Final answer:
The probability of the first letter being 'A' or 'Z' in the word ARIZONA is 0.143, the probability of it being a vowel is 0.429, and for the letter 'H', which is not present in the word, the probability is 0.
Explanation:
The probability that the first letter is 'A' in a random arrangement of the letters in the word ARIZONA is simply the number of 'A's divided by the total number of letters. Since there is one 'A' out of seven letters, the probability is 1/7, which in decimal form is approximately 0.143, rounded to the nearest thousandth.
Similarly, for the letter 'Z', since there's one 'Z' in the word ARIZONA, the probability is also 1/7, which is about 0.143 when rounded to the nearest thousandth.
The probability that the first letter is a vowel (A, I, or O in ARIZONA) involves adding the probabilities of each individual vowel being the first letter. There are three vowels out of seven letters, so the probability is 3/7, which is approximately 0.429, rounded to the nearest thousandth.
Since the letter 'H' is not in the word ARIZONA, the probability that the first letter is 'H' is 0.
PLEASE HELP I HAVE A FEW MORE LIKE THESE TO GO
Answer:
h(-4) = -3
h(-2) = -3
h(0) = -2
Step-by-step explanation:
We can see that the function's value is -3 for all numbers less than or equals to -2 and the function is (x-1)^2-3 between -1 and 1
So,
h(-4) = -3
As -4 is less than -2, so the value of function will be equal to -3.
h(-2) = -3
Similarly, on x=-2 the value of function will be -3.
And for h(0)
h(0) = (0-1)^2-3
= (-1)^2-3
=1-3
=-2
..
The process by which the bureaucracy produces what is effectively legislation is called
A.
rule-making.
B.
administrative discretion.
C.
activism.
D.
selective implementation.
E.
the formal hearing procedure.
Answer:
Step-by-step explanation:
Great question, it is always good to ask away and get rid of any doubts that you may be having.
This process is called Rule-Making. It is basically when the Federal Government makes regulations on certain topics. These Regulations help advance and set strict boundaries on projects so that people know what they can and cannot do, as well as protect others from being scammed or robbed by these projects. This is all done by the administrative process known as Rule Making.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Which of the following is true for the number of trials for a binomial experiment. Explain your answer. The number of trials in binomial experiment a) can be infinite, b) is unlimited, c) must be fixed.
Answer:
The correct option is C.
Step-by-step explanation:
Consider the provided information.
Four conditions of a binomial experiment:
There should be fixed number of trials
Each trial is independent with respect to the others
The maximum possible outcomes are two
The probability of each outcome remains constant.
Now, observe the provided options:
Option A and B are not possible as they doesn't satisfy the conditions of binomial experiment which is there must be fixed number of trials.
Now observe the option C which state that there must be fixed number of trials, it satisfy the condition of a binomial experiment.
Therefore, the correct option is C.
The number of trials in a binomial experiment must be fixed, with two possible outcomes and independent and identical trial conditions.
The number of trials in a binomial experiment must be fixed as one of the key characteristics of a binomial experiment is that there is a fixed number of trials (n). This characteristic distinguishes it from other types of experiments.
In a binomial experiment, there are only two possible outcomes (success and failure) for each trial, and these probabilities do not change from trial to trial. The trials are independent and identical in conditions.
For example, tossing a fair coin multiple times or conducting a series of independent Bernoulli trials involve a fixed number of trials, making them fit the criteria for a binomial experiment.
Complete each of the following metric relationships:
a. 1 m = cm
b. 1 m = nm
c. 1 mm = m
d. 1 L = mL
Answer:
a. 1 m = 100 cm
b. 1 m = 1.000.000.000 nm
c. 1 mm = 0,001 m
d. 1 L = 1.000 mL
Step-by-step explanation:
1 m means 1 meter, that will be our reference word.
a. A cm is a centimeter, “centi” is the prefix that means “one hundred”. So, 1 meter equals 100 centimeters.
b. It happens the same when we have nanometers. “nano” means “one billion” (nine ceros behind the number one), so 1 meter equals 1.000.000.000 nanometers.
c. In this case, it goes the other way around. If we follow the logic that we use before, 1 meter equals 1.000 millimeters (“milli” means "one million"). But the problem says we have 1 mm, so we have to do a direct rule of 3. If 1000 millimeters equals 1 meter, 1 millimeter equals 1/1000 (or 0,01) meters.
d. Now we have liters instead of meters but is the same logic. 1 liter equals 1000 milliliters.
