True, A P-value of 0.07 is less than that significance requirement of 0.1, suggesting that there is sufficient evidence to conclude that people who have a significant other are happier than single people.
The hypothesis being examined was that those who have a significant other are happier on average than those who do not. To test this hypothesis, data from a sample size of persons were collected and evaluated to compute the P-value.
The P-value of 0.07 was determined to be less than the level of significance established at 0.1, implying that there is sufficient evidence to support the hypothesis that persons with a significant other are happier on average than solitary people.
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78. the probability that a marksman will hit a target each time he shoots is 0.89. if he fires 15 times, what is the probability that he hits the target at most 13 times?
The probability that he hits a target at most 13 times is 0.5031 when a marksman has a 0.89 percent chance of hitting the target each time he fires.
Given that,
A marksman has a 0.89 percent chance of hitting the target each time he fires.
We have to find what is the probability that he will only hit the target 13 times out of 15 shots.
We know that,
The probability of getting exactly k successes in n trials is given by the probability mass function that is,
P(X=k)=[tex](\frac{n}{k} )p^{k}(1-p)^{n-k}[/tex]
Where [tex](\frac{n}{k})= n!/k!(n-k)![/tex]
The probability that he hits the target at most 13 times is,
1-[P(he hits 14 times)+P(he hits 15 times)]
So, calculate the probability that he hits a target 14 times as follows:
P(X=14)=[tex](\frac{15}{14} )0.89^{14}(1-0.89)^{1}[/tex]
P(X=14)=15×(0.89)¹⁴×(0.11)
P(X=14)=0.3228
And the probability that he hits a target 15 times is
P(X=15)=[tex](\frac{15}{15} )0.89^{15}(1-0.89)^{0}[/tex]
P(X=15)=1×(0.89)¹⁵×(1)
P(X=15)=0.1741
And the required probability is
=1-[P(he hits 14 times)+P(he hits 15 times)]
=1-[0.3228+0.1741]
=0.5031
Therefore, The probability that he hits a target at most 13 times is 0.5031 when a marksman has a 0.89 percent chance of hitting the target each time he fires.
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how do i do Write the prime factorization of 6. Use exponents when appropriate and order the factors from least to greatest (for example, 2235).
Answer:
2*3
Step-by-step explanation:
divide by smallest number possible
6/2=3
3/3=1
2*3=6
so 2*3
Select the simplification that accurately explains the following statement. √9=9 1/2
The simplification that explains the statement is [tex](9^{\frac{1}{2} })^{2} = (9^{\frac{1}{2} } )(9^{\frac{1}{2} } ) = 9^{\frac{1}{2} + \frac{1}{2} } = 9^{\frac{2}{2} } = 9^{1} = 9[/tex]
How to simplify an expression?The law of indices(exponent) can be used to explain the following statement:
[tex]\sqrt{9} = 9^{\frac{1}{2} }[/tex]
Therefore, using the law of indices.
[tex](a^{x}) (a^{y} ) = a^{x+y}[/tex]
Hence, applying this law of exponents.
Exponents of numbers are added when the numbers are multiplied, subtracted when the numbers are divided, and multiplied when raised by still another exponent:
[tex](9^{\frac{1}{2} })^{2}[/tex]
[tex](9^{\frac{1}{2} } )(9^{\frac{1}{2} } )[/tex]
[tex]9^{\frac{1}{2} + \frac{1}{2} }[/tex]
[tex]9^{1}[/tex]
9
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Point d is on line segment \overline{ce} ce. Given de=x+2,de=x+2, cd=3x+2,cd=3x+2, and ce=5x+2,ce=5x+2, determine the numerical length of \overline{ce}. Ce.
Answer:
Segment CE is 12 units.
Step-by-step explanation:
CD and DE add together to make CE. See image.
Use this idea to write an equation. Combine like terms and solve for x. Use the x value you find to calculate the length of CE. See image.
an individual is teaching a class on excel macros. the individual plans to break the class up into groups of 4 and wants each group to have a different exercise to practice on. write an equation that expresses the situation, let x be the independent variable and y be the dependent variable.
The required relation between independent and dependent variables is y = x/4.
Explain dependent and independent variables?A simple method for considering independent and dependent variables is, the point at which you're leading an examination, the free factor is what you change, and the reliant variable changes hence. You can likewise consider the autonomous variable the reason and the reliant variable as the impact
According to question:The independent variable x is the total number of students in the class.
