doing the test for any other combination of Y1 and Y2 values will give unequal figures as well.
The probability that an individual without a college education earns more than $100,000 is 0.4, whereas the probability that a person with a B.S. or higher degree earns more than $100,000 is 0.6. The probability that a person chosen at random has a B.S. degree is 0.5. What is the probability that a person has at least a B.S. degree if it is known that he or she earns more than $100,000?
Answer:
0.6
Step-by-step explanation:
Given:
P( Person chosen at random has a B.S. degree), P(C) = 0.5
P( Person chosen at random does not have a B.S. degree), P(C') = 1 - 0.5 = 0.5
P(Student earns more than $100,000) = P(E)
P(Student earns more than $100,000, without going college) = P(E | C') = 0.4
P(Student earns more than $100,000, with college degree) = P(E | C) = 0.6
Now,
P(at least a B.S. degree | earns more than $100,000), P(C | E)
using Baye's theorem
we have
P(C | E) = [tex]\frac{P(C)\timesP(E | C)}{P(C)\timesP(E | C)+P(C')\timesP(E | C')}[/tex]
or
P(C | E) = [tex]\frac{0.5\times0.6}{0.5\times0.6+0.5\times0.4}[/tex]
or
P(C | E) = [tex]\frac{0.3}{0.5}[/tex]
or
P(C | E) = 0.6
To find the probability that a person has at least a B.S. degree if it is known that he or she earns more than $100,000, we can use Bayes' Theorem and the given probabilities.
Explanation:To find the probability that a person has at least a B.S. degree if it is known that he or she earns more than $100,000, we can use Bayes' Theorem. Let A be the event that the person has at least a B.S. degree, and let B be the event that the person earns more than $100,000. The probability of A given B can be calculated as:
P(A|B) = (P(B|A) * P(A)) / P(B)
Given that P(B|A) = 0.6, P(A) = 0.5, and P(B) = 0.4, we can substitute the values into the formula:
P(A|B) = (0.6 * 0.5) / 0.4 = 0.75
Therefore, the probability that a person has at least a B.S. degree if it is known that he or she earns more than $100,000 is 0.75.
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Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
Answer:
92
Step-by-step explanation:
She is taking four tests. She wants to average a 95 on those tests. Therefore, she must score 95*4=380 total. She scored 97 and 91, or 188 total. Therefore, she must score 380-188 = 192 total on her next two tests. Her maximum score possible on the fourth test is 100. Therefore, the lowest score possible for the third test would be 192-100=92
The lowest possible score Isabella could have made on the third test
is 92 marks.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, Isabella must take four 100-point tests in her math class and her goal is to achieve an average grade of 95 on the tests.
So, The total marks she needs to score in four games is (4×95).
= 380.
Now, She scores 97 and 91 in her first and second tests.
So, She needs to score
380 - (97 + 91)
= 380 - 188.
= 192 in her two tests.
Now, The highest marks she can score in the fourth test is 100.
Therefore, The lowest possible marks she can score is,
= 192 - 100.
= 92.
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The manager of a grocery store has taken a random sample of 100 customers. The average length of time it took the customers in the sample to check out was 3.1 minutes. The population standard deviation is known to be 0.5 minutes. We want to see if we can find significant evidence to prove that the mean waiting time of all customers is significantly more than 3 minutes. The test statistic is 2. What is the p-value? Round your answer to three decimal places.
Answer: The critical value is: insufficient.
Step-by-step explanation: Refer to Exhibit 9-2. The p-value is:
a. .0500
b. .0228
c. .0456
d. .0250
In order to test the hypotheses H0: μ ≤ 100 and Ha: μ > 100 at an α level of significance, the null hypothesis will be rejected if the test statistic z is:
The practice of concluding "do not reject H0" is preferred over "accept H0" when we:
a. have an insufficient sample size.
b. have not controlled for the Type II error.
c. are testing the validity of a claim.
d. are conducting a one-tailed test.
In hypothesis testing, the critical value is: insufficient
The populations, P, of six towns with time t in years are given by
1) P=2400(0.8)^t
2) P=900(0.77)^t
3)P=2100(0.98)^t
4)P=600(1.18)^t
5)P=1100(1.08)^t
6)P=1700(1.191)^t
Answer the following questions regarding the populations of the six towns above. Whenever you need to enter several towns in one answer, enter your answer as a comma separated list of numbers. For example if town 1, town 2, town 3, and town 4, are all growing you could enter 1, 2, 3, 4 ; or 2, 4, 1, 3 ; or any other order of these four numerals separated by commas.
