The probability density function (PDF) of a chi-squared distribution is used to describe the probability of a certain outcome occurring when a continuous random variable is sampled. The PDF is a mathematical formula that gives the probability of a given value of the random variable, x, occurring. It is typically expressed as f(x).
The chi-squared distribution is a type of continuous probability distribution that is used to describe the sum of the squares of multiple normally distributed random variables. Its PDF is represented as:
f(x) = 1/2k/2π-1/2 * xk/2-1 * e-x/2
Where x is the value of the chi-squared random variable and k is the number of degrees of freedom.
The PDF of the chi-squared distribution is used to calculate the probability of a particular outcome. It is used in hypothesis testing to determine the probability that a given set of data fits a certain distribution. It is also used to estimate the confidence intervals of parameters in various models.
The chi-squared distribution is widely used in statistical applications, such as testing for goodness of fit, tests for independence, tests for homogeneity, and tests for normality. It is also used in the analysis of variance, analysis of covariance, regression analysis, and other advanced statistical techniques.
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Please help this is my last question on my unit test and I do not know it
Find and explain the error in the student’s work below.
Solve 2x² - 5x -12 = 0 using the Quadratic Formula.
The roots of the given quadratic equation are x= 4, [tex]\frac{-3}{2}[/tex]
The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. It also goes by the name quadratic equations. The quadratic equation has the following generic form:
ax² + bx + c = 0The roots of a quadratic equation are found using the quadratic formula. In place of the factorization method, this formula aids in evaluating the quadratic equations' solutions. The quadratic formula aids in identifying the problem's fictitious roots when a quadratic equation lacks actual roots. Shreedhara Acharya's formula is another name for the quadratic formula.
2x² - 5x -12 = 0
The Shridharacharya formula or quadratic formula -
[tex]x=\frac{-b\ +-\sqrt{b^2-4ac}}{2a}[/tex]
we have a=2, b=-5 and c=-12
[tex]x=\frac{5+-\sqrt{5^2-4*2*12}}{2*2}\\\\x=\frac{5+-\sqrt{25+96}}{4}\\\\x=\frac{5+\sqrt{121}}{4}\\\\x=\frac{5+-11}{4}\\now, \\x=\frac{5+11}{4}=\frac{16}{4}\\x=4\\x=\frac{5-11}{4}\\x=-6/4\\x=-3/2[/tex]
The mistake in your solution is we have the value of b=-5 so , when we put the value of b in a formula then -5 will be 5.
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in a survey, 69% of americans said they own an answering machine. if 15 americans are selected at random, find the probability that exactly 7 own an answering machine. round your answer to three decimal places.
Rounding to three decimal places, the probability that exactly 7 Americans out of a sample of 15 own an answering machine is approximately 0.170.
To find the probability that exactly 7 of the 15 Americans selected at random own an answering machine, we can use the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where P(X = k) is the probability of getting exactly k successes, n is the sample size, p is the probability of success, and (n choose k) is the binomial coefficient which represents the number of ways to choose k successes out of n trials.
In this case, n = 15, p = 0.69 (the proportion of Americans who own an answering machine), and k = 7. Thus, the probability of getting exactly 7 Americans who own an answering machine out of a sample of 15 is:
P(X = 7) = (15 choose 7) * 0.69^7 * (1 - 0.69)^(15 - 7)
Using a calculator or statistical software, we can evaluate this expression to find:
P(X = 7) ≈ 0.170
This means that if we were to repeat this survey many times with samples of size 15, we would expect approximately 17% of the samples to have exactly 7 Americans who own an answering machine.
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solve the following polynomial inequality: x^2-5x-36>0
include all steps and place the answer in interval notation form.
please explain the steps and how you got the answer
Answer:
To solve the inequality x^2-5x-36>0, we need to find the values of x for which the expression is greater than zero.
One way to do this is by factoring the quadratic expression:
x^2-5x-36 = (x-9)(x+4)
The expression is positive when either both factors are positive or both factors are negative.
When both factors are positive: x-9 > 0 and x+4 > 0
Solving for x, we get x > 9 and x > -4. Therefore, x > 9.