And the dependent variable y is number of students in each group.
Therefore,
y = x / 4
Here, when we change the value of x then value of y automatically changed.
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1) Brian is playing a game of dice. He gets 10 points for each 6 he rolls. He loses 4 points
every time he does not roll a 6. After 20 tries, Brian's score was 144 points. How many
times did he not roll a 6?
Answer:
He rolled a 6, 16 times
He rolled not a 6, 4 times
Brian did not roll 6 in the dice 14 times.
What is subtraction?Subtraction is mathematic operation. Which is used to remove terms or objects in expression.
Given, Brian is playing a game of dice. He gets 10 points for each 6 he rolls. He loses 4 points every time he does not roll a 6.
After 20 tries his score is 144 points.
If he rolled 6 each time,
his score would be 200.
Subtracting the number 144 to 200.
200 - 144
= 56
And 56/4
= 14
That means he did not roll 6, 14 times.
Therefore, 14 times he did not roll 6.
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the following data were collected for a two-factor anova with two treatments and three blocks. treatment block 1 2 a 46 31 b 37 26 c 44 35 using the 0.05 significance level conduct a test of hypothesis to determine whether the block or the treatment means differ. required: a. state the null and alternate hypotheses for treatments. b. state the decision rule for treatments. (round your answer to 1 decimal place.) c. state the null and alternate hypotheses for blocks. also, state the decision rule for blocks. (round your answer to 1 decimal place.)
The test hypothesis using 0.05 significance level has the following results:The treatment means differ, The block means do not differ.
What is hypothesis testing?A poll or study's findings can be tested using the statistical procedure known as hypothesis testing to see if the findings are significant. By calculating the likelihood that the findings happened by chance, you are basically evaluating if the findings are genuine. If it's possible that your results were a coincidence, the experiment won't be reproducible and won't be of much help.
Hypothesis testing may be one of the most perplexing components for students, mostly because you must first determine what your null is so that you can conduct a test. These challenging word problems that you encounter might sometimes be challenging to understand.
Make a plan for your null hypothesis.
Write down your null hypothesis
Determine the type of test you must do,
How to solve?
Describe the null and alternate treatment hypotheses:
Null hypothesis: No difference in treatment means
Alternative interpretation: The means differ
Indicate the treatment decision-making process:
F(1,2)=18.51 as seen in the table
Reject the null hypothesis if the estimated F exceeds 18.51.
Treatments Calculated F=43.75 > 18.51; hence, the null hypothesis is rejected.
As a result, we find that the treatment means differ.
Write out the null and alternative hypotheses for blocks.
Block means are not different under the null hypothesis.
A different theory There exists at least one pair of distinct meanings.
State the block decision rule as well:
Table F(2,2)=19.0 value
If the estimated F is greater than 19, reject the null hypothesis.
Blocks determined that F=8.14 19.0, thus we did not reject the null hypothesis.
Our conclusion is that the block means are equivalent.
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a population of 10,000 people has an annual growth rate of 6.5%. use the exponential growth equation to determine the size of this population after five years if the growth rate stays the same. the size of the population after five years if the growth rate stays the same is closest to:
The Total Population after five years will be 13,700 people
What is Compound Interest?
Compound interest is the adding of interest to the principle sum of a loan or deposit, or interest on interest plus interest. It is the outcome of reinvesting interest, or adding it to the lent capital, rather than paying it out or requesting payment from the borrower, so that interest is received on the principle sum plus previously accrued interest in the next month. In finance and economics, compound interest is the norm.
Solution:
To solve the problem mentioned we need to use the concept of compound interest:
Principal = 10,000
Annual Growth Rate = 6.5%
Time = 5 years
Amount = Principal * ((100+Rate)/100)^Time
Total Population after 5 years = 10,000 * (106.5/100)^5
= 10,000 * (1.065)^5
= 10,000 * 1.37008
= 13,700
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(-4a^5b^3)^2•(-3a^2b)^2
Answer: 144a^14 b^8
Step-by-step explanation:
(-4a^(5)b^(3))^(2)* (-3a^(2)b)^(2) = 144a^14 b^8
Find the differences between 3,428 and 8,254
Answer: 4826
Step-by-step explanation: 8254 - 3248 = 4826
Caculate the simple interest earned on $5000 at 5% for 3.5 years
Answer:
56.0
Step-by-step explanation:
This equation has one solution. 5(x – 1) 3x = 7(x + 1) what is the solution?