Find the solution of the initial value problem
dy/dx=(-2x+y)^2-7 ,y(0)=0
Substitute [tex]v(x)=-2x+y(x)[/tex], so that [tex]\dfrac{\mathrm dv}{\mathrm dx}=-2+\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then the ODE is equivalent to
[tex]\dfrac{\mathrm dv}{\mathrm dx}+2=v^2-7[/tex]
which is separable as
[tex]\dfrac{\mathrm dv}{v^2-9}=\mathrm dx[/tex]
Split the left side into partial fractions,
[tex]\dfrac1{v^2-9}=\dfrac16\left(\dfrac1{v-3}-\dfrac1{v+3}\right)[/tex]
so that integrating both sides is trivial and we get
[tex]\dfrac{\ln|v-3|-\ln|v+3|}6=x+C[/tex]
[tex]\ln\left|\dfrac{v-3}{v+3}\right|=6x+C[/tex]
[tex]\dfrac{v-3}{v+3}=Ce^{6x}[/tex]
[tex]\dfrac{v+3-6}{v+3}=1-\dfrac6{v+3}=Ce^{6x}[/tex]
[tex]\dfrac6{v+3}=1-Ce^{6x}[/tex]
[tex]v=\dfrac6{1-Ce^{6x}}-3[/tex]
[tex]-2x+y=\dfrac6{1-Ce^{6x}}-3[/tex]
[tex]y=2x+\dfrac6{1-Ce^{6x}}-3[/tex]
Given the initial condition [tex]y(0)=0[/tex], we find
[tex]0=\dfrac6{1-C}-3\implies C=-1[/tex]
so that the ODE has the particular solution,
[tex]\boxed{y=2x+\dfrac6{1+e^{6x}}-3}[/tex]
A particle is moving along a projectile path at an initial height of 80 feet with an initial speed of 112 feet per second. This can be represented by the function H(t) = −16t2 + 112t + 80. What is the maximum height of the particle? 196 feet 276 feet 392 feet 472 feet
Answer:
276 feet
Step-by-step explanation:
The best way to do this is to complete the square, which puts the quadratic in vertex form. The vertex of a negative parabola, which is what this is, is the highest point of the function...the max value. The k coordinate of the vertex will tell us that highest value. To complete the square, we will first set the quadratic equal to 0, then move the constant over to the other side:
[tex]-16t^2+112t=-80[/tex]
The rule for completing the square is that the leading coefficient must be a positive 1. Ours is a negative 16, so we have to factor out -16:
[tex]-16(t^2-7t)=-80[/tex]
Now the next thing is to take half the linear term, square it, and then add it to both sides. Our linear term is 7, half of that is 7/2. Squaring 7/2 gives you 49/4. So we add 49/9 into the parenthesis on the left. However, we can't forget that there is a -16 out front there that refuses to be ignored. We have to add then (-16)(49/4) onto the right:
[tex]-16(t^2-7t+\frac{49}{4})=-80-196[/tex]
The purpose of this is to create a perfect square binomial that serves as the h value of the vertex (h, k). Stating that perfect square on the left and doing the addition on the right:
[tex]-16(t-\frac{7}{2})^2=-276[/tex]
Now we finalize by moving the constant back over and setting it back equal to y:[tex]y=-16(t-\frac{7}{2})^2+276[/tex]
The vertex is [tex](\frac{7}{2},276)[/tex]
That translates to "at 3.5 seconds the particle is at its max height of 276 feet".
The correct answer is b)276 feet.
The standard form of a quadratic equation is[tex]H(t) = at^2 + bt + c.[/tex]For the given equation, a = -16, b = 112, and c = 80. The vertex (h, k) of the parabola can be found using the formula h = -b/(2a).
Let's calculate h:
h = -b/(2a) = -112 / (2 * -16) = -112 / -32 = 3.
Now we substitute h back into the original equation to find k, the maximum height:
k = H(h) = -16(3.5)^2 + 112(3.5) + 80
Calculating k:
k = -16(12.25) + 392 + 80
k = -196 + 392 + 80
k = 196 + 80
k = 276
So the maximum height is 276 feet, which corresponds to option 2.
How many solutions does the system have? y=3x+2 y=3x-6
Answer:
None
Step-by-step explanation:
The slopes are the same. The y intercepts are 2 and - 6. These lines are parallel which means they never intersect. No intersection means no solution. Just to show you what this means, I graphed this on Desmos for you.
Red: y = 3x - 6
Blue: y = 3x + 2
These two never meet.
Answer:
Step-by-step explanation:
Explanation:
Graph both the equations and read the coordinates of the point of intersection as shown in the graph below:
Assume the readings on thermometers are normally distributed with a mean of 0degreesC and a standard deviation of 1.00degreesC. Find the probability that a randomly selected thermometer reads between negative 1.52 and negative 0.81 and draw a sketch of the region.
Answer:
Step-by-step explanation:
Given : The readings on thermometers are normally distributed with
Mean : [tex]\mu=\ 0[/tex]
Standard deviation : [tex]\sigma= 1[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = -1.52
[tex]z=\dfrac{-1.52-0}{1}=-1.52[/tex]
For x = -0.81
[tex]z=\dfrac{-0.81-0}{1}=-0.81[/tex]
The p-value = [tex]P(-1.52<z<-0.81)=P(z<-0.81)-P(z<-1.52)[/tex]
[tex]0.2089701-0.0642555=0.1447146\approx0.1447[/tex]
Hence, the probability that a randomly selected thermometer reads between negative 1.52 and negative 0.81 = 0.1447
To find the probability, standardize the values using z-scores and find the area under the normal curve between the z-scores.
