Answer:
a) Yes
b) (2,0.8)
c)[tex](2\sqrt2,1), (-2\sqrt2,1)[/tex]
d) [tex]x \in (-\infty,\infty)[/tex]
e) (0,0)
f) (0,0)
Step-by-step explanation:
We are given the following function in the question:
[tex]f(x) = \displaystyle\frac{16x^2}{x^4 + 64}[/tex]
a) We have to check whether given point lies on the function or not.
[tex](-2\sqrt2,1)\\\\f(-2\sqrt2) = \displaystyle\frac{16(-2\sqrt2)^2}{(-2\sqrt2)^4 + 64} = \frac{128}{128} = 1[/tex]
b) Find value of f(x) at x = 2
[tex]f(2) = \displaystyle\frac{16(2)^2}{(2)^4 + 64} =\frac{64}{80}= 0.8[/tex]
Thus, (2,0.8) lies on the graph of given function.
c) We have to find the value of x, when f(x) = 1
[tex]1 = \displaystyle\frac{16x^2}{x^4 + 64}\\\\x^4 -16x^2 + 64 = 0\\(x^2-8)^2 = 0\\x^2 - 8 = 0\\x = \pm 2\sqrt2[/tex]
[tex](2\sqrt2,1), (-2\sqrt2,1)[/tex] lies on he graph of function.
d) Domain is the collection of all values of x for which the function is defined.
[tex]x \in (-\infty,\infty)[/tex]
e) x-intercepts
This is the value of x such that the function is zero.
[tex]0 = \displaystyle\frac{16x^2}{x^4 + 64}\\\\16x^2 = 0\\x = 0[/tex]
f) y-intercept
It is the value of function when x is zero.
[tex]f(0) = \displaystyle\frac{16(0)^2}{(0)^4 + 64} = 0[/tex]
The function passes trough origin.
The solution involves determining if a specific coordinates exist on the graph, calculating the function value for specific x-values, determining x-values for a specific function value, finding the domain of the function, and identifying the x and y intercepts of the function.
Explanation:The function is f(x) = 16x²/(x⁴ + 64).
To check if the point (-2√2, 1) is on the graph, substitute x = -2√2 into f(x). If f(-2√2) equals 1, the point is on the graph.For x = 2, substitute x = 2 into f(x) to get f(2). The point on the graph for this x-value will be (2, f(2)).To find x when f(x) = 1, set f(x) equal to 1 and solve for x. The points on the graph will be (x, 1), where x are the solutions to the equation.The domain of the function f is all real numbers except for x-values that make the denominator zero. Solve x⁴ + 64 = 0 to find x-values to exclude from the domain.To find the x-intercepts of the graph, set f(x) equal to zero and solve for x.The y-intercept of the graph is the value of f(x) at x = 0, which is f(0).Learn more about Function Analysis here:https://brainly.com/question/34156091
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We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric Group of answer choices True False
Answer:
We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.
And if the distribution is not bell shaped or symmetric then we can't use it.
Step-by-step explanation:
Previous concepts
The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). "Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ)".
Solution to the problem
We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.
And if the distribution is not bell shaped or symmetric then we can't use it.
The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False. The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.
The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False.
The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.
It states that approximately 68% of the data is within one standard deviation of the mean, 95% is within two standard deviations, and more than 99% is within three standard deviations.
Therefore, the Empirical Rule is not applied when the data is not bell-shaped or symmetric.
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Delbert wants to make 200 ml of a 5% alcohol solution by mixing a 3% alcohol solution with a 10% alcohol solution. What quantities of each of the two solutions does he need to use?
Answer:
Step-by-step explanation:
a newborn is treated for pulmonary valve stenosis; stretching of the valve opening is accomplished via a percutaneous balloon pulmonary valvuloplasty. what is the root operation?
Answer:
dilation
Step-by-step explanation:
When the pulmonary valve does not work properly, it can interfere with blood flow from the heart to the lungs, as well as force the heart to work harder to carry the blood that is needed to the rest of the body. Some children have heart conditions present at the time of birth and may require repair or replacement of the pulmonary valve, this option has a lower risk of infection, preserves the strength and functioning of the valve, and eliminates the need to take medication.
Final answer:
The root operation for treating pulmonary valve stenosis in newborns using balloon valvuloplasty is the valvuloplasty itself, which is a non-surgical procedure to widen the stenosed heart valve.
Explanation:
The root operation for treating newborn pulmonary valve stenosis with percutaneous balloon pulmonary valvuloplasty is the procedure known as valvuloplasty.
This procedure involves the insertion of a specialized catheter with a balloon at its tip into a blood vessel, usually via the leg, and navigating it to the valve.
The balloon is then inflated to widen the stenosed valve and allow better blood flow. Subsequently, the balloon is deflated and removed, completing the valvuloplasty.
How many three-digit phone prefixes that are used to represent a particular geographic area are possible that have no 0 or 1 in the first or second digits?
There are 640 possible three-digit phone prefixes that do not have 0 or 1 in the first or second digits.
The number of three-digit phone prefixes are possible that have no 0 or 1 in the first or second digits.
Since each digit in a phone number can be a number from 0-9, but the first two digits cannot be 0 or 1, we have 8 choices (2-9) for the first digit, 8 choices (2-9) for the second digit, and 10 choices (0-9) for the third digit because the third digit has no such restriction.
