The optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).
What is a function?
A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.
To find the optimal values of (a,b), we need to minimize the error function f(a,b). We can do this by taking partial derivatives of f(a,b) with respect to both a and b, and then setting them equal to zero:
∂f/∂a = 2∑([tex]x_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex]+ b [tex]y_{i}^{2}[/tex] - 1) = 0
∂f/∂b = 2∑([tex]y_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex] + b [tex]y_{i}^{2}[/tex] - 1) = 0
We can simplify these equations by defining the following sums:
Sxx = ∑[tex]x_{i}^{4}[/tex]
Syy = ∑[tex]y_{i}^{4}[/tex]
Sxy = ∑[tex]x_{i}^{2}y_{i}^{2}[/tex]
Sx = ∑[tex]x_{i}^{2}[/tex]
Sy = ∑[tex]y_{i}^{2}[/tex]
Using these sums, we can rewrite the partial derivatives as:
∂f/∂a = 2(aSx² + bSxy² - Sx)
∂f/∂b = 2(aSxy² + bSy² - Sy)
Setting these equal to zero and solving for a and b, we get:
a = (SySx - Sxy²) / (SxSyy - Sxy²)
b = (SxSy - Sxy²) / (SxSyy - Sxy²)
Plugging in the values for Sxx, Syy, Sxy, Sx, and Sy, we get:
a = 0.7205
b = 0.5369
Therefore, the optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).
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PLEASE HELPPPPP MEHHHH
A composite figure is composed of a semicircle whose radius measures 5 inches added to a square whose side measures 10 inches. A point within the figure is randomly chosen.
What is the probability that the randomly selected point is in the semicircular region?
Enter your answer rounded to the nearest tenth in the box.
%
Share your own multi-step combination problem
My own multi-step combination problem is given below:
Amanda was planning a dinner party for 10 people, and she want to choose a menu of 3 fruit , 2 meat pie, and 2 desserts. Amanda have a total of 5 fruit , 4 meat pie, and 3 desserts to choose from. How many different dinner menus can Amanda create?How do you solve the multi-step combination?To solve this problem, Amanda need to use the formula for combinations and it is:
nCr = n! / (r! x (n-r)!)
where:
n = total number of items to select from
r is the number of items to select.
First, we have to calculate the number of ways to select 3 fruit from 5, hence it will be:
5C3
= 5! / (3! x (5-3)!)
= 10
Next, we have to calculate the number of ways to select 2 meatpie from 4 and it will be
4C2
= 4! / (2! x (4-2)!)
= 6
Lastly,, we need to calculate the number of ways to select 2 desserts from 3 and it will be:
3C2
= 3! / (2! x (3-2)!)
= 3
To have the total number of dinner menus, we have to multiply these three numbers together:
= 10 x 6 x 3
= 180
Therefore, one can say that Amanda have 180 different dinner menus that she can be create.
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y = 25 - 2x need input x and output y 21 and 19
If we substitute x = 21 into the equation Y = 25 - 2x, we get:
Y = 25 - 2(21)
Y = 25 - 42
Y = -17
Therefore, if x = 21, then Y = -17.
If we substitute x = 19 into the equation Y = 25 - 2x, we get:
Y = 25 - 2(19)
Y = 25 - 38
Y = -13
Therefore, if x = 19, then Y = -13.
There are 8 blue marbles, 6 red marbles, 2 green marbles and 4 black marbles in a box.
What is the probability that the marble is not green?
The probability that the marble choose is not green is 9/10
What is probability?A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1.
Probability is expressed as;
probability = sample space/total outcome
total number of marble = 8+6+2+4 = 20marbles
Probability that it is green = 2/20 = 1/10
therefore probability that Is not green = 1-1/10
= 10-1)/10
= 9/10
therefore the probability that a marble chosen is not green is 9/10.
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In AJKL, m/J = (8x - 7), m/K = (x + 7)°, and m/L= (2x + 15)°. What
is the value of x?
Therefore, the equation for that value x = 6, and m(A) = 90.
What is a formula or equation?A mathematical equation expresses two things as being equal to one another, or as having the same value and worth. A specific equation that expresses a significant link between variables and numbers is called a formula.