The quadratic equation 15[tex]x^{2}[/tex]-22x -7 = 0 has solution x = 1.7355 or -0.26886
In algebra, a quadratic equation (from Latin quadratus 'square') is any equation that can be rearranged in standard form as
a[tex]x^{2}[/tex] + bx + c =0
where x represents an unknown value, and a, b, and c represent known numbers. One supposes generally that a ≠ 0; those equations with a = 0 are considered degenerate because the equation then becomes linear or even simpler. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent equation
given that
5(x-1)3x = 7(x+1)
15[tex]x^{2}[/tex]-15x = 7x +7
15[tex]x^{2}[/tex]-22x -7 = 0
x = -b ± [tex]\sqrt{b^{2}-4ac }[/tex]/2a
x = -(-22) ± [tex]\sqrt{(-22)-4(15)(-7)}[/tex] /2(15)
x = 1.7355 or -0.26886
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Answer: x = 12
Step-by-step explanation:
Distribute the parenthesis on both sides of the equation
5x - 5 + 3x = 7x + 7
8x - 5 = 7x + 7 ( subtract 7x from both sides )
x - 5 = 7 ( add 5 to both sides )
x = 12
Therefore, your answer is: x = 12
The graph of a linear function is shown on the grid.
#6. Find the slope
#7. Find the y-intercept:
#8. Equation: y=
The slope and y-intercept of the given linear equation are [tex]-\frac{5}{4}[/tex] and [tex]5[/tex] respectively. The equation of the given graph is [tex]y=-\frac{5}{4} x+5[/tex]
What is the Slope of a graph?
The ratio of rise to run is known as a line's slope.When given a line's graph and asked to determine its equation, the first step is to determine the slope.The slope formula is used in the procedure for determining the slope from a graph.If there are two points [tex](x_{1} , y_{1} )[/tex] and [tex](x_{2} , y_{2} )[/tex] on the line, then the equation for the slope (m) is given as, [tex]m=\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]What is a y-intercept?
The location where a graph crosses the y-axis is known as the y-intercept (c). Or to put it another way, it is the value of y when x=0.Here, from the graph, we can find a line passing through the two points, [tex](4,0)[/tex] and [tex](0,5)[/tex].
The equation for the slope formula is given as, [tex]m=\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
Using the slope formula, we get the slope of the given graph as,
[tex]m=\frac{ 5-0 }{0-4 }\\\implies m=-\frac{5}{4}[/tex]
The y-intercept (c) is the value of y when x equals zero.
The line is passing through the point [tex](0,5)[/tex]
Here, the value of x is equal to 0 and the value of y is equal to 5.
So, the y-intercept of the given graph, [tex]c=5[/tex]
We know that the equation of a straight line is expressed as, [tex]y=mx+c[/tex]
Substituting the values of m and c in the straight line equation form, we get the equation of the given line as,
[tex]y=-\frac{5}{4} x+5[/tex]
Therefore, the slope of the given graph, [tex]m=-\frac{5}{4}[/tex] and the y-intercept, [tex]c=5[/tex].
Also, the equation of the given line is [tex]y=-\frac{5}{4} x+5[/tex]
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A book sold 32,100 copies in its first month of release. Suppose this represents 8.2% of the number of copies sold to date. How many copies have
been sold to date?
Round your answer to the nearest whole number.
Answer:
2,632 copies
Step-by-step explanation:
32100× 0.082= 2632.2
About 2,632
if a researcher analyzed data using linear regression correlating a performance test and gpa and calculated a slope of 0.028 and y intercept of 0.63. what would be the predicted student's gpa if she scored 63 on the performance test?
The GPA of the student who scored 63 in the performance test is 2.394.
Given, a researcher analyzed data using linear regression correlating a performance test and GPA and calculated a slope of 0.028 and y-intercept of 0.63.
we have find the GPA of a student who scored 63 in the performance test.
Let the score of the student in performance test be, x
and the GPA of the student be, y
So, y = mx + c
as, m = 0.028 and c = 0.63
y = 0.028x + 0.63
at x = 63
y = 0.028×63 + 0.63
y = 2.394
So, the GPA of the student who scored 63 in the performance test is 2.394.
