Final answer:
The domain of f(x) = 8/(1 + eˣ) is all real numbers, or (-∞, ∞). For f(x) = 5/(1 - eˣ), the domain is all real numbers except x = 0, which in interval notation is (-∞, 0) ∪ (0, ∞).
Explanation:
Finding the Domain of Functions
To find the domain of a function, we look for all possible values of x for which the function is defined. The exponential function eˣ is defined for all real numbers, so we mainly need to be concerned with the denominators in these functions not being equal to zero.
For the function f(x) = 8/(1 + eˣ), since 1 + eˣ is always greater than zero for all real x (because the exponential function eˣ is always positive), the function is defined for all real numbers. Therefore, the domain is all real numbers, or in interval notation, (-∞, ∞).For the function f(x) = 5/(1 - eˣ), we must exclude any value of x that makes the denominator zero. The denominator 1 - eˣ equals zero when eˣ = 1. This occurs when x = 0. Therefore, the domain of this function is all real numbers except x = 0, or in interval notation, (-∞, 0) ∪ (0, ∞).Given f(x)= a×e−bx , where a = 1 and b = 6,
calculate g(x)=dfdx and obtain g(1) (that is, evaluate the derivative of f(x) at x = 1).
Report your answer with three significant figures.
Answer:
g(1) = -0.015
Step-by-step explanation:
We are given he following in the question:
[tex]f(x) = ae^{-bx}[/tex]
For a = 1 and b = 6, we have,
[tex]f(x) = e^{-6x}[/tex]
We have to find the the derivative of f(x) with respect to x.
[tex]g(x) = \dfrac{d(f(x))}{dx} = \dfrac{d(e^{-6x})}{dx}\\\\g(x) = -6e^{-6x}\\\\g(1) = \dfrac{d(f(x))}{dx}\bigg|_{x=1} = -6e^{-6} = -0.015[/tex]
Thus, g(1) = -0.015
Determine whether the data described are qualitative or quantitative. The maximum speed limit on interstate highways.
a. Qualitative
b. Quantitative
The data of the maximum speed limit on interstate highways is quantitative because it is expressible as a numerical value, which is an amount or quantity.
Explanation:The data described, 'the maximum speed limit on interstate highways', is quantitative. This is because data is quantitative when it is expressible as an amount or quantity. In this case, a speed limit is typically represented as a numerical value in miles or kilometers per hour, which is an amount or quantity. Therefore, the data of the maximum speed limit on interstate highways is quantitative.
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The share of aggregate income held by middle-income households in 1970 was 62%, whereas that held by upper-income households was 29%. The corresponding figures in 2014 were 43% and 49%, respectively. The models describing the fall and the rise in the share of the aggregate incomes of these two groups are approximately linear over the period under consideration.† (a) Find the mathematical models describing the percent share of aggregate income held by each group from 1970 through 2014. (Let t denote the time, in years, with t = 0 corresponding to the beginning of 1970.) middle-income households y = Incorrect: Your answer is incorrect. upper-income households y = Incorrect: Your answer is incorrect. (b) Find the time when the aggregate income held by upper-income households first exceeded that held by middle-income households. (Give the calendar year in which the change occurred.)
Answer:
a) y = -19/44t +62; y = 5/11t +29
b) 2007
Step-by-step explanation:
(a) The two-point form of the equation of a line is useful when two data points are given.
y = (y2 -y1)/(x2 -x1)(x -x1) +y1
Middle Income
The two given points are (0, 62), (44, 43), so the equation is ...
y = (43 -62)/(44 -0)(t -0) +62
y = -19/44t +62
__
Upper Income
The two given points are (0, 29), (44, 49), so the equation is ...
y = (49 -29)/(44 -0)(t -0) +29
y = 20/44t +29
y = 5/11t +29
__
(b) The year in which the shares are equal is found by setting the y-values equal:
-19/44t +62 = 5/11t +29
33 = 39/44t
t = (33)(44)/39 ≈ 37.2
The upper income share exceeded the middle income share in the year 1970 +37.2 = 2007.
The mathematical models for the middle-income and upper-income households are y1(t) = 62% - 0.38% * t and y2(t) = 29% + 0.40% * t, respectively. The aggregate income held by upper-income households first exceeded that held by middle-income households in the year 2014.
Explanation:In order to find the mathematical model, we will first determine the slope for each data set. The slope is calculated as the change in y divided by the change in x. Here, 'y' represents the share of aggregate income and 'x' represents time.
For the middle-income households, the slope (m1) is: m1 = (43% - 62%)/(2014 - 1970) = -0.38% per year. So, the equation becomes: y1(t) = 62% - 0.38% * t.
For the upper-income households, the slope (m2) is: m2 = (49% - 29%)/(2014 - 1970) = 0.40% per year. So, the equation becomes: y2(t) = 29% + 0.40% * t.
To find out when the aggregate income held by the upper-income households first exceeded that held by middle-income households, we set the two equations equal to each other and solve for 't'. This gives us t = (62% - 29%)/(0.40% + 0.38%) = 43.7, or roughly 44 years after 1970 which is the year 2014.
