Answer:
It hit the ground at t=2
It reaches its highest point at t=1
Its maximum height =16
Step-by-step explanation:
The graph of the function of the height will be a parapola opening downward where it's x intercepts = the time where the object hit the ground and its maximum point = the point with the maximum point
The x intercepts are 0 when the object was thrown and 2 when it landed
The x coordinate of the vertex is in the middle of the two x intercepts so it will be 1
Substituting the x coordinate of the vertex gives us the y coordinate of the vertex which is the maximum hight 16
The pomegranate hits the ground at time t=2 seconds (tg=2) and reaches its maximum height at time t=1 second (th=1). The maximum height (h) is 16 feet.
Explanation:The given function f(t) = -16t2 + 32t represents the height of the pomegranate at any time t. The pomegranate hits the ground when its height is zero, so set f(t) = 0 and solve for t.
0 = -16t2 + 32t
0 = t(-16t + 32)
Thus, t = 0 (the initial time) and t = 2 (when it hits the ground). Therefore, tg = 2.
The pomegranate reaches its maximum height when the derivative of f(t) is zero. This will give us the time th.
f'(t) = -32t + 32
Set f'(t) = 0 and solve for t. Then -32t + 32 = 0 implies t = 1. So, th = 1.
The maximum height h is simply f(th) = -16(1)2 + 32(1) = 16 feet.
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What are the solutions to the inequality (x-3)(x+5) <_ 0
(x-3)(x-5)<_0
If any individual factor on the left side of the equation is equal to 00, the entire expression will be equal to 00.
x−3=0x-3=0
x+5=0x+5=0
Set the first factor equal to 00 and solve.
x=3x=3
Set the next factor equal to 00 and solve.
x=−5x=-5
Consolidate the solutions.
x=3,−5x=3,-5
Use each root to create test intervals.
x<−5x<-5
−5<x<3-5<x<3
x>3x>3
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
x<−5x<-5 False
−5<x<3-5<x<3 True
x>3x>3 False
The solution consists of all of the true intervals.
−5<x<3-5<x<3
The result can be shown in multiple forms.
Inequality Form:
−5<x<3-5<x<3
Interval Notation:
(−5,3)
The following are the number of birth per year per 1,000 population for 20 countries: 34, 24, 10, 15, 22, 15, 17, 22, 10, 17, 25, 32, 15, 20, 31, 18, 37, 12, 15, 18.The mean birth per year per 1,000 population is:A.22.3B.19.5C.22.45D.None of the aboveThe median birth per year per 1,000 population isA.15B.18C.22D.None of the aboveThe mode birth per year per 1,000 population isA.15B.18C.22D.None of the aboveThe range birth per year per 1,000 population isA.15B.18C.28D.None of the above
Answer:
A. D = None of the above
B. B = 18
C. A = 15
D. D = None of the above
Step-by-step explanation:
Rearranging the population from the highest to the lowest.
10, 10, 12, 15, 15, 15, 15, 17, 17, 18, 18, 20, 22, 22, 24, 25, 31, 32, 34, 37
Mean = (summation of all the 20 samples)/ no of samples
Mean per year per 1000 population = 409/20
= 20.45
Median = the middle value (for odd numbered samples)
= the sum of the middle value ÷ 2 ( for even numbered samples)
Median birth per year per 1000 population = (18 + 18)/2
= 18
Mode = the sample that has the number of frequency
Mode per year per 1000 population = 15 (frequency = 4)
Range = highest value sample - lowest value sample
Range per year per 1000 population = 37 - 10
= 27
Assume the radius of an atom, which can be represented as a hard sphere, is r 1.95 Å.The atom is placed in a ( a ) simple cubic, ( b ) fcc, ( c ) bcc, and ( d ) diamond lattice. As-suming that nearest atoms are touching each other, what is the lattice constant of eachlattice
Answer: Simple cubic=0.39nm
Face centred cubic
=0.55nm
Body centered cubic
=0.45nm
Diamond lattice= 0.9nm
Step-by-step explanation: The lattice constant (a)
for SC=2*r
Fcc=4*r/√2
Bcc= 4*r/√3
Diamond lattice=8*r/√3
Here,
r is the atomic radius measured in nm
r = 1.95Å * 1nm/10Å
=0.195nm
Now let's calculate (a)
SC = 2*r = 2*0.195 nm=0.39nm
Fcc = 4*r/√2 =4*0.195nm/√2
= 0.55nm
Bcc = 4*r/√3 =4*0.195nm/√3
= 0.45nm
Diamond lattice = 8*r/√3
=8*0.195nm/√3
= 0.9nm
What size sample should we draw from a population with standard deviation of 15 to have a sampling distribution standard error of 1.875?1564288
Answer: n = 64
Step-by-step explanation:
Standard error (SE) of a statistical data is the standard deviation of its sampling distribution.
S.E = r/√n. .....1
Where,
SE = standard error
r = standard deviation
n = number of samples
Given:
r = 15
SE = 1.875
From equation 1, making n the subject of formula:
n = (r/SE)^2
n = (15/1.875)
n = 64
A deer population grows at a rate of four percent per year. How many years will it take for the population to double?
Answer:
It is going to take 17.65 years for the population to double.
Step-by-step explanation:
The population of the deer is after t years is given by the following equation
[tex]P(t) = P_{0}(1 + r)^{t}[/tex]
In which [tex]P_{0}[/tex] is the initial population and r is the decimal growth rate.
A deer population grows at a rate of four percent per year. This means that [tex]r = 0.04[/tex]
How many years will it take for the population to double?
