The interval of t-values such that c(t) = (cos t, sin t) traces the upper half of the unit circle (in the counter-clockwise direction) is [0, pi].
To see why this is the case, recall that the unit circle is given by the equation x^2 + y^2 = 1, where (x,y) are the coordinates of a point on the circle. The upper half of the unit circle corresponds to the set of points (x,y) where y is positive or zero. We want to find the values of t for which c(t) lies on the upper half of the unit circle.
Using the definition of c(t), we have c(t) = (cos t, sin t). The y-coordinate of c(t) is sin t, so we want, sin t to be positive or zero. Since sin t is positive in the first and second quadrants of the unit circle, and zero at t = 0 and t = pi, we have that c(t) traces the upper half of the unit circle when t is in the interval [0, pi].
To see that c(t) traces the upper half of the unit circle in the counter-clockwise direction, note that as t increases from 0 to pi, c(t) moves counterclockwise around the unit circle, starting at (1,0) and ending at (-1,0). Thus, the interval [0, pi] corresponds to one-half of a full counterclockwise rotation around the unit circle, which is exactly the upper half of the circle.
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Word Users According to a survey by Olsten Staffing Services, the percentage of companies reporting usage of Microsoft Word years since 1984 is given by $$ P(t)=\fra…
Word Users According to a survey by Olsten Staffing Services, the percentage of companies reporting usage of Microsoft Word years since 1984 is given by P(t) = 99.774/1+3.014e
(a) What is the growth rate in the percentage of Microsoft Word users?
(b) Use a graphing utility to graph P=P(t)
(c) What was the percentage of Microsoft Word users in 1990
(d) During what year did the percentage of Microsoft Word users reach 90%
(e) Explain why the numerator given in the model is reasonable. What does it imply?
(a) The growth rate in the percentage of Microsoft Word users is not explicitly provided in the given equation. To find the growth rate, we need the derivative of the function P(t) with respect to time (t).
(b) Graphing the function P(t) requires a graphing utility. The graph will show the percentage of Microsoft Word users over time.
(c) To find the percentage of Microsoft Word users in 1990, substitute t = 1990 into the equation P(t) = 99.774/(1 + 3.014e).
(d) To determine the year when the percentage of Microsoft Word users reached 90%, we need to solve the equation P(t) = 90 for t.
(e) The numerator in the model, 99.774, represents the initial percentage of companies reporting usage of Microsoft Word in 1984. It implies that almost 100% of the surveyed companies were already using Microsoft Word at that time. The model assumes a high initial adoption rate for the software.
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employees at an antique store are hired at a wage of $15 per hour, and they get a $0.75 raise each year. write an equation that shows how a worker's hourly wage, y, depends on the number of years he or she has worked at the store,
To represent the hourly wage of an employee at the antique store, we can use the following equation:
y = 15 + 0.75x
where y represents the worker's hourly wage, and x represents the number of years the employee has worked at the store. In this equation, 15 is the initial hourly wage, and 0.75 is the annual raise.
The equation that shows how a worker's hourly wage, y, depends on the number of years he or she has worked at the store can be written as:
y = 15 + 0.75x
where x represents the number of years the employee has worked at the antique store.
This equation takes into account the starting wage of $15 per hour and the $0.75 raise that the employee receives each year they work at the store.
So, for example, if an employee has worked at the store for 5 years, their hourly wage would be:
y = 15 + 0.75(5) = 18.75
where y represents the worker's hourly wage, and x represents the number of years the employee has worked at the store.
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Example 2
Gabriela plans to carpet her living room, except for the quarter-circle shown in the
corner. That area will be a wood floor where she will put her piano. The radius of a
quarter circle is 8 feet. If carpeting costs $9.55 per square foot, what is the cost of the
carpeting she will use in her living room?
The cost of the carpeting Gabriela will use in her living room is $3,305.47 and Area of living room is 346.41 sq ft
Area of rectangular room = length x width = 25 ft x 16 ft = 400 sq ft
Area of quarter-circle = (1/4) x pi x r^2 = (1/4) x pi x 8^2 = 16 pi sq ft
So the area of the living room is:
Area of living room = Area of rectangular room - Area of quarter-circle
= 400 sq ft - 16 pi sq ft
= 346.41 sq ft
The cost of carpeting this area is:
Cost of carpeting = Area of living room x Cost per square foot
= 346.41 sq ft x $9.55/sq ft
≈ $3,305.47
Therefore, the cost of the carpeting Gabriela will use in her living room is $3,305.47.