When both factors are negative: x-9 < 0 and x+4 < 0
Solving for x, we get x < 9 and x < -4. Therefore, x < -4.
Now, we have two intervals: x < -4 and x > 9. To check whether the expression is positive within these intervals, we can pick a value within each interval and plug it into the expression.
Let's choose x = -5 (within x < -4) and x = 10 (within x > 9).
For x = -5:
x^2-5x-36 = (-5)^2-5(-5)-36
= 25+25-36
= 14
Since 14 is greater than zero, the expression is positive when x = -5.
For x = 10:
x^2-5x-36 = 10^2-5(10)-36
= 100-50-36
= 14
Since 14 is greater than zero, the expression is also positive when x = 10.
Therefore, the solution to the inequality x^2-5x-36>0 is x < -4 or x > 9, which can be written in interval notation as (-∞,-4) ∪ (9,∞).
Which of the following equations could be the function pictured in the graph?
A. y= (x-1)(x+3)
B. y= (x+1)(x-3)
C. y= (x+1)(x+3)
D. y= (x+1)(x-1)
Answer:
B. y = (x+1)(x-3)
Step-by-step explanation:
Bbecause when y intersect the x axis, y = 0
so (x+1)(x-3) = 0, which we get x = -1 and x = 3
As shown in the graph this is true, because the function does indeed also intersect with the x axis at x = -1 and x = 3 as well.
What is the value of x given the following image?
The numerical value of x in the expression (x+5) and 2(x-4) is 61.
What is the numerical value of x?The sum of angles on a straight line equals 180 degrees.
From the figure in the image:
Angle GDC = x + 5Supplemetary angle to GDC = 2( x - 4 )We know that the sum of angles on a straight line equals 180 degrees.
Since angle GDC and its supplement are on a straight line.
Hence;
Angle GDC + Supplemetary angle to GDC = 180
Plug in the given values and solve for x.
( x + 5 ) + 2( x - 4 ) = 180
Apply distributive property.
x + 5 + 2x - 8 = 180
Collect like terms
x + 2x - 8 + 5 = 180
3x - 3 = 180
3x = 180 + 3
3x = 183
x = 183/3
x = 61
Therefore, the value of x is 61.
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pleaseee help super confused, and if you can please explain it i don’t understand
Answer:
China's top ten famous flowers are: the queen of flowers - plum blossom, the king of flowers - peony, frost blooming - chrysanthemum, gentleman's flower - orchid, the queen of flowers - rose, blooming flowers - Rhododendron, delicate flowers - camellia, water hibiscus - lotus, ten miles of fragrance - osmanthus, Lingbo fairy - daffodil 10 kinds of precious and beautiful local flowers.
Plum blossom is known as the "top ten famous flowers in China". These ten kinds of flowers respectively contain different levels of Chinese spiritual and cultural deposits, with profound and strong historical connotation. They are unique in the flower industry, marking the extraordinary significance of Chinese traditional culture. However, among China's top 10 traditional famous flowers, only the plum blossom, osmanthus and lotus have international registration rights.
someone please help meee!! ASAPPP
Hi so you have to do grid for the equation
Compare the amount of sand in the top cone of the hourglass to the amount there will be when the height of the sand in the top cone is only 1 inch.
HINT: The cones are similar
the amount of sand in the top cone when the height of the sand is only 1 inch is (h-1)/h times the amount of sand in the top cone originally.
the cones are similar, their volumes are proportional to the cube of their heights. Let's denote the height of the top cone as h, and the radius of the top and bottom bases as r. Then, the volume of the top cone can be expressed as:
V₁ = (1/3)π[tex]r^2[/tex]h
If the height of the sand in the top cone is reduced to 1 inch, then the height of the remaining sand in the top cone is (h-1) inches. The volume of the remaining sand in the top cone can be expressed as:
V₂ = (1/3)π[tex]r^2[/tex](h-1)
To compare the amount of sand in the top cone in these two scenarios, we can take the ratio of their volumes:
V₂/V₁ = [(1/3)π[tex]r^2[/tex](h-1)] / [(1/3)π[tex]r^2[/tex]h] = (h-1)/h
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if customer interarrival times are exponentially distributed with rate 6 customers per hour, then to simulate the minutes between customer arrives in excel, one would use: group of answer choices
To simulate the minutes between customer arrivals in Excel, one would use the Exponential distribution function in Excel.
