A health insurance company wants to know the proportion of admitted hospital patients who have a type 2 diabetes at orlando health hospitals. how large of a sample should be taken to estimate the proportion within 6% at 93% confidence?
Answers
To estimate the proportion of admitted hospital patients who have type 2 diabetes at Orlando Health Hospitals within a certain margin of error and confidence level,
we can use the following formula to determine the necessary sample size:
\(n = [(Z^2 * p * q) / E^2] / [1 + ((Z^2 * p * q) / E^2N)]\)
where:
n = sample size
Z = z-score for the desired confidence level (in this case, 1.81 for a 93% confidence level)
p = estimated proportion of patients with type 2 diabetes (unknown)
q = 1 - p
E = margin of error (0.06)
N = population size (unknown)
Since we do not know the estimated proportion of patients with type 2 diabetes or the population size,
we can assume a conservative estimate of p = q = 0.5, which maximizes the sample size required.
Plugging in the values, we get:
\(n = [(1.81^2 * 0.5 * 0.5) / 0.06^2] / [1 + ((1.81^2 * 0.5 * 0.5) / 0.06^2N)]\)
Simplifying, we get:
\(n = 1242.95 / [1 + (2.4 / N)]\)
To satisfy the requirements of the problem, we need to round up to the nearest whole number, so we need a sample size of at least 1243 patients.
Note that if we had a better estimate for p or N, we could use those values in the formula to get a more precise sample size.
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Related Questions
Evaluate the indefinite integral
∫11x (In(8x))2dx.
(Use C as an arbitrary constant.)
Answers
The indefinite inregral solution is `∫11x (In(8x))2dx = 704/3 * ln^3(8x) + C`
To evaluate the indefinite integral `∫11x (In(8x))2dx`, using integration by substitution with u = ln(8x), the following steps should be taken:
Let u = ln(8x) Differentiate both sides of the equation to obtain: `du/dx = 8/x`
Multiply both sides by x to obtain: `x du/dx = 8`
Rewrite the integral in terms of u as follows: `∫ln^2(8x)11xdx = ∫ln^2(u)11x(x du/dx)dx`
Since `x du/dx = 8`, the integral can be rewritten as:`∫ln^2(u)88dx`
Simplifying, we obtain:`88∫ln^2(u)dx` Let `v = ln(u)`, then:`dv/dx = 1/u * du/dx = 1/ln(8x) * 8/x = 8/(x ln(8x))`
Multiply both sides by `dx` to obtain:`dv = 8/(x ln(8x)) dx`
The integral can be rewritten as:`88∫v^2(1/v) * (8/(ln(8x))) dv`
Simplifying further, we obtain:`88 * 8∫v^2 dv`
Evaluating the integral, we obtain:`88 * 8 * v^3/3 + C = 704/3 * ln^3(8x) + C`
Therefore, the answer to the problem is: `∫11x (In(8x))2dx = 704/3 * ln^3(8x) + C`
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3 ounces for every 3/4 cup
Answers
A student scored 120 marks out of 200.
Write her marks scored as a fraction in lowest terms.
Answers
solve the equation 18=n-18
n=_____
Answers
Answer:
36
Step-by-step explanation:
"Using Stokes's Theorem, evaluate the line integral Where C is
the Circle of radius 1 on the z = 1 plane with counterclockwise
orientation when viewed from the positive z axis and centered on
the z ax
Answers
The line integral ∫C F · dr can be evaluated using Stokes's Theorem, which relates line integrals to surface integrals.
Stokes's Theorem states that the line integral of a vector field F around a closed curve C is equal to the surface integral of the curl of F over any surface S bounded by C. Mathematically, it can be written as:
∫C F · dr = ∬S curl(F) · dS
In this case, we have a circle C on the z = 1 plane with a radius of 1 and a counterclockwise orientation when viewed from the positive z-axis. To evaluate the line integral, we need to find the curl of the vector field F and the corresponding surface S.