Therefore, the total number of possible phone prefixes is calculated by multiplying the number of choices for each digit,
8 (choices for the first digit) × 8 (choices for the second digit) × 10 (choices for the third digit) = 640 possible three-digit phone prefixes.
You spend $40 for 8 hamburgers and 4 hotdogs at a ballgame. The next game you spend $32 for 3 hamburgers and 10 hotdogs. Write a system of linear equations to represent this scenario.
Answer: the system of linear equations to represent this scenario are
8x + 4y = 40
3x + 10y = 32
Step-by-step explanation:
Let x represent the cost of one hamburger.
Let y represent the cost of one hotdog.
You spend $40 for 8 hamburgers and 4 hotdogs at a ballgame. This means that
8x + 4y = 40 - - - - - - - - - - - - 1
The next game you spend $32 for 3 hamburgers and 10 hotdogs. This means that
3x + 10y = 32- - - - - - - - - - - - 2
Multiplying equation 1 by 3 and equation 2 by 8, it becomes
24x + 12y = 120
24x + 80y = 256
- 68y = - 136
y = - 136 /- 68
y = 2
Substituting y = 2 into equation 2, it becomes
3x + 10y = 32
3x + 10 × 2 = 32
3x = 32 - 20 = 12
x = 12/3 = 4
What is the volume of a cylinder with a height of 2 feet and a radius of 6 feet? Use 3.14 for pi. Enter your answer in the box. ft³
Answer:
[tex]V=226.08\ ft^3[/tex]
Step-by-step explanation:
we know that
The volume of a cylinder is equal to
[tex]V=\pi r^{2} h[/tex]
where
r is the radius of the base of the cylinder
h is the height of the cylinder
we have
[tex]r=6\ ft\\h=2\ ft\\\pi=3.14[/tex]
substitute the given values in the formula
[tex]V=(3.14)(6)^{2}(2)\\ V=226.08\ ft^3[/tex]
When the General Social Survey asked subjects of age 18-25 in 2004 how many people they were in contact with at least once a year, the responses had the following summary statistics: mean: 20.2 mode: 10 standard deviation: 28.7 minimum: 0 Q1: 5 median: 10 Q3: 25 maximum: 300
Answer:
They are in contact with 20 people at least once in a year.
Step-by-step explanation:
We are given the basic summary statistics:
Mean = 20.2
Mode = 10
Median = 10
Standard deviation = 28.7
1st quartile = 5
3rd quartile = 25
Minimum = 0
Maximum = 300.
Out of all these statistics, we know that the median is the middle value or mid-value. Thus, by formula, we have that:
median = n/2. Thus,
==> 10 = n/2
==> n = 2*10 = 20
NB: From the distribution of the summary statistics, we can clearly see that there is evidence of outlier in the series. Thus, median is the most appropriate statistic (because is not affected by outlier) to give a true picture of the data series or sets.
Suppose a test for a virus has a false-positive rate of 0.009 and a false-negative rate of 0.002. Assume that 1.5% of the population has the virus. (a) What is the chance someone from this population will test positive? (Enter exact answer.) (b) If someone tests positive, what is the chance he actually has the virus? (Answer correct to four decimal places.)
Answer:
(a) 0.023835
(b) 0.6281
Step-by-step explanation:
(a) The chance someone from this population will test positive is given by the percentage of people who have the virus multiplied by the change of testing positive (1 - false-negative rate) added to the percentage of people who do not have the virus multiplied by the change of testing positive (false-positive rate)
[tex]P(+) = 0.015*(1-0.002)+(1-0.015)*0.009\\P(+) = 0.023835[/tex]
(b) The probability that someone actually has the virus given that they have tested positive is determined as the probability of having the virus and testing positive divided by the probability of testing positive:
[tex]P(V|+) = \frac{ 0.015*(1-0.002)}{0.023835}\\P(V|+) = 0.6281[/tex]
PLEASE SHOW WORK
Area of the triangle: 1 1/3 yards and 6 yards
[tex]\boxed{A=4yd^2}[/tex]
Explanation:I'll assume the dimensions are:
[tex]base \ (b)=1 \frac{1}{3}yards \\ \\ height \ (h)=6yards[/tex]
First of all, let's convert mixed fraction into improper fraction:
[tex]Add \ whole \ part \ and \ fractional \ part: \\ \\ 1\frac{1}{3}=1+\frac{1}{3} \\ \\ \\ Simplifying: \\ \\ 1\frac{1}{3}=\frac{3+1}{3} =\frac{4}{3}[/tex]
The formula for the area of a triangle is:
[tex]A=\frac{1}{2}b\times h \\ \\ A=\frac{1}{2}(\frac{4}{3})(6) \\ \\ \boxed{A=4yd^2}[/tex]
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(19.-2).(-11, 10) find the slope
Answer:
[tex]\large\boxed{\large\boxed{slope=-2/5}}[/tex]
Explanation:
The problem is: given the points (19,−2) and (−11,10) find the slope of the line that joins them.
