The sample variance computed differently from the population variance is the calculation in the numerator is different and the calculation in the denominator is different
The sample variance is computed differently from the population variance in that the calculation in the numerator is different and the calculation in the denominator is different. Specifically, in the numerator, the sample variance formula includes a computation for SS (sum of squared deviations from the mean), while the population variance formula does not.
Additionally, in the denominator, the sample variance formula divides by n-1 (sample size minus one) instead of by the denominator (population size) in the population variance formula.
The sample variance is computed differently from the population variance in the following ways:
1. The calculation in the numerator is the same for both sample and population variance, as they both involve computing the sum of squared differences (SS) between each data point and the mean.
2. The calculation in the denominator is different. For the population variance, the denominator is the number of data points in the population (N), while for the sample variance, the denominator is the number of data points in the sample (n) minus 1.
So, the correct answer is: the calculation in the denominator is different (Option C).
Here are the formulas for each variance:
Population variance: σ² = Σ(x - μ)² / N
Sample variance: s² = Σ(x - X)² / (n-1)
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(b) Prove that if the sequence (4) satisfies lim = L = 0, then a) is unbounded. 71
We have proved that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded.
To prove that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded, we will use proof by contradiction.
Assume that (a_n) is bounded. Then, there exists a positive number M such that |a_n| ≤ M for all n in the natural numbers.
Since lim a_n = L = 0, we can choose an ε > 0 such that if n is sufficiently large, then |a_n - L| < ε. In other words, there exists a natural number N such that for all n ≥ N, |a_n - L| < ε.
Consider the case when n ≥ N and a_n > 0 (the case when a_n < 0 is similar). Then, we have:
a_n = L + (a_n - L) > L - ε
Since a_n ≤ M, we have:
0 ≤ a_n < M
Combining these inequalities, we get:
0 ≤ L - ε < a_n < M
This implies that a_n is bounded between two positive numbers, which contradicts the assumption that (a_n) is unbounded. Therefore, our initial assumption that (a_n) is bounded must be false, and hence (a_n) is unbounded when lim a_n = L = 0.
Therefore, we have proved that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded.
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consider the -matrix and . we want to find the least-squares solution of the linear system using the projection onto the column space of . the projection of onto is 0 -1 -2 the least-square solution of is the solution of the linear system . thus is
The least-square solution H' is given by the solution vector in, resulting in H' = x = [0.6]. This solution minimizes the squared error between Ax and b and represents the best approximation for the given linear system.
The least-square solution of the linear system Ax = b can be found by projecting b onto the column space of A. Given the matrix A as [1 -1] and the vector b as [-2], the projection projcol(A)(b) of b onto Col(A) is approximately -0.3.
The least-square solution H' of Ax = b is obtained by solving the linear system Aîn = projcol(A)(b). In this case, the solution vector în is approximately [0.6]. Therefore, the least-square solution Ĥ for the given system is x = [0.6].
In order to find the least-square solution, we first compute the projection projcol(A)(b) of b onto the column space of A. This projection represents the closest point in the column space of A to the vector b. In this case, the projection is approximately -0.3. Next, we solve the linear system Aîn = projcol(A)(b), where A is the given matrix and în is the solution vector. By substituting the projection value, we get the equation [1 -1]în = -0.3. Solving this equation yields the value of în as approximately [0.6].
Therefore, the least-square solution H' is given by the solution vector în, resulting in H' = x = [0.6]. This solution minimizes the squared error between Ax and b and represents the best approximation for the given linear system.
Complete Question:
Finding the least square solution via projection ſi 1 0 Consider the 3 x 2-matrix A= 1 -1 and b= -2 We want to find the least-squares solution of the 1 0 -2 linear system Ax = b using the projection onto the column space of A. The projection projcol(A)(b) of b onto Col(A) is -0.3 projcol(A)(b) -2.3 x 0% 0.6 The least-square solution Ĥ of Ax = b is the solution of the linear system Aîn = projcol(A)(b). Thus în is 0.6 Â= ? x 0% 1.