This is t when [tex]P(t) = 2P_{0}[/tex]
[tex]P(t) = P_{0}(1.04)^{t}[/tex]
[tex]2P_{0} = P_{0}(1.04)^{t}[/tex]
[tex](1.04)^{t} = 2[/tex]
Here, we apply the log 10 to both sides of the equation.
It is important to note the following property of logarithms.
[tex]\log{a^{t}} = t\log{a}[/tex]
[tex]\log{(1.04)^{t}} = \log{2}[/tex]
[tex]t\log{1.04} = 0.3[/tex]
[tex]0.017t = 0.3[/tex]
[tex]t = \frac{0.3}{0.017}[/tex]
[tex]t = 17.65[/tex]
It is going to take 17.65 years for the population to double.
Answer:it will take approximately 18 years
Step-by-step explanation:
A deer population grows at a rate of four percent per year. The growth rate is exponential. We would apply the formula for exponential growth which is expressed as
A = P(1 + r/n)^ nt
Where
A represents the population after t years.
n represents the period of growth
t represents the number of years.
P represents the initial population.
r represents rate of growth.
From the information given,
A = 2P
P = P
r = 4% = 4/100 = 0.04
n = 1
Therefore
2P = P(1 + r/n)^ nt
2P/P = (1 + 0.04/1)^1 × t
2 = (1.04)^t
Taking log of both sides to base 10
Log 2 = log1.04^t = tlog1.04
0.3010 = t × 0.017
t = 0.3010/0.017 = 17.7 years
You research commute times to work and find that the population standard deviation is 9.3 minutes. Repeat Exercise, using the standard normal distribution with the appropriate calculations for a standard deviation that is known. Compare the results. In a random sample of eight people, the mean commute time to work was 35.5 minutes and the standard deviation was 7.2 minutes.
Answer:
Case [tex] s =7.2[/tex]
[tex]35.5-2.36\frac{7.2}{\sqrt{8}}=29.49[/tex]
[tex]35.5+2.36\frac{7.2}{\sqrt{8}}=41.51[/tex]
So on this case the 95% confidence interval would be given by (29.49;41.51)
Case [tex] \sigma =9.3[/tex]
[tex]35.5-1.96\frac{9.3}{\sqrt{8}}=29.06[/tex]
[tex]35.5+1.96\frac{9.3}{\sqrt{8}}=41.94[/tex]
So on this case the 95% confidence interval would be given by (29.06;41.94)
And we conclude that the intervals are very similar.
Step-by-step explanation:
If we assume that for this question we need to find a confidence interval for the population mean. We have the following procedure:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X= 35.5[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=7.2 represent the sample standard deviation
n=8 represent the sample size
Solution to the problem
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=8-1=7[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,7)".And we see that [tex]t_{\alpha/2}=2.36[/tex]
Now we have everything in order to replace into formula (1):
[tex]35.5-2.36\frac{7.2}{\sqrt{8}}=29.49[/tex]
[tex]35.5+2.36\frac{7.2}{\sqrt{8}}=41.51[/tex]
So on this case the 95% confidence interval would be given by (29.49;41.51)
If we assume that the real population standard deviation is [tex] \sigma =9.3[/tex] the confidence interval is given by:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]
[tex]35.5-1.96\frac{9.3}{\sqrt{8}}=29.06[/tex]
[tex]35.5+1.96\frac{9.3}{\sqrt{8}}=41.94[/tex]
So on this case the 95% confidence interval would be given by (29.06;41.94)
And we conclude that the intervals are very similar.
Sketch the region of integration for the following integral. ∫π/40∫6/cos(θ)0f(r,θ)rdrdθ
Answer:
The graph is sketched by considering the integral. The graph is the region bounded by the origin, the line x = 6, the line y = x/6 and the x-axis.
Step-by-step explanation:
We sketch the integral ∫π/40∫6/cos(θ)0f(r,θ)rdrdθ. We consider the inner integral which ranges from r = 0 to r = 6/cosθ. r = 0 is located at the origin and r = 6/cosθ is located on the line x = 6 (since x = rcosθ here x= 6)extends radially outward from the origin. The outer integral ranges from θ = 0 to θ = π/4. This is a line from the origin that intersects the line x = 6 ( r = 6/cosθ) at y = 1 when θ = π/2 . The graph is the region bounded by the origin, the line x = 6, the line y = x/6 and the x-axis.
The region of integration is a rectangular region in the polar coordinate system, bounded by the angles 0 and π/40, and the radii 0 and 6/cos(θ).
Explanation:The region of integration for the given integral is a rectangular region in the polar coordinate system, bounded by the angles 0 and π/40, and the radii 0 and 6/cos(θ).
To sketch this region, draw a rectangle in the polar coordinate plane, where the horizontal sides represent the radii and the vertical sides represent the angles. The bottom of the rectangle is at radius 0 and the top is at radius 6/cos(θ). The left side of the rectangle is at angle 0 and the right side is at angle π/40.
Therefore, the region of integration can be represented by the rectangle with sides defined by the radii 0 and 6/cos(θ), and the angles 0 and π/40.
LetAandBbe two events in a sample space for which P(A)=2/3,P(B)=1/6, and P(A∩B)=1/9. What is P(A∪B)?
Answer:
13/18
Step-by-step explanation:
P(A∪B)=P(A)+P(B)−P(A∩B)
P(A∪B)=2/3 + 1/6 − 1/9 = 13/18
How many people have to be in a room in order that the probability that at least two of them celebrate their birthday on the same day is at least 0.06? (Ignore leap years, and assume that all outcomes are equally likely.)