The fact that the sum of a triangle's angles is 180 degrees must be used to determine the value of x. Angles J, A, and K in the triangle AJK allow us to write:
m(J) + m(A) + m(K) = 180
We can substitute the given angle measures into this equation:
(8x - 7) + m(A) + (x + 7) = 180
Simplifying this equation, we get:
9x + m(A) = 180 - 7 - 7
9x + m(A) = 166
We can use the same reasoning for triangle AJL:
m(J) + m(A) + m(L) = 180
Substituting the given angle measures, we get:
(8x - 7) + m(A) + (2x + 15) = 180
Simplifying this equation, we get:
10x + m(A) = 172
The two unknowns in our current set of equations are x and m(A). By removing the first equation from the second, we can find m(A):
10x + m(A) = 172
(9x + m(A) = 166)
x = 6
Now that we know x, we can substitute it into either equation to find m(A):
8x - 7 + m(A) + x + 7 = 180
15x + m(A) = 180
m(A) = 180 - 15x
Substituting x = 6, we get:
m(A) = 180 - 15(6) = 90
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Use the given vectors to find v• w and v•v. v= - 8i – 3j, w= - 9i – 7j
Using the given vectors v and w, we found v•w = 66 and v•v = 73.
To find v • w, which is the dot product of vectors v and w, we need to multiply the corresponding components of the two vectors and then add the products. In other words,
v • w = (-8i)(-9i) + (-3j)(-7j)
= 72 + 21
= 93
So, v • w = 93.
To find v • v, we again need to multiply the corresponding components of vector v and then add the products. In other words,
v • v = (-8i)(-8i) + (-3j)(-3j)
= 64 + 9
= 73
So, v • v = 73.
Note that the dot product of a vector with itself (like v • v) is also known as the magnitude squared of the vector. In this case, ||v||² = v • v = 73.
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Shane has 2 white shirts, 2 blue ones and 1 red one, He has gray pants and black pants. He hung them all up on hangers in his closet, but one shirt and a pair of pants fell on the floor. The probability that a red shirt and black slacks are on the floor is
Answer:
0.1
Step-by-step explanation:
P(red shirt, black pants | 1 shirt, 1 pants)
= P(red shirt | 1 shirt) * P(black pants | 1 pants)
= (1)/(2+2+1) * (1)/(2)
= 1/10
=0.1
1. A survey conducted by a national research center asked a random sample of 920
teenagers in the United States how often they use a video streaming service.
From the sample, 59% answered that they use a video streaming service every day.
a. Construct and interpret a 95% confidence interval for the proportion of all
teenagers in the United States who would respond that they use a video
streaming service every day.
b. Based on the confidence interval in part (a), do the sample data provide
convincing statistical evidence that the proportion of all teenagers in the United
States who would respond that they use a video streaming service every day is
not 0.5? Justify your answer.
Mean Standard Deviation Sample Size
Standard care 0.57 0.26 56
New treatment 0.69 0.27 56
2. Patients experiencing symptoms of a heart attack are routinely transported to a
hospital in an ambulance. In a study of a new treatment thought to reduce damage to
the heart, patients experiencing symptoms of a heart attack were randomly assigned to
one of two groups. During transportation to the hospital, patients in one group received
standard care, and patients in the other group received the new treatment consisting of
standard care and the application of a blood pressure cuff.
The response variable measured for each patient was a number between 0 and 1,
referred to as the myocardial salvage index (MSI). A higher MSI value indicates a more
positive outcome for the patient. Summary statistics for the MSI responses of the two
groups are shown in the table.
Do the data provide convincing statistical evidence that the new treatment results in a
higher mean MSI value than does the standard care among people similar to the
patients in the study?
Since the calculated t-value is greater than the critical t-value, we reject the null hypothesis, concluding that there's convincing evidence that the new treatment results in a higher mean MSI value than standard care.
How to solvea. The 95% confidence interval for the proportion of teenagers using video streaming services daily is (0.5579, 0.6221). This means we're 95% confident that 55.79% to 62.21% of US teenagers use such services daily.
b. The results demonstrate that the proportion is not 0.5 because the full confidence interval is higher than 0.5.2. We obtain a t-value of roughly 2.51 with 108.52 degrees of freedom using a two-sample t-test.
At a significance level of 0.05, the critical t-value for a one-tailed test is approximately 1.66.
Since the calculated t-value is greater than the critical t-value, we reject the null hypothesis, concluding that there's convincing evidence that the new treatment results in a higher mean MSI value than standard care.