(a) Which of the towns are growing?
(b) Which of the towns are shrinking?
(c) Which town is growing the fastest?
What is the annual percentage growth RATE of that town? %
(d) Which town is shrinking the fastest?
What is the annual percentage decay RATE of that town? %
(e) Which town has the largest initial population?
(f) Which town has the smallest initial population?
Answer:
Since, in the population function,
[tex]P = ab^t[/tex]
a = initial population,
b = population change factor,
If 0 < b < 1, then population will shrink,
While, if b > 1, then the population will grow,
(a) Since, 1.18, 1.08 and 1.191 is greater than 1,
Thus, town 4), 5) and 6) are growing.
(b) Since, 0.8, 0.77 and 0.98 are less than 1,
Thus, town 1), 2) and 3) are shrinking.
(c) An exponential growth function with highest change factor grows fastest.
∵ 1.191 > 1.18 > 1.08
⇒ town 6) is growing fastest.
(d) An exponential decay function with lowest change factor shrinks fastest,
∵ 0.77 < 0.8 < 0.98 < 1.08 < 1.18 < 1.191,
⇒ Town 2) shrinks fastest.
(e) Since,
2400 > 2100 > 1700 > 1100 > 900 > 600
⇒ town 1) has the largest initial population.
(f) Similarly,
Town 4) has the smallest initial population.
Answer:
a.
i, ii, and iv
Step-by-step explanation:
The physical plant at the main campus of a large state university recieves daily requests to replace florecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 55 and a standard deviation of 4. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 43 and 55?
Answer:
50%
Step-by-step explanation:
68-95-99.7 rule
68% of all values lie within the 1 standard deviation from mean [tex](\mu-\sigma,\mu+\sigma)[/tex]
95% of all values lie within the 1 standard deviation from mean [tex](\mu-1\sigma,\mu+1\sigma)[/tex]
99.7% of all values lie within the 1 standard deviation from mean [tex](\mu-3\sigma,\mu+3\sigma)[/tex]
The distribution of the number of daily requests is bell-shaped and has a mean of 55 and a standard deviation of 4.
[tex]\mu = 55\\\sigma = 4[/tex]
68% of all values lie within the 1 standard deviation from mean [tex](\mu-\sigma,\mu+\sigma)[/tex] = [tex](55-4,55+4)[/tex]= [tex](51,59)[/tex]
95% of all values lie within the 2 standard deviation from mean [tex](\mu-1\sigma,\mu+1\sigma)[/tex]= [tex](55-2(4),55+2(4))[/tex]= [tex](47,63)[/tex]
99.7% of all values lie within the 3 standard deviation from mean [tex](\mu-3\sigma,\mu+3\sigma)[/tex]= [tex](55-3(4),55+3(4))[/tex]= [tex](43,67)[/tex]
Refer the attached figure
P(43<x<55)=2.5%+13.5%+34%=50%
Hence The approximate percentage of light bulb replacement requests numbering between 43 and 55 is 50%
obesity: The National Center for Health Statistics conducted the National Health Interview Survey (NHIS) for 27,787 U.S. civilian noninstitutionalized adults in January – September 2014. According to an early release report, an estimated 29.9% of U.S. adults aged 20 and over were obese. Obesity is defined as a body mass index (BMI) of 30 kg/m 2 or more. True or false? The 29.9% is a parameter representing a population of 27,787 adults.
Answer:
False.
Step-by-step explanation:
Hello!
29.9% is the proportion of adults aged 20 and over that were obese in January - September 2014.
This percentage was obtained from a sample of 27 787 adults, it's not a parameter but an estimation of the population proportion obtained from this sample.
I hope it helps!
False, the 29.9% does not represent a parameter for the specific population of 27,787 adults surveyed in the National Health Interview Survey (NHIS).
The 29.9% figure representing U.S. adults aged 20 and over who were estimated to be obese based on the NHIS data is a statistic, not a parameter of the surveyed 27,787 adults, as it aims to generalize the obesity rate to the broader U.S. adult population.
Instead, this percentage is a statistic representing an estimate of the entire U.S. adult population aged 20 and over, based on the sample surveyed.
In statistics, a parameter refers to a numerical measure that describes a characteristic of a population, whereas a statistic is a numerical measure that describes a characteristic of a sample from the population.