Hence, the GPA of the student who scored 63 in the performance test is 2.394.
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Which set of ordered pairs represents a function?
O {(-6,-4), (2, -1), (-6, 0), (-9, 8)}
O {(7,-7), (8,9), (7, 7), (-4,9)}
O {(-4,-1), (-6, -7), (7,6), (-8,6)}
O {(8,-8), (2, -7), (9,-4), (8,-5)}
Submit Answer
Answer:Answer:
D.
{(-9,8), (-5,-8), (-7,8), (-6,-8)}
Step-by-step explanation:
HLP PLLSSSSSS!!! TYSM!!!
Solve 2(3m + 1) − 2m = −14.
m = −4
m equals negative 14 over 3
m equals negative 16 over 6
m = −0.06
Answer:
m = -4
Step-by-step explanation:
2(3m + 1) - 2m = -14
Use distributive property.
6m + 2 - 2m = -14
Combine like terms.
4m + 2 = -14
Subtract 2
4m = -16
Divide by 4.
m = -4
Given the function f(x) =2x-1 what is the value of f(-7)
Answer:
Step-by-step explanation:
hi
a frog starts at location a and wants to get to location b that is 500 meters away. starting at a, it sits at a given spot on the path from a to b for a time that is exponen- tially distributed with parameter 1 (minute). after that, it jumps 1 meter towards b, sits in the new spot for an exponentially distributed time, etc. all the times are independent. what is the probability (approximately) that it will get to b within 7 hours?
The probability that it will get to b within 7 hours is approximately 0.014.
Probability:
In math, the favorable way of happening the event is known as probability.
Given,
A frog starts at location a and wants to get to location b that is 500 meters away. starting at a, it sits at a given spot on the path from a to b for a time that is exponentially distributed with parameter 1 (minute). after that, it jumps 1 meter towards b, sits in the new spot for an exponentially distributed time, etc. all the times are independent.
Here we need to find the the probability (approximately) that it will get to b within 7 hours.
While we looking at the given question we have identified that,
starting point = a
ending point = b ( 500 meters)
Time take for one jump = 1 minute
Distance covered in one jump = 1 meter.
Here they also said that the time is independent but the distance remains constant.
So, the probability that it will get to b within 7 hours is calculated as,
=> 7/500
=> 0.014
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The table shows the proportional relationship between the price for a certain number of buckets of golf balls at a driving range.
Buckets 3 5 7
Price (dollars) 28. 50 47. 50 66. 50
Determine the constant of proportionality.
9. 5
10. 5
18. 25
28. 5
The constant of proportionality is 9.5.
The correct option is (a)
Given,
In the question:
The proportional relationship between the price for a certain number of buckets of golf balls at a driving range.
Buckets 3 5 7
Price (dollars) 28. 50 47. 50 66. 50
To determine the constant of proportionality.
Now, According to the question:
We know that:
Proportional relationship equation:
y = kx, where k-constant of proportionality
In the given table, number of buckets represent input values (x) and the price represents output values (y).
We have for the first column:
x = 3, y = 28.50
Find the value of k by plugging in the values of x and y:
28.50 = 3k
k = 28.50/3
k = 9.5
The options are:
a) 9. 5
b) 10. 5
c) 18. 25
d) 28. 5
Hence, The constant of proportionality is 9.5.
So, The correct option is (a).
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an architect designs a diagonal path across a rectangular patio. the path is 29 meters long. the width of the patio is x meters, and the length of the path is 5 meters more than the width. which equation can be used to find the dimensions of the patio? 0.5(x)(x 5)
The equation of the path to find the dimensions of the patio is x² + (x + 5)² = 841.
Here we have to find the equation of the dimensions of the patio.
Data provided:
the path = 29 meters
length of the path is 5 meters more than the width of the path.
So let us assume that the length of the path is x
with this, the width of the path is x + 5
The formula for the diagonal:
Diagonal² = length² + width²
Putting the data we have:
29² = x² + (x +5)²
841 = x² + (x+5)²
Therefore we get the equation as x² +(x + 5)² = 841.