The slope of a line is the change in the y-coordinate over the change of the x-coordinate:
slope = rise / run = Δy / ΔxThus:
[tex]slope=[10-(-2)]/[-11-19]\\\\slope=12/(-30)\\\\slope=-12/30[/tex]
Simplify, dividing both numerator and denominator by 6:
[tex]slope=-2/5\leftarrow answer[/tex]
Pretend today is your birthday, and you're hoping for some money. But your grandma is a finance professor and likes making things difficult for you. She tells you that she'll either give you $1,500 today, or give you $550 each year at the end of the year for the next 3 years. If the applicable discount rate is 6%, should you take the $1,500?
Answer:
no, the 550 each year is best offer
Step-by-step explanation:
Answer:
I’ll just stay with the 550 a year
Step-by-step explanation:
A 11-inch candle is lit and burns at a constant rate of 1.3 inches per hour. Let t represent the number of hours since the candle was lit, and suppose f is a function such that f ( t ) represents the remaining length of the candle (in inches) t hours after it was lit. Write a function formula for f .
Answer:
f(t)= 11 in - 1.3 in/h *t
Step-by-step explanation:
defining the length of the candle as L , then since the candle burns at a constant rate , then
-dL/dt = 1.3 in/h = a
therefore
-∫dL = a∫dt
-L(t)=a*t + C , C=constant
at t=0 , the length of the candle is L₀= 11 in ,thus
-L₀=a*0 + C → C= -L₀
replacing the value of C
-L(t)=a*t - L₀
L(t) = L₀ - a*t = 11 in - 1.3 in/h *t
then
f(t)= 11 in - 1.3 in/h *t
Seventy percent of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities. (Enter your answers to three decimal places.) (a) P(all of the next three vehicles inspected pass) (b) P(at least one of the next three inspected fails) (c) P(exactly one of the next three inspected passes)
Based on the given information the required probabilities are as follows:
(a) Probability that all of the next three vehicles pass: 0.343
(b) Probability that at least one of the next three vehicles fails: 0.657
Given that,
70% of all vehicles examined at a certain emissions inspection station pass the inspection.
Successive vehicles pass or fail independently of one another.
(a) To find the probability that all three vehicles pass,
Multiply the individual probabilities.
Given that 70% of vehicles pass,
The probability that a single vehicle passes is 0.7.
So, the probability that all three vehicles pass is 0.7³ = 0.343.
(b) To find the probability that at least one of the next three vehicles fails, We can find the complement probability and subtract it from 1.
The complement probability is the probability that all three vehicles pass, which we calculated in the previous part.
So, the probability that at least one vehicle fails is 1 - 0.343 = 0.657.
(c) To find the probability that exactly one of the next three vehicles passes,
Use the binomial probability formula:
[tex]P(X=k) = ^nC_k p^k (1-p)^{(n-k)}[/tex],
Where n is the number of trials,
k is the number of successes,
p is the probability of success, and
C(n,k) is the combination of n and k.
In this case,
n = 3,
k = 1,
p = 0.7
Plugging these values into the formula,
We get [tex]P(X=1) = ^3C_1\times 0.7 \times (1-0.7)^2[/tex]
P(X=1) = 3x0.7x0.3²
P(X=1) = 0.189
So, the probabilities are:
(a) P(all of the next three vehicles inspected pass) = 0.343
(b) P(at least one of the next three inspected fails) = 0.657
(c) P(exactly one of the next three inspected passes) = 0.189
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To calculate the probabilities, we use the binomial probability formula. The probability of all three vehicles passing is 0.343, the probability of at least one vehicle failing is 0.657, and the probability of exactly one vehicle passing is 0.189.
Explanation:To calculate the probabilities, we can use the binomial probability formula. Let's solve each part step by step:
(a) P(all of the next three vehicles inspected pass):
The probability of each vehicle passing is 70%, so the probability of all three passing is 0.7 x 0.7 x 0.7 = 0.343.
(b) P(at least one of the next three inspected fails):
The probability of a vehicle failing is 30%, so the probability of all three passing is 1 - (0.7 x 0.7 x 0.7) = 0.657.
(c) P(exactly one of the next three inspected passes):
There are three possible scenarios where exactly one vehicle passes: (Pass, Fail, Fail), (Fail, Pass, Fail), (Fail, Fail, Pass). The probability of each scenario is 0.7 x 0.3 x 0.3 = 0.063. Since there are three scenarios, the total probability is 0.063 x 3 = 0.189.
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A forensic scientist uses the functions
G() = 2.56f+47.24 and H(t) = 2.74t+61.22
to find the height of a woman if the scientist is given the length of the woman's
femur bone for the length of the woman's tibia bone t in centimeters. Find the height of a woman whose femur measures 49 centimeters
The height of a woman whose femur measures 49 centimeters is
(Simplify your answer.)
Since we're given the femour's length, we'll have to use the first function.
If we substitute [tex]f=49[/tex] in the expression we have
[tex]g(f)=2.56f+47.24 \implies g(49)=2.56\cdot 49+47.24=125.44+47.24=172.68[/tex]
The height of the woman with a femur length of 49 cm is 172.68 cm.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
G(f) = 2.56f + 47.24
f = 49 cm
G(49) = 2.56 x 49 + 47.24
G(49) = 125.44 + 47.24
G(49) = 172.68 cm
Thus,
The height of the woman is 172.68 cm.
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Tristan and Iseult play a game where they roll a pair of dice alternatingly until Tristan wins by rolling a sum 9 or Iseult wins by rolling a sum of 6.