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when applying the integral test, we can use differential calculus to check that the function is decreasing: if is a continuous function on , and is differentiable on with , then is decreasing on .
When applying the integral test for the convergence of a series, we can use differential calculus to check if the function being integrated is decreasing. The integral test is a method for determining the convergence or divergence of a series by comparing it to an integral of a related function. If the integral of the function converges, then the series also converges, and if the integral diverges, then the series also diverges.
To apply the integral test, we need to first identify a function that is continuous, positive, and decreasing on the interval of interest. We then integrate this function from the starting point of the series to infinity. If the integral converges, then the series also converges, and if the integral diverges, then the series also diverges.
Differential calculus can be used to check that the function being integrated is decreasing. Specifically, we can use the first derivative of the function to determine if it is decreasing on the interval. If the derivative is negative, then the function is decreasing, and if the derivative is positive, then the function is increasing. If the derivative is zero, then the function may or may not be decreasing, depending on its behavior at that point.
Overall, the integral test and the use of differential calculus provide powerful tools for determining the convergence or divergence of a series. By identifying a suitable function and checking it's decreasing behavior using the derivative, we can use the integral test to evaluate the convergence of a wide range of series.
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The bent rod is supported by a smooth surface at B and by a collar at A, which is fixed to the rod and is free to slide over the fixed inclined rod. Suppose that F = 160 lb and M = 400 lb. Ft.
a). Determine the magnitude of the reaction force on the rod at B.
b). Determine the magnitude of the reaction force on the rod at A.
c). Determine the moment of reaction on the rod at A
The bent rod is supported by a smooth surface at B and by a collar at A, which is fixed to the rod and is free to slide over the fixed inclined rod, Then the magnitude of the reaction force on the rod at B is 160 lb, the magnitude of the reaction force on the rod at B is 161.11 lb, the moment of reaction on the rod at A -400 lb.
a). To decide the size of the response drive on the pole at B, ready to consider the strengths acting on the pole. Since the bar is in static balance, the net drive acting on the bar within the vertical course must be zero. Subsequently, ready to compose:
B_y - F =
B_y = F = 160 lb
Therefore, the greatness of the response drive on the bar at B is 160 lb.
b). To decide the size of the response constraint on the bar at A, able to consider the powers acting on the collar at A. Since the collar is free to slide over the settled slanted bar, the response drive at A must be opposite the pole. In this manner, ready to compose: A_x + Fsin(30°) =
A_y - Fcos(30°) =
A_x = -Fsin(30°) = -80 lb
A_y = Fcos(30°) = 138.56 lb
In this manner, the size of the response drive on the bar at A is:
|A| = sqrt(A_x2 + A_y2) = sqrt((-80)2 + (138.56)2) ≈ 161.11 lb
c). To decide the minute of response on the bar at A, ready to consider the minutes acting on the collar at A. Since the collar is settled to the pole, the minute of the response constrain at A must balance the minute of the outside drive M. Subsequently, we will type in:
M + A_y*d =
|Ma| = A_y*d = -M = -400 lb.ft
Therefore, the minute of response on the bar at A is -400 lb. ft (counterclockwise).