This function is part of the Statistical functions category and can be found under the "Statistical" tab in Excel.
An exponential distribution is a probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.
The Exponential distribution function in Excel takes two arguments: Lambda and x.
Lambda is the rate parameter, which describes the average number of events per unit of time.
In this case, the rate is 6 customers per hour, so Lambda would be equal to 6.
X is the time interval, and in this case, it would be the time between customer arrivals in minutes.
Therefore, to simulate the minutes between customer arrivals, one would use the following formula in Excel: =EXPON.DIST(x/60,1/6,TRUE)
The "x/60" part of the formula is used to convert the time interval from minutes to hours, as the rate is given in customers per hour.
The "1/6" part of the formula is used to set the Lambda value to 6.
The "TRUE" part of the formula is used to specify that we want the cumulative distribution function (CDF), which gives the probability that the time between events is less than or equal to x.
To simulate the minutes between customer arrivals in Excel, we would use the Exponential distribution function with Lambda = 6 and x being the time interval between customer arrivals in minutes.
The formula would be = EXPON.DIST(x/60,1/6,TRUE).
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When George Washington became president in 1789, the army he had commanded in the American Revolution had
disbanded
ARMY DISBANDED MEANS SOLDIERS WERE RELEASED FROM SERVICE
When George Washington became president in 1789, the army he had
commanded in the American Revolution had disbanded. This means that
the soldiers were released from service and the military units were
dissolved after the end of the war.He pursued two intertwined interests:
military arts and western expansion. At 16 he helped survey Shenandoah
lands for Thomas, Lord Fairfax. Commissioned a lieutenant colonel in 1754,
he fought the first skirmishes of what grew into the French and Indian War.
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A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. Which of the following would result in the smallest margin of error in estimating the mean salt content, u? A. 90% confidence, n = 25 B. 90% confidence, n = 50 C. 95% confidence, n = 25 D. 95% confidence, n = 50 E. n = 100 at any confidence level
The option that would result in the smallest margin of error in estimating the mean salt content, u, is 95% confidence, n = 50. The correct answer is Option D.
What is the margin of error?The margin of error is the amount by which a statistic is expected to differ from the true value of the population parameter. The interval estimate is calculated with the help of a margin of error. The margin of error and the interval estimate are inversely related to each other. If we want a small margin of error, we must increase the sample size.
What is the confidence level?The confidence level is the likelihood that a population parameter will fall within a specified range of values. The confidence level is determined by the sample size and margin of error. The sample size and margin of error are directly related to each other. When the sample size is smaller, the margin of error is larger. When the sample size is larger, the margin of error is smaller.
How to determine the smallest margin of error?The margin of error is the highest at a confidence level of 50%. In general, as the confidence level increases, the margin of error decreases, and vice versa. As the sample size increases, the margin of error decreases. It follows that a 95% confidence level, n = 50 would yield the smallest margin of error in estimating the mean salt content, u. Hence, option D) 95% confidence, n = 50 would result in the smallest margin of error in estimating the mean salt content, u.
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how far will 560J raise a block weighing 8 N
a force of 560J can raise a block weighing 8 N to a height of 70 meters. It's important to note that this calculation assumes no energy is lost due to friction or other factors, so in reality, the block may not reach exactly 70 meters.
How to calculate work?
To calculate the height to which a block weighing 8 N can be raised by a force of 560J, we need to use the formula for work done:
Work (W) = force (F) x distance (d) x cos(theta)
where theta is the angle between the force and the displacement. In this case, the force is the weight of the block (8 N), the distance is the height the block is raised (h), and theta is 0 degrees because the force is acting vertically upward and the displacement is in the same direction.
So, we can rearrange the formula to solve for the height (h):
h = W / (F x cos(theta))
Substituting the values we have, we get:
h = 560 J / (8 N x cos(0 degrees))
Since the cosine of 0 degrees is 1, we can simplify to:
h = 560 J / 8 N
h = 70 meters
Therefore, a force of 560J can raise a block weighing 8 N to a height of 70 meters. It's important to note that this calculation assumes no energy is lost due to friction or other factors, so in reality, the block may not reach exactly 70 meters.