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Choose the correct description of the following quadratic formula hen compared to the parent function (x^2)
Answer:
is the answer C is correct bro
Answer: B
Step-by-step explanation:
B. The negative in front indicates direction. It's a quadratic opening down. and the 6 is the stretch
Instead of over 1 down 1 it goes over 1 down 6 from the vertex. so it's skinnier
The length of the shorter leg of a 30-60-90 Special Right Triangle is 17 yd long. How long is the longer leg of the triangle? Question 3 options:
The longer leg of the triangle is 17✓3 yards based on the information of triangle and shorter leg length.
We will use law of sines relating an angle and it's opposite side -
sin shorter angle/shorter side = sin longer angle/longer side
The formula is as per the known fact that wide angle will have wider side opposite to it
Keep the values in formula -
sin 30/17 = sin 60/longer side
Longer side = 17 sin 60/sin 30
Substitute the values of sin 30 and sin 60
Longer side = (17 × ✓3/2)/(1/2)
Longer side = 17✓3 yards.
Thus, the longer leg of the right triangle is 17✓3 yards.
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Question 6 (1 point) CT scans were taken of the brains of Jimmy and 10 members of his family. We want to know if the volume of Jimmy's hippocampus, as measured by the scan, is significantly smaller than those of his family members. Which test should we use? A. one-tailed single-sample t-test B. two-tailed dependent samples t-test C. one-tailed dependent samples t-test D. two-tailed single-sample t-test
The correct answer is option D, the two-tailed single-sample t-test.
To determine which test should be used in this scenario, we need to consider the following factors:
Type of data: The data collected from the CT scans are continuous data.
Sample size: The sample size is small (11 in total).
Relationship between samples: The data from Jimmy's hippocampus is independent from that of his family members.
Based on these factors, we can eliminate options C and B, which both involve dependent samples.
Next, we need to determine whether we are comparing Jimmy's hippocampus volume to a known value or to the average volume of his family members. If we were comparing Jimmy's hippocampus to a known value (e.g. the population average), we would use a one-sample t-test (option A). However, since we are comparing Jimmy's hippocampus volume to the average volume of his family members, we need to use a two-sample t-test.
Therefore, the correct answer is option D, the two-tailed single-sample t-test.
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Write an expression in terms of x, for the perimeter of the quadrilateral. Express your answer in its simplest form
The expression in terms of x, for the perimeter of the quadrilateral is:
22x + 12
How to write an expression in terms of x, for the perimeter of the quadrilateral?The perimeter of an object is the sum of the sides of the the object. Thus, the perimeter of the quadrilateral can be found by adding all the four sides of the quadrilateral. That is:
Perimeter = (3x-5) + (2x+7) + (15x-2) + (2x-3)
Perimeter = 3x-5 + 2x+7 + 15x-2 + 2x-3
Perimeter = 22x + 12
Therefore, the expression in terms of x, for the perimeter is 22x + 12.
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Complete question
Check the image
uppose you have a set with k elements. set up a recurrence relation to count the number of subsets of the set (alternatively, the cardinality of its power set). don't forget your initial condition.
Therefore, we have the recurrence relation: |P(S_k)| = 2 * |P(S_{k-1})|
with the initial condition |P(S_0)| = 1 (since the empty set is the only subset of the empty set).
Sure! To count the number of subsets of a set with k elements, we can use the fact that each element can either be in a subset or not. This gives us two options for each element, so there are 2^k total subsets.
To set up a recurrence relation for this, let S_k denote the set with k elements and P(S_k) denote its power set (the set of all subsets of S_k). Then, we can relate P(S_k) to P(S_{k-1}) by considering whether or not to include the kth element in each subset.
If we don't include the kth element, then each subset of S_{k-1} is also a subset of S_k, so there are |P(S_{k-1})| subsets that don't include the kth element.
If we do include the kth element, then each subset of S_{k-1} can be extended by including the kth element, giving us |P(S_{k-1})| more subsets.
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Statistics from the Port Authority of New York and New Jersey show that 83% of the vehicles using the Lincoln Tunnel that connects New York City and New Jersey use E-ZPass to pay the toll rather than stopping at a toll booth. Twelve cars are randomly selected.