Since the circle C lies on the z = 1 plane, we can consider the surface S to be the disk bounded by C. The normal vector of this surface points in the positive z-direction. The curl of F can be computed, and then the surface integral can be evaluated over S.
Without knowing the specific vector field F, it is not possible to provide the exact calculations for the line integral. However, by applying Stokes's Theorem, you can use the given information to set up the integral and evaluate it using the appropriate techniques.
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18. In AABC, 44 is a right angle, and m4B 45°. What is the length of BC? If the answer is not an integer, leave it in simplest radical form. The
diagram is not drawn to scale.
054 ft
17√3 ft
17√2 ft
017
Answers
Answer:
17√3
Step-by-step explanation:
In a 45-45-90 triangle, the hypotenuse is √3 times larger than the legs.
17*√3 = 17√3
The reliability factor table provides factors for as many as
three computations when planning and evaluating the results of a
PPS sample. Describe in general terms each of these
computations
Answers
The three computations covered by the reliability factor table are sample size, index of reliability, and index of precision. Sample size deals with the size of the sample being used in order to achieve a desirable level of reliability.
Index of reliability is used to measure the consistency of results achieved over multiple trials. It does this by calculating the total number of items that contribute significantly to the final result. Finally, the index of precision measures the effect size of the sample, which is determined by comparing the results from the sample with the expected results.
The sample size computation gives the researcher an idea of the number of items that should be included in a sample in order to get the most reliable results. This is done by taking into account a number of factors including the variability of the population, the type of measurements used, and the desired level of accuracy.
The index of reliability is commonly calculated by finding the ratio of the number of items contributing significantly to the total result to the total number of items in the sample. This ratio is then multiplied by 100 in order to get a final score.
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Simplify by combining like terms: 3x + 2 + x + 7
A. 5x + 7
B. 3x + 9
C. 4x + 9
D. 6x + 7
Answers
3x and x are like terms
And 2 and 7 are like terms
Hope this helps :D
You read a newspaper article that describes a study of whether stress management can help reduce heart attacks. The 107 subjects all had reduced blood flow to the heart and so were at risk of a heart attack. They were assigned at random to three groups. The article goes on to say:
One group took a four-month stress management program, another underwent a four-month exercise program and the thirst received usual heart care from their personal physicians. In the next three years, only three of the 33 people in the stress management group suffered "cardiac events", defined as a fatal or non-fatal heart attack or a surgical procedure such as a bypass or angioplasty. In the same period, seven of the 34 people in the exercise group and 12 out of the 40 patients in usual care suffered such events.
a. Use the information in the news article to make a two-way table that describes the study results.
b. What are the success rates of the three treatments in preventing cardiac events?
Answers
The success rates of the three treatments in preventing cardiac events are approximately 90.91% for the stress management program, 79.41% for the exercise program, and 70% for usual heart care.
a. Two-way table:
| Treatment | Cardiac Events | No Cardiac Events |
|-----------|-----------------|----------------------|
| Stress Management | 3 | 30 |
| Exercise | 7 | 27 |
| Usual Care | 12 | 28 |
b. The success rates of the three treatments in preventing cardiac events can be calculated as the percentage of people in each group who did not experience a cardiac event.
- Stress management: 90.9% success rate (30/33)
- Exercise: 79.4% success rate (27/34)
- Usual care: 70% success rate (28/40)
Therefore, the stress management program had the highest success rate in preventing cardiac events, followed by the exercise program and then usual care.
Stress management and its impact on reducing heart attacks. Let's analyze the study results and create a two-way table, as well as calculate the success rates of the three treatments in preventing cardiac events.