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Two boats depart from a port located at (–10, 0) in a coordinate system measured in kilometers, and they travel in a positive x-direction. The first boat follows a path that can be modeled by a quadratic function with a vertex at (0, 5), and the second boat follows a path that can be modeled by a linear function and passes through the point (10, 4). At what point, besides the common starting location of the port, do the paths of the two boats cross?
(–6, 0.8)
(–6, 3.2)
(6, 3.2)
(6, 0.8)
The points (-10, 0) and (0, 5), on the path of the first boat that can be modeled by a quadratic function and the point (10, 4) on the path of the linear function modelling the second boat, indicates that besides the common starting location, the solution for the point at which the paths of the boats cross again at (6, 3.2)
What is a solution for a system of equations?The solution to a system of equation is the value of the input variable at which the output of the equations in the system of equations are the same.
Location of the port = (-10, 0)
The vertex of the path of the first boat that can be modeled by a quadratic function = (0, 5)
Point through which the boat that follows a path that can be modeled as a linear function passes = (10, 4)
The vertex form of the equation of a parabola is; y = a·(x - h)² + k
Where;'
(h, k) = The coordinates of the vertex
(h, k) = (0, 5)
Therefore;
y = a·(x - 0)² + 5 = a·x² + 5
When x = -10, y = 0, therefore;
y = 0 = a·(-10)² + 5 = 100·a + 5
0 = 100·a + 5
a = -5/100 = -1/20
a = -0.05
The equation is therefore; y = -0.05·x² + 5
The slope of the straight line path of the other boat is; m = (4 - 0)/(10 - (-10)) = 1/5 = 0.2
The equation is therefore; y - 0 = 0.2·(x - (-10)) = 0.2·x + 2
y = 0.2·x + 2
The point where the two paths cross is therefore;
-0.05·x² + 5 = 0.2·x + 2
-0.05·x² - 0.2·x + 5 - 2
-0.05·x² - 0.2·x + 3 = 0
x = (0.2 ± √((-0.2)² - 4 × (-0.05) × 3))/(2×(-0.05))
Therefore;
x = -10 and x = 6
The x-coordinate of the point the two paths cross, besides the starting location is x = 6
Therefore;
y = 0.2 × 6 + 2 = 3.2
The point besides the common starting location, the paths of the boats cross is; (6, 3.2)
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Answer:
(–6, 3.2)
Step-by-step explanation:
The answer above is correct.
Marlene wants to determine which podcast adults prefer.
She surveys 30 adults entering a park one morning. Which
type of sample does this represent?
The type of sample that the sample taken by Marlene represents is a Convenience sample.
What is a Convenience sample ?Researchers who use convenience sampling obtain market research data from a pool of respondents who are conveniently accessible. Because it is so rapid, easy, and economical, it is the sampling procedure that is used the most frequently. If members choose to be part of the sample, they are frequently simple to get in touch with.
Therefore, Marlene employed a convenience sampling technique by going to the park, which was close by, and taking a sample of the people present.
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which of these coefficients are statistically significant at a significance level of 0.05? which of these coefficients are statistically significant at a significance level of 0.01?
On solving the provided question, we got - 0.001 is a stronger significance level,
What is Coefficient?any of a product's characteristics taken into account in respect to a certain characteristic. especially: a term's constant component as opposed to its variable. a value that represents a quality or attribute by a number (as of a substance, device, or process)
Describe a coefficient example.Whenever a variable is multiplied, a coefficient is used. In this case, "z" is a variable, making 6 a coefficient. 6z stands for 6 times z. Coefficient 1 applies to variables without a value.
0.001 is a stronger significance level, It means that the probability of error for the statistic is greater than 5 in 100, therefore the researchers concluded that the research is not statistically significant.
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Gina reads 120 pages in 4 days. Which reading rate is equivalent?
1: 74 pages in 3 days
2: 60 pages in 2 days
3: 105 pages in 5 days
4:80 pages in 3 days
Answer:
60
Step-by-step explanation:
she read 60 pages in just 2 days
on average, an american professional football game lasts about five hours, even though the ball is actually in play only 11 minutes. assume that game times are normally distributed with a standard deviation of 0.9 hour.
The probability the game lasts less than 4.5 hours is 0.2877.
How to calculate the probability?
μ = mean population = 5
σ = standard deviation = 0.9
The probability for the game lasts less than 4.5 hours can be calculated using z test formula.