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Draw a line for the axis of symmetry of function f. Also mark the x-intercept(s), y-intercept, and vertex of the function. f(x) = -(x + 1)2 + 4
Answer:
Part A: vertex at (-1,4)
Part B: line of symmetry x =-1
Part C: x-intercept x = -3 and x = 1
Part D: y-intercept y = 3
Step-by-step explanation:
Given f(x) = - (x+1)² + 4
The given equation represents a parabola.
The general equation of the parabola with a vertex at (h,k)
f(x) = (x-h)² + k
Part A: To find the vertex, compare the general equation with the given function.
So, the vertex of the function will be at (-1,4)
Part B: the line for the axis of symmetry of function will be at x =-1
Part C: To find x-intercept, put y = 0
So, - (x+1)² + 4 = 0
- (x+1)² = -4
(x+1)² = 4
x + 1 = ±√4 = ±2
x + 1 = 2 OR x + 1 = -2
x = 1 OR x =-3
x-intercept at x = 1 and x = -3
Part D:To find y-intercept, put x = 0
So, y = - (0+1)² + 4 = -1 + 4 = 3
y-intercept at y = 3
See the attached figure.
Answer:
Vertex at (-1,4)
Line of symmetry x =-1
x-intercept x = -3 and x = 1
y-intercept y = 3
49. If a statistician specifies a 5% level of significance, then she will reject the null hypothesis only if her sample result differs from her hypothesized value by an amount that would occur by chance a) Less than 5% of the time. b) More than 5% of the time. c) 95% of the time or more. d) 2.5% of the time or less.
Answer:
The answer is a) less than 5% of the time
Step-by-step explanation:
By definition, the level of significance is the probability that one rejects a null hypothesis of a test when it is true. Denoted as alpha α, it shows how convincing the sample data is and concludes if it is statistically significant.
A level of significaance of 5% or 0.05, shows that you need more convincing power before determining whether you willreject the null hypothesis
By way of rule,
-If p-value ≤ level of significance α , then reject the null hypothesis.
-If p-value ≥ level of significance α , then do not reject the null hypothesis.
This implies that a statistician with a 5% level of significance will reject the null hypothesis if her hypothesized value falls below this mark.
Answer:
c) 95% of the time or more.
Step-by-step explanation:
The level of significance in a statistical research is the probability of rejecting the null hypothesis when it is actually true. A possible example is a significance level of 0.02 which indicates a 2% risk of concluding that a difference exists when there is no actual difference. The lower the significance level, the stronger the evidence required before you can reject the null hypothesis. 5% significance level may imply 95% confidence interval which means the range of values that occurs 95% of the time.
Therefore, rejecting the null hypothesis at 5% significance level means that the alternative hypothesis differs from the null hypothesis by 95% or more of its possible values (95% of the time or more).
Simplify (4x - 6) – (3x + 6).
1) ×- 12
2)-x - 12
3) x + 12
4) -x + 12
Answer:
1) x-12
Step-by-step explanation:
(4x-6)-(3x+6)
4x-6-3x-6
x-12
Suppose that you have measured a length of 6 cm on one board and 8 cm on the other. You would adjust the two boards until the length of the string had value c to ensure that the boards made a right angle. What is c? Express your answer in centimeters to three significant figures.
Answer:
c = 10.000cm
Step-by-step explanation:
If the 2 boards made the right angle, c would be the hypotenuse with 2 sides of 6 cm and 8 cm. We can then use Pythagorean formula to solve for c
[tex]c^2 = 6^2 + 8^2 = 36 + 64 = 100[/tex]
[tex]c = \sqrt{100} = 10.000 cm[/tex]
Data is collected from a:
A. sample
B. population
C. variable
D. None of the above answers is correct
Data is collected from a sample, which is a smaller subset of the population.
Explanation:The correct answer is A. sample. When data is collected, it is usually infeasible or impractical to collect information from an entire population. Therefore, a sample, which is a smaller subset of the population, is selected for data collection and analysis. For example, if we want to know the average height of students in a school, it would be time-consuming and costly to measure the height of every student. Instead, we can select a sample of students from different grades or classrooms to represent the entire population and collect data from them.
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Evaluate the line integral Z C z2 dx 2y dy 2xz dz where C is any closed curve in the space.
Answer:
Some other information should be given, I have added those necessary information, should in case you come across a similar question on line integral.
Step-by-step explanation:
To evaluate line integral, some other information has to be given, Assume (so when you see a similar question, you will know how to evaluate line integral) as part of the question X =t2, Y = 2t and Z = 3t and at interval 0≤t≤1
A step by step explanation and calculation has been attached.
Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.
a) A student in your class has a cat, a dog and a Ferret is Ex(C(x)∧D(x)∧F(x)).
b) All students in your class have a cat, a dog or a Ferret is ∀x(C(x)∨D(x)∨F(x)).
c) Some student in you class has cat and Ferret but not a dog is Ex(C(x)∧D(x)∨¬F(x)).