Click here for the Excel Data File
a. How many of the 12 vehicles would you expect to use E-ZPass? (Round your answer to 4 decimal places.)
b. What is the mode of the distribution (the mode is the value with the highest probability)? What is the probability associated with the mode? (Round your answer to 4 decimal places.)
c. What is the probability eight or more of the sampled vehicles use E-ZPass? (Round your answer to 4 decimal places.)
Rounding to 4 decimal places, we expect 9.9600 of the 12 vehicles to use E-ZPass.
Rounding to 4 decimal places, the mode is 10 and the probability associated with the mode is 0.2822.
Rounding to 4 decimal places, the probability of eight or more of the sampled vehicles using E-ZPass is 0.9975.
a. We can use the binomial distribution to model the number of vehicles out of 12 that use E-ZPass. The probability of a vehicle using E-ZPass is 0.83, so the expected number of vehicles out of 12 that use E-ZPass is:
E(X) = np = 12 x 0.83 = 9.96
Rounding to 4 decimal places, we expect 9.9600 of the 12 vehicles to use E-ZPass.
b. The mode of the binomial distribution is the value of X that maximizes the probability mass function (PMF). In this case, the PMF is given by:
P(X = x) = (12 choose x) * 0.83^x * 0.17^(12-x)
We can find the mode by calculating the PMF for each possible value of X and identifying the value(s) with the highest probability. Alternatively, we can use the formula for the mode of the binomial distribution, which is:
mode = floor((n+1)p)
where n is the number of trials and p is the probability of success.
In this case, the mode is:
mode = floor((12+1) x 0.83) = floor(10.08) = 10
The probability associated with the mode is:
P(X = 10) = (12 choose 10) * 0.83^10 * 0.17^2 = 0.2822
Rounding to 4 decimal places, the mode is 10 and the probability associated with the mode is 0.2822.
c. The probability of eight or more of the sampled vehicles using E-ZPass can be found using the binomial distribution:
P(X >= 8) = 1 - P(X < 8) = 1 - P(X <= 7)
Using a binomial calculator or a cumulative binomial table, we can find:
P(X <= 7) = 0.0025
Therefore:
P(X >= 8) = 1 - 0.0025 = 0.9975
Rounding to 4 decimal places, the probability of eight or more of the sampled vehicles using E-ZPass is 0.9975.
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pls help ASAP 20 points
Answer:
A. Bay Side: mean = 17.1, median = 16; Seaside: mean = 19.5, median = 18
B. Bay Side: σ = 8.96, IQR = 12, range = 37; Seaside: σ = 9.03, IQR = 16, range = 31
C. Bay Side has lower center values and less variation.
Step-by-step explanation:
Given stem and leaf plots for 15 class sizes at each of two schools, you want to know (a) their measures of center, (b) their measures of variation, and (c) which would be preferred for lower class size.
A. CenterThe first attachment shows the statistics for Bay Side School. It tells you the measures of center for Bay Side are ...
Mean: 17.1Median: 16The second attachment shows the statistics for Seaside School. The measures of center there are ...
Mean: 19.5Median: 18We note the measures of center indicate smaller classes at Bay Side.
B. VariationFor Bay Side School, the measures of variation are ...
Standard deviation: 8.96IQR: 22 -10 = 12Range: 42 -5 = 37; with outlier removed, 25 -5 = 20For Seaside School, the measures of variation are ...
Standard deviation: 9.03IQR = 27 -11 = 16Range: 36 -5 = 31The measures of variation are generally smaller for Bay Side.
C. Smaller ClassesThe measures of center and the measures of variation both favor Bay Side School as the school of choice for smaller classes.
__
Additional comment
The arithmetic for these descriptive statistics can be tedious and error-prone. It is convenient to let a calculator do it. The lists of data points are given as L1 and L2 for the calculator screens attached. L1 is Bay Side data, and the result of the 1-Var Stats calculation is shown in the first attachment. Seaside data was put in L2, which was used for the calculations shown in the second attachment.
The Q1 and Q3 data values are the 4th lowest and 4th highest data values in each of the lists. The median is the 8th data value, counted from either end.
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Which of the following is equivalent to
5x²+2=-7x
The equivalent expression is (5x + 2)(x + 1)
What are algebraic expressions?Algebraic expressions are mathematical expressions that comprises of variables, terms, coefficients, factors and constants.
Also, note that equivalent expressions are defined as expressions having the same solution but differ in the arrangement of its variables.