Answer:
At least 8 people are needed in the room
Step-by-step explanation:
The probability of n people celebrating their birthday in different days is
1*364/365*363/365*....*(365-(n-1))/365
(for the first person any of the 365 days is suitable, for the second person only 364 persons is suitable, for the third only 364 and so on)
At least two people celebratig their birthday on the same day is the complementary event, thus its probability is
1- 364/365*....*(365-n+1)/365
Lets compute the probabilities for each value of n:
for n = 2: 1-364/365 = 0.00273for n = 3: 1-364/365*363/365 = 0.008for n = 4: 1-364/365*363/365*362/365 = 0.016for n = 5: 1-364/365*363/365*362/365*361/365 = 0.027for n = 6: 1-364/365*363/365*362/365*361/365*360/365 = 0.04for n = 7: 1-364/365*363/365*362/365*361/365*360/365*359/365 = 0.056for n = 8: 1-365/365*363/365*362/365*361/365*360/365*359/365*358/365 = 0.074 > 0.06We need at least 8 people in the room.
If we get the 100 from number 1 each year for ten years and invest each payment in an account that earns 8% how much will be there at the end? (1448.66)
Answer:
$1,448.66
Step-by-step explanation:
The future value of an annuity with yearly deposits 'P' at an interest rate of 'r' invested for 'n' years is determined by:
[tex]FV = P[\frac{(1+r)^n-1}{r}][/tex]
For P = $100, r = 0.08 and n = 10 years:
[tex]FV = 100[\frac{(1+0.08)^{10}-1}{0.08}]\\FV=\$1,448.66[/tex]
The amount at the end of the ten years is $1,448.66
Laura and Phillip each fire one shot at a target, Laura has aprobability 0.5 of hitting the target, and Phillip has aprobability 0.3. The shots are independent.
a. Find the probability that the target is hit.
b.Find the probability that the target is hit by exactly oneshot.
c.Given that the target was hit by exactly one shot, find theprobabilty that Laura hit the target.
a. The probability that the target is hit is 0.65.
b. The probability that the target is hit by exactly one shot is 0.5.
c. Given exactly one shot hits, the probability Laura hit is 0.7.
a. To find the probability that the target is hit, we can calculate the probability that at least one shot hits the target. Since the shots are independent, we can use the complement rule:
[tex]\[ P(\text{Target hit}) = 1 - P(\text{Both miss}) \][/tex]
Laura misses with probability 0.5 and Phillip misses with probability 0.7 (since the probability of hitting the target is 0.3). Therefore:
[tex]\[ P(\text{Both miss}) = 0.5 \times 0.7 = 0.35 \][/tex]
[tex]\[ P(\text{Target hit}) = 1 - 0.35 = 0.65 \][/tex]
So, the probability that the target is hit is 0.65.
b. To find the probability that the target is hit by exactly one shot, we need to consider the cases where Laura hits and Phillip misses, and where Laura misses and Phillip hits. These events are mutually exclusive, so we can add their probabilities:
[tex]\[ P(\text{Exactly one shot hits}) = P(\text{Laura hits, Phillip misses}) + P(\text{Laura misses, Phillip hits}) \][/tex]
[tex]\[ = (0.5 \times 0.7) + (0.5 \times 0.3) \][/tex]
= 0.35 + 0.15
= 0.5
So, the probability that the target is hit by exactly one shot is 0.5.
c. Given that the target was hit by exactly one shot, we need to find the probability that Laura hit the target. This is the conditional probability:
[tex]\[ P(\text{Laura hit} | \text{Exactly one shot hit}) = \frac{P(\text{Laura hit and exactly one shot hit})}{P(\text{Exactly one shot hit})} \][/tex]
From part b, we found [tex]\( P(\text{Exactly one shot hit}) = 0.5 \)[/tex]. And from part b, we found [tex]\( P(\text{Laura hit and exactly one shot hit}) = 0.35 \)[/tex].
[tex]\[ P(\text{Laura hit} | \text{Exactly one shot hit}) = \frac{0.35}{0.5} = 0.7 \][/tex]
So, the probability that Laura hit the target, given that the target was hit by exactly one shot, is 0.7.
Which of these are propositions? What are the truth values of those that are propositions?
a) Do not pass go.
b) What time is it?
c) There are no black flies in Maine.
d) 4 + x = 5.
e) The moon is made of green cheese.
f) 2 n ≥ 100
Answer:
c) There are no black flies in Maine -> False
e) The moon is made of green cheese -> False
Explanation:
A proposition is a statement that poses an idea that can be True or False.
Do not pass go, is not a proposition is an order, and there is not an answer to that statement.What time is it? Is not a proposition is a question and it's response goes beyond True or False. There are no black flies in Maine is indeed a proposition because is a statement that can either be True or False. 4 + x = 5 is not a proposition because is a statement with an infinite number of answers, in contrast, 4 + 1 = 5 is a proposition because is either True or False. The moon is made of green cheese is a proposition because is a statement that can either be True or False. 2 n ≥ 100 is not a proposition because is a statement with an infinity number of answer and is not clear whether is True or False, in contrast, 2 ≥ 100 is a proposition because is either True or False.Only 'There are no black flies in Maine' and 'The moon is made of green cheese.' are propositions. The truth values depend on the accuracy of the statements. The others are not propositions.