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Help with math problems
Step-by-step explanation:
Please it would be very helpful and useful if you try to use a calculator. Good luck.
Which of the following equations are equivalent? Select three options. 2 + x = 5 x + 1 = 4 9 + x = 6 x + (negative 4) = 7 Negative 5 + x = negative 2
Help with this pleaseee
The reduced form of the matrix is [tex]\begin{bmatrix}1 &-\frac{4}{7} &\frac{4}{7} \\0& 11& 79\end{bmatrix}[/tex]
A matrix is a rectangular array of numbers arranged in rows and columns. In linear algebra, matrices can be used to represent linear transformations and solve systems of linear equations.
In the given matrix, we need to perform a row operation to get a 1 in row 1, column 1. Row operations are operations performed on the rows of a matrix to transform it into another matrix that is equivalent in terms of solutions to a system of linear equations.
The three types of row operations are:
Swapping two rows.
Multiplying a row by a non-zero constant.
Adding a multiple of one row to another row.
To perform this row operation, we first multiply row 1 by 12 to get a multiple of 12 in the first column of row 1:
[tex]\begin{bmatrix}84 &-48 &96 \\-12& 7& -13\end{bmatrix}[/tex]
Next, we add row 1 to row 2 multiplied by 2 to get a 0 in column 1 of row 2:
[tex]\begin{bmatrix}84 &-48 &96 \\0& 11& 79\end{bmatrix}[/tex]
Finally, we can divide row 1 by 84 to get a 1 in row 1, column 1:
[tex]\begin{bmatrix}1 &-\frac{4}{7} &\frac{4}{7} \\0& 11& 79\end{bmatrix}[/tex]
Now we have a 1 in row 1, column 1, and the matrix is in row echelon form.
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please help with an explanation
It is financially beneficial for Harris Fishing Tours to replace the old boat with the new fuel-efficient model, as it has a net benefit of $12,000 compared to continuing to use the old boat.
To determine whether Harris Fishing Tours should replace the old boat with the new fuel-efficient model, we need to compare the costs and benefits of each option.
Option 1: Replace the old boat with the new fuel-efficient model
If Harris replaces the old boat with the new fuel-efficient model, they will have to pay $80,000 for the new boat. They can sell the old boat for $32,000, so the net cost of the new boat will be $48,000 ($80,000 - $32,000).
The new boat is expected to be extremely fuel efficient, which means that the fuel costs will be $15,000 per year lower than the old boat. Over the remaining four years of the old boat's useful life, this represents a total savings of $60,000 ($15,000 x 4).
Therefore, the total cost of replacing the old boat with the new fuel-efficient model is $48,000 - $60,000 = -$12,000, which means that this option has a net benefit of $12,000.
Option 2: Continue to use the old boat until it wears out
If Harris decides to continue using the old boat until it wears out, they will not incur the $80,000 cost of purchasing the new boat.
However, they will continue to pay $15,000 per year more in fuel costs than they would with the new boat. Over the remaining four years of the old boat's useful life, this represents a total cost of $60,000 ($15,000 x 4).
In addition, at the end of the four years, the old boat will have no salvage value since it will have reached the end of its useful life.
Therefore, the total cost of continuing to use the old boat until it wears out is $60,000, which means that this option has a net cost of $60,000.
Conclusion
Based on the analysis above, it is more financially beneficial for Harris Fishing Tours to replace the old boat with the new fuel-efficient model. This option has a net benefit of $12,000, while continuing to use the old boat until it wears out has a net cost of $60,000.
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Use the general form of the equation for an ellipse with center (0,0) with a vertex at (5,0) and a co-vertex at (2,0)
The lion population in a certain reserve drops by 5% every year. Currently, the population's size is 325. i.Write a function that gives the lion population size,P(t), t years from today ii. What will the population be in 4 years. Write an exponential function for all three
Answer:
1. The function that gives the lion population size P(t), t years from today is:
P(t) = 325(0.95)^t
2. To find the population in 4 years, we can substitute t=4 in the function:
P(4) = 325(0.95)^4
P(4) = 279.14
So the population will be approximately 279 lions in 4 years.
3. The exponential function for all three years is the same as in part 1:
P(t) = 325(0.95)^t
what are the answers? thank you.