If Tristan rolled the dice first, what is the probability that Tristan wins?
Answer:
If Tristan rolled the dice first the probability that Tristan wins is 0.474.
Step-by-step explanation:
The probability of an event E is computed using the formula:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}[/tex]
Given:
Tristan and Iseult play a game where they roll a pair of dice alternatively until Tristan wins by rolling a sum 9 or Iseult wins by rolling a sum of 6.
The sample space of rolling a pair of dice consists of a total of 36 outcomes.
The favorable outcomes for Tristan winning is:
S (Tristan) = {(3, 6), (4, 5), (5, 4) and (6, 3)} = 4 outcomes
The favorable outcomes for Iseult winning is:
S (Iseult) = {(1, 5), (2, 4), (3, 3), (4, 2) and (5, 1)} = 5 outcomes
Compute the probability that Tristan wins as follows:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}\\ P(Tristan\ wins)=\frac{4}{36}\\P(T)=\frac{1}{9} \\\approx0.1111[/tex]
Compute the probability that Iseult wins as follows:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}\\ P(Iseult\ wins)=\frac{4}{36}\\P(I)=\frac{1}{9} \\\approx0.1111[/tex]
If Tristan plays first, then the probability that Tristan wins is:
= P(T) + P(T')P(I')P(T) + P(T')P(I')P(T')P(I')P(T)+...
=P(T) + [(1-P(T))(1-P(I))P(T)]+[(1-P(T))(1-P(I))(1-P(T))(1-P(I))P(T)]+...
[tex]=0.1111+(0.8889\times0.8611\times0.1111)+(0.8889\times0.8611\times0.8889\times0.8611\times0.1111))+...\\=0.1111[1+(0.8889\times0.8611)+(0.8889\times0.8611)^{2}+...]\\[/tex]This is an infinite geometric series.
The first term is, a = 0.1111 and the common ratio is, r = (0.8889×0.8611).
The sum of infinite geometric series is:
[tex]S_{\infty}=\frac{a}{1-r}\\ =\frac{0.1111}{1-(0.8889\times0.8611}\\ =0.47364\\\approx0.474[/tex]
Thus, the probability that Tristan wins if he rolled the die first is 0.474.
A company produces a certain product, and each unit of this product may have 3 different types of defects. Let Di, D2,Ds represent the three different kinds of defects.
Suppose further that for each unit produced P(D) = .07 P(D) = .12 P(Ds) = .05 P(D, U Ds) = .14 P(Din D2nDs) = .01
(a) What is the probability that a unit does not have a type 1 defect?
(b) What is the probability that a unit has both a type 2 and 3 defect?
(c) What is the probability that a unit has both a type 2 and 3 defect, but not a type 1 defect?
(d) What is the probability that a unit has at most two defects?
Complete Question
The complete question is shown on the first uploaded image
Answer:
a) 0.88
b) 0.02
c) 0.01
d) 0.99
Step-by-step explanation:
Step one: State the given parameters
[tex]P(D_{1} ) = 0.12[/tex] [tex]P(D_{2} ) = 0.07[/tex]
[tex]P(D_{3} ) = 0.05[/tex] [tex]P (D_{1} U D_{2} ) = 0.13[/tex]
[tex]P(D_{1}n D_{2}n D_{3}) = 0.01[/tex] [tex]P(D_{1} U D_{3}) = 0.14[/tex]
Step 2 : Obtain the probability that a unit does not have a type 1 defect
[tex]P(\frac{}{D_{1} })[/tex] = [tex]1 -P(D_{1} )[/tex]
= [tex]1 - 0.12[/tex]
= 0.88
Step 3 : Obtain the probability that a unit has both type 2 and 3 defect?
The probability of the unit having both type 2 and type 3 defect is denoted as [tex]P(D_{2} n D_{3} )[/tex]
This is calculated as
[tex]P(D_{2}n D_{3}) =P(D_{2} ) + P(D_{3}) - P(D_{2} U D_{3})\\\\ = 0.07 + 0,05 - 0.13[/tex]
= 0.02
Therefore P(D_{2} n D_{3} ) = 0.02
Step 4 : Obtain the probability that the unit has both a type 2 and type 3 ,but not a type 1 defect
Let [tex]P(\frac{}{D_{1}} n D_{2} n D_{3} )[/tex] denote the probability that the unit has both a type 2 and type 3 ,but not a type 1 defect.