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11. if 3 men can dig a 3 x 2 x 4 meter hole in 20 minutes, how long will it take 6 men to dig a 4 x 4 x 3 meter hole?
it would take 6 men 20 minutes to dig a hole that is 4 x 4 x 3 meters
To solve this problem, we can use the formula that relates the time taken to do a job to the number of workers available, assuming that all workers have the same productivity. This formula is given by the equation:Work Done = Productivity x Time Taken x Number of Workers available. If we let the work done be W, the productivity be P, and the time taken be T, then the equation becomes:W = P x T x N. We are given that 3 men can dig a hole that is 3 x 2 x 4 meters in 20 minutes. We can find the volume of this hole by multiplying its dimensions:V = 3 x 2 x 4 = 24 cubic metersThis means that 3 men can dig a volume of 24 cubic meters in 20 minutes.
To find their productivity, we can use the formula: Productivity = Work Done / (Time Taken x Number of Workers)P = W / (T x N) = 24 / (20 x 3) = 0.4 cubic meters per minute per worker. Now we can use this productivity to find the time taken for 6 men to dig a hole that is 4 x 4 x 3 meters. We can find the volume of this hole by multiplying its dimensions:V = 4 x 4 x 3 = 48 cubic meters.
To dig this hole, the amount of work done would be 48 cubic meters. The productivity per worker is still 0.4 cubic meters per minute, but the number of workers is now 6. Therefore, the time taken would be:T = W / (P x N) = 48 / (0.4 x 6) = 20 minutes.Therefore, it would take 6 men 20 minutes to dig a hole that is 4 x 4 x 3 meters, given that 3 men can dig a hole that is 3 x 2 x 4 meters in 20 minutes.
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19% of 43
Help please!!!
Answer:
8.17
Step-by-step explanation:
43x19/100 equals 8.17
Find the new y-intercept by writing an equation in slope-intercept form that
is parallel to the line y = -3x + 8 and goes through the point (5, 3).
Answer:
y- intercept = 18
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = - 3x + 8 ← is in slope- intercept form
with slope m = - 3
• Parallel lines have equal slopes , then
y = - 3x + c ← is the partial equation
to find c substitute (5, 3 ) into the partial equation
3 = - 3(5) + c = - 15 + c ( add 15 to both sides )
18 = c
y = - 3x + 18 ← equation of parallel line
with y- intercept c = 18
in a deli, the ratio of ham subs to cheese subs sold in a day was 9:4. If 36 cheese subs were sold, how many ham subs were sold?
Answer: 81 ham subs
Step-by-step explanation:
4 times 9 is 36. That means you have to multiply 9 by 9, to get 81 ham subs. The ratio is 81:36. :)
(20points) phone camera took the pictures in the aspect ratio of 3:2. Luckily, Naomi can enlarge, shrink or rotate the pictures, but she doesn't want to have to crop the pictures at all or leave any extra space on the sides.
Which print sizes will she be able to order without leaving any extra space or having to cut off any extra material?
Which print sizes will she be able to order without leaving any extra space or having to cut off any extra material?
2x3
4x4
4x5.3
4x6
5x7
D 8x8
8x10
11x14
16x20
20x30
24x36
Como resolver ecuaciones que tienen diferentes incógnitas
Ej: -x-4z=-15
To solve an equation with multiple unknowns, you need to have as many equations as there are unknowns.the solution to the system of equations is [tex]x = 13/11 and z = 37/11.[/tex]
What is the system of equations?For the equation you provided, [tex]-x - 4z = -15[/tex] , there are two unknowns: x and z. To solve for both, you need another equation that involves x and z.
If you have another equation that involves x and z, you can use a method like substitution or elimination to solve for both variables simultaneously.
For example, let's say you have the equation [tex]2x + 3z = 7.[/tex] You can use substitution to solve for one variable in terms of the other and substitute the result into the other equation.