a. Two-way table:
| Treatment | Cardiac Events | No Cardiac Events | Total |
|--------------------|----------------|-------------------|-------|
| Stress Management | 3 | 30 | 33 |
| Exercise Program | 7 | 27 | 34 |
| Usual Heart Care | 12 | 28 | 40 |
| Total | 22 | 85 | 107 |
b. Success rates of the three treatments in preventing cardiac events:
1. Stress Management:
Success rate = (No Cardiac Events / Total) * 100
Success rate = (30 / 33) * 100
Success rate ≈ 90.91%
2. Exercise Program:
Success rate = (No Cardiac Events / Total) * 100
Success rate = (27 / 34) * 100
Success rate ≈ 79.41%
3. Usual Heart Care:
Success rate = (No Cardiac Events / Total) * 100
Success rate = (28 / 40) * 100
Success rate = 70%
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Whats his mistake and the correct awnser
Answers
Answer:
θ ≈ 35.7°
Step-by-step explanation:
His mistake was in using the wrong trig. ratio
Instead of using cosine he should have used the sine ratio
sinθ = \(\frac{opposite}{hypotenuse}\) = \(\frac{7}{12}\) , then
θ = \(sin^{-1}\) (\(\frac{7}{12}\) ) ≈ 35.7° ( to the nearest tenth )
PLEASE HELPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
MEEEEEEEEEEEE
Answers
Answer:
Part A.
Sample 1 mean is 3.75
Sample 2 mean is 3.5
Sample 3 mean is 4.25
Part B.
The range of sample means is 0.75
Part C.
A.
C.
What would be the result of executing the following code?
int[] x = {0, 1, 2, 3, 4, 5};
Group of answer choices
A-An array of 6 values, all initialized to 0 and referenced by the variable x will be created.
B-An array of 6 values, ranging from 0 through 5 and referenced by the variable x will be created.
C-The variable x will contain the values 0 through 5.
D-A compiler error will occur.
Answers
The result of executing the given code is (B) an array of 6 values, ranging from 0 through 5, will be created and referenced by the variable x.
The code `int[] x = {0, 1, 2, 3, 4, 5};` is initializing an array of integers named `x`. The values inside the curly braces represent the elements of the array. In this case, the values are 0, 1, 2, 3, 4, and 5.
Option (A) is incorrect because the values in the array are not all initialized to 0. Instead, each value corresponds to its respective position in the array.
Option (C) is also incorrect because the variable `x` does not directly store the values 0 through 5. Instead, `x` is a reference to the array that contains those values.
Option (D) is not applicable as the code provided is syntactically correct and will not result in a compiler error.
Therefore, the correct answer is (B) - an array of 6 values, ranging from 0 through 5, will be created and referenced by the variable `x`.
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select all of the following functions for which the extreme value theorem guarantees the existence of an absolute maximum and minimum.Select all that applyf(x) = ln(1 - x) over [0,2]g(x) = ln(1 + x) over [0,2]h(x) = √1 - x over [1,4]k(x) = 1/√1 - x over [1,4]None of the above
Answers
Out of the given functions, g(x), h(x), and k(x) satisfy the extreme value theorem, and guarantee the existence of an absolute maximum and minimum value over the given interval.
Now, let's apply the extreme value theorem to the given functions:
f(x) = ln(1 - x) over [0,2]: This function is not continuous on the given interval since it is undefined at x = 1. Therefore, the extreme value theorem does not apply to this function.
g(x) = ln(1 + x) over [0,2]: This function is continuous on the given interval, and its derivative is always positive, which means it is an increasing function. Therefore, the absolute minimum value occurs at x = 0 and the absolute maximum value occurs at x = 2. Hence, the extreme value theorem applies to this function.
h(x) = √1 - x over [1,4]: This function is continuous on the given interval, and its derivative is always negative, which means it is a decreasing function. Therefore, the absolute maximum value occurs at x = 1 and the absolute minimum value occurs at x = 4. Hence, the extreme value theorem applies to this function.
k(x) = 1/√1 - x over [1,4]: This function is continuous on the given interval, and its derivative is always negative, which means it is a decreasing function. Therefore, the absolute maximum value occurs at x = 1 and the absolute minimum value occurs at x = 4. Hence, the extreme value theorem applies to this function.