This can be calculated as follows :
[tex]P(\frac{}{D_{1}} n D_{2} n D_{3} ) = P(D_{2} n D_{3}) - P(D_{1} n D_{2}nD_{3})[/tex]
= 0.02 - 0.01
= 0.01
Step 4 : Obtain the probability that a unit has at most two defects
P(at most 2 defects) = 1 - P(all three defects)
= [tex]1- P(D_{1} n D_{2}nD_{3})[/tex]
= 1 - 0.01
= 0.99
A publication released the results of a study of the evolution of a certain mineral in the Earth's crust. Researchers estimate that the trace amount of this mineral x in reservoirs follows a uniform distribution ranging between 55 and 1010 parts per million
a. Find E(x) and interpret its value
b. Compute P(2.875 x35)
c. Computn Plx<4.125)
Answer:
a) [tex]E(A)=\frac{1+6}{2}=3.5 ppm[/tex]
b) [tex] P(2.875 <X < 3.5) = F(3.5) -F(2.875) = \frac{3.5-1}{5}- \frac{2.875-1}{5}= \frac{1}{8}= 0.125[/tex]
c) [tex] P(X<4.125) = F(4.125) = \frac{4.125-1}{5}= 0.625[/tex]
Step-by-step explanation:
If we work with the limits defined from 5 to 10 then part b and c from this question not makes sense. If we work with the limits 1 and 6 all the parts for the question makes sense because if we work with 5 and 10 the only thing that we can find is the expected value [tex] E(A) = \frac{5+10}{2}= 7.5[/tex]
Assuming the following correct question : "A publication released the results of a study of the evolution of a certain mineral in the Earth's crust. Researchers estimate that the trace amount of this mineral x in reservoirs follows a uniform distribution ranging between 1 and 6 parts per million"
Solution to the problem
Let A the random variable that represent " amount of the mineral x ". And we know that the distribution of A is given by:
[tex]A\sim Uniform(1 ,6)[/tex]
Part a
For this uniform distribution the expected value is given by [tex]E(X) =\frac{a+b}{2}[/tex] where X is the random variable, and a,b represent the limits for the distribution. If we apply this for our case we got:
[tex]E(A)=\frac{1+6}{2}=3.5 ppm[/tex]
Part b
For this case we can use the cumulative distribution function for the uniform distribution given by:
[tex] F(X=x)= \frac{x-a}{b-a} = \frac{x}{6-1} =\frac{x-1}{5} , 1 \leq X \leq 6[/tex]
And we want this probability:[tex] P(2.875 <X < 3.5) = F(3.5) -F(2.875) = \frac{3.5-1}{5}- \frac{2.875-1}{5}= \frac{1}{8}= 0.125[/tex]Part c
For this case we want this probability:
[tex] P(X<4.125) = F(4.125) = \frac{4.125-1}{5}= 0.625[/tex]
Examine the diagram below and find the value of angle a and angle b
Answer:
50°, 105°
Step-by-step explanation:
a + 130 = 180 ( sum of angles on a straight line)
a = 180 - 130 = 50°
a + b + 45 = 180 ( sum of angles in a Δ)
30 + b + 45 = 180
75 + b = 180
b = 180 - 75 = 105°
On an assembly line that fills 8-ounce cans, a can will be rejected if its weight is less than 7.90 ounces. In a large sample, the mean and the standard deviation of the weight of a can is measured to be 8.05 and 0.05 OZ, respectively. (a) Calculate the percentage of the cans that is expected to be rejected on the basis of the given criterion. (b) If the filling equipment is adjusted so that the average weight becomes 8.10 OZ, but the standard deviation remains 0.05 OZ, calculate the rejection rate (% of cans being rejected) . (c) If the filling equipment is adjusted so that the average weight remains 8.05 OZ, but the standard deviation is reduced to 0.03 OZ, calculate the rejection rate.
The percentage of cans expected to be rejected based on given mean and standard deviation are calculated using the Z score and standard normal distribution table. By adjusting the mean and standard deviation, the rejection rates will change accordingly.
Explanation:This question is about calculating the expected rejection rate of cans based on different conditions using statistical concepts like mean and standard deviation.
(a) The Z score for 7.9 is (7.9 - 8.05) / 0.05 = -3. We use the standard normal distribution table to find the probability of a can having weight less than 7.9 ounces. That's almost 0.1% (0.001), so about 0.1% of cans are expected to be rejected.
(b) After adjusting the average weight to 8.1 oz, the Z score for 7.9 becomes (7.9 - 8.1) / 0.05 = -4. Again, find the probability in the standard normal distribution table, it is almost 0, so the rejection rate will drastically decrease.
(c) When the standard deviation is reduced to 0.03 but mean remains 8.05, the Z score becomes (7.9 - 8.05) / 0.03 = -5. The rejection rate will be extremely close to 0 as per standard normal distribution table reference.
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Find the instantaneous rate of change for the function at the given value. f (x )equals x squared plus 3 x at xequalsnegative 3 The instantaneous rate of change at xequalsnegative 3 is nothing.
Answer:
the instantaneous rate of change of f(x) at x=(-3) is f'(x=(-3))= (-3)
Step-by-step explanation:
for f(x)=x²+3*x
the rate of change of f(x) is
f'(x)=df(x)/dx = 2x + 3
since the derivative of x² is 2x and the derivative of 3*x is 3.
Then at x=(-3)
f'(x=(-3))= 2*(-3) +3 = (-3)
then the instantaneous rate of change of f(x) at x=(-3) is f'(x=(-3))= (-3)
Time spent using e-mail per session is normally distributed, with mu equals 11 minutes and sigma equals 3 minutes. Assume that the time spent per session is normally distributed. Complete parts (a) through (d). a. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes? . 259 (Round to three decimal places as needed.) b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.5 and 11 minutes? . 297 (Round to three decimal places as needed.) c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes? . 68 (Round to three decimal places as needed.)
Answer:
a) 0.259
b) 0.297
c) 0.497
Step-by-step explanation:
To solve this problem, it is important to know the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 11, \sigma = 3[/tex]
a. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?
Here we have that [tex]n = 25, s = \frac{3}{\sqrt{25}} = 0.6[/tex]
This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.