Explanation:To find the probability that a randomly selected thermometer reads between -1.52 and -0.81, we need to find the area under the normal curve between these two values. First, we need to standardize the values by finding the z-scores for these values using the formula z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation. After finding the z-scores, we can then use the normal distribution table or a calculator to find the area between these z-scores.
The z-score for -1.52 is z = (-1.52 - 0) / 1.00 = -1.52 and the z-score for -0.81 is z = (-0.81 - 0) / 1.00 = -0.81. Using a normal distribution table or a calculator, we can find the area to the left of -1.52 and the area to the left of -0.81. The probability that a randomly selected thermometer reads between -1.52 and -0.81 is the difference between these two areas: P(-1.52 < X < -0.81) = P(X < -0.81) - P(X < -1.52).
Using the normal distribution table or a calculator, we can find that P(X < -0.81) is approximately 0.2123 and P(X < -1.52) is approximately 0.0655. Therefore, the probability that a randomly selected thermometer reads between -1.52 and -0.81 is approximately 0.2123 - 0.0655 = 0.1468, or 14.68%. The sketch of the region would be a shaded area under the standard normal curve between -1.52 and -0.81.
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Write a short description of the pattern in this sequence of six numbers and then follow that pattern to write the next three numbers in the sequence 1/5 2/8 3/11 4/14 5/17 6/20
Answer: The next three terms are [tex]\dfrac{7}{23},\dfrac{8}{26},\dfrac{9}{29}[/tex]
Step-by-step explanation:
Since we have given that
[tex]\dfrac{1}{5},\dfrac{2}{8},\dfrac{3}{11},\dfrac{4}{14},\dfrac{5}{17},\dfrac{6}{20}[/tex]
After analyzing the above pattern, we get that
In numerator , numbers are run in consecutive manner.
In denominator, number is added to 3 to get the next term.
So, the general form would be [tex]\dfrac{n}{3n+2}[/tex]
So, the next three terms would be
[tex]\dfrac{7}{3\times 7+2}=\dfrac{7}{23}\\\\\dfrac{8}{3\times 8+2}=\dfrac{8}{26}\\\\\dfrac{9}{3\times 9+2}=\dfrac{9}{29}[/tex]
Hence, the next three terms are
[tex]\dfrac{7}{23},\dfrac{8}{26},\dfrac{9}{29}[/tex]
We have 7 boys and 3 girls in our church choir. There is an upcoming concert in the local town hall. Unfortunately, we can only have 5 youths in this performance. This performance team of 5 has to be picked randomly from the crew of 7 boys and 3 girls. What is the probability that exactly 4 boys are picked in this team of 5?
Answer:
105/252 = 0.41666...
Step-by-step explanation:
There are (7C4)(3C1) = (35)(3) = 105 ways to choose exactly 4 boys. There are 10C5 = 252 ways to choose 5 youths, so the probability that a randomly chosen team will consist of exactly 4 boys is ...
105/252
_____
nCk = n!/(k!(n-k!))
Answer:
There is a 41.67% probability that exactly 4 boys are picked in this team of 5.
Step-by-step explanation:
The order is not important, so we use the combinations formula.
[tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
Number of desired outcomes.
Four boys and one girl: So
[tex]C_{7,4}*C_{3,1} = \frac{7!}{4!(7-4)!}*\frac{3!}{1!(3-1)!} = 35*3 = 105[/tex]
Number of total outcomes:
Combination of five from a set of 10.
So
[tex]C_{10,5} = \frac{10!}{5!(10-5)!} = 252[/tex]
What is the probability that exactly 4 boys are picked in this team of 5?
[tex]P = \frac{105}{252} = 0.4167[/tex]
There is a 41.67% probability that exactly 4 boys are picked in this team of 5.
. A recent report in a women magazine stated that the average age for women to marry in the United States is now 25 years of age, and that the standard deviation is assumed to be 3.2 years. A sample of 50 U.S. women is randomly selected. Find the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years.
Answer: 0.0136
Step-by-step explanation:
Given : Mean : [tex]\mu=25[/tex]
Standard deviation : [tex]\sigma=3.2[/tex]
Sample size : [tex]n=50[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 24
[tex]z=\dfrac{24-25}{\dfrac{3.2}{\sqrt{50}}}=-2.21[/tex]
The p-value = [tex]P(z\leq-2.21)= 0.0135526\approx0.0136[/tex]
Hence, the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years = 0.0136
Consider the function V=g(x), where g(x) =x(6-2x)(8-2x), with x being the length of a cutout in cm and V being the volume of an open box in cm³, where x must between 0 and 3 cm. Determine the maximum volume of the open box in cm³. Round your answer to two decimal places. Also, indicate or show what process you used to obtain this answer. (Hint: graphing)
Answer:
The maximum volume of the open box is 24.26 cm³
Step-by-step explanation:
The volume of the box is given as [tex]V=g(x)[/tex], where [tex]g(x)=x(6-2x)(8-2x)[/tex] and [tex]0\le x\le3[/tex].