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Table A tracks the date when 100 players played a game, where one row corresponds to a player id and the date they opened and played the game. The same player id can have multiple rows if they have played the game across multiple days. In Table A, 90 played for 1 day and 10 played across 2 days. Table B tracks the install date of 99 players who were in Table A for the same game. In Table B, one row corresponds to a player id and the date they installed the game. The same player id can have multiple rows if they have installed the game multiple times. Of the 90 players who played for 1 day, 82 installed once, 7 installed twice and 1 was not in Table B. Of the 10 players who played across 2 days, 9 installed once and 1 installed twice. How many rows would the resulting table have if you took Table A and left (outer) joined with Table B using the player id?
Answers
According to the question the resulting table would have 109 rows.
What is table?Table is a piece of furniture with a flat top and one or more legs, used as a surface for working at, eating from or on which to place things. Tables come in a variety of materials, shapes, and sizes and are used in many different settings, including homes, offices, and commercial establishments.
This is because all 90 players who played for 1 day would be joined, plus the 10 players who played across 2 days. Of the 90 players who played for 1 day, 82 would have 1 row each from the join, 7 would have 2 rows from the join, and 1 would have 0 rows from the join (since it was not in Table B). Of the 10 players who played across 2 days, 9 would have 1 row each from the join, and 1 would have 2 rows from the join. This adds up to a total of 109 rows.
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rework problem 25 in section 4.2 of your text, involving the rolling of a die. assume that the die is weighted so that the probability of a 1 is 0.1, the probability of a 2 is 0.3, the probability of a 3 is 0.1, the probability of a 4 is 0.1, the probability of a 5 is 0.2, and the probability of a 6 is 0.2. you roll the die until the sum of all numbers which have appeared exceeds 3. a random variable x is defined to be the number of rolls. how many different values are possible for the random variable x ?
Answers
Answer: There are 10 different values possible for the random variable x.
Step-by-step explanation:
In this modified problem, the probability of rolling each number is different from a standard fair die. To determine how many different values are possible for the random variable x, we can use the fact that the maximum sum of three rolls is 18 (if all three rolls are 6s), so x cannot be greater than 18.
To find the possible values for x, we can use a probability tree or a table to list all possible outcomes and their probabilities. Here is a table listing the possible values for x and their probabilities:
x Probability
1 0.4
2 0.49
3 0.581
4 0.6639
5 0.73951
6 0.809561
7 0.8746051
8 0.93414459
9 0.987730103
10 1
To calculate these probabilities, we can use the following recursive formula:
P(x=1) = P(sum=1) = 0.1
P(x=k) = P(x=k-1) * (1 - P(sum≤k-1)), for k = 2 to 9
P(x=10) = 1 - P(sum≤9)
Here, P(sum≤k) is the probability of the sum of the first k rolls being less than or equal to 3. We can calculate this probability recursively as well:
P(sum≤1) = P(1) = 0.1
P(sum≤k) = P(sum≤k-1) + P(k), for k = 2 to 18
Here, P(k) is the probability of rolling a k, which is given in the problem statement.
Using these formulas, we can calculate the probabilities for each value of x and list them in the table above. We can see that there are 10 possible values for x, ranging from 1 to 10.
Therefore, there are 10 different values possible for the random variable x.
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Which expressions are equivalent to one-third 8 d startfraction 5 over 7 endfraction? check all that apply. 1 3 8 d 5 7 8 d one-third startfraction 5 over 7 endfraction startfraction 1 over 7 endfraction 8 d startfraction 5 over 3 endfraction one-third startfraction 5 over 7 endfraction 8 d 3 8 d 7 8 d one-third startfraction 7 over 5 endfraction startfraction 5 over 7 endfraction one-third 8 d
Answers
multiply Start Fraction 5 Over 7 End Fraction by 3.
What do math fractions mean?
An element of a whole is a fraction. In mathematics, the number is represented as a quotient, where the numerator and denominator are divided. Both are integers in a simple fraction.
Whether it is in the numerator or denominator, a complex fraction contains a fraction.
The numerator and denominator of a correct fraction are opposite each other.
What is basic fraction theory?