From the information given, we have that;
5x²+2=-7x
Put into standard form
5x² + 7x + 2 = 0
To solve the quadratic equation, we have to find the pair factors of the product of 5 and 2 that sum up to 7, we have;
Substitute the values
5x² + 5x + 2x + 2 = 0
group in pairs
(5x² + 5x) + ( 2x + 2 ) = 0
factorize the expression
5x(x + 1) + 2(x + 1) = 0
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Suppose that X~ unif(-1,2), and define Z = e. First, find pdf of Z, and use it to calculate E [Z]. Then, use
the formula for the expected value of a function of RV to find E [Z], and compare with your previous answer.
In order to get an upvote, use legible handwriting
The value of E(Z)=[tex]=e^{\frac{2}{3} }[/tex]
To find the pdf of Z, we need to use the transformation formula for pdfs:
[tex]f_Z(z) = f_X((g)^{(-1)}z ) * |(\frac{d}{dz}) (g)^{-1} (z)|,[/tex]
where [tex]g(x) = e^x[/tex] and [tex](g)^{-1} (z) = ln(z)[/tex] since [tex](e)^{(ln(z)} = z[/tex].
So, we have:
[tex]f_Z(z) = f_X(ln(z)) * |\frac{d}{dz} ln(z)|[/tex]
[tex]=\frac{1}{3z} (for 0 < z < e^2)[/tex]
To find E[Z], we can use the definition of expected value:
[tex]E(Z) = \int\limits {0^{e^{2} } } z f_Z(z) dz \,[/tex]
[tex]E(Z) = \int\limits {0^{e^{2} } } z (\frac{1}{3z} ) dz \,[/tex]
[tex]= (\frac{1}{3} ) \int\limits {0^{e^{2} } } dz \,[/tex]
[tex]= (\frac{1}{3} ) {e^{2} -0 }[/tex]
[tex]=e^{\frac{2}{3} }[/tex]
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12 1 point Suppose P(A) = 0.8, P(B) = 0.5 and P(AUB) = 0.9. Which one of the following statements is true? Events A and B are independent. - Events A and B are both mutually exclusive and independent. The probability of the intersection of A and B is 0.1. Events A and B are mutually exclusive.
Only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
None of the statements is true.
If events A and B were independent, then P(A∩B) = P(A)P(B) = 0.4, which is not equal to 0.1.
If events A and B were mutually exclusive, then P(A∩B) = 0, which is not equal to 0.1.
Therefore, neither of the first two statements is true.
Since P(A∪B) = P(A) + P(B) - P(A∩B), we have P(A∩B) = 0.4, which is not equal to 0.1. Therefore, the third statement is not true.
The only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
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Random samples of 6 male students and 14 female students were asked how many hours a week they exercise with the following results: Males: Sample mean is 5.24 and sample standard deviation is 3.01. Females: Sample mean is 4.22 and sample standard deviation is 2.33. (a) Find a 95% confidence interval for true mean hours of exercise per week in each group. Males. ( ), ( ) Females.( ), ( )
A 95% confidence interval for the true mean hours of exercise per week is (2.08, 8.40) for males and (2.95, 5.49) for females
To find a 95% confidence interval for the true mean hours of exercise per week for each group, we will use the following formula:
[tex]Confidence interval = Sample mean± (t \frac{Sample standard deviation}{\sqrt{n} } )[/tex]
where t is the t-score based on the degrees of freedom and the desired confidence level (95%).
(a) Males:
1. Determine the t-score: For a 95% confidence interval and 6 - 1 = 5 degrees of freedom, the t-score is approximately 2.571.
2. Calculate the margin of error: [tex]2.571 (\frac{3.01}{\sqrt{6} }) = 3.16[/tex]
3. Find the confidence interval: 5.24 ± 3.16 = (2.08, 8.40)
Males: (2.08, 8.40)
(b) Females:
1. Determine the t-score: For a 95% confidence interval and 14 - 1 = 13 degrees of freedom, the t-score is approximately 2.160.
2. Calculate the margin of error: [tex]2.160 (\frac{2.33}{\sqrt{14} }) = 1.27[/tex]
3. Find the confidence interval: 4.22 ± 1.27 = (2.95, 5.49)
Females: (2.95, 5.49)
In conclusion, a 95% confidence interval for the true mean hours of exercise per week is (2.08, 8.40) for males and (2.95, 5.49) for females.