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WHAT IS THE AREA OF A TRAPEZOID WITH COORDINATES (1,4) (1,-3) (6,6) (6,-5)
Answer:
THESE NUTS
Step-by-step explanation:
what is th answer to this question
The total surface area of the trapezoidal prism is S = 3,296 inches²
Given data ,
Let the total surface area of the trapezoidal prism is S
Now , the measures of the sides of the prism are
Side a = 10 inches
Side b = 32 inches
Side c = 10 inches
Side d = 20 inches
Length l = 40 inches
Height h = 8 inches
Lateral area of prism L = l ( a + b + c + d )
L = 40 ( 10 + 32 + 10 + 20 )
L = 2,880 inches²
Surface area S = h ( b + d ) + L
On simplifying the equation , we get
S = 2,880 inches² + 8 ( 52 )
S = 3,296 inches²
Hence , the surface area of prism is S = 3,296 inches²
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The generic metal A forms an insoluble salt AB(s) and a complex AC5(aq). The equilibrium concentrations in a solution of AC5 were found to be [A] = 0. 100 M, [C] = 0. 0360 M, and [AC5] = 0. 100 M. Determine the formation constant, Kf, of AC5. The solubility of AB(s) in a 1. 000-M solution of C(aq) is found to be 0. 131 M. What is the Ksp of AB?
Algibra 1, unit 1, PLEASE HELP!
Step-by-step explanation:
Let's find out:
ax - bx + y = z subtract 'y' from both sides of the equation
ax-bx = z-y reduce L side
x ( a-b) = z-y divide both sides by (a-b)
x = (z-y) / (a-b) Done.
The function y=f(x) is graphed below. Plot a line segment connecting the points on f where x=−5 and x=−3. Use the line segment to determine the average rate of change of the function f(x) on the interval −5≤x≤−3.
Use the line segment to determine the average rate of change of the function f(x) on the interval. The function y=f(x) is graphed below. Plot a line segment.:
Step-by-step explanation:
The value of a phone when it was purchased was $500. It loses 1/5 of its value a year. What is the value of the phone after 1 year?
Answer:
[tex]\huge\boxed{\sf \$400}[/tex]
Step-by-step explanation:
Value of phone = $500
Loss in price = 1/5 of total price
Loss in price:= 1/5 × 500 (of means to multiply)
= 1 × 100
= $100
Value of phone after one year:= Actual price - loss
= 500 - 100
= $400[tex]\rule[225]{225}{2}[/tex]
please state the appropriate statistical test being used: I.e : (t - test for independent samples, z scores, single sample t test, t test for related samples, pearson correlation, Chi-square goodness of fit, Chi-square test for independence).
A graduate student in developmental psychology believes that there may be a relationship between birth weight and subsequent IQ. She randomly samples seven psychology majors at her university and gives them an IQ test. Next, she obtains the weight at birth of the seven majors from the appropriate hospitals (after obtaining permission from the students, of course).
The data are shown in the following table:
Student 1 2 3 4 5 6 7
Birth Weight (lbs) 5.8 6.5 8.0 5.9 8.5 7.2 9.0
IQ 122 120 129 112 127 116 130
What can the graduate student conclude? Use a = 0.05
State the appropriate statistical test:
H0:
H1:
df (if appropriate) and Critcal Value :
State Results, Decision, and Conclusions:
The graduate student cannot reject the null hypothesis that there is no significant correlation between birth weight and subsequent IQ among psychology majors at the university.
The appropriate statistical test to use in this scenario is the Pearson correlation coefficient.
H0: There is no significant correlation between birth weight and subsequent IQ.
H1: There is a significant correlation between birth weight and subsequent IQ.
df = n-2 = 7-2 = 5 (where n is the sample size)
Critical value (at alpha = 0.05 and df = 5) = ±2.571
Using a statistical software or calculator, we can find that the sample correlation coefficient is 0.758, with a p-value of 0.076.
Since the p-value is greater than the alpha level of 0.05, we fail to reject the null hypothesis. Therefore, we cannot conclude that there is a significant correlation between birth weight and subsequent IQ among psychology majors at the university.
In conclusion, the graduate student cannot reject the null hypothesis that there is no significant correlation between birth weight and subsequent IQ among psychology majors at the university.
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let t be the linear transformation corresponding to a 2 x 2 matrix a. how can we tell geometrically that a is diagonal
If we can observe this behavior in the data or the context of the problem, we can infer that a is diagonal.