To solve for x, rearrange the first equation to get x in terms of z:
[tex]-x - 4z = -15[/tex]
[tex]x = -15 + 4z[/tex]
Substitute the expression for x into the second equation:
[tex]2x + 3z = 7[/tex]
[tex]2(-15 + 4z) + 3z = 7[/tex]
Distribute the 2:
[tex]-30 + 8z + 3z = 7[/tex]
Simplify:
[tex]11z = 37[/tex]
Solve for z:
[tex]z = 37/11[/tex]
Substitute the value of z back into the equation for x:
[tex]x = -15 + 4z[/tex]
[tex]x = -15 + 4(37/11)[/tex]
[tex]x = -15 + 148/11[/tex]
Simplify:
[tex]x = 13/11[/tex]
Therefore, the solution to the system of equations is x = 13/11 and z = 37/11.
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1. Make a box plot of the ticket prices shown and identify the median price.
$25.50 $45.00
$24.00
$32.50 $32.00 $20.00
$38.50 $50.00 $45.00
Answer:
Step-by-step explanation:
karla paid $200 for an item that was originally priced at $350. what percent of the original price did karla pay? round your answer to the nearest tenth of a percent.
Karla paid 57.1% of the original price for an item that was originally priced at $350.
The explanation is that we can find the percentage that Karla paid by dividing the amount she paid by the original price and then multiplying by 100.
Percentage is a fraction out of 100. The formula for finding the percentage is as follows: (part / whole) × 100.
So, if Karla paid $200 for an item that was originally priced at $350, we can find the percentage she paid as follows: (200/350) × 100 = 57.1%.
Therefore, Karla paid 57.1% of the original price. To round it to the nearest tenth of a percent, we can simply round the answer to one decimal place, which is 57.1%.
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if n > 30 , then the sampling distribution of the mean will have a shape of symmetric , the mean will be equal to [ select ] and the standard error will be equal to
When n > 30, the sampling distribution of the mean will have a shape of symmetric, the mean will be equal to the population mean (μ), and the standard error will be equal to σ/√n.
The shape of a sampling distribution of the mean is always symmetric when n > 30. This is because the Central Limit Theorem (CLT) states that the sampling distribution of the mean is approximately normal when the sample size is greater than 30. Therefore, when n > 30, the shape of the sampling distribution of the mean will be symmetric.
The mean of the sampling distribution of the mean is equal to the population mean, μ. This is because the sampling distribution of the mean is a probability distribution of all the possible sample means, and it is centered around the population mean. Therefore, the mean of the sampling distribution of the mean is equal to μ.
The standard error (SE) of the sampling distribution of the mean is equal to the population standard deviation (σ) divided by the square root of the sample size (n). This is because the standard error is a measure of how accurately the sample mean estimates the population mean. Since the sample size is greater than 30, the sample mean will accurately estimate the population mean and the standard error will be equal to σ/√n.
In summary, when n > 30, the sampling distribution of the mean will have a shape of symmetric, the mean will be equal to the population mean (μ), and the standard error will be equal to σ/√n.
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suppose men always married women who were exactly 3 years younger. what would the correlation between their ages be?
The correlation between the ages of men and women in this situation would be +1. This means that for every one-year increase in the age of the man, the age of the woman would increase by one year as well. The correlation coefficient of +1 signifies a perfect positive linear correlation between the two variables.
In other words, if a man is 40 years old, the woman he is married to would be 37. If the man was 50, then the woman would be 47, and so on. This would remain true for any age, regardless of the difference in their ages. Therefore, the correlation between their ages would always remain +1.
It is important to note that this correlation coefficient is not necessarily reflective of traditional or even natural social or biological behavior. Rather, it is simply a mathematical relationship that would exist in this specific situation. In reality, the age gap between married couples can vary widely depending on various social and cultural factors.
To summarize, the correlation between the ages of men and women in this scenario would be +1, signifying a perfect positive linear correlation. This means that for any given age of the man, the age of the woman would be exactly three years younger.
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Find the areas of the figures whose vertices are
(a) (0,0), (3,7), (5,1)
(c) (-4,2), (0,-8), (5,11)
(e) (-2,-4), (3,1), (-1,5), (6,-3)
(b) (-1,-2), (-2,3), (4,-4)
(a) The area of the triangle is 16 square units.