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Tickets for a summer concert cost $81.50 each. Starting next week, the tickets will be on sale for 30% off. What will the sale price of the tickets be?
The sale price of the ticket is $57.05
How to calculate the sale price of the ticket?The ticket for the summer concert is $81.50
The ticket will be on sale for 30% off
The sale price can be calculated as follows
81.50 × 30/100
= 81.50 × 0.3
= 24.45
Sale price= 81.50-24.45
= 57.05
Hence the sale price is $57.05
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Cuánto es 234 entre 14?
12) Suppose f : [a, b] → R point set). is exactly two-to-one (Vy ER, f-'(y) = 0 or f-(y) is a 2 a) Give an example of such a function b) Prove that no such function can be continuous
There is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range.
a) One example of such a function is f(x) = x² on the interval [-1, 1]. Note that f(-1) = f(1) = 1 and for any other value y in the range (0, 1], the preimage [tex]f^{(-1)}[/tex] (y) consists of exactly two points, namely sqrt(y) and -sqrt(y). Similarly, for any y in the range [-1, 0), the preimage f^(-1)(y) consists of exactly two points, namely sqrt(-y) and -sqrt(-y).
b) To prove that no such function can be continuous, suppose for contradiction that f is a continuous function that is exactly two-to-one. Let y be a value in the range of f, and let x1 and x2 be the two distinct points in the preimage f^(-1)(y). Without loss of generality, we can assume that x1 < x2.
Since f is continuous, it must satisfy the intermediate value theorem. This means that for any value z between f(x1) and f(x2), there exists a point x in the interval [x1, x2] such that f(x) = z. In particular, this holds for the value y, since f(x1) = f(x2) = y.
Since f is exactly two-to-one, the preimage [tex]f^{(-1)}[/tex] (y) must consist of exactly two points. Therefore, there is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.
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How much is this? Pls help me I’ll make u brainlesss!
The actual sum of the quarter dollars is determined as $ 1.25.
How much is the quarter dollars?The actual sum of the quarter dollars is calculated by summing up all the individual coins.
For the diagram given, we have a total of;
1 quarter dollar + 1 quarter dollar + 1 quarter dollar + 1 quarter dollar + 1 quarter dollar = 5 quarter dollars
A quarter dollar or 1 quarter dollar = 25 cent = $0.25
The total sum of the quarter dollars is calculated as;
= $0.25 + $0.25 + $0.25 + $0.25 + $0.25
= $ 1.25
Thus, the total sum of the quarter dollars has been obtained in dollar equivalent.
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find the value of x
1. 2x+5=11
2. 3x-6=9
The value ox in each of the equations given are:
1. x = 3 2. x = 5
How to Solve for the Value of x in an Equation?For each of the equation given, we can solve to find the value of x by isolating the variable to one side of the equation while applying the properties of equality.
1. 2x + 5 = 11
2x = 11 - 5 [subtraction property of equality]
2x = 6
x = 6/2 [division property]
x = 3
2. 3x - 6 = 9
3x = 9 + 6 [addition property]
3x = 15
3x/3 = 15/3 [division property]
x = 5
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what is the mass of x divided by 12
The value of expression is,
⇒ x ÷ 12
We have to given that;
The algebraic expression is,
⇒ x divided by 12
Hence, We can formulate;
The value of correct expression is,
⇒ x ÷ 12
⇒ x / 12
Thus, The value of expression is,
⇒ x ÷ 12
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The volume of a spherical balloon is 3054 cm^3. Find the surface area of the balloon to the nearest whole number.
Imagine a sequence of three independent Bernouli trials with success probability p = 1/4. We define the random vector X = [X1, X2, X3]^T, where the three components Xi are independent, identically distributed Bernouli(p = 1/4) random variables. (a) Determine the PMF px(x1, X2, X3) (b) Calculate the covariance matrix Cx. Now suppose Y [Y1, Y2, Y3]^T is a related random vector, whose components are described by: • Y = number of successes in the first trial • Y2 = number of successes in the first two trials . • Y3 = number of successes among all three trials (c) We can express Y as a linear function Y = AX. Determine the matrix A. (d) Calculate the covariance matrix Cx.