If the matrix a is diagonal, it means that its eigenvectors are orthogonal to each other. Geometrically, this means that the linear transformation t corresponding to a scales the input vector along the direction of the eigenvectors without rotating it.
More specifically, let λ1 and λ2 be the eigenvalues of a, and let v1 and v2 be the corresponding eigenvectors. If a is diagonal, then we have:
a * v1 = λ1 * v1
a * v2 = λ2 * v2
This means that the linear transformation t scales the input vector v1 by a factor of λ1 along the direction of v1, and scales the input vector v2 by a factor of λ2 along the direction of v2. Since v1 and v2 are orthogonal, this scaling does not rotate the input vector.
Geometrically, this means that the linear transformation t corresponding to a stretches or shrinks the input vector along the direction of the eigenvectors without rotating it. If we can observe this behavior in the data or the context of the problem, we can infer that a is diagonal.
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Please help me proof/solve the following question: Consider the subset of real numbers: A = {x ER: (x – 1)<1} = 1. Prove by contradiction that 2 is the least upper bound for A. 2. Prove by contradiction that 2 is an upper bound for A. 3. Does max(A) exist? If so, what is max(A)? Either way, briefly justify your answer.
Max(A) exists and is equal to 2.
To prove that 2 is the least upper bound for A, we will assume the opposite, i.e., there exists a smaller upper bound for A, say c < 2. Then, by definition of an upper bound, we have x ≤ c for all x ∈ A. In particular, we can choose x = 1 + (c - 1)/2, which satisfies (x - 1) < 1 and x > c, contradicting the assumption that c is an upper bound for A. Therefore, 2 is the least upper bound for A.
To prove that 2 is an upper bound for A, we need to show that x ≤ 2 for all x ∈ A. By definition of A, we have (x - 1) < 1, which implies x < 2. Therefore, 2 is an upper bound for A.
Since 2 is the least upper bound for A and 2 is in A, we have max(A) = 2. This follows from the fact that max(A) is the smallest number that is an upper bound for A, and we have already shown that 2 is the least upper bound for A. Therefore, max(A) exists and is equal to 2.
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When tossing a two-sided, fair coin with one side colored yellow and the other side colored green, determine P(yellow).
yellow over green
green over yellow
2
one half
The calculated value of the probability P(yellow) is 0.5 i.e. one half
How to determine P(yellow).From the question, we have the following parameters that can be used in our computation:
Sections = 2
Color = yellow, and green
Using the above as a guide, we have the following:
Yellow = 1
When the yellow section is selected, we have
P(yellow) = yellow/section
The required probability is
P(yellow) = 1/2
Evaluate
P(yellow) = 0.5
Hence, the value is 0.5
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HighTech Inc. randomly tests its employees about company policies. Last year in the 430 random tests conducted, 20 employees failed the test. (Use t Distribution Table & z Distribution Table.) Required: a. What is the point estimate of the population proportion? (Round your answer to 1 decimal place.) Point estimate of the population proportion % b. What is the margin of error for a 95% confidence interval estimate? (Round your answer to 3 decimal places.) Margin of error c. Compute the 95% confidence interval for the population proportion. (Round your answers to 3 decimal places.) Confidence interval for the population proportion is between and d. Is it reasonable to conclude that 4% of the employees cannot pass the company policy test? No Yes
Answer:
a. The point estimate of the population proportion is calculated as the proportion of employees who failed the test in the sample , which is 20/430. Thus, the point estimate is 4.7%.
b. The margin of error for a 95% confidence interval estimate can be calculated using the following formula:
ME = z*sqrt((p*(1-p))/n)
where: ME = margin of error z = z-score for the desired level of confidence (1.96 for 95% confidence) p = point estimate of the population proportion (0.047) n = sample size (430)
Plugging these values into the formula yields:
ME = 1.96*sqrt((0.047*(1-0.047))/430) = 0.038
Rounding this to 3 decimal places gives the margin of error as 0.038.