Every fraction has two components: the numerator, which is the actual number of parts, and the denominator, which is the total number of parts.
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Answer:
c is the answer i took the test
Step-by-step explanation:
One supermarket sold 72 cases of Zing soap, but only one-third as many cases of Essence soap. If each case holds 156 bars of soap, how many bars were sold in all
Answers
96*156 =^
There will be 1334 bars sold in all if each case holds 156 bars of soap.
From the question, we are told that one supermarket sold 72 cases of Zing soap, if there is only one-third as many cases of Essence soap, then;
Amount of cases of Essence soap = one-third of 72
Amount of cases of Essence soap = \(\frac{1}{3} \times 72\)
Amount of cases of Essence soap = 24cases
If each case holds 156 bars of soap, then we can write:
1 case = 56 bars of soap
To get the total bars that were sold. we can write:
24cases = x
DIvide both expressions:
\(\frac{1}{24} = \frac{56}{x} \\cross \ multiply\\1 \times x=24 \times 56\\x=1334 bars\)
This shows that 1334 bars were sold in all
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Calculate X, can anyone help please
Answers
answer:
40°
step by step:
93°+47° (i think the circle at the bottom says 47)=140°
180° ( as the angles in a triangle must add up too 180°) - 140° = 40°
What is the surface area of this complex shape?
408 ft
458 ft
545 ft
720 ft
680 ft
1000 ft
GIVING BRAINLIEST TO WHO ANSWERS IT CORRECTLY
Answers
The surface area of the figure will be 458 square feet. Then the correct option is B.
What is the area?The quantity area demonstrates the amount of a sector on a planar or hemispherical cup. Surface area refers to the area of an open surface or the border of a multi-object, whereas the area of a plane region or plane field refers to the area of a form or planar material.
The surface area of the figure is given as,
\(\sf SA = 2 [(12 \times 5) + (12 \times 5) + (7 \times 7) + (5 \times 12)]\)
\(\sf SA = 2 (60 + 60 + 49 + 60)\)
\(\sf SA = 2 (229)\)
\(\sf SA = 458 \ square \ feet\)
The surface area of the figure will be 458 square feet. Then the correct option is B.
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schools in a certain state receive funding based on the number of students who attend the school. to determine the number of students who attend a school, one school day is selected at random and the number of students in attendance that day is counted and used for funding purposes. the daily number of absences at high school a in the state is approximately normally distributed with mean of 120 students and standard deviation of 10.5 students. (a) if more than 140 students are absent on the day the attendance count is taken for funding purposes, the school will lose some of its state funding in the subsequent year. approximately what is the probability that high school a will lose some state funding?
Answers
The probability that high school A will lose some state funding is approximately 0.0287 or 2.87%.
What is probability?Probability is a way to gauge how likely something is to happen. Many things are difficult to predict with absolute certainty.
We can use the normal distribution to approximate the probability that high school A will lose some state funding. Let X be the number of absent students on the selected school day. We know that X follows a normal distribution with mean µ = 120 and standard deviation σ = 10.5.
We need to find the probability that X is greater than 140. To do this, we standardize X by subtracting the mean and dividing by the standard deviation:
Z = (X - µ) / σ = (140 - 120) / 10.5 = 1.90
We can now use a standard normal distribution table or calculator to find the probability that a standard normal variable is greater than 1.90. The result is approximately 0.0287.
Therefore, the probability that high school A will lose some state funding is approximately 0.0287 or 2.87%.
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Math algerbra I believe I have to simplify the equation but I have no idea we’re to start
Answers
Given the equation:
\(2=(1.07)^x\)To solve this equation for x, we need to take the binary logarithm on both sides of the equation:
\(\log _22=\log \lbrack(1.07)^x\rbrack\)There are two important properties of logarithms (binary or any other kind):
\(undefined\)Noah′s family travels 5.26 miles multiplied by 102 miles each day. How many miles do they travel each day?