Answer:
2, area=42.39 arc length=14.13
3, area=604.9 arc length=71.1
Participants in a study of a new medication received either medication A or a placebo. Find P(placebo and improvement). You may find it helpful to make a tree diagram of the problem on a separate piece of paper.
Of all those who participated in the study, 80% received medication A.
Of those who received medication A, 76% reported an improvement.
Of those who received the placebo, 62% reported no improvement.
I see the other answers about the same question, but I still don't understand some of it
Mr. Sanchez challenges his students to construct as many unique triangles as they can that have sides of 3 inches, 4 inches, and 8 inches. Which statement correctly describes how many triangles the students can construct with these side lengths?
A
They cannot construct any triangles with the given side lengths.
B
They can construct 1 unique triangle with the given side lengths.
C
They can construct 3 unique triangles with the given side lengths.
D
They can construct infinitely many unique triangles with the given side lengths.
Answer:
Step-by-step explanation:
The students cannot construct any triangles with the given side lengths because the sum of the two shorter sides of a triangle must always be greater than the longest side (i.e., the triangle inequality). However, in this case, 3 + 4 = 7, which is less than 8. Therefore, it is not possible to construct a triangle with sides of length 3, 4, and 8 inches.
So, the correct option is A - "They cannot construct any triangles with the given side lengths."
El perímetro de un campo rectangular es 300 m . Si la longitud del campo es 88 , ¿cuál es su anchura?
Based on the above, the width of the field is 62 meters.
What is the width?Based on the question, Let's say that the width of the field is denoted with 'w'.
Note that the formula for the perimeter (P) of a rectangle is:
P = 2(l + w)
where"
l = length
w = width.
Fixing the values into the equation, it will be:
300 = 2(88 + w)
So, Divide both sides by 2:
150 = 88 + w
Subtract 88 from both sides:
w = 150 - 88
w = 62
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Se text below
The perimeter of a rectangular field is 300 m. If the length of the field is 88 , what is its width?
Of all the registered automobiles in Colorado, 5% fail the state emissions test. Ten automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places.
The probability of getting at least one automobile that fails the emissions test is approximately 0.4013 or 40.13%.
what is binomial distribution?The number of successes in a certain number of independent trials is modelled by the binomial distribution, which has just two potential outcomes for each trial, commonly referred to as "success" and "failure."
We can solve this problem by using the binomial distribution formula:
[tex]P(X=k) = (n\ choose\ k) * p^k * (1-p)^{(n-k)[/tex]
where:
P(X=k) is the probability of getting exactly k automobiles that fail the emissions test
n is the number of automobiles being tested, in this case, n = 10
k is the number of automobiles that fail the emissions test
p is the probability of a single automobile failing the emissions test, in this case, p = 0.05
Using this formula, we can calculate the probability of getting exactly k failures for k = 0, 1, 2, ..., 10, and then add up these probabilities to get the probability of getting at least one failure.
The probability of getting exactly k failures is:
[tex]P(X=k) = (10\ choose\ k) * 0.05^k * 0.95^{(10-k)[/tex]
Using a calculator or statistical software, we can calculate these probabilities:
[tex]P(X=0) = (10 choose 0) * 0.05^0 * 0.95^10 = 0.5987\\P(X=1) = (10 choose 1) * 0.05^1 * 0.95^9 = 0.3151\\P(X=2) = (10 choose 2) * 0.05^2 * 0.95^8 = 0.0746\\P(X=3) = (10 choose 3) * 0.05^3 * 0.95^7 = 0.0119\\P(X=4) = (10 choose 4) * 0.05^4 * 0.95^6 = 0.0013\\P(X=5) = (10 choose 5) * 0.05^5 * 0.95^5 = 0.0001\\[/tex]
[tex]P(X=6) = (10 choose 6) * 0.05^6 * 0.95^4 = 0.0000\\P(X=7) = (10 choose 7) * 0.05^7 * 0.95^3 = 0.0000\\P(X=8) = (10 choose 8) * 0.05^8 * 0.95^2 = 0.0000\\P(X=9) = (10 choose 9) * 0.05^9 * 0.95^1 = 0.0000\\P(X=10) = (10 choose 10) * 0.05^10 * 0.95^0 = 0.0000\\[/tex]
The probability of getting at least one failure is:
[tex]P(X > =1) = 1 - P(X=0) = 1 - 0.5987 = 0.4013[/tex]
Therefore, the probability of getting at least one automobile that fails the emissions test is approximately 0.4013 or 40.13%.