c. To compute the 95% confidence interval for the population proportion , you start by finding the bounds of the interval:
Lower bound = point estimate - margin of error
Upper bound = point estimate + margin of error
Plugging in the values gives:
Lower bound = 0.047 - 0.038 = 0.009
Upper bound = 0.047 + 0.038 = 0.085
Rounding these values to 3 decimal places, the 95% confidence interval is between 0.009 and 0.085.
d. No, it is not reasonable to conclude that 4% of the employees cannot pass the company policy test, because the 95% confidence interval for the population proportion includes values below 4%. We can only conclude that it is plausible that less than 4% of the employees cannot pass the test, but we cannot reject the possibility that the proportion is actually higher than 4%.
Step-by-step explanation:
We cannot reject the null hypothesis that the proportion of employees who cannot pass the test is equal to 4%.
a. The point estimate of the population proportion is the sample proportion, which is calculated as the number of employees who failed the test divided by the total number of tests conducted:
point estimate of population proportion = 20/430 = 0.0465 or 4.65%
Therefore, the point estimate of the population proportion is 4.65%.
b. To find the margin of error for a 95% confidence interval estimate, we need to first calculate the standard error of the proportion:
standard error of proportion = sqrt[(point estimate of population proportion) * (1 - point estimate of population proportion) / sample size]
standard error of proportion = sqrt[(0.0465) * (1 - 0.0465) / 430] = 0.020
Then, we can find the margin of error using the z-table for a 95% confidence level:
margin of error = z * (standard error of proportion)
For a 95% confidence level, the z-value is 1.96.
margin of error = 1.96 * 0.020 = 0.039
Therefore, the margin of error for a 95% confidence interval estimate is 0.039.
c. To compute the 95% confidence interval for the population proportion, we use the formula:
point estimate of population proportion ± margin of error
Substituting the values we obtained in parts a and b, we get:
95% confidence interval = 0.0465 ± 0.039
95% confidence interval = (0.008, 0.085)
Therefore, the 95% confidence interval for the population proportion is between 0.008 and 0.085.
d. It is not reasonable to conclude that 4% of the employees cannot pass the company policy test because the lower bound of the confidence interval is 0.008, which is significantly lower than 4%. The confidence interval suggests that the true proportion of employees who cannot pass the test could be as low as 0.8%. Additionally, the point estimate of the population proportion is 4.65%, which is higher than the hypothesized 4%. Therefore, we cannot reject the null hypothesis that the proportion of employees who cannot pass the test is equal to 4%.
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This sentence is from the passage.
"With an estimated 100 billion galaxies in the universe,
each outfitted with some 100 billion to 200 billion
stars, we have a stellar inventory of 10 far-flung suns:
so many stars to yearn toward, so many ways to get
lost in the dark." (Paragraph 5)
What does the comment "so many stars to yearn
toward, so many ways to get lost in the dark" suggest
about efforts to understand the universe?
1. Trying to understand the universe is unlikely
to produce any meaningful results.
O2. Trying to understand the universe is as
tempting to human curiosity as it is daunting.
3. Trying to understand the universe should
begin with getting an accurate count of the
number of stars.
O4. Trying to understand the universe should
become easier when humans are able to
travel greater distances in space.
The meaning of the statement is : Trying to understand the universe is as tempting to human curiosity as it is daunting. Option 2
How to explain the phraseThe use of "so many stars to yearn toward" suggests a multitude of stars existing in the universe, sparking fascination amongst humankind. Herein lies an implication of our innate desire for exploration and comprehension of the cosmos, urging us to seek knowledge about countless celestial bodies.
Yet, the phrase "so many ways to get lost in the dark" hints at the vastness of the universe, teeming with infinite stellar objects that present significant challenges in their study and analysis. It is this immense scope that gives rise to the difficulty of bridging the gaps in our understanding of space.