Explanation:In logic, a proposition is any declarative sentence that is either true or false, but not both. Using this definition:
a) 'Do not pass go.' is not a proposition as it is a command.b) 'What time is it?' is not a proposition as it is a question. c) 'There are no black flies in Maine.' is a proposition. Its truth value can be true or false, depending on whether or not there are black flies in Maine.d) '4 + x = 5.' is not a proposition because its truth value depends on the value of x.e) 'The moon is made of green cheese.' is a proposition. That proposition is false.f) '2 n ≥ 100' is not a proposition because its truth value depends on the value of n.Learn more about Proposition here:
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Determine if the statement is true or false.
Different sequences of row operations can lead to different reduced echelon forms for the same matrix.
True False
Justify your answer.
Answer:
False
Step-by-step explanation:
There are two concepts:
1. Row Echelon Form: There can be more than two row echelon forms of a single matrix, so different sequences of row operations can lead to different row echelon forms of a single matrix.
2. Reduced Row Echelon Form: It's unique for each matrix, so different sequences of row operations always lead to the same reduced row echelon form for the same matrix.
every 55 students there were 2 professor's
How many professors for 1650 students
Answer:
60 professors
Step-by-step explanation:
for every 55 students there were 2 professors
for 1650 students, there will be (1650 x 2) / 55 =3300/55 = 60 professors
Final answer:
To find the number of professors needed for 1650 students when the ratio is 2 professors for every 55 students, you calculate 2 / 55 = x / 1650, resulting in x = 60. Therefore, 60 professors are needed for 1650 students.
Explanation:
The question asks how many professors would be needed for 1650 students if there is a ratio of 2 professors for every 55 students. To solve this, you can set up a proportion where the number of students is directly proportional to the number of professors. The proportion can be expressed as 2 professors / 55 students = x professors / 1650 students. Solving for x involves cross-multiplying and dividing.
Step 1: Cross-multiply to get 2 × 1650 = 55 × x.
Step 2: Simplify the equation: 3300 = 55x.
Step 3: Divide both sides by 55 to solve for x: x = 3300 / 55.
Step 4: Calculate the result: x = 60. Thus, you would need 60 professors for 1650 students.
the value V, of a 300000 investment that earns 2% annual interest is given by V=f(t) where t is in years
How much is the investment worth in 3 years
Answer:
$318,362.40
Step-by-step explanation:
The equation that describes the future value of investments with interests compounded annually is:
[tex]V = P*(1+r)^t[/tex]
Where P is the initial investment (300,000), r is the interest rate (2%) and t is the time of investment, in years (3):
[tex]V = 300,000*(1+0.02)^3\\V=\$318,362.40[/tex]
The investment is worth $318,362.40 in 3 years.
On any given day, the probability that a large local river is polluted by carbon tetrachloride is 0.12. Each day, a test is conducted to determine whether the river is polluted by carbon tetrachloride. This test has proved correct 84 percent of the time.1. What proportion of days does the test indicate carbon tetrachloride pollution? 2. Suppose that on a particular day the test indicated carbon tetrachloride pollution What is the probability that such pollution actually exists? 3. What percentage of days where the test is negative will the river actually be polluted?
Answer:
a) 0.2416
b) 0.4172
c) 0.0253
Step-by-step explanation:
Since the result of the test should be independent of the time , then the that the test number of times that test proves correct is independent of the days the river is correct .
denoting event a A=the test proves correct and B=the river is polluted
a) the test indicates pollution when
- the river is polluted and the test is correct
- the river is not polluted and the test fails
then
P(test indicates pollution)= P(A)*P(B)+ (1-P(A))*(1-P(B)) = 0.12*0.84+0.88*0.16 = 0.2416
b) according to Bayes
P(A∩B)= P(A/B)*P(B) → P(A/B)=P(A∩B)/P(B)
then
P(pollution exists/test indicates pollution)=P(A∩B)/P(B) = 0.84*0.12 / 0.2416 = 0.4172
c) since
P(test indicates no pollution)= P(A)*(1-P(B))+ (1-P(A))*P(B) = 0.84*0.88+ 0.16*0.12 = 0.7584
the rate of false positives is
P(river is polluted/test indicates no pollution) = 0.12*0.16 / 0.7584 = 0.0253
Determine if the function f(x) = 4√x − 3x satisfies the Mean Value Theorem on [4, 49]. If so, find all numbers c on the interval that satisfy the theorem.
a) c = 812b) c = −814c) c = 818d) c = 814
Answer:
c
Step-by-step explanation:
The mean value theorem states that for a continuous and differentiable function on a interval there exists a number c from that interval, that
[tex]f'(c)=(f(b)-f(a))/(b-a)[/tex]
First we must determine the endpoints:
[tex]f(49)=-119[/tex]
[tex]f(4)=-4[/tex]
We find the derivative of the function, f'(c)
[tex]f'(c)=3+2\sqrt{c}[/tex]
Therefore:
[tex]3+2\sqrt{c}=((-119)-(-4))/((49)-(4))[/tex]
Simplify:
[tex]3+2\sqrt{c}=-23/9[/tex]
[tex]c=81/4[/tex]
Which of the following statement is true about k-NN algorithm?
k-NN performs much better if all of the data have the same scale
k-NN works well with a small number of input variables (p), but struggles when the number of inputs is very large
k-NN makes no assumptions about the functional form of the problem being solved
A) 1 and 2
B) 1 and 3
C) Only 1
D) All of the above
In this exercise we have to use the knowledge of algorithm to write the correct alternative that best matches, thus we can say that:
Letter D
The computational complicatedness of K-NN increases as the extent or bulk of some dimension of the training basic document file increase and the treasure gets considerably unhurried as the number of examples and free variables increase.
Also, K-NN happen a non-parametric machine intelligence treasure and as such form no assuming possession about the working form of the question at hand.