X = 11.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{11.2 - 11}{0.6}[/tex]
[tex]Z = 0.33[/tex]
[tex]Z = 0.33[/tex] has a pvalue of 0.6293.
X = 10.8
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{10.8 - 11}{0.6}[/tex]
[tex]Z = -0.33[/tex]
[tex]Z = -0.33[/tex] has a pvalue of 0.3707.
0.6293 - 0.3707 = 0.2586
0.259 probability, rounded to three decimal places.
b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 10.5 and 11 minutes?
Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So
X = 11
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{11 - 11}{0.6}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5.
X = 10.5
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{10.5 - 11}{0.6}[/tex]
[tex]Z = -0.83[/tex]
[tex]Z = -0.83[/tex] has a pvalue of 0.2033.
0.5 - 0.2033 = 0.2967
0.297, rounded to three decimal places.
c. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 10.8 and 11.2 minutes?
Here we have that [tex]n = 100, s = \frac{3}{\sqrt{100}} = 0.3[/tex]
This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.
X = 11.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{11.2 - 11}{0.3}[/tex]
[tex]Z = 0.67[/tex]
[tex]Z = 0.67[/tex] has a pvalue of 0.7486.
X = 10.8
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{10.8 - 11}{0.3}[/tex]
[tex]Z = -0.67[/tex]
[tex]Z = -0.67[/tex] has a pvalue of 0.2514.
0.7486 - 0.2514 = 0.4972
0.497, rounded to three decimal places.
To find the probability that the sample mean is between eight minutes and 8.5 minutes, calculate the z-scores for both values and find the area under the standard normal distribution curve between these z-scores.
Explanation:To find the probability that the sample mean is between eight minutes and 8.5 minutes, we need to calculate the z-scores for both values and then find the area under the standard normal distribution curve between these two z-scores.
The formula to calculate the z-score is: z = (x - mu) / (sigma / sqrt(n))
where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.
Using the given information, we can calculate the z-scores as follows:
z1 = (8 - 11) / (3 / sqrt(25))
z2 = (8.5 - 11) / (3 / sqrt(25))
Next, we use a standard normal distribution table or a calculator to find the area between these two z-scores, which represents the probability that the sample mean is between eight minutes and 8.5 minutes.
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Howard collected data from a random sample of 600 people in his department asking whether or not they use the company's healthcare . Based on the results, he reports that 48% of the people in his company use the company's healthcare. Why is this statistic misleading
Answer:
This statistic is misleading because Howard surveys only his department, and not membes of all the departments that the company has.
Step-by-step explanation:
This is a common statistics practice, when we want to study something from a population, we find a sample of this population.
However, the sample has to be representative
For example:
I want to estimate the proportion of New York state residents who are Buffalo Bills fans. So i ask, lets say, 1000 randomly selected Buffalo residents wheter they are Buffalo Bills fans, and expand this to the entire population of New York State residents. This is not representative of all New York State residents, just Buffalo residents.
In this problem, we have that:
Howard wants to know the proportion of employees of a company who use the company's healthcare. He asks only his department. However, a company as multiple departments, which leads to the statistics found in Howard's survey being misleading.
Final answer:
Howard's statistic that 48% of people in his company use the healthcare may be misleading due to potential issues like non-representative sampling, biased survey questions, response bias, and lack of current context.
Explanation:
The question about Howard reporting that 48% of the people in his company use the company's healthcare is misleading may have several underlying reasons. Firstly, the sample size and how the sample was obtained are critical factors that determine the reliability and representation of the poll. A random sample of 600 people is generally a good size, but if the sample is not representative of the entire company's demographics, the statistic could be misleading.
Another potential issue could be the phrasing of the survey question. Questions that are biased or leading can influence the way people respond and thus skew the results. In addition, respondents might have reasons to provide socially desirable answers rather than true ones, or they might misremember their actual usage of healthcare. This is an example of response bias, which can lead to inaccurate data.
Finally, it's vital to consider the context, such as whether the data is recent and if there have been any significant changes in the company's healthcare policy or overall employee demographics since the survey. All these factors can contribute to why a statistic might be considered misleading, as it may not accurately reflect the true situation.
A group of 8 friends (5 girls and 3 boys) plans to watch a movie, but they have only 5 tickets. If they randomly decide who will watch the movie, what is the probability that there are at least 3 girls in the group that watch the movie? A. 0.018 B. 0.268 C. 0.536 D. 0.821
Answer:
D. 0.821
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The combinations formula is important to solve this question:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
Desired outcomes
The order is not important. For example, Elisa, Laura and Roze is the same outcome as Roze, Elisa and Laura. This is why we use the combinations formula.
At least 3 girls.
3 girls
3 girls from a set of 5 and 2 boys from a set of 3. So
[tex]C_{5,3}*C_{3,2} = 30[/tex]
4 girls
4 girls from a set of 5 and 1 boy from a set of 3. So
[tex]C_{5,4}*C_{3,1} = 15[/tex]
5 girls
5 girls from a set of 5
[tex]C_{5,5} = 1[/tex]
[tex]D = 30+15+1 = 46[/tex]
Total outcomes
5 from a set of 8. So
[tex]T = C_{8,5} = 56[/tex]
Probability
[tex]P = \frac{D}{T} = \frac{46}{56} = 0.821[/tex]
So the correct answer is:
D. 0.821
Answer:
I got answer choice D
Hope this helps :)
Step-by-step explanation:
Find two values of c in (− π/ 4 , π /4) such that f(c) is equal to the average value of f(x) = 2 cos(2x) on ( − π/ 4 , π/ 4 ). Round your answers to three decimal places.