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A teacher gave a 5 question multiple choice
quiz. Each question had 4 choices to select
from. If the a student completely guessed
on every problem, what is the probability
that they will have less than 3 correct
answers? (CDF)
A)0.896
B)0.088
C)0.984
D)0.264
You want to estimate the average gas price in your city for a gallon of regular gas you take a random sample of the prices from 15 gas stations and find the average costs is $2:42 with a standard deviation of $0.017 create a 99% confidence interval for the mean price for a gallon of gasoline
Answer:
if being beautiful was a crime you would be innocent
Step-by-step explanation:
I NEED HELP WHAT IS 1/4 x 20
21 I NEED HELP QUICK
Answer:
Step-by-step explanation:
the answer is 5/21
Answer: 5/21, I multiplied the two fractions on a piece of paper then simplified the answer.
Solve the system by substitution
y=-4x
y=x-5
Answer:
Point form:
(1,-4)
Equation form:
x=1,y=-4
Step-by-step explanation:
Answer:
Step-by-step explanation:
The solution to the system of equations by substitution is x = 1 and y = -4.
To solve the system of equations by substitution, we can substitute the expression for y from the first equation (-4x) into the second equation (y = x - 5), resulting in -4x = x - 5. By rearranging the equation and solving for x, we get x = 1. Substituting this value back into the first equation, we find y = -4.
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100p + brainliest: TRUE OR FALSE, y=[tex]4^{x}[/tex] and y=[tex]log_{4}[/tex]x are inverses of each other.
Answer:
True
Step-by-step explanation:
If you graph the two equations, you'll notice that they are reflections about the line [tex]y =x[/tex]
Question 7
7. Terrance needs to find the lateral surface area of the box shown below. * 10 points
Assuming that the base is the bottom of the prism, which of the
expressions below will give him the correct lateral surface area?
14.5
A. (14.5)(7)(8.6)
OB. (14.5+7)(8.6)
O C. (14.5+14.5+7+7)(8.6)
O D. (14.5+14.5+7+7)(8.6) + 2(14.5)(7)
8.6
The expression that will give the correct lateral surface area of the rectangular prism = (14.5 + 14.5 + 7 + 7)(8.6)
How to find the Lateral surface area?The lateral surface of an object is for all the sides of the object, excluding its base and top (when they exist). The lateral surface area is defined as the area of the lateral surface. This is different from the total surface area, which is the lateral surface area together with the areas of the base and top.
The lateral surface area is given by the formula here as:
(LSA) = 2(l + w)h
Given the following:
l = 14.5
w = 7
h = 8.6
Thus:
Lateral surface area of the prism = 2(l + w)h = 2(14.5 + 7)8.6
Lateral surface area of the prism = (14.5 + 14.5 + 7 + 7)(8.6)
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What is an obtuse angle?
Answer: An obtuse angle is an angle that measures between 90 and 180 degrees. An obtuse angle is wider than a right angle but narrower than a straight angle.
Step-by-step explanation:
Answer:
Obtuse angle is any angle greater than 90°: Straight angle is an angle measured equal to 180°: Zero angle is an angle measured equal to 0°: Complementary angles are angles whose measures have a sum equal to 90°: Supplementary angles are angles whose measures have a sum equal to 180°.
Find the absolute extrema of f(x) = x^6/7 on the interval (-2, -1]
The absolute maximum of f(x) =
[tex]x^( \frac{6}{7} )[/tex]
on the interval (-2, -1] occurs at x = -1, and the absolute maximum value is 1.
To find the absolute extrema of f(x) on the given interval, we need to evaluate the function at the endpoints and at the critical points within the interval. However, since the function is continuous and differentiable on the interval, the only potential critical point is where its derivative is equal to zero.
Taking the derivative of f(x), we get f'(x) =
[tex](6/7)x^( \frac{1}{7} )[/tex]
Setting this equal to zero, we get x = 0, which is outside the given interval.