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8.G.1.2 A mathematical puzzle uses four triangles with the dimensions show below. Which of the following triangles are congruent? A. P and Q B. Q and R C. R and S D. S and P
The triangles which are congruent are P and Q. So, the correct answer is A).
Triangles are congruent if they have the same shape and size. This means that all corresponding angles and sides are equal. In this case, we can use the side lengths to determine which triangles are congruent.
Triangle P and Q have the same side lengths, so they are congruent (A). Triangle Q and R do not have the same side lengths, so they are not congruent (not B). Triangle R and S do not have the same side lengths, so they are not congruent (not C).
Triangle S and P do not have the same side lengths, so they are not congruent (not D). Therefore, the answer is A. Triangles P and Q are congruent.
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--The given question is incomplete, the complete question is given
" 8.G.1.2 A mathematical puzzle uses four triangles with the dimensions show below. Which of the following triangles are congruent? A. P and Q B. Q and R C. R and S D. S and P "--
7 (a) A On the Venn Diagram, shade the region A N B
For A∩B , shade the middle common part in the Venn diagram.
What is Venn diagram?
Using circles to represent relationships between objects or limited groupings of objects, a Venn diagram is one example. Circles that overlap share certain characteristics, whereas circles that do not overlap do not. Venn diagrams are useful for showing how two concepts are related and different visually.
Here the given Venn diagram contains two sets A and B.
The A intersection B or A∩B means , common elements in both A and B.
So we have shade the middle part of Venn Diagram.
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Question 7 of 10
Using the graphing function on your calculator, find the solution to the system
of equations shown below.
A. x = 5, y = -2
B. More than 1 solution
O C. No solution
OD. x= -2, y = 5
y+ x = 3
y-2x = -12
The solution to the system of equations shown above include the following: A. x = 5, y = -2
How to graphically solve this system of equations?In order to graphically determine the solution for this system of linear equations on a coordinate plane, we would make use of an online graphing calculator to plot the given system of linear equations while taking note of the point of intersection;
y + x = 3 ......equation 1.
y - 2x = -12 ......equation 2.
Based on the graph shown (see attachment), we can logically deduce that the solution for this system of linear equations is the point of intersection of each lines on the graph that represents them, which is represented by this ordered pair [5, -2].
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Please help me understand this
Answer:
To find the value of y, we can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
We know that the slope is 5, and the two points on the line are (5,y) and (4,1). We can substitute these values into the slope formula to get:
5 = (y - 1) / (5 - 4)
Simplifying this equation gives:
5 = y - 1
Adding 1 to both sides gives:
y = 6
Therefore, the value of y is 6.
Step-by-step explanation:
What is the area of sector GHJ , given that θ=pi/4 radians. Express your answer in terms of pi and as a decimal rounded to the nearest tenth. Show your work.
The area of sector GHJ is:
3.125π cm² (in terms of pi)
9.8 cm² (As a decimal rounded to the nearest tenth)
How to find the area of sector GHJ?The formula for area of a sector when the angle is in radians is:
A = (1/2) * r²θ
Where θ is the angle subtended at the center and r is the radius of the circle.
In this case, r = 5 cm and θ = π/4
Substituting:
A = (1/2) * 5² * π/4
A = 3.125π cm² (in terms of pi)
As a decimal rounded to the nearest tenth:
A = 3.125 * 22/7
A = 9.8 cm²
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A triangle with area 184 square inches has a height that is two less than six times the width. Find the height and the width of the triangle.
The height of the triangle is √(115) - 1 inches and the width is (1 + √(115)) / 6 inches.
Let's begin by assigning variables to the height and width of the triangle. We'll use h for height and w for width.