Cy = [1 1 1; 0 1 1; 0 0 1] [p(1-p) 0 0; 0 p(1-p) 0; 0 0 p(1-p)] [1 0 0; 1 1 0; 1 1 1]
Cy = [
(a) The probability mass function (PMF) for X is:
px(x1, x2, x3) = P(X1 = x1, X2 = x2, X3 = x3) = P(X1 = x1) * P(X2 = x2) * P(X3 = x3) = (1-p)^(1-x1) * p^(x1) * (1-p)^(1-x2) * p^(x2) * (1-p)^(1-x3) * p^(x3) = p^(x1+x2+x3) * (1-p)^(3-x1-x2-x3)
where p=1/4 is the probability of success and (x1,x2,x3) can take values in {0,1}.
(b) The covariance matrix Cx can be calculated using the formula:
Cx = E[(X - mu)(X - mu)^T]
where mu is the mean vector of X, which is [p, p, p]^T in this case, and E denotes the expected value.
Using the fact that X1, X2, X3 are independent, we have:
E[X1X2] = E[X1]E[X2] = p^2
E[X1X3] = E[X1]E[X3] = p^2
E[X2X3] = E[X2]E[X3] = p^2
E[X1] = E[X2] = E[X3] = p
E[X1^2] = E[X2^2] = E[X3^2] = p
E[(X1-p)(X2-p)] = E[X1X2] - p^2 = 0
E[(X1-p)(X3-p)] = E[X1X3] - p^2 = 0
E[(X2-p)(X3-p)] = E[X2X3] - p^2 = 0
Therefore, the probability matrix Cx is:
Cx = E[(X - mu)(X - mu)^T] = E[X X^T] - mu mu^T
Cx = [p^2+p(1-p) p^2 p^2;
p^2 p^2+p(1-p) p^2;
p^2 p^2 p^2+p(1-p)]
- [p^2 p^2 p^2;
p^2 p^2 p^2;
p^2 p^2 p^2]
Cx = [p(1-p) 0 0;
0 p(1-p) 0;
0 0 p(1-p)]
(c) Y can be expressed as a linear combination of X:
Y = [1 0 0] X1 + [1 1 0] X2 + [1 1 1] X3
Therefore, the matrix A is:
A = [1 0 0;
1 1 0;
1 1 1]
(d) The covariance matrix Cy of Y can be calculated as:
Cy = A Cx A^T
Substituting the values of A and Cx, we get:
Cy = [1 1 1; 0 1 1; 0 0 1] [p(1-p) 0 0; 0 p(1-p) 0; 0 0 p(1-p)] [1 0 0; 1 1 0; 1 1 1]
Cy = [
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A committee of 7 people, which must consist at least 4 men and at least 1 woman, is to be chosen from 10 men and 9 women. (a) Calculate the number of possible committees that can be chosen. (b) 1 woman refuses to be on the committee with a particular man. Calculate the number of possible committees that can be chosen. (c) The committee that is chosen consists of 4 men and 3 women. They queue up randomly in a line for refreshments. Calculate the probability that the no women are next to each other in the queue.
a. The total number of possible committees is [tex]{10\choose 4} \times {14\choose 3} = 1001 \times 364 = 364364[/tex].
b. The number of committees on which the specific man and woman who refuses to serve on the committee are [tex]{8\choose 3} \times {9\choose 4} = 5040[/tex].
c. There are a limited number of combinations in which no two ladies are consecutive that is [tex]4! \times {4\choose 3} 3! = 288[/tex].
(a) To form a committee of 7 people from 10 men and 9 women, we can choose 4 men out of 10 in [tex]{10\choose 4}[/tex] ways, and choose 3 more people from the remaining 5 men and 9 women in [tex]{14\choose 3}[/tex] ways. Therefore, the total number of possible committees is [tex]{10\choose 4} \times {14\choose 3} = 1001 \times 364 = 364364[/tex].
(b) If 1 woman refuses to be on the committee with a particular man, we can count the number of committees that do not include that particular man and that woman, and subtract that number from the total number of possible committees.
There are [tex]{9\choose 1}[/tex] ways to choose the woman who refuses to be on the committee with the particular man, and [tex]{8\choose 3}[/tex] ways to choose the remaining 3 women. There are [tex]{9\choose 4}[/tex] ways to choose 4 men out of the 9 remaining men.