Therefore, we only need to evaluate the function at the endpoints of the interval. Plugging in x = -2 and x = -1, we get f(-2) =
[tex](-2)^( \frac{6}{7})[/tex]
≈ 1.419 and f(-1) =
[tex](-1)^( \frac{6}{7} )[/tex]
= 1.
Since f(-1) = 1 is greater than f(-2), we have found the absolute maximum value of the function on the interval (-2, -1], and it occurs at x = -1.
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Determine the equilibrium point for the supply and demand functions given below. D(x) and S(x) represent a price per item and x the quantity of items. Write your answer as an order pair in the form (x,y).p=D(x)=3200/√xp=S(x)=2x√
The equilibrium point is (1600, 80) in the form (x, y).
We need to find the point where the demand function D(x) is equal to the supply function S(x).
The functions are given as follows:
D(x) = 3200/√x
S(x) = 2x√
To find the equilibrium point, we need to set D(x) equal to S(x):
3200/√x = 2x√
Now, let's solve for x:
1. Isolate x by multiplying both sides by √x:
3200 = 2x√ * √x
2. Simplify by squaring both sides:
(3200)^2 = (2x√)^2
3. Perform the squaring:
10,240,000 = 4x^2
4. Divide both sides by 4 to isolate x^2:
2,560,000 = x^2
5. Take the square root of both sides:
x = √2,560,000
x = 1600
Now that we have x, we can find the corresponding price y by plugging x into either D(x) or S(x):
y = D(1600) = 3200/√1600
y = 3200/40
y = 80
So, the equilibrium point is (1600, 80) in the form (x, y).
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Solving by elimination
5x+2y= -3
3x+3y=9
please help me solve x and y
After solving by elimination, the value of x and y are -3 and 6 respectively
Elimination refers to the process in which the variables are calculated by eliminating one of the variables.
Given the equations are:
5x + 2y = -3
3x + 3y = 9
For solving by elimination, we multiply the first equation by 3 and the second by 2.
15x + 6y = -9
6x + 6y = 18
Subtract both equations and we get
9x = -27
x = -3
Put the value in one of the given equation
5 (-3) + 2y = -3
-15 + 2y = -3
2y = -3 + 15
2y = 12
y = 6
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HELP need help ASAP (!!!!!!)
The value of component form and the magnitude of the vector v is,
v = √52
We have to given that;
Two points on the graph are, (3, 5) and (- 1, - 1)
Hence, We can formulate value of component form and the magnitude of the vector v is,
v = √ (x₂ - x₁)² + (y₂ - y₁)²
v = √(- 1 - 3)² + (- 1 - 5)²
v = √16 + 36
v = √52
Thus, The value of component form and the magnitude of the vector v is,
v = √52
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what is the median of this data set 60,70,69,65,62,70,72
Answer:
69
Step-by-step explanation:
first you need to know that median is middle of the data set so put the numbers in order from lowest to highest.
60, 62, 65, 69, 70, 70, 72 now find the number in the middle which is 69. and if there is ever 2 numbers in the middle find the number in between them.
Hope this helps!! Good luck
Exercise 1. Consider a Bernoulli statistical model, where the probability of a success is the parameter of interest and there are n independent observations x =\ x 1 ,...,x 1 \ where x_{i} = 1 with probability 0 and x_{i} = 0 with probability 1 - theta Define the hypotheses H_{0} / theta = theta_{0} and H_{A} / theta = theta_{A} and assume alpha = 0.05 and theta_{0} < theta_{A}
(a) Use Neyman-Pearson's lemma to define the rejection region of the type n overline x > kappa
(b) Let n = 20 theta_{0} = 0.45 , theta_{A} = 0.65 and sum i = 1 to n x i =11 Decide whether or not H_{0} should be iid rejected. Hint: use the fact that n overline X sim Bin(n, theta) when Bernoulli (0). [5]
(a) the rejection region is n overline x > kappa.