Expand the function to obtain:
[tex]g(x)=4x^3-28x^2+48x[/tex]
Differentiate wrt x to obtain:
[tex]g'(x)=12x^2-56x+48[/tex]
To find the point where the maximum value occurs, we solve
[tex]g'(x)=0[/tex]
[tex]\implies 12x^2-56x+48=0[/tex]
[tex]\implies x=1.13,x=3.54[/tex]
Discard x=3.54 because it is not within the given domain.
Apply the second derivative test to confirm the maximum critical point.
[tex]g''(x)=24x-56[/tex], [tex]g''(1.13)=24(1.13)-56=-28.88\:<\:0[/tex]
This means the maximum volume occurs at [tex]x=1.13[/tex].
Substitute [tex]x=1.13[/tex] into [tex]g(x)=x(6-2x)(8-2x)[/tex] to get the maximum volume.
[tex]g(1.13)=1.13(6-2\times1.13)(8-2\times1.13)=24.26[/tex]
The maximum volume of the open box is 24.26 cm³
See attachment for graph.
Write this trinomial in factored form.
8x² - 9x + 1
Enter the correct answar.
Answer:
(8x-1) (x-1)
Step-by-step explanation:
8x² - 9x + 1
(8x - ) (x - )
We know it is minus because we have -9x
We have +1 so both have to be -
To fill in the blanks we put 1
The only combination is 1*1 =1
(8x-1) (x-1)
Lets check
8x^2 -x -8x +1
(8x^2 -9x+1
Problem 3.2.14a
Show that 2^2x+1 +1 is divisible by 3.
Answer:
The given expression is divisible by 3 for all natural values of x.
Step-by-step explanation:
The given expression is
[tex]2^{2x+1}+1[/tex]
For x=1,
[tex]2^{2(1)+1}+1=2^{3}+18+1=9[/tex]
9 is divisible by 3. So, the given statement is true for x=1.
Assumed that the given statement is true for n=k.
[tex]2^{2k+1}+1[/tex]
This expression is divisible by 3. So,
[tex]2^{2k+1}+1=3n[/tex] .... (1)
For x=k+1
[tex]2^{2(k+1)+1}+1[/tex]
[tex]2^{2k+2+1}+1[/tex]
[tex]2^{(2k+1)+2}+1[/tex]
[tex]2^{2k+1}2^2+1[/tex]
Using equation (1), we get
[tex](3n-1)2^2+1[/tex]
[tex](3n)2^2-2^2+1[/tex]
[tex](3n)2^2-4+1[/tex]
[tex](3n)4-3[/tex]
[tex]3(4n-1)[/tex]
This expression is also divisible by 3.
Therefore the given expression is divisible by 3 for all natural values of x.
Show why it's not and find the 10 combinations that will be in both groups?
rolling 3 dice, are the events A: sum divisible by 3 and B: sum divisible 5 mutually exclusive?
Answer:
Step-by-step explanation:
Given that there are 3 events as
rolling 3 dice, are the events A: sum divisible by 3 and B: sum divisible 5
The sample space will have
(1,1,1)...(6,6,6)
Sum will start with 3 and end with 18
Sum divisible by 3 are 3,6,9..18
Sum divisible by 5 are 5,10,....15
The common element is 15
Hence these two events are not mutually exclusive.
You are designing a rectangular poster to contain 100 in2 of printing with a 4-in margin at the top and bottom and a 1-in margin at each side. What overall dimensions will minimize the amount of paper used?
Answer:
28 inches high by 7 inches wide
Step-by-step explanation:
Let x represent the width of the poster with margins. Then the printable width is (x -2). The printable height will be 100/(x-2), so the overall poster height is ...
height = 100/(x -2) +8 = (8x +84)/(x -2)
The poster's overall area is the product of its width and height, so is ...
A = x(8x +84)/(x -2)
The derivative of this with respect to x is ...
A' = ((16x +84)(x -2) -(8x^2 +84x)(1))/(x -2)^2
This is zero when the numerator is zero, so ...
8x^2 -32x -168 = 0
x^2 -4x -21 = 0 . . . . . . divide by 8
(x +3)(x -7) = 0 . . . . . . . factor
The values of x that make these factors be zero are -3 and +7. The height corresponding to a width of 7 is ...
height = 100/(7 -2) +8 = 28
The amount of paper is minimized when the poster is 7 inches wide by 28 inches tall.