The invention everything better accompanying information in visible form of the same scale, therefore standard the information in visible form superior to applying the invention happen urged.
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All the statements provided about k-NN (scale importance, efficiency with small input variables, and no assumptions about the functional form of the problem) are true.
Explanation:The question asks which of the given statements is true about the k-NN (k-Nearest Neighbors) algorithm. The correct answer would be 'D) All of the above.' This is because all three statements accurately describe different aspects of how the k-NN algorithm works:
k-NN performs much better if all of the data have the same scale. Applying a suitable scaling method is important as it ensures all features contribute equally. k-NN works well with a small number of input variables (p), but struggles when the number of inputs grows large. This is due to the 'curse of dimensionality,' as the volume of the input space increases exponentially with the number of dimensions, impacting performance and accuracy. k-NN makes no assumptions about the functional form of the problem being solved. This is a key strength of k-NN; it's a non-parametric method that doesn’t make explicit assumptions about the functional form of the data.Learn more about k-NN algorithm here:
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The breaking strength of a fiber is required to be at least 150 psi. Past experience has indicated that the standard deviation of breaking strength is σ= 3 psi. A random sample of four specimens is tested. The
results are y1=145, y2=153, y3=150 and y4=147.
(a) State the hypotheses that you think should be tested in this experiment
(b) Test these hypotheses using α
= 0.05. What are your conclusions?
(c) Find the
P-value for the test in part (b).
(d) Construct a 95 percent confidence interval on the mean breaking strength.
The 95 percent confidence interval on the mean breaking strength is (143.9764, 153.5236).
We are given that;
Strength for breaking the fiber= 150 psi
σ= 3 psi
Now,
(a) The hypotheses that should be tested in this experiment are:
[tex]$H_0: \mu = 150$[/tex] (The mean breaking strength of the fiber is 150 psi)
[tex]$H_a: \mu < 150$[/tex] (The mean breaking strength of the fiber is less than 150 psi)
This is a one-tailed test, since we are interested in testing whether the mean breaking strength is below a certain value.
(b) To test these hypotheses using [tex]$\alpha = 0.05$[/tex], we need to calculate the test statistic and compare it with the critical value or the p-value. The test statistic is given by:
[tex]$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$[/tex]
where [tex]$\bar{x}$[/tex] is the sample mean, [tex]$\mu_0$[/tex] is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Plugging in the given values, we get:
[tex]$t = \frac{148.75 - 150}{3/\sqrt{4}} = -1.6667$[/tex]
The critical value for a one-tailed test with[tex]$\alpha = 0.05$[/tex] and n-1 = 3 degrees of freedom is -2.3534, which can be found from a t-table.
Since t > -2.3534 or p > 0.05, we fail to reject the null hypothesis at the 0.05 level of significance. There is not enough evidence to conclude that the mean breaking strength of the fiber is less than 150 psi.
(c) The p-value for the test in part (b) is 0.0834, as mentioned above.
(d) A 95 percent confidence interval on the mean breaking strength is given by:
[tex]$\bar{x} \pm t_{\alpha/2,n-1} \frac{s}{\sqrt{n}}$[/tex]
where [tex]$t_{\alpha/2,n-1}$[/tex] is the critical value for a two-tailed test with [tex]$\alpha = 0.05$[/tex] and n-1 = 3 degrees of freedom, which is 3.1824.
Plugging in the given values, we get:
[tex]$148.75 \pm 3.1824 \frac{3}{\sqrt{4}}$[/tex]
Simplifying, we get:
[tex]$148.75 \pm 4.7736$[/tex]
Therefore, by statistics answer will be (143.9764, 153.5236).
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use gcf to factor the expression 40x+24y-56
Suppose that 20% of adults practices a gluten-free diet, 15% of adults practice a dairy-free diet, and 8% of adults practice both of these diets. One adult is selected at random.
a. Given a random individual practices a glute-free diet, what is the probability they practice a dairy-free diet?
b. Are events "practices a gluten-free diet" and "practices a dairy-free diet" independent?
c. Given a random individual practices a dairy-free diet, what is the probability they do not practice a gluten free diet?
Answer:
a. 0.4
b. Not independent events
c. 0.47
Step-by-step explanation:
Let A= Practices Gluten free diet
B= Practices Dairy free diet
A and B= Practices Both diets
P(A)=0.20
P(B)=0.15
P(A and B)=0.08
a.
[tex]P(B/A)=\frac{P(A and B)}{P(A)}[/tex]
[tex]P(B/A)=\frac{0.08}{2}[/tex]
[tex]P(B/A)=0.40[/tex]
b.
The two events are independent if P(B/A) =P(B) or P(A/B)=P(A)
As, P(B/A) ≠P(B)
0.4≠0.15
So, the event gluten free diet and dairy free diet are dependent events.
c.
[tex]P(A'/B)=\frac{P(A' and B)}{P(B)}[/tex]
[tex]P(A'and B)= P(B)-P(A and B)[/tex]
[tex]P(A' and B)=0.15-0.08=0.07[/tex]
[tex]P(A'/B)=\frac{P(A' and B)}{P(B)}[/tex]
[tex]P(A'/B)=\frac{0.07}{0.15}[/tex]
[tex]P(A'/B)=0.47[/tex]
Answer:
a. 0.4
b. Not independent
c. 0.47
Step-by-step explanation:
Consider the concentration, C, in mg/liter, of a drug in the blood as a function of x, of the amount, in mg, of the drug given and t, the time in hours since the injection. For 0 < = x < = 4 mg and t > = 0 hours, we have C = f(x, t) = te^-t(5-x). Find f(1, 2).Interpret in terms of drug concentration. a. The change in concentration of a 1 mg dose in the blood 2 hours after injection. b. The amount of a 1 mg dose in the blood 2 hours after injection. c. The concentration of a 1 mg dose in the blood 2 hours after injection. d. The concentration of a 2 mg dose in the blood 1 hour after injection. e. The change in concentration of a 2 mg dose in the blood 1 hour after injection.