Answers
Answer:
536.52
Step-by-step explanation:
5.26x102=536.52
Given the diagram below, find the measure of each missing angle. type only the number in the blank
Answers
The measure of angle 1 is 50 degrees
The measure of angle 2 is 40 degrees
The measure of angle 3 is 40 degrees
The measure of angle 4 is 100 degrees
The measure of angle 5 is 40 degrees
What are the measures of the angles?The shapes formed are triangles. A triangle is a three-sided polygon. The sum of the angles is a triangle is 180 degree. A right triangle is a triangle in which one of its angle measure 90 degrees. An isosceles triangle has two opposite sides of equal lengths and equal angles.
1 = 180 - 130 = 50 degrees (this is because angle on a straight line is equal to 180 degrees)
2 = 180 - 90 - 50 = 40 degrees (this is because sum of angles in a triagnle is equal)
3 = 40 degrees ( this is because alternate angles are equal)
5 = 40 degrees ( this is because the triangle is an isosceles triangle)
4 = 180 - 40 - 40 = 100 degrees
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Which angle is formed by a secant and tangent line?ANGA) GSEB) EGSC) NGL
Answers
In the image line segment SE is the tangent. The secant line making an angle with tangent line SE is NS. So the angle is ∠GSE. So option B is the correct answer.
Tangent is a line which intersect with only a single point on the curve. The point where the line meets is the point of tangency. Tangent line is always perpendicular to the radius drawn to the point of tangency. Here the tangent line is SE, point of tangency is S.
Secant lines are the lines which passes through two points of a curve. A chord drawn to the circle is a secant line. Here there are two secant lines NS and NA. Here only NS makes an angle with the tangent line.
So the angle made by a tangent and secant line is ∠GSE.
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The image for the question is attached below.
rearrange this equation to isolate c. a=b(1c−1d)
Answers
The equation, rearranged to isolate c, is: c = (a + bd) / b
In order to isolate c, we need to get c by itself on one side of the equation. Here's how we can do that:
First, we can distribute the b to get:
a = bc - bd
Next, we can add bd to both sides of the equation:
a + bd = bc
Finally, we can divide both sides by b to isolate c:
(a + bd) / b = c
The equation, rearranged to isolate c, is: c = (a + bd) / b
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for a standard normal distribution, find: p(-1.62 < z < 2.01)
Answers
The probability of the interval -1.62 < z < 2.01 in a standard normal distribution is approximately 0.9262 or 92.62%.
In a standard normal distribution, the mean is 0 and the standard deviation is 1. The z-score represents the number of standard deviations a data point is from the mean. To find the probability of a specific interval, we calculate the area under the curve between the corresponding z-values.
Given the interval -1.62 < z < 2.01, we need to find the area under the standard normal curve between these two z-values. This can be done using a standard normal distribution table or by using a statistical software or calculator.
By looking up the z-values in the table or using software, we find the corresponding probabilities: P(z < -1.62) = 0.0526 and P(z < 2.01) = 0.9788.
To find the probability of the interval -1.62 < z < 2.01, we subtract the probability of the lower bound from the probability of the upper bound: P(-1.62 < z < 2.01) = P(z < 2.01) - P(z < -1.62 = 0.9788 - 0.0526 = 0.9262.
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Help me solve these math problems please. I really need help!
Answers
The value of x and y in the equations are are -2 and -1, 1 and 6, 10/6 and -1/3
What is simultaneous equation?Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time.
y=x+1 and y=4x +7
x+1=4x +7
Collecting like terms
x-4x = 7-1
-3x=6
x=-2
For the value y
y=-2+1
y = -1
For equation 2
3x -y =-1
y=-x+5
3x-(-x+5)=-1
3x+x-5=-1
4x=-1+5
4x=4
Therefore; x=1
For the value y
y= 1+5
y=6
For equation 3
x+2x=3
-x+3y = 7
x=3-2y
-(3-2y) +3y =7
-3+2y+3y=7
6y = 7+3
6y==10
y=10/6
For the value of x
x = 3+20/6
Simplifying to have -2/6 -1/3
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hydrostatic weighing uses which statistic to predict the percentage of body fat?