Answer:
c₁ = 1/2 cos⁻¹ (2/π) = 0.44
c₂ = -1/2 cos⁻¹ (2/π) = -0.44
Step-by-step explanation:
the average value of f(x)=2 cos(2x) on ( − π/ 4 , π/ 4 ) is
av f(x) =∫[2*cos(2x)] dx /(∫dx) between limits of integration − π/ 4 and π/ 4
thus
av f(x) =∫[cos(2x)] dx /(∫dx) = [sin(2 * π/ 4 ) - sin(2 *(- π/ 4 )] /[ π/ 4 - (-π/ 4)]
= 2*sin (π/2) /(π/2) = 4/π
then the average value of f(x) is 4/π . Thus the values of c such that f(c)= av f(x) are
4/π = 2 cos(2c)
2/π = cos(2c)
c = 1/2 cos⁻¹ (2/π) = 0.44
c= 0.44
since the cosine function is symmetrical with respect to the y axis then also c= -0.44 satisfy the equation
thus
c₁ = 1/2 cos⁻¹ (2/π) = 0.44
c₂ = -1/2 cos⁻¹ (2/π) = -0.44
The two values are,
[tex]c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi}) or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi}) [/tex]
Given that,
[tex]f(x)=2cos(2x)[/tex]
[tex]f_{avg}=\frac{1}{\frac{\pi}{2} }\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}2cos(2x)dx\\ =\frac{8}{2\pi} sin\frac{\pi}{2} \\ =\frac{4}{\pi} [/tex]
[tex]f(c)=2cos(2c)=\frac{4}{\pi} \\ cos2c=\frac{2}{\pi} \\ c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\[/tex]
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Answer if you have a big brain 96 POINTS
Answer:
1) 2x+7
2) -3x+11
3) 0.75x-2
4) -2x+0
5) -1.5x+2
6) -4x+16
Step-by-step explanation:
1) y = mx + c
m = 2 when x=1 , y=9
9 = 2(1)+c
c = 7
y = 2x + 7
2) m = -3
When x=4, y= -1
-1 = -3(4) + c
c = -1+12 = 11
y = -3x + 11
3) m = 0.75
When x= -4, y= -5
-5 = 0.75(-4) + c
-5 = -3 + c
c = -2
y = 0.75x - 2
4) m = (y2-y1)/(x2-x1)
m = (2-(-6))/(-1-3) = 8/-4 = -2
y = -2x + c
When x= -1, y= 2
2 = -2(-1) + c
2 = 2 + c
c = 0
y = -2x + 0
5) m = (-10-(-4))/(8-4)
m = (-10+4)/4 = -6/4 = -1.5
y = -1.5x + c
When x= 4, y= -4
-4 = -1.5(4) + c
-4 = -6 + c
c = 2
y = -1.5x + 2
6) m = (-4-4)/(5-3) = -8/2 = -4
When x= 3, y= 4
4 = -4(3) + c
4 = -12 + c
c = 16
y = -4x + 16
Answer:
1) 2x+7
2) -3x+11
3) 0.75x-2
4) -2x+0
5) -1.5x+2
6) -4x+16
Step-by-step explanation:
A particular electronic component is produced at two plants for an electronics manufacturer. Plant A produces 60% of the components used and the remainder are produced by plant B. The proportion of defective components produced at plant A is 1% and the proportion of defective components produced at plant B is 2%.If a component received by the manufacturer is defective, the probability that it was produced at plant A isA. 3/7.
B. 2/7.
C. 4/7.D. 1/7.
Answer:
A) 3/7
Step-by-step explanation:
We start by calculating the following probabilities:
P(produced by A) = 0.6
P(produced by A and defective) = P(A ∩ def) = 0.6*0.01 = 0.006
P(produced by A and not defective) = P(A ∩ not def) = 0.6*0.99 = 0.594
P(produced by B and defective) = P(B ∩ def) = 0.4*0.02 = 0.008
P(produced by B and not defective) = P(B ∩ not def) = 0.4*0.98 = 0.392
The probability that it was produced by A given that it is defective is:
P(A|def) = P(A ∩ def) / P(def) = P(A ∩ def) / (P(A ∩ def)+P(B ∩ def)) = 0.006 / (0.006+0.008) = 6/14 = 3/7
Final answer:
The probability that a defective electronic component was produced by Plant A is 3/7. This was found using conditional probability and the information on production percentages and defect rates from both plants.
Explanation:
The question involves applying the concept of conditional probability to determine the probability that a defective electronic component was produced by Plant A. We need to calculate this using Bayes' theorem with the given probabilities for production and defect rates at both plants.
Calculate the probability of a component being defective, considering both plants.Compute the conditional probability that a defective component comes from Plant A.To answer the question: The probability of a component being defective from either Plant A or Plant B is calculated as follows:
P(Defective) = P(Defective | A)P(A) + P(Defective | B)P(B)
= (0.01)(0.60) + (0.02)(0.40)
= 0.006 + 0.008
= 0.014
Next, the probability that the component was produced at Plant A given that it is defective is:
P(A | Defective) = P(Defective | A)P(A) / P(Defective)
= (0.01)(0.60) / 0.014
= 0.006 / 0.014
= 3/7
Therefore, the correct answer is A. 3/7.