Answer:
c. the concentration of 1 mg dose in the blood 2 hours after injection.
Step-by-step explanation:
We have, C = f(x, t) = t*e^-t(5-x)
then
C = f(1, 2) = 2*e^-2(5-1)
C = 2*e^-2(4)
C = 2*e^-8 OR C = 2/e^8
The concentration of a drug in blood depends on the dose and the time since administration. The given function allows calculating these concentrations, resulting in a concentration of a 1 mg dose 2 hours after injection of 8e^-2 mg/liter and a 2 mg dose 1 hour after injection of 3e^-1 mg/liter. The change in concentration cannot be determined without the previous concentration information.
Explanation:To find the concentration of the drug, we need to plug the values of x and t into the given function C = f(x, t) = te^-t(5-x). For f(1,2), we replace x with 1 and t with 2. Therefore, C = 2e^-2(5-1) = 2e^-2*4 = 2*4*e^-2 = 8e^-2.
a. The change in concentration of a 1 mg dose in the blood 2 hours after injection. To find the change in concentration, we would need the previous concentration. However, as this information is not provided, we cannot determine the change in concentration for this case.
b. The amount of a 1 mg dose in the blood 2 hours after injection. The amount of the dose remains constant at 1 mg; what changes is its concentration due to distribution and elimination processes.
c. The concentration of a 1 mg dose in the blood 2 hours after injection. With the values provided, we already calculated this to be C = 8e^-2 mg/liter.
d. The concentration of a 2 mg dose in the blood 1 hour after injection. Plain the value of x = 2 and t = 1 into the equation, we get C = 1e^-1(5-2) = 3e^-1 mg/liter.
e. The change in concentration of a 2 mg dose in the blood 1 hour after injection. Similar to part a, without previous concentration information, it is not possible to determine the change in this case.
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A computer password consists of eight characters. How many different passwords are possible if each character may be any lowercase letter or digit? How many different passwords are possible if each character may be any lowercase letter or digit, and at least one character must be a digit? A computer system requires that passwords contain at least one digit. If eight characters are generated at random, and each is equally likely to be any of the 26 letters or 10 digits, what is the probability that a valid password will be generated?
Answer:
1. 2,821,109,907,456 passwords
2. 2,612,282,842,880 passwords
3. 0.93
Step-by-step explanation:
Number of lower case letters = 26
Number of digits = 10
Total Characters = 26 + 10 = 36
Length of password = 8
1. Each character can be chosen from 36 objects (26 lowercase letters and 10 digits).
Since there's no restriction on repetition of characters, there are
(26+10)^8 different passwords
= 36^8
= 2,821,109,907,456 passwords
2.
For atleast one character to be a must be a digit means 1 character or 2 character or 3 characters... and so on
So we calculate by looking at no character being a digit
and then subtract the result from the total(36^8).
No character being a digit = All the characters are only letters
I.e. 26^8
Atleast one character being a digit = 36^8 - 26^8
= 2,821,109,907,456 - 208,827,064,576
= 2,612,282,842,880 passwords
3.
Probability = Number of possible outcomes/Number of Total outcomes
Number of possible outcomes = Number of Atleast 1 digit = 36^8 - 26^8 (as in question 2)
Total outcomes = 36^8
Probability = (36^8 - 26^8) / 36^8
= 2,612,282,842,880 / 2,821,109,907,456
= 0.92597698373108952
= 0.93 (approximated)
From the probability, the number of different passwords that are possible if each character may be any lowercase letter or digit is 2,821,109,907,456 passwords
How to calculate the probabilityThe following information can be deduced:
Number of lower case letters = 26Number of digits = 10Total Characters = 26 + 10 = 36Length of password = 8The number of different passwords are possible if each character may be any lowercase letter or digit will be:
= (26+10)⁸ different passwords
= 2,821,109,907,456 passwords
The number of different passwords that are possible if each character may be any lowercase letter or digit, and at least one character must be a digit will be:
= 36⁸ - 26⁸
= 2,612,282,842,880 passwords
Lastly, the probability that a valid password will be generated will be:
= (36⁸ - 26⁸) / 36⁸
= 0.93
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*8. Consider a LTI system with unit impulse response, h(t) = e-3tu(t). Using direct integration technique for finding convolution, find its zero-state response due to an input, x(t) = u(t) (which is called unit step response of the system). Also, from your answer above, write down its response due to an input of the form, x(t) = 2δ(t) – 4u(t). [Hint: Use principle of superposition]
Answer:
Step-by-step explanation:
Check the attachment for solution
MVA Harper industries has $900 million of common equity on its balance sheet; its stock price is $80 per share; and its market value added (MVA) is $50 million. How many common shares are currently outstanding?
Answer:
11,875,000 common shares outstanding
Step-by-step explanation:
The number of common shares outstanding of Harper industries is determined by the company's common equity added to its MVA and then divided by the stock price per share.