Therefore, the number of committees that include the particular man and the woman who refuses to be on the committee is [tex]{8\choose 3} \times {9\choose 4} = 5040[/tex]. The total number of possible committees is [tex]{10\choose 4} \times {9\choose 3} = 12600[/tex]. Thus, the number of possible committees that do not include the particular man and the woman who refuses to be on the committee is 12600 - 5040 = 7560.
(c) There are [tex]{10\choose 4}[/tex] ways to choose 4 men out of the 10 men, and {9\choose 3} ways to choose 3 women out of the 9 women. The total number of possible ways to form a committee of 4 men and 3 women is [tex]{10\choose 4} \times {9\choose 3} = 12600[/tex]. To calculate the probability that no women are next to each other in the queue, we can count the number of arrangements where no two women are consecutive and divide it by the total number of possible arrangements. We can arrange the 4 men in a line in 4! ways, and arrange the 3 women in the 4 spaces between the men in [tex]{4\choose 3} 3![/tex] ways. Therefore, the number of arrangements where no two women are consecutive is [tex]4! \times {4\choose 3} 3! = 288[/tex]. The probability that no women are next to each other in the queue is 288/12600 = 0.0229, or about 2.29%.
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There are 5 quadratics below. Four of them have two distinct roots each. The other has only one distinct root; find the value of that root.a. 4x^2 + 16x − 9b. 2x^2 + 80x + 400c. x^2 − 6x − 9d. 4x^2 − 12x + 9e. −x^2 + 14x + 49
Answer:
x = 3/2 or 1.5
Step-by-step explanation:
All 5 of the quadratics are in standard form, whose general form is[tex]ax^2+bx+c[/tex]
One of the ways in which we solve quadratic equations is through the quadratic formula which is[tex]x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}[/tex], where x is the root(s)
We can find the total number of solutions a quadratic equation has using the discriminant from the quadratic formula which is[tex]b^2-4ac[/tex]
When the discriminant is greater than 0, there is 2 distinct rootsWhen the discriminant is equal to 0, there is 1 distinct rootWhen the discriminant is less than 0, there are 0 distinct/"real" roots(a.) For 4x^2 + 16x - 9b, 4 is our a value, 16 is our b value and -9 is our c value:
[tex]16^2-4(4)(-9)\\256+144\\400 > 0[/tex]
(b.) For 2x^2 + 80x + 400, 2 is our a value, 80 is our b value, and 400 is our c value:
[tex]80^2-4(2)(400)\\6400-3200\\3200 > 0[/tex]
(c.) For x^2 - 6x - 9, 1 is our a value, -6 is our b value and -9 is our c value
Quick fact: for x^2 or -x^2, there's a 1 or -1 in front of the variable, but it's usually not written because it's a well known mathematical effect and it's assumed we already know this)[tex](-6)^2-4(1)(-9)\\36+36\\72 > 0[/tex]
(d.) For 4x^2 - 12x + 9, 4 is our a value, -12 is our b value, and 9 is our c value:
[tex](-12)^2-4(4)(9)\\144-144\\0=0[/tex]
We don't have to do (e.) because we see that since the discriminant for (d.) equals 0, this is the quadratic with only one distinct solution/rootWe can now solve for this root using the quadratic formula[tex]x=\frac{-(-12)+/-\sqrt{(-12)^2-4(4)(9)} }{2(4)}\\ \\x=\frac{12+/-\sqrt{0} }{8} \\\\x=12/8=3/2\\or\\x=1.5[/tex]
A mower retails for $425. It is put on sale for 23% off. The store manager discounted the mower another $10. To the nearest tenth of a percent, what is the percent decrease in the mower's price?
The percent decrease in the mower's price to the nearest tenth of a percent is 25.3%.
We have,
We need to calculate the initial discount given by the 23% off sale:
Discount
= 0.23 x $425
= $97.75
After the first discount, the mower's price is:
New price
= $425 - $97.75
= $327.25
Then, the store manager discounted it by another $10, so the final price is:
Final price
= $327.25 - $10
= $317.25
The total decrease in price is:
= $425 - $317.25
= $107.75
The percent decrease in the mower's price is:
Percent decrease
= (107.75 / 425) x 100%
= 25.3%
Therefore,
The percent decrease in the mower's price to the nearest tenth of a percent is 25.3%.
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whats the answer
x-y=5
3x-2y=12
need to find the x and the y.