_____
Comment on the problem and solution
You will notice that the poster is 4 times as high as it is wide. It is no accident that this ratio is the ratio of the vertical margin to the horizontal margin. That is, the fraction of the poster devoted to margin is the same in each direction. This is the generic solution to this sort of problem.
Knowing that the margins have a ratio of 4:1 tells you the printable area will have a ratio of 4:1, hence is equivalent to 4 squares, each with an area of 100/4 = 25 square inches. That means the printable area is √25 = 5 inches wide by 4×5 = 20 inches high, so the overall poster area is 28 inches high by 7 inches wide. This arithmetic can be all mental and does not involve derivatives.
To minimize the amount of paper used, the overall dimensions of the rectangular poster should be 102 inches in width and 108 inches in height.
Explanation:To minimize the amount of paper used, we need to find the dimensions of the rectangle that will enclose 100 in2 of printing. Since there is a 4-inch margin at the top and bottom, the height of the rectangle will be the printing area plus the margins, which is 100 + 4 + 4 = 108 inches. Similarly, there is a 1-inch margin on each side, so the width of the rectangle will be the printing area plus the margins, which is 100 + 1 + 1 = 102 inches. Therefore, the overall dimensions of the rectangle that will minimize the amount of paper used are 102 inches in width and 108 inches in height.
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The time required for an automotive center to complete an oil change service on an automobile approximately follows a normal distribution, with a mean of 19 minutes and a standard deviation of 3 minutes. (a) The automotive center guarantees customers that the service will take no longer than 20 minutes. If it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half-price? (b) If the automotive center does not want to give the discount to more than 2% of its customers, how long should it make the guaranteed time limit?
Answer:
We have a normal distribution with a mean of 19 minutes and a standard deviation of 3 minutes. To solve the problem we're going to need the help of a calculator:
P(z>20) = 0.3694
Therefore, the percentage of costumbers that will receive the service for half-price is: 36.94%.
Also, we've found that p(z>25.16) = 0.02. Therefore, if they only want to offer half-price discount to only 2% of its costumber, the time limit should be 25.16 minutes.
Times for a surgical procedure are normally distributed. There are two methods. Method A has a mean of 33 minutes and a standard deviation of 8 minutes, while method B has a mean of 37 minutes and a standard deviation of 4.0 minutes. (a) Which procedure is preferred if the procedure must be completed within 34 minutes?
Answer:
Method A.
Step-by-step explanation:
For solving this question we need to find out the z-scores for both methods,
Since, the z-score formula is,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where, [tex]\mu[/tex] is mean,
[tex]\sigma[/tex] is standard deviation,
Given,
For method A,
[tex]\mu = 33[/tex]
[tex]\sigma=8[/tex]
Thus, the z score for 34 is,
[tex]z_1=\frac{34-33}{8}=0.125[/tex]
While, for method B,
[tex]\mu = 37[/tex]
[tex]\sigma = 4[/tex]
Thus, the z score for 34 is,
[tex]z_2=\frac{34-37}{4}=-0.75[/tex],
Since, [tex]z_1 > z_2[/tex]
Hence, method A is preferred if the procedure must be completed within 34 minutes.
Comparison of two normal distribution can be done via intermediary standard normal distribution. The procedure to be preferred for getting the procedure completed within 34 minutes is: Method A
How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] )
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
Firstly, we need to figure out what the problem is asking, and a method which we can apply. Two normal distributions have to be compared. We can convert them to standard normal distribution for comparison. Then, we will get the p-value for 34 minutes(converted to standard normal variate's value) which will tell about the probability of obtaining time as 34 minutes(or under it)(this can be obtained with p value) in method A or B. The more the probability is there, the more chances for that method would be for getting completed within 34 minutes (compared to other method).
For method A:Let X = time taken for completion of surgical procedure by method A,
Then, by given data, we have: [tex]X \sim N(\mu = 33, \sigma = 8)[/tex]
The probability that X will fall within value 34 is [tex]P(X \leq 34)[/tex]
Converting this whole thing to standard normal distribution, we get the needed probability as:
[tex]P(X \leq 34) = P(Z = \dfrac{X - \mu}{\sigma} \leq \dfrac{34 - 33}{8} ) = P(Z \leq 0.15)[/tex]
From the z-tables, the p value for Z = 0.15 is 0.5596
Thus, we get:
[tex]P(X \leq 34) = P(Z \leq 0.15 ) \approx 0.5596[/tex]
For method B:Let Y = time taken for completion of surgical procedure by method B,
Then, by given data, we have: [tex]Y \sim N(\mu = 37, \sigma = 4)[/tex]
The probability that X will fall within value 34 is [tex]P(Y \leq 34)[/tex]
Converting this whole thing to standard normal distribution, we get the needed probability as:
[tex]P(Y \leq 34) = P(Z = \dfrac{Y - \mu}{\sigma} \leq \dfrac{34 - 37}{4} ) = P(Z \leq -0.25)[/tex]
From the z-tables, the p value for Z = -0.25 is 0.4013
Thus, we get:
[tex]P(Y \leq 34) = P(Z \leq -0.25 ) \approx 0.4013[/tex]
Thus, we see that:
P(method A will make surgical procedure last within 34 minutes) = 0.5596 > P(method B will make surgical procedure last within 34 minutes) = 0.4013
Thus, method A should be preferred, as there is higher chances for method A to get the surgery completed within 34 minutes than method B.