If common equity is $900 million, MVA is $50 million and the price per share is $50, the number of shares outstanding is:
[tex]n=\frac{900,000,000+50,000,000}{80}\\n=11,875,000[/tex]
Answer:
N = 11,875,000
Therefore, the number of common shares that are currently outstanding is 11,875,000
Step-by-step explanation:
Oustanding shares can be defined as the company's stock currently held by all its shareholders, it also including share blocks held by institutional investors and restricted shares owned by the company’s officers and insiders. The outstanding common shares can be defined mathematically with the equation below.
MVA = (P ×N) - BV .....1
Where;
MVA = market value added
P = price per share
N = Number of outstanding shares
BV = balance sheet value of common equity
From equation 1, we can make N the subject of formula.
N = (MVA + BV)/P .....2
Since;
MVA = $50,000,000
BV = $900,000,000
P = $80
Substituting into equation 2
N = (50,000,000 + 900,000,000)/80
N = 11,875,000
Therefore, the number of common shares that are currently outstanding is 11,875,000
Which of the following are considered cylinders? Hint : Try to do this as if it were on the exam. Use the traces (or cross-sections) method to draw a picture to help you classify. (Click all that apply to get credit.)
A. 5y^{2}+2z^{2}=2
B. y=-\left(2x^{2}+2\right)
C. 5x^{2}+2y^{2}+4z^{2} = 5
D. 4x^{2}-4y^{2}+4z^{2} = 2
E. -\left(\cos\!\left(-4y\right)\right)=1-z
F. 2x^{2}+4y^{2}-4z = -1
G. None of the above.
Answer:
Step-by-step explanation:
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given curve in some plane.
a)[tex]5y^{2}+2z^{2} = 2 [/tex]
this is cylinder , this is ellipse in y-z plane ,but along x-axis ,we shall get the same curve in every possible plane parallel to the yz-plane.
b)[tex]y = -2x^{2} - 2 [/tex]
We have to be careful not to draw interpret it as an equation in 2-space even though there is no z - that just means it does not depend upon z. If we sketch this graph in the xy-plane (so z = 0, we obtain the parabola [tex]y = -2x^ 2 - 2[/tex]) . Since there is no z-value in the equation, we shall get the same curve in every possible plane parallel to the x-y-plane. In particular, the graph of this surface will be all vertical lines passing through the curvey = -2x^ 2 - 2 . in the xy-plane. By definition, this makes the graph a cylinder.
C)[tex]5x^{2}+2y^{2}+4z^{2} = 5 [/tex]
this is ellipsoid
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/tex]
d)4x^(2)-4y^(2)+4z^(2) = 2
hyperboloid
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1[/tex]
{x,y,z can be exchanged to each other}
e)1-z = - cos (-4y)
this is some curve in y-z plane ,so it is cylinder.
f)
2x^{2}+4y^{2}-4z = -1
elliptical paraboloid
Find an explicit solution of the given initial-value problem.dy/dx = ye^x^2, y(3) = 1.
Answer:
[tex]y=exp(\int\limits^x_4 {e^{-t^{2} } } \, dt)[/tex]
Step-by-step explanation:
This is a separable equation with an initial value i.e. y(3)=1.
Take y from right hand side and divide to left hand side ;Take dx from left hand side and multiply to right hand side:
[tex]\frac{dy}{y} =e^{-x^{2} }dx[/tex]
Take t as a dummy variable, integrate both sides with respect to "t" and substituting x=t (e.g. dx=dt):
[tex]\int\limits^x_3 {\frac{1}{y} } \, \frac{dy}{dt} dt=\int\limits^x_3 {e^{-t^{2} } } dt[/tex]
Integrate on both sides:
[tex]ln(y(t))\left \{ {{t=x} \atop {t=3}} \right. =\int\limits^x_3 {e^{-t^{2} } } \, dt[/tex]
Use initial condition i.e. y(3) = 1:
[tex]ln(y(x))-(ln1)=\int\limits^x_3 {e^{-t^{2} } } \, dt\\ln(y(x))=\int\limits^x_3 {e^{-t^{2} } } \, dt\\[/tex]
Taking exponents on both sides to remove "ln":
[tex]y=exp (\int\limits^x_3 {e^{-t^{2} } } \, dt)[/tex]
Given the following data: X 1 4 6 7 Y 9 7 8 1 a) Find the coefficient of correlation. b) Find the equation of the regression line. c) What is the predicted value for Y, if X = 3?
a. [tex]\[ r \approx -0.736 \][/tex]
b. The equation of the regression line is [tex]\(Y = 11.075 - 1.127X\)[/tex].
c. The predicted value for [tex]\(Y\)[/tex] when [tex]\(X = 3\)[/tex] is approximately 7.794.
To find the coefficient of correlation [tex](\(r\))[/tex] and the equation of the regression line, we can follow these steps:
a) Find the Coefficient of Correlation [tex](\(r\))[/tex]:
The formula for the coefficient of correlation [tex](\(r\))[/tex] is given by:
[tex]\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \][/tex]
where \(n\) is the number of data points, [tex]\(\sum xy\)[/tex] is the sum of the product of corresponding values of [tex]\(X\)[/tex] and [tex]\(Y\), \(\sum x\)[/tex] is the sum of [tex]\(X\)[/tex], and [tex]\(\sum y\)[/tex] is the sum of [tex]\(Y\)[/tex].