(b) kappa = 13/20 = 0.65. Since n overline x = 11 > kappa, we reject the null hypothesis and conclude that there is evidence in favor of the alternative hypothesis that theta > 0.45.
What is hypothesis?
A hypothesis is a proposed explanation or tentative answer to a research question or phenomenon. The null hypothesis is the default position that there is no significant difference between two groups or variables, while the alternative hypothesis proposes that there is a significant difference.
(a) According to Neyman-Pearson's lemma, the likelihood ratio is the most powerful test for a simple vs. a composite hypothesis. The likelihood function for the Bernoulli distribution is:
[tex]L(\theta | x) = \theta^k (1 - \theta)^{(n-k)[/tex]
where k is the number of successes in n trials. The likelihood ratio is:
[tex]\Lambda(x) = L(\theta_A | x) / L(\theta_0 | x)[/tex]
[tex]= (\theta_A^k (1 - \theta_A)^{(n-k)}) / (\theta_0^k (1 - \theta_0)^{(n-k)})[/tex]
Taking the logarithm and simplifying, we get:
[tex]log \Lambda(x) = k log(\theta_A / \theta_0) + (n-k) log((1 - \theta_A) / (1 - \theta_0))[/tex]
To define the rejection region, we need to find the value of kappa such that [tex]P(n overline x > kappa | \theta = \theta_0)[/tex] = alpha, where overline x is the sample mean. Since n overline x sim Bin(n, theta_0), we have:
[tex]P(n overline x > kappa | \theta = \theta_0) = 1 - P(n overline x < = kappa | \theta = \theta_0)\\= 1 - F(n overline x < = kappa | \theta = \theta_0)\\= 1 - sum from i=0 to floor(kappa*n) (n choose i) (\theta_0^i) ((1-\theta_0)^(n-i))[/tex]
where F is the cumulative distribution function of the binomial distribution. We can use a numerical method or a table to find kappa such that [tex]P(n overline x > kappa | \theta = \theta_0) = \alpha.[/tex]
Therefore, the rejection region is n overline x > kappa.
(b) Using the given values, we have k = 11, n = 20, [tex]\theta_0 = 0.45[/tex], and [tex]\theta_A = 0.65[/tex]. The sample mean is overline x = k/n = 0.55. To find kappa, we need to solve:
[tex]P(n overline x > kappa | \theta = \theta_0) = alpha\\1 - F(n overline x < = kappa | \theta = \theta_0) = 0.05\\F(n overline x < = kappa | \theta = \theta_0) = 0.95[/tex]
Using a binomial table, we find that the 0.95th percentile of the binomial distribution with n = 20 and theta = 0.45 is 13. Therefore, kappa = 13/20 = 0.65. Since n overline x = 11 > kappa, we reject the null hypothesis and conclude that there is evidence in favor of the alternative hypothesis that theta > 0.45.
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Hello, pls help. I can't figure out how to do this.
Using the derivative, the expression for f(x) = 8x - 16
How to find the function given the derivative?Since the graph of the derivative of f is shown, The domain of f is the set of all x such that 0 < x < 4. Given that f(2) = 0, write an expression for f(x) in terms of x.
To do this , we proceed as follows.
Now, the f(x) is the area under the curve of f'(x)
So, f(x) = ∫f'(x)dx
So, f'(x) = ∫₀⁴f''(x)dx
Now, ∫₀⁴f''(x)dx = area under the curve of f'(x)
= 1/2 × 4 × 4
= 2 × 4
= 8
So, f'(x) = 8
Now, f(x) = ∫f'(x)dx
f(x) = ∫8dx
f(x) = 8x + c
Now, we have that f(2) = 0
So, substituting this into the equation, we have that
f(2) = 8x + c
0 = 8(2) + c
0 = 16 + c
c = - 16
So, substituting c into f(x), we have that
f(x) = 8x - 16
So, f(x) = 8x - 16
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