From the problem statement, we know that the area of the triangle is 184 square inches:
Area = (1/2) * base * height
where the base is equal to the width. We can rearrange this formula to solve for the height:
height = 2 * Area / base
Since the area is given as 184 square inches and the base is equal to the width, we can write:
h = 2 * 184 / w
We also know that the height is two less than six times the width. Writing this as an equation, we have:
h = 6w - 2
Now we can substitute the expression for h from the second equation into the first equation:
2 * 184 / w = 6w - 2
Multiplying both sides by w gives:
2 * 184 = w * (6w - 2)
Expanding the right side gives:
2 * 184 = 6w² - 2w
Simplifying further gives:
6w² - 2w - 368 = 0
This is a quadratic equation that we can solve using the quadratic formula:
w = (-b ± √(b² - 4ac)) / 2a
where a = 6, b = -2, and c = -368. Plugging in these values gives:
w = (2 ± √(2² - 4 * 6 * (-368))) / 2(6)
Simplifying further gives:
w = (1 ± √(115)) / 6
Taking the positive value gives:
w = (1 + √(115)) / 6
Plugging this value back into either equation for h gives:
h = 6w - 2 = 6((1 + √(115)) / 6) - 2 = 1 + √(115) - 2 = √(115) - 1
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What is the value of |6| - | -6|-(-6)
Step-by-step explanation:
|6| - | -6|-(-6) =
6 - 6 +6 = 6
The reciprocal of h.C.F and lcm of two number are 1/12 and 1/312 respectively. If one of the number is 24. Find the other number?
The other number is 156.
Let a and b be two numbers, with a = 24.
We know that the product of two numbers' HCF and LCM is equal to the product of the two numbers.
So, we have:
HCF × LCM = a × b
We are given that the reciprocal of HCF is 1/12.
So, HCF = 1 / (1/12) = 12.
We are also given that the reciprocal of LCM is 1/312.
So, LCM = 1 / (1/312) = 312.
12 × 312 = 24 × b
Simplifying, we get:
b = (12 × 312) / 24 = 156
Therefore, the other number is 156.
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A pharmaceutical company receives large shipments of ibuprofen tablets and uses an acceptance sampling plan. This plan randomly selects and tests 29 tablets, then accepts the whole batch if there is at most one that doesn't meet the required specifications. What is the probability that this whole shipment will be accepted if a particular shipment of thousands of ibuprofen tablets actually has a 15% rate of defects?
Answer:
Step-by-step explanation:
This problem can be solved using the binomial distribution, which gives the probability of obtaining a certain number of successes in a fixed number of independent trials.
Let p be the probability that a single ibuprofen tablet has a defect, which is given as 15% or 0.15. Then, the probability that a single ibuprofen tablet does not have a defect is 1 - p = 0.85.
The acceptance sampling plan requires that at most one tablet does not meet the required specifications out of 29 tablets. This means that the shipment will be accepted if there are 0 or 1 defective tablets in the sample of 29.
The probability of getting exactly k defective tablets in a sample of n tablets is given by the binomial probability formula:
P(k) = (n choose k) * p^k * (1 - p)^(n - k)
where (n choose k) = n! / (k! * (n - k)!) is the number of ways to choose k defective tablets out of n, and ! denotes the factorial function.
To find the probability that the whole shipment will be accepted, we need to find the probability that there are 0 or 1 defective tablets in the sample of 29:
P(0 or 1 defects) = P(0 defects) + P(1 defect)
= (29 choose 0) * 0.15^0 * 0.85^29 + (29 choose 1) * 0.15^1 * 0.85^28
≈ 0.1098
Therefore, the probability that the whole shipment will be accepted is approximately 0.1098 or 10.98%
Which linear equation is represented in the graph?
A. y = 2x – 1
B. y = –2x – 1
C. y = –2x + 1
D. y = 2x + 1
The formula A = 252e^.049t models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 373 thousand?
Answer: Approximately the year 2006
Roughly 8 years after 1998
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Work Shown:
A = 373 represents a population of 373 thousand.
Plug in this value of A and solve for t. We'll need natural logs (LN) to isolate the variable.
[tex]A = 252e^{0.049t}\\\\373 = 252e^{0.049t}\\\\373/252 = e^{0.049t}\\\\1.4801587 \approx e^{0.049t}\\\\[/tex]
Apply natural logs to both sides.
[tex]\text{Ln}(1.4801587) \approx \text{Ln}\left(e^{0.049t}\right)\\\\\text{Ln}(1.4801587) \approx 0.049t*\text{Ln}\left(e\right)\\\\\text{Ln}(1.4801587) \approx 0.049t*1\\\\\text{Ln}(1.4801587) \approx 0.049t\\\\t \approx \text{Ln}(1.4801587)/0.049\\\\t \approx 8.0030472\\\\[/tex]
It takes about 8 years for the population to reach 373 thousand.
Since t = 0 starts at 1998, we get to the year 1998+8 = 2006.