[tex]\[ n = 4 \][/tex]
[tex]\[ \sum x = 1 + 4 + 6 + 7 = 18 \][/tex]
[tex]\[ \sum y = 9 + 7 + 8 + 1 = 25 \][/tex]
[tex]\[ \sum xy = (1 \times 9) + (4 \times 7) + (6 \times 8) + (7 \times 1) = 9 + 28 + 48 + 7 = 92 \][/tex]
[tex]\[ \sum x^2 = 1^2 + 4^2 + 6^2 + 7^2 = 1 + 16 + 36 + 49 = 102 \][/tex]
[tex]\[ \sum y^2 = 9^2 + 7^2 + 8^2 + 1^2 = 81 + 49 + 64 + 1 = 195 \][/tex]
Now, substitute these values into the formula:
[tex]\[ r = \frac{4 \times 92 - 18 \times 25}{\sqrt{[4 \times 102 - (18)^2][4 \times 195 - (25)^2]}} \][/tex]
[tex]\[ r = \frac{368 - 450}{\sqrt{[408 - 324][780 - 625]}} \][/tex]
[tex]\[ r = \frac{-82}{\sqrt{84 \times 155}} \][/tex]
[tex]\[ r \approx -0.736 \][/tex]
b) Find the Equation of the Regression Line:
The equation of the regression line [tex](\(Y = a + bX\))[/tex] is given by:
[tex]\[ b = r \times \frac{s_y}{s_x} \][/tex]
[tex]\[ a = \bar{y} - b \bar{x} \][/tex]
where [tex]\(s_y\)[/tex] and [tex]\(s_x\)[/tex] are the standard deviations of [tex]\(Y\)[/tex] and [tex]\(X\)[/tex] respectively, and [tex]\(\bar{y}\)[/tex] and [tex]\(\bar{x}\)[/tex] are the means of [tex]\(Y\)[/tex] and [tex]\(X\)[/tex] respectively.
[tex]\[ s_x = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}} \][/tex]
[tex]\[ s_y = \sqrt{\frac{\sum (y - \bar{y})^2}{n-1}} \][/tex]
[tex]\[ \bar{x} = \frac{\sum x}{n} \][/tex]
[tex]\[ \bar{y} = \frac{\sum y}{n} \][/tex]
[tex]\[ s_x \approx 2.160 \][/tex]
[tex]\[ s_y \approx 3.299 \][/tex]
Now, calculate [tex]\(b\)[/tex] and [tex]\(a\)[/tex]:
[tex]\[ b = -0.736 \times \frac{3.299}{2.160} \][/tex]
[tex]\[ b \approx -1.127 \][/tex]
[tex]\[ \bar{x} = \frac{18}{4} = 4.5 \][/tex]
[tex]\[ \bar{y} = \frac{25}{4} = 6.25 \][/tex]
[tex]\[ a = 6.25 - (-1.127 \times 4.5) \][/tex]
[tex]\[ a \approx 11.075 \][/tex]
So, the equation of the regression line is [tex]\(Y = 11.075 - 1.127X\)[/tex].
c) Predicted Value for Y when X = 3:
[tex]\[ Y = 11.075 - 1.127 \times 3 \][/tex]
[tex]\[ Y \approx 7.794 \][/tex]
So, the predicted value for [tex]\(Y\)[/tex] when [tex]\(X = 3\)[/tex] is approximately 7.794.
Two students each use a random number generator to pick an integer between 1 and 7.
(a) What is the probability that they pick the same number?
(b) What is the probability that they pick different numbers?
Answer:
a) 14.29% probability that they pick the same number.
b) 85.71% probability that they pick the different numbers.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
The outcomes in this problem is as follows:
(Student A Number, Student B Number)
(1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (7,1)
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (7,2)
(1,3), (2,3), (3,3), (4,3), (5,3), (6,3), (7,3)
(1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (7,4)
(1,5), (2,5), (3,5), (4,5), (5,5), (6,5), (7,5)
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6), (7,6)
(1,7), (2,7), (3,7), (4,7), (5,7), (6,7), (7,7)
So there are 49 total outcomes.
(a) What is the probability that they pick the same number?
There are 7 possible outcomes in which they pick the same number:
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7)
So there is a 7/49 = 0.1429 = 14.29% probability that they pick the same number
(b) What is the probability that they pick different numbers?
There are 49-7 = 42 possible outcomes in which they pick different numbers.
So there is a 42/49 = 0.8571 = 85.71% probability that they pick the different numbers.
We run a linear regression and the slope estimate is 0.5 with estimated standard error of 0.2. What is the largest value of for which we would NOT reject the null hypothesis that ?
Answer: (0.1, 0.9) We cannot reject the null hypothesis.
Step-by-step explanation:
First, calculate with the negative. Next, with the positive.
B1 - 2 x SE(B1)
0.5 - 2 x 0.2 = 0.1
B1 + 2 x SE(B1)
0.5 + 2 x 0.2 = 0.9
Final answer:
The student's question about hypothesis testing in linear regression focuses on whether to reject the null hypothesis based on a slope estimate, its standard error, and the p-value. Since the p-value exceeds the significance level, the null hypothesis is not rejected, indicating insufficient evidence of a nonzero slope at the 5 percent level.
Explanation:
The student is asking about hypothesis testing in the context of linear regression. Specifically, the question pertains to whether the null hypothesis, which posits no effect (i.e., a slope of zero), should be rejected based on a given slope estimate, its standard error, and a provided p-value. In hypothesis testing, if the p-value is greater than the chosen level of significance (alpha), you do not reject the null hypothesis.
In the provided scenario, if the p-value is indeed 0.2150 and the level of significance (alpha) is set to 5 percent (0.05), the decision would be not to reject the null hypothesis. This is because the p-value is greater than the alpha level (0.2150 > 0.05). The conclusion is that there is insufficient evidence at the 5 percent significance level to suggest that the true slope is different from zero.