Therefore, since the 29.9% figure estimated from the NHIS sample aims to generalize to the broader U.S. adult population, it is classified as a statistic, and not a parameter of the 27,787 adults directly surveyed.
g A university computer breaks down on average 2.1 times a month. Find the probability that during the next month this computer will break down at least six times. Use the Poisson probability formula. (Round to 4 digits, ex. 0.1234)
Answer:
Step-by-step explanation:
A regression and correlation analysis resulted in the following information regarding a dependent variable ( y) and an independent variable ( x). Σx = 90 Σ(y - )(x - ) = 466 Σy = 170 Σ(x - )2 = 234 n = 10 Σ(y - )2 = 1434 SSE = 505.98 The sum of squares due to regression (SSR) is
Answer:
The sum of squares due to regression(SSR)=928.02
Step-by-step explanation:
We are given that
Dependent variable=y
Independent variable=x
[tex]\sum x=90[/tex]
[tex]\sum(y-\bar y)(x-\bar x)=466[/tex]
[tex]\sum y=170[/tex]
[tex]\sum(x-\bar x)^2=234[/tex]
n=10
[tex]\sum(y-\bar y)^2=1434[/tex]
SSE=505.98
We have to find the sum of squares due to regression.
It means we have to find SSR.
SST=[tex]\sum(y-\bar y)^2=1434[/tex]
[tex]SSR=SST-SSE=1434-505.98=928.02[/tex]
Hence, the sum of squares due to regression(SSR)=928.02
The government of Preon (a small island nation) was voted in at the last election with 68% of the votes. That was 2 years ago, and ever since then the government has assumed that their approval rating has been the same. Some recent events have affected public opinion and the government suspects that their approval rating might have changed. They decide to run a hypothesis test for the proportion of people who would still vote for them.The null and alternative hypotheses are:H0: Pi symbol = 0.68HA: Pi symbol ≠ 0.68The level of significance used in the test is α = 0.1. A random sample of 102 people are asked whether or not they would still vote for the government. The proportion of people that would is equal to 0.745. You may find this standard normal table useful throughout this question.Calculate the test statistic (z) for this hypothesis test.
Answer: 1.41
Step-by-step explanation:
Test statistic(z) for proportion is given by :-
[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
, where p=population proportion.
[tex]\hat{p}[/tex]= sample proportion
n= sample size.
As per given , we have
[tex]H_0:\mu=0.68\\\\ H_a: \mu\neq0.68[/tex]
n= 102
[tex]\hat{p}=0.745[/tex]
Then, the test statistic (z) for this hypothesis test will be :-
[tex]z=\dfrac{0.745-0.68}{\sqrt{\dfrac{0.68(1-0.68)}{102}}}\\\\=\dfrac{0.065}{\sqrt{\dfrac{0.2176}{102}}}\\\\=\dfrac{0.065}{\sqrt{0.0021333}}\\\\=\dfrac{0.065}{0.04618802}=1.40729132792\approx1.41[/tex]
[Rounded to the two decimal places]
Hence, the test statistic (z) for this hypothesis test = 1.41
What does the central nervous system use to determine the strength of a stimulus?
- origin of the stimulus
- frequency of action potentials
- size of action potentials
- type of stimulus receptor
Answer:
- frequency of action potentials
The central nervous system determines the strength of a stimulus based on the frequency of action potentials. The intensity of the stimulus is interpreted not by the size, but by the frequency at which these potentials are produced.
Explanation:The central nervous system determines the strength of a stimulus based on the frequency of action potentials. The central nervous system uses action potentials to transfer and process information. An action potential is a brief electrical charge that travels along an axon. The strength or intensity of the stimulus is interpreted not by the size of the action potentials, but by the frequency at which they are produced. For instance, a slightly warm temperature may produce action potentials at a lower frequency than a very hot temperature, which would produce action potentials at a high frequency. The central nervous system decodes this frequency to understand the intensity of the stimulus.
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How would you graph -x plus 2 thx
A researcher would like to estimate p, the proportion of U.S. adults who support recognizing civil unions between gay or lesbian couples. If the researcher would like to be 95% sure that the obtained sample proportion would be within 1.5% of p (the proportion in the entire population of U.S. adults), what sample size should be used?
(a) 17,778(b) 4,445(c) 1,112(d) 67(e) 45
Answer:
(b) 4,445
Step-by-step explanation:
If the researcher would like to be 95% sure that the obtained sample proportion would be within 1.5% of p (the proportion in the entire population of U.S. adults), what sample size should be used?
Given a=0.05, |Z(0.025)|=1.96 (check standard normal table)
So n=(Z/E)^2*p*(1-p)
=(1.96/0.015)^2*0.5*0.5
=4268.444
Take n=4269
Answer:(b) 4,445