(b) The area of the triangle is 54 square units.
(c) the area of the quadrilateral is 20 square units.
(d) The area of the triangle is 12.5 square units.
What is the area of the figures?(a) To find the area of the triangle with vertices (0,0), (3,7), and (5,1), we can use the formula:
A = 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Plugging in the values, we get:
A = 1/2 | 0(7 - 1) + 3(1 - 0) + 5(0 - 7) |
= 1/2 | 0 + 3 - 35 |
= 1/2 |-32|
= 16
(b) To find the area of the triangle with vertices (-4,2), (0,-8), and (5,11), we can again use the same formula:
A = 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Plugging in the values, we get:
A = 1/2 | (-4)(-8 - 11) + (0)(11 - 2) + (5)(2 - (-8)) |
= 1/2 | 108 |
= 54
(c) To find the area of the quadrilateral with vertices (-2,-4), (3,1), (-1,5), and (6,-3), we can divide it into two triangles and find the area of each triangle.
We can use the formula for the area of a triangle:
A = 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
For the triangle with vertices (-2,-4), (3,1), and (-1,5), we get:
A1 = 1/2 | (-2)(1 - 5) + (3)(5 + 4) + (-1)(-4 - 1) |
= 1/2 | (-8 + 35 + 5) |
= 16
For the triangle with vertices (3,1), (-1,5), and (6,-3), we get:
A2 = 1/2 | (3)(5 + 3) + (-1)(-3 - 1) + (6)(1 - 5) |
= 1/2 | (24 + 4 - 20) |
= 4
Therefore, the area of the quadrilateral is:
A = A1 + A2
= 16 + 4
= 20 square units
(d) To find the area of the triangle with vertices (-1,-2), (-2,3), and (4,-4), we can use the same formula:
A = 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Plugging in the values, we get:
A = 1/2 | (-1)(3 + 4) + (-2)(-4 + 2) + (4)(-2 - 3) |
= 1/2 | (-7 - 4 - 14) |
= 12.5
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. In which step does a mistake first occur?
(24+3 + 10)-14 + 2
Step 1: (8 + 10) -14 + 2
Step 2: 18 -14+2
Step 3: 4 +2
Step 4: 2
NEED HELP ASAP !!
TY
Answer:
The first one...............
I need it simplified help
The answer is 23/40
To simplify this expression, we need to find a common denominator for each pair of fractions that are being added or subtracted.
The common denominator for 1/4 and 1/5 is 20, so we can rewrite 1/4 as 5/20 and 1/5 as 4/20. Then, we can subtract these fractions to get 1/20.
The common denominator for -3/4 and 1/8 is 8, so we can rewrite -3/4 as -6/8. Then, we can add these fractions to get -5/8.
Now, we can combine the two simplified fractions by finding a common denominator of 40. We can do this by multiplying 20 and 8, which gives us 160.
Then, we can rewrite 1/20 as 2/40 and -5/8 as -25/40. Adding these fractions together gives us:
2/40 - 25/40 = -23/40.
The absolute value of any number is always positive so it becomes 23/40.
The simplified version of (1/4 - 1/5) + (-3/4 + 1/8) is 23/40.
the fact that the overall frequency of an event can be determined by multiplying together the frequencies of the independently occurring factors related to that event is called what?
The fact that the overall frequency of an event can be determined by multiplying together the frequencies of the independently occurring factors related to that event is called a frequency multiplier. Also called as multiplication rule.
What is the multiplication rule?
The multiplication rule states that the overall probability of a sequence of events happening jointly is the product of the probabilities of the individual events.
In probability, the multiplication rule is used to calculate the joint probability of two independent events that occur together.
What are the factors in a probability scenario?In probability scenarios, the factors are the set of conditions that influence the probability of a certain occurrence, event, or outcome.
These factors are used to calculate the probability of an event. The likelihood of the occurrence of an event is based on these factors.
The multiplication rule can be used to calculate the overall frequency of an event by multiplying the frequency of each factor independently related to the event.
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Helppp.. also show work pls
Answer:
Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
100 points please help!