The values of x and y in the system of equations, x - y = 5 and 3x - 2y = 12, are: x = 2 y = -3
How to Solve a System of Equations?Given the system of equations:
x - y = 5 --> eqn. 1
3x - 2y = 12 --> eqn. 2
Rewrite equation 1:
x = 5 + y --> eqn. 3
Substitute x for 5 + y into equation 2:
3(5 + y) - 2y = 12
15 + 3y - 2y = 12
15 + y = 12
y = 12 - 15 [subtraction property]
y = -3
Substitute y = -3 into equation 3:
x = 5 + (-3)
x = 2
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regression statistics multiple r 0.717752328 r square 0.515168405 adjusted r square 0.494754443 standard error 8.735082924 observations 100 anova df ss ms f significance f regression 4 7702.221 1925.555 25.2361 2.9621e-14 residual 95 7248.659 76.302 total 99 14,950.880 step 1 of 2 : how many independent variables are included in the regression model?
Based on the information provided, it appears that there are 4 independent variables included in the regression model.
This is indicated by the "regression" row in the ANOVA table, which shows that there are 4 degrees of freedom (df) for the regression. This is determined by the degrees of freedom (df) for the regression, which is 4. The df for regression represents the number of independent variables in the model.
A statistical method called regression links a dependent variable to one or more independent (explanatory) variables. A regression model can demonstrate whether changes in one or more of the explanatory variables are related to changes in the dependent variable.
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7. Use the following figure to answer questions a-d.
a. Find the perimeter of the figure if the following is true:
a = x + 7
b = 2x + 2
C = 8x
d = x + 10
b. What is the perimeter of the figure in part
(a) if x = 4
c. What is the perimeter of the figure in part (b)
if x = 2 ?
d. Find the perimeter of the figure if the following
is true:
a = x2
b = 4x + 8
c = 2x²
d = x
a) The perimeter of the figure [tex]p = 12x + 19[/tex]
b) The perimeter of the figure when [tex]x = 4[/tex] is 67 units.
c) The perimeter of the figure when[tex]x = 2[/tex] is 43 units
d) The perimeter of the figure is [tex]3x^2 + 5x + 8[/tex] units.
a) The perimeter P of the figure is the sum of the lengths of its sides:
[tex]P = a + b + c + d[/tex]
Substituting the given expressions for a, b, c, and d in terms of x, we get:
[tex]P = (x + 7) + (2x + 2) + (8x) + (x + 10)[/tex]
[tex]= 12x + 19[/tex]
b) Substituting x = 4, we get:
[tex]P = 12x + 19[/tex]
[tex]= 12(4) + 19[/tex]
[tex]= 67[/tex]
Therefore, the perimeter of the figure when [tex]x = 4[/tex] is 67 units.
c) Substituting [tex]x = 2,[/tex] we get:
[tex]P = 12x + 19[/tex]
[tex]= 12(2) + 19[/tex]
[tex]= 43[/tex]
Therefore, the perimeter of the figure when[tex]x = 2[/tex] is 43 units.
d) Substituting the given expressions for a, b, c, and d in terms of x, we get:
[tex]P = x^2 + (4x + 8) + 2x^2 + x[/tex]
[tex]= 3x^2 + 5x + 8[/tex]
Therefore, the perimeter of the figure when[tex]a = x^2, b = 4x + 8, c = 2x^2,[/tex] and [tex]d = x[/tex] is [tex]3x^2 + 5x + 8[/tex] units.
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A pair of standard since dice are rolled. Find the probability of rolling a sum of 12 with these dice.
P(D1 + D2 = 12) = ------
The probability of rolling a sum of 12 with these dice.
P(D1 + D2 = 12) = 1/36.
When two standard six-sided dice are rolled, there are 36 conceivable results (6 x 6 = 36). To calculate the likelihood of rolling an entirety of 12, we ought to decide how numerous of these 36 conceivable results result in a sum of 12.
As it were a way to induce an entirety of 12 is to roll two sixes, so there's as it were one conceivable result that comes about in a whole of 12. Hence, the likelihood of rolling an entirety of 12 with two dice is 1/36, or roughly 0.0278 (adjusted to the closest thousandth).
This is often because the likelihood of rolling a particular number on one pass-on is 1/6, and since we have two dice, we duplicate 1/6 by 1/6 to induce the likelihood of a particular combination, which is 1/36.
In other words, the likelihood of rolling an entirety of 12 is exceptional moo, which makes it an uncommon event when rolling two dice.
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789,506 round to ten thousand