In a box containing 25 cherries, 2 of them are rotten. Susan randomly picks cherries in the box. How many cherries should be picked so that the probability of having exactly 2 rotten cherries among them equals 1/20?
Answer:
Susan should pick 6 cherries from box, so the probability of picking the 2 rotten cherries is 1/20
Step-by-step explanation:
assuming that each cherry is equally probable to be chosen , since each cherry is independent from the others and sampling is done without replacement , the random variable X= number of cherries that are rotten from the picked ones follows a hyper geometrical distribution , where
P(X=k)= C(M,k) * C(N-M, n-k) / C(N,n)
where
N= population size = 25
n= number of picks
M = total number of rotten cherries =2
k = number of rotten cherries picked =2
C( ) = combination
then
1/20=C(2,2)*C(25-2,n-2)/C(25,n) = 1 * (23!/(n-2)!*(25-n)! / (25!/(n!*(25-n)!
1/20 = n!/(n-2)! * 1/(24*25)
24*25/20 = n*(n-1)
n²-n-30 =0
n= (1 +√(1+4*1*30))/2 = 12/2= 6
n=6
then Susan should pick 6 cherries from box, so the probability of picking the 2 rotten cherries is 1/20
To find the number of cherries Susan should pick, use the combination formula and probability calculation. After setting up an equation with probability equal to 1/20, solve for 'x' using trial and error methods.
Explanation:To answer the question, we need to use the combination formula. This formula in the field of statistics is used to find the number of possible combinations that can be obtained by taking 'r' elements from a set of 'n' elements.
The formula is: C(n, r) = n! / [(n - r)! * r!]
Given 25 cherries, 2 of which are rotten, Susan wants to choose some cherries such that the probability of getting exactly 2 rotten cherries is 1/20. Let's assume she needs to pick 'x' cherries.
Now we can write the probability equation: Probability = [C(2, 2) * C(23, x - 2)] / C(25, x) = 1/20
Unfortunately, we can't explicitly solve this equation because it would require checking different values of 'x'. However, it can be solved manually or through trial and error using software or a calculator. After checking different values, you can find the 'x' that satisfies the equation.
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Create a profile for an election with 4 candidates such that, for each of the 4 candidates, there is a positional voting method that selects that candidate as the unique winner.
Answer:
In the profile here, candidate A wins plurality, B wins anti plurality, C wins Borda count, and D wins vote-for-two.
Step-by-step explanation:
2 3 2 4 3
A A B C D
D C D D C
B B C B B
C D A A A
An election profile can be created where each of 4 candidates can win a unique voting method: For plurality where the most top ranked votes win, for Borda count where points are given based on positions, for a positional voting method where points awarded to lower place candidates changes, thereby awarding the win to different candidates.
Explanation:In the context of voting theory and social choice theory, we can create a profile for an election with 4 candidates (let's call them A, B, C and D) where each candidate can win based on different positional voting methods.
Consider this profile for the 20 voters:
6 voters prefer A > B > C > D5 voters prefer B > C > D > A5 voters prefer C > D > A > B4 voters prefer D > A > B > C
For plurality method (where the candidate ranked first by most voters wins), A would win with 6 votes. For Borda count method (where points are given based on ranks), Candidate B would gain the most points and win. For positional method where points are assigned 3 for 1st place, 2 for 2nd place, 1 for last two places, C would win. For positional method where points are assigned differently - 4 points for 1st position, 2 for 2nd, 1 for 3rd and 0 points for fourth, D wins. The unique winner for each method thus satisfies the given condition.
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Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution. What is the probability that sales will be less than 4,300 oranges?
The probability that sales will be less than 4,300 oranges is 21.19%
The z score is used to determine by how many standard deviations the raw score is above or below the mean. The z score is given by:
[tex]z=\frac{x-\mu}{\sigma} \\\\where\ x=raw\ score, \mu=mean, \sigma=standard\ deviation\\\\Given\ that:\\\mu=4700,\sigma=500\\\\For\ x=4300:\\\\z=\frac{4300-4700}{500} =-0.8[/tex]
From the normal distribution table:
P(x < 4300) = P(z < -0.8) = 0.2119 = 21.19%
The probability that sales will be less than 4,300 oranges is 21.19%
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The probability that sales will be less than 4,300 oranges is approximately 21.23%.
Explanation:To find the probability that sales will be less than 4,300 oranges, we need to standardize the value using the z-score formula.
The z-score formula is given by z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 4,300, μ = 4,700, and σ = 500.
Substituting these values into the formula, we get z = (4,300 - 4,700) / 500 = -0.8. We can then use a z-table or a calculator to find the probability associated with a z-score of -0.8, which is approximately 0.2123 or 21.23%.
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5 3/10 + 3 9/10 simplify or mixed number
The answer is 9 1/5 simplified.
Too lazy to make up an explanation or change it to a mixed number even tho it is easy...
Answer:
Step-by-step explanation:
5³/10 +3 9/10
Convert to improper fraction
53/10 +39/10
L. C. M =10
=92/10
=9²/10
=9¹/5