Answers
Hydrostatic weighing uses the statistic known as density to predict the percentage of body fat.
Hydrostatic weighing, also known as underwater weighing or densitometry, is a method used to estimate body composition by measuring the density of an individual's body.
The principle behind hydrostatic weighing is that fat tissue is less dense than lean tissue, such as muscle and bone.
During a hydrostatic weighing test, the individual is submerged in water while their body volume is measured. By comparing the weight on land with the weight in water, the density of the body can be calculated.
The measured density is then used in equations or regression models to estimate the percentage of body fat.
The statistic of density is crucial in this process because it represents the ratio of an individual's mass to their volume.
By accounting for the density of fat and lean tissue, hydrostatic weighing provides an estimate of body fat percentage based on the differences in density between these components.
It is worth noting that while hydrostatic weighing has been widely used as a reference method for body composition assessment, more modern techniques such as dual-energy X-ray absorptiometry (DXA) and bioelectrical impedance analysis (BIA) have gained popularity due to their convenience and accessibility.
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Arian is making bracelets. For each bracelet, it takes 1/10 hour to pick out materials and 1/4 hour to braid it together. How many bracelets can Arian make in 5 hours? Step by step pls
Answers
Answer:
14 Bracelets
Step-by-step explanation:
1/10 is 6 minutes
1/4 is 15 minutes
6+15=21 minutes
60 minutes x5 is 300 minutes
300/21 = 14.3 but you cannot leave one in the middle so the answer is 14
Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = (x - 1)^2 + (y + 2)^2 subject to the constraint x^2 + y^2 = 80, if such values exist
Answers
The global maximum and minimum values of f(x, y) subject to the constraint x^2 + y^2 = 80 are both approximately 15.314, and they occur at the points (2√5, -4√5) and (-2√5, 4√5), respectively.
To use Lagrange multipliers, we need to define the Lagrangian function as follows:
L(x, y, λ) = f(x, y) - λg(x, y)
where g(x, y) is the constraint function, which in this case is g(x, y) = x^2 + y^2 - 80.
So, we have:
L(x, y, λ) = (x - 1)^2 + (y + 2)^2 - λ(x^2 + y^2 - 80)
Next, we need to find the partial derivatives of L with respect to x, y, and λ, and set them equal to 0:
∂L/∂x = 2(x - 1) - 2λx = 0
∂L/∂y = 2(y + 2) - 2λy = 0
∂L/∂λ = x^2 + y^2 - 80 = 0
Simplifying the first two equations, we get:
x(1 - λ) = 1
y(1 - λ) = -2
Dividing these two equations, we get:
x/y = 1/-2 => x = -y/2
Substituting this into the constraint equation, we get:
(y/2)^2 + y^2 = 80 => 5y^2 = 320 => y = ±4√5
Substituting this back into x = -y/2, we get:
x = ∓2√5
So, the critical points are (2√5, -4√5) and (-2√5, 4√5).
To determine whether these are maximum or minimum values, we need to compute the second partial derivatives of L:
∂^2L/∂x^2 = 2 - 2λ
∂^2L/∂y^2 = 2 - 2λ
∂^2L/∂x∂y = 0
At the critical points, we have λ = 1/5, so:
∂^2L/∂x^2 = -2/5 < 0, so (2√5, -4√5) is a local maximum.
∂^2L/∂y^2 = -2/5 < 0, so (-2√5, 4√5) is a local maximum.
To determine whether these are global maximum or minimum values, we need to compare the values of L at these critical points and at the endpoints of the constraint region:
L(2√5, -4√5) ≈ 15.314
L(-2√5, 4√5) ≈ 15.314
L(4√2, 0) = 18
L(-4√2, 0) = 18
So, the global maximum and minimum values of f(x, y) subject to the constraint x^2 + y^2 = 80 are both approximately 15.314, and they occur at the points (2√5, -4√5) and (-2√5, 4√5), respectively.
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Lagrange multipliers : brainly.com/question/30776684
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