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A market research firm conducts telephone surveys with a 40% historical response rate. What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions? In other words, what is the probability that the sample proportion will be at least 150/400 = .375 (to 4 decimals)?
Answer:
100%
Step-by-step explanation:
If there is a 40% response rate and there are 400 numbers, that means that 160 people will respond.
Suppose 56% of the registered voters in a country are Republican. If a sample of 447 voters is selected, what is the probability that the sample proportion of Republicans will be less than 60%? Round your answer to four decimal places.
Answer: 0.9554
Step-by-step explanation:
Given : The proportion of the registered voters in a country are Republican = P=0.56
The number of voters = 447
The test statistic for proportion :-
[tex]z=\dfrac{p-P}{\sqrt{\dfrac{P(1-P)}{n}}}[/tex]
For p= 0.60
[tex]z=\dfrac{0.60-0.56}{\sqrt{\dfrac{0.56(1-0.56)}{447}}}\approx1.70[/tex]
Now, the probability that the sample proportion of Republicans will be less than 60% (by using the standard normal distribution table):-
[tex]P(x<0.60)=P(z<1.70)=0.9554345\approx0.9554[/tex]
Hence, the probability that the sample proportion of Republicans will be less than 60% = 0.9554
ydx+(y-x)dy=0
Please be as thorough as possible when explaining this, I'm struggling very much trying to solve ODE's
Answer: The required solution of the given differential equation is
[tex]x+y\log y=Cy.[/tex]
Step-by-step explanation: We are given to solve the following ordinary differential equation :
[tex]ydx+(y-x)dy=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We will be using the following formulas for integration and differentiation :
[tex](i)~d\left(\dfrac{x}{y}\right)=\dfrac{ydx-xdy}{y^2},\\\\\\(ii)~\int\dfrac{1}{y}dy=\log y.[/tex]
From equation (i), we have
[tex]ydx+(y-x)dy=0\\\\\Rightarrow ydx+ydy-xdy=0\\\\\\\Rightarrow \dfrac{ydx+ydy-xdy}{y^2}=\dfrac{0}{y^2}~~~~~~~~~~~~~~~~~~~~[\textup{dividing both sides by }y^2]\\\\\\\Rightarrow \dfrac{ydx-xdy}{y^2}+\dfrac{1}{y}dy=0\\\\\\\Rightarrow d\left(\dfrac{x}{y}\right)+d(\log y)=0.[/tex]
Integrating the above equation on both sides, we get
[tex]\int d\left(\dfrac{x}{y}\right)+\int d(\log y)=C~~~~~~~[\textup{where C is the constant of integration}]\\\\\\\Rightarrow \dfrac{x}{y}+\log y=C\\\\\Rightarrow x+y\log y=Cy.[/tex].
Thus, the required solution of the given differential equation is
[tex]x+y\log y=Cy.[/tex].
You wish to test the following claim ( H 1 ) at a significance level of α = 0.025 . H o : μ = 50.6 H 1 : μ > 50.6 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 10 with a mean of ¯ x = 54.6 and a standard deviation of s = 10.5 . What is the critical value for this test
Answer: 1.205
Step-by-step explanation:
Given : Significance level : [tex]\alpha=0.025[/tex]
[tex]H_0:\mu=50.6\\\\H_1:\mu>50.6[/tex]
We assume that population is normally distributed.
The sample size : [tex]n=10[/tex], which is less than 30 , so we apply t-test.
Mean : [tex]\overline{x}=54.6[/tex]
Standard deviation : [tex]\sigma=10.5[/tex]
The test statistic for population mean is given by :-
[tex]t=\dfrac{\overline{x}-\mu_0}{\dfrac{\sigma}{\sqrt{n}}}\\\\=\dfrac{54.6-50.6}{\dfrac{10.5}{\sqrt{10}}}=1.20467720387\approx1.205[/tex]
Hence, the critical value = 1.205
If S is countable and nonempty, prove their exist a surjection g: N --> S
Math: Analysis and Proof
Answer with Step-by-step explanation:
We are given S be any set which is countable and nonempty.
We have to prove that their exist a surjection g:N[tex]\rightarrow S[/tex]
Surjection: It is also called onto function .When cardinality of domain set is greater than or equal to cardinality of range set then the function is onto
Cardinality of natural numbers set =[tex]\chi_0[/tex]( Aleph naught)
There are two cases
1.S is finite nonempty set
2.S is countably infinite set
1.When S is finite set and nonempty set
Then cardinality of set S is any constant number which is less than the cardinality of set of natura number
Therefore, their exist a surjection from N to S.