The quadratic equation x² + 2 - 4x = -4 in standard form is x² - 4x + 6 = 0
What is a quadratic equation in standard form?A quadratic equation in standard form is given by ax² + bx + c = 0
Now, given the equation x² + 2 - 4x = -4, we desire to write it in standard form, we proceed as follows.
Since we have the quadratic equation x² + 2 - 4x = -4 writing it in standard form, we have
x² + 2 - 4x = -4
Adding 4 to both sides of the equation, we have that
x² + 2 - 4x + 4 = -4 + 4
x² + 2 + 4 - 4x = 0
x² + 6 - 4x = 0
Re-arranging, we have that
x² - 4x + 6 = 0
So, in equation in standard form is x² - 4x + 6 = 0
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a statistics professor wants to see if more than 80% of her students enjoyed taking her class. at the end of the term, she takes a random sample of students from her large class and asks, in an anonymous survey, if the students enjoyed taking her class. which set of hypotheses should she test?
To test if more than 80% of her students enjoyed taking her class, the statistics professor should use a hypotheses test.
The null hypotheses (H0) would be that the proportion of students who enjoyed taking her class is equal to or less than 80%, while the alternative hypothesis (Ha) would be that the proportion is greater than 80%.
H0: p <= 0.80
Ha: p > 0.80
Where p: proportion of students who enjoyed taking the class.
To test this hypothesis, the professor would need to collect a random sample of students from her class and ask them if they enjoyed taking the class. She could then use a test statistic, such as the z-score, to determine the probability of obtaining the observed proportion of students who enjoyed the class, assuming the null hypothesis is true.
If the probability of obtaining the observed proportion is very low (typically, less than 0.05), then the professor would reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that more than 80% of her students enjoyed taking her class.
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An airship flies 150 km with the wind and then turns around and flies back, taking 6 hr and 15 min for the round trip. Find the speed of the wind if the speed of the airship in calm weather is 50 km/hr.
Answer:
10km/hr
Step-by-step explanation:
The speed of the wind is approximately 6.93 km/hr.
We have,
Let's call the speed of the wind "w" (in km/hr).
When the airship is flying with the wind, its speed relative to the ground is the sum of its speed in calm weather (50 km/hr) and the speed of the wind (w km/hr). So the airship's speed with the wind is 50 + w km/hr.
When the airship is flying against the wind, its speed relative to the ground is the difference between its speed in calm weather (50 km/hr) and the speed of the wind (w km/hr).
So the airship's speed against the wind is 50 - w km/hr.
The distance flown by airship in each direction is the same (150 km).
Let's use the formula for distance, rate, and time:
distance = rate x time
For the first leg of the trip (flying with the wind), we have:
distance = (50 + w) x time
distance = (50 + w) x t(1)
where t(1) is the time taken for the first leg of the trip.
For the second leg of the trip (flying against the wind), we have:
distance = (50 - w) x time
distance = (50 - w) x t(2)
where t(2) is the time taken for the second leg of the trip.
The total time for the round trip is given as 6 hours and 15 minutes, or 6.25 hours.
So we have:
t(1) + t(2) = 6.25
We also know that the distance flown in each direction is 150 km. So we have:
(50 + w) x t(10 = 150
(50 - w) x t(2) = 150
Now we can solve this system of equations for w:
t(1) = 150 / (50 + w)
t(2) = 150 / (50 - w)
t(1) + t(2) = 6.25
Substituting the first two equations into the third equation, we get:
150 / (50 + w) + 150 / (50 - w) = 6.25
Multiplying both sides by (50 + w)(50 - w), we get:
150(50 - w) + 150(50 + w) = 6.25(50 + w)(50 - w)
Expanding and simplifying, we get:
7500 - 150w + 7500 + 150w = 15625 - 312.5w²
Simplifying further, we get:
15000 = 312.5w²
Dividing both sides by 312.5, we get:
w² = 48
Taking the square root of both sides, we get:
w = ±√(48)
Since the speed of the wind cannot be negative, we take the positive square root:
w = √(48) ≈ 6.93 km/hr
Therefore,
The speed of the wind is approximately 6.93 km/hr.
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