P(x < 4.5) = P(z < [tex]\frac{x-\mu}{\sigma}[/tex] )
= P(z < [tex]\frac{4.5-5}{0.9}[/tex])
= P(z < -0.56)
Using z table for P(z < -0.56). So,
= 0.2877
Thus, the probability is 0.2877 for the game lasts less than 4.5 hours.
Your question is incomplete, but most probably your full question was
on average, an american professional football game lasts about five hours, even though the ball is actually in play only 11 minutes. assume that game times are normally distributed with a standard deviation of 0.9 hour. 1. Find the probability that a game lasts less than 4.5 hours.
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in a random sample of 200 items, 42 are defective. if the null hypothesis is that 23% of the items in the population are defective, what is the value of zstat?
The value of z-stat is -0.6721.
What is z stat?
The relationship between a value and the mean of a group of values is quantified by a Z-stat. The Z-stat is calculated using standard deviations from the mean. When a data point's Z-stat is 0, it means that it has the same score as the mean.
One standard deviation from the mean would be indicated by a Z-stat of 1.0. Z-stats can be positive or negative; a positive value means the score is above the mean, while a negative value means it is below the mean.
Solution Explained:
We use the formula,
[tex]z = \frac{P - \pi }{\sqrt{\frac{\pi (1-\pi )}{200} }}[/tex], where P is the observed proportion, π is the hypothesized proportion
Therefore, P = 42/200 = 0.21 & π = 23/100 = 0.23
Putting the values in
[tex]z = \frac{0.21 - 0.23 }{\sqrt{\frac{0.23 (1-0.23)}{200} }}[/tex]
After calculating, z-stat is therefore equal to -0.6721
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the length of a rectangle is four times its width. if the perimeter is at most 130 centimeters, what is the greatest possible value for the width? question 3 options: 2w + 2 • (4w) < 130 2w + 2 • (4w) > 130 2w + 2 • (4w) ≤ 130 2w + 2 • (4w) ≥ 130
Answer:
Step-by-step explanation:
Let L be the Length and W be the Width of a rectangle.
We are told that: L = 4W [the length of a rectangle is four times its width]
Perimeter = 2L + 2W
We learn that P ≤ 130 cm
2L + 2W ≤ 130 cm
Substitute L = 4W:
2L + 2W ≤ 130 cm
2(4W) + 2W ≤ 130 cm
10W ≤ 130 cm
W ≤ 13 cm
Options:
2w + 2 • (4w) < 130 Not correct since the < sign does not allow for the "at most 130 cm.")
2w + 2 • (4w) > 130 Must be ≤, not >
2w + 2 • (4w) ≤ 130 This option works since the ≤ sign is correct.
I don’t understand this question , it’s so confusing
what is the probability of these events when we randomly select a permutation of {1, 2, 3, 4}? a) 1 precedes 4. b) 4 precedes 1. c) 4 precedes 1 and 4 precedes 2. d) 4 precedes 1, 4 precedes 2, and 4 precedes 3. e) 4 precedes 3 and 2 precedes 1.
The probability that these events when we randomly select a permutation is 24, 1/2, 1/2, 1/3, 1/4, 1/4
What is a probability simple definition?A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to express probabilities.
What is the probability formula?P(A) = n(A)/n (S)
The probability of an event "A" is P(A). The number of successful outcomes is n(A). The total number of events in the sample space is denoted by n(S).
1) From the set we can select 4*3*2*1=24 permutations.
2) There are six permutations that begin with 1, three that begin with 2, three that begin with 3, and no permutations that begin with 4 when 1 comes before 4. So, P1 ={ 6+3+3+0}/{24} = 1/2
3) No permutations begin with 1, three permutations begin with 2, three permutations begin with 3, and six permutations begin with 4, where 4 comes before 1. P2=1/2
4) There are no permutations that begin with 1, no permutations that begin with 2, no permutations that begin with 2, no permutations that begin with 3, and six permutations that begin with 4, where 4 comes before both 1 and 2. Therefore, P 3 = 1/3
5) No permutations begin with 1, no permutations begin with 2, no permutations begin with 3, and there are six permutations that begin with 4, where 4 comes before 1, 2, and 3. So, P4= 1/4
6) There are no combinations that begin with 1, three combinations that begin with 2, no combinations that begin with 3, and three combinations that begin with 4, where 4 comes before 3 and 2 comes before 1. So, P5= 1/4
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