2.When S is countably infinite set and cardinality with aleph naught
Then cardinality of set S is equal to cardinality of set of natural .Therefore, their exist a surjection from N to S.
Hence, proved
Calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. (Round your answers to four decimal places.) f(x) = cos x, x1 = 1.6
Answer:
x₃=1.599997=1.6
Step-by-step explanation:
f(x)=cos x, x₁=1.6
[tex]f'(x)=\frac{d}{dx}cos\ x\\=-sin\ x[/tex]
First iteration
At x₁=1.6
f(x₁)=f(1.6)
=cos(1.6)
=-0.0292
f'(x₁)=f'(1.6)
=-sin(1.6)
=-0.9996
[tex]\frac{f(x_1)}{f'(x_1)}=\frac{-0.0292}{-0.9996}\\=0.0292\\x_2=x_1-\frac{f(x_1)}{f'(x_1)}=1.6-(0.0292)\\\therefore x_2=1.5708[/tex]
[tex]f(x_2)=f(1.5708)\\=cos 1.5708\\=0.000003\\f'(x_2)=-sin1.5708\\=-1\\x_3=x_2-\frac{f(x_2)}{f'(x_2)}=1.6-\frac{0.000003}{-1}\\\therefore x_3=1.599997=1.6[/tex]
The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal. Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 0. Is H a subspace of the vector space V?
Let [tex]\vec h[/tex] and [tex]\vec\eta[/tex] be two vectors in [tex]H[/tex].
[tex]H[/tex] is a subspace of [tex]V[/tex] if (1) [tex]\vec h+\vec\eta\in H[/tex] and (2) for any scalar [tex]k[/tex], we have [tex]k\vec h\in H[/tex].
(1) True;
[tex]\mathrm{tr}(\vec h+\vec\eta)=\mathrm{tr}(\vec h)+\mathrm{tr}(\vec eta)=0[/tex]
so [tex]\vec h+\vec\eta\in H[/tex].
(2) Also true, since
[tex]\mathrm{tr}(k\vec h)=0k=k[/tex]
Therefore [tex]H[/tex] is a subspace of [tex]V[/tex].
Answer: Yes, H is a subspace of V
Step-by-step explanation:
We know that V is the space of all the 2x2 matrices with real entries.
H is the set of all 2x2 matrices with real entries that have trace equal to 0.
Obviusly the matrices that are in the space H also belong in the space V (because in H you have some selected matrices and in V you have all of them). The thing we need to prove is if H is an actual subspace.
Suppose we have two matrices that belong to H, A and B.
We must see that:
1) if A and B ∈ H, then (A + B)∈H
2) for a scalar number k, k*A ∈ H
lets write this as:
[tex]A = \left[\begin{array}{ccc}a1&a2\\a3&a4\\\end{array}\right] B = \left[\begin{array}{ccc}b1&b2\\b3&b4\\\end{array}\right][/tex]
where a1 + a4 = 0 = b1 + b4
then:
[tex]A + B = \left[\begin{array}{ccc}a1 + b1&a2 + b2\\a3 + b3&a4 + b4\\\end{array}\right][/tex]
the trace is:
a1 + b1 - (a4 + b4) = (a1 - a4) + (b1 - b4) = 0
then the trace is nule, and (A + B) ∈ H
and:
[tex]kA = \left[\begin{array}{ccc}k*a1&k*a2\\k*a3&k*a4\end{array}\right][/tex]
the trace is:
k*a1 - k*a4 = k(a1 - a4) = 0
so kA ∈ H
then H is a subspace of V
12. Evaluate the function rule for the given value.
y = 4×2* for x = -6
Answer:
1/16
Step-by-step explanation:
Substituting the given value of x into the equation, we get ...
[tex]y = 4\times 2^x=4\times 2^{-6}=\dfrac{4}{2^6}=\dfrac{4}{64}=\dfrac{1}{16}[/tex]
What is 2 to the power of three halves equal to?
Final answer:
2 to the power of three halves is equivalent to the square root of 2 cubed, which is approximately 2.83.
Explanation:
In mathematics, when we raise a number to a fraction exponent, we are essentially taking the root of that number. In this case, 2 to the power of three halves is equivalent to the square root of 2 cubed.
2 to the power of three halves = [tex]\sqrt{(2^3)}[/tex] = [tex]\sqrt{8}[/tex] = 2.83
The average age of residents in a large residential retirement community is 69 years with standard deviation 5.8 years. A simple random sample of 100 residents is to be selected, and the sample mean age of these residents is to be computed. The probability that the average age of the 100 residents selected is less than 68.5 years is_______.
The probability that the average age of the 100 residents selected is less than 68.5 years is approximately 0.1949 or 19.49%.
Explanation:The subject of this problem refers to statistics, specifically the concept of the sampling distribution of sample means. It is related to the central limit theorem, which states that if you take sufficiently large random samples from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population's distribution. We know that the population average (mean) is 69 and the standard deviation is 5.8.
We are given a sample size of 100, and hence we calculate its standard deviation as 5.8/√100 = 0.58. With this value, we can use the z-score formula (Z = (X - μ) / σ), where X is the sample mean, μ is the population mean, and σ is the standard deviation of the sample mean, to find the z-score for a sample mean of 68.5 years: Z = (68.5 - 69) / 0.58 ≈ -0.86.
Finally, this Z score is used to find the probability that the sample mean age is less than 68.5 years, by referring to a standard normal distribution table, also known as the Z-table. It should be taken into account that this table provides the probability that a value is less than the given Z score, which is exactly what we need in this case. Consulting the Z table with Z=-0.86, we find that the probability is approximately 0.1949 or 19.49%.
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What is the Laplace Transform of 7t^3 using the definition (and not the shortcut method)
Answer:
Step-by-step explanation:
By definition of Laplace transform we have
L{f(t)} = [tex]L{{f(t)}}=\int_{0}^{\infty }e^{-st}f(t)dt\\\\Given\\f(t)=7t^{3}\\\\\therefore L[7t^{3}]=\int_{0}^{\infty }e^{-st}7t^{3}dt\\\\[/tex]
Now to solve the integral on the right hand side we shall use Integration by parts Taking [tex]7t^{3}[/tex] as first function thus we have
[tex]\int_{0}^{\infty }e^{-st}7t^{3}dt=7\int_{0}^{\infty }e^{-st}t^{3}dt\\\\= [t^3\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(3t^2)\int e^{-st}dt]dt\\\\=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\[/tex]
Again repeating the same procedure we get
[tex]=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt= \frac{3}{s}[t^2\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t^2)\int e^{-st}dt]dt\\\\=\frac{3}{s}[0-\int_{0}^{\infty }\frac{2t^{1}}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{2}}[\int_{0}^{\infty }te^{-st}dt]\\\\[/tex]
Again repeating the same procedure we get
[tex]\frac{3\times 2}{s^2}[\int_{0}^{\infty }te^{-st}dt]= \frac{3\times 2}{s^{2}}[t\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t)\int e^{-st}dt]dt\\\\=\frac{3\times 2}{s^2}[0-\int_{0}^{\infty }\frac{1}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{3}}[\int_{0}^{\infty }e^{-st}dt]\\\\[/tex]
Now solving this integral we have
[tex]\int_{0}^{\infty }e^{-st}dt=\frac{1}{-s}[\frac{1}{e^\infty }-\frac{1}{1}]\\\\\int_{0}^{\infty }e^{-st}dt=\frac{1}{s}[/tex]
Thus we have
[tex]L[7t^{3}]=\frac{7\times 3\times 2}{s^4}[/tex]
where s is any complex parameter
Find all of the zeros of the function f(x) = x^3 + 5x^2 + 6x. If there is more than one answer, enter your answers as a comma separated list. If there are no zeros, enter NONE. Enter exact answers, not decimal approximations. x =
There are more than one zero i.e. three zeros of the given polynomial.
The zeros are:
[tex]x=0,-2,-3[/tex]
Step-by-step explanation:We are given a function f(x) by:
[tex]f(x)=x^3+5x^2+6x[/tex]
We know that the zeros of the function f(x) are the possible values of x at which the function is equal to zero.
Hence, when f(x)=0 we have:
[tex]x^3+5x^2+6x=0\\\\i.e.\\\\x^3+3x^2+2x^2+6x=0\\\\i.e.\\\\x^2(x+3)+2x(x+3)=0\\\\i.e.\\\\(x^2+2x)(x+3)=0\\\\x(x+2)(x+3)=0\\\\i.e.\\\\x=0\ or\ x=-2\ or\ x=-3[/tex]
The zeros of the function f(x) = x^3 + 5x^2 + 6x are x = 0, x = -3, x = -2.
Explanation:To find the zeros of the function f(x) = x^3 + 5x^2 + 6x, we first need to rewrite it in a factorized form in order to identify the roots. This is done by factoring out the common factor, which is 'x' in this equation, giving us x(x^2 + 5x + 6) = 0.
Therefore, x = 0 gives the first zero of this function. Now we are left with the quadratic equation x^2 + 5x + 6 = 0. Using the quadratic formula (-b ± sqrt[b^2 - 4ac]) / 2a, we can figure out the other roots of the equation. For this equation, a=1, b=5 and c=6.
Plugging these values into the quadratic formula, we get: x = [ -5 ± sqrt( (5)^2 - 4*1*6) ] / 2*1 = [ -5 ± sqrt(25 - 24) ] / 2 = -5 ± 1 / 2. Therefore, x = -3 and x = -2 are the other roots of this equation.
So, the zeros of the function f(x) = x^3 + 5x^2 + 6x are: x = 0, x = -3, x = -2.
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