The various types of logical connectives include conjunction (“and”), disjunction (“or”), negation (“not”), conditional (“if . . . then”), and biconditional (“if and only if”).
Let C(x) : x has a cat
D(x) : x has a dog
F(x) : x has a Ferret
a) A student in your class has a cat, a dog and a Ferret.
Ex(C(x)∧D(x)∧F(x))
b) All students in your class have a cat, a dog or a Ferret.
∀x(C(x)∨D(x)∨F(x))
c) Some student in you class has cat and Ferret but not a dog
Ex(C(x)∧D(x)∨¬F(x))
Therefore,
a) A student in your class has a cat, a dog and a Ferret is Ex(C(x)∧D(x)∧F(x)).
b) All students in your class have a cat, a dog or a Ferret is ∀x(C(x)∨D(x)∨F(x)).
c) Some student in you class has cat and Ferret but not a dog is Ex(C(x)∧D(x)∨¬F(x)).
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The statements can be expressed using logical connectives and quantifiers in mathematical logic. They express different combinations of students owning cats, dogs, and ferrets.
Explanation:The statements can be expressed in terms of C(x), D(x), F(x), quantifiers, and logical connectives as follows:
a) ∃x (C(x) ∧ D(x) ∧ F(x)): This states that there exists a student 'x' such that 'x has a cat, a dog, and a ferret'. b) ∀x (C(x) ∨ D(x) ∨ F(x)): This states that for all students 'x', 'x has a cat, a dog, or a ferret'. c) ∃x (C(x) ∧ F(x) ∧ ¬D(x)): This states that there exists a student 'x' such that 'x has a cat and a ferret, but not a dog'. d) ¬∃x (C(x) ∧ D(x) ∧ F(x)): This states that there does not exist a student 'x' such that 'x has a cat, a dog, and a ferret'. e) ∃x C(x) ∧ ∃x D(x) ∧ ∃x F(x): This states that for each animal (cats, dogs, ferrets), there exists a student 'x' that has this animal as a pet. Learn more about Mathematical Logic here:https://brainly.com/question/30928478
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what is the total area of this prism?
T.A=
Answer:
The answer to your question is 61.8 in²
Step-by-step explanation:
Process
1.- Calculate the height of the triangle
3² = h² + 1.5²
h² = 3² - 1.5²
h² = 9 - 2.25
h² = 6.75
h = 2.6 in
2.- Calculate the area of the triangles
Area = [tex]\frac{base x height}{2}[/tex]
Area = [tex]\frac{3 x 2.6}{2}[/tex]
Area = 3.9 in²
There are 2 triangles = 2(3.9) = 7.8 in²
3.- Calculate the area of the rectangles
Area = base x height
Area = 6 x 3 = 18 in²
There are 3 rectangles = 3(18) = 54 in²
4.- Total area
At = 7.8 + 54
= 61.8 in²
Answer:
(104 +16√13) ft
Step-by-step explanation:
If a population mean is 300 and the sample mean is 400, the difference of 100 is called:
A.Standard error
B.Sampling error
C.Allowable error
D.None of the above.
Answer:
B. Sample error.
Step-by-step explanation:
Such type of error is called Sample error.
Sample error occurs when
Sample error is a statistical error when an expert fails to choose a sample which symbolizes the whole data population and the outcomes of the sample do not reflect the outcomes from the whole population.
If a population mean is 300 and the sample mean is 400, the difference of 100 is called sampling error. So, option b is correct.
Sampling error represents the difference between the population mean and the sample mean, which arises purely by chance because a sample rather than the entire population is observed.
The other options are incorrect in this context:
Standard error: This measures the spread of the sample means around the population mean.Allowable error: This term refers to the acceptable range of error in forecasting or measurement but is not applicable in this context.If r1 , r2 , and r3 represent rotations from Dn and f 1 , f 2 , and f 3 represent reflections from Dn , determine whether r1 r2 f 1 r3 f 2 f 3 r3 is a rotation or a reflection.
The combined transformation operation r1 r2 f1 r3 f2 f3 r3 is a reflection. This is because a sequence of transformations combining rotations and reflections, when arranged and executed in order, still produces a reflection.
Explanation:The problem is understanding whether the composition of certain transformations, namely rotations (r1, r2, r3) and reflections (f1, f2, f3) from Dn, results in another rotation or a reflection. In mathematics, a composition of transformations means applying several transformations in sequence.
Here, think about the properties of rotations and reflections. A rotation followed by another rotation will result in a rotation. Similarly, a reflection followed by another reflection is equivalent to a rotation. However, a reflection followed by a rotation or vice versa results in a reflection.
The sequence provided is r1 r2 f1 r3 f2 f3 r3. Assuming each transformation in the sequence is executed in order, we can see that it is divided into four segments: rr, rf, fr, and r. The first segment, rr, will result in a rotation. The second segment, rf, will result in a reflection. That reflection followed by the next r, resulting in another reflection. Finally, the last segment is just a rotation. Therefore, reflection followed by a rotation results in a reflection.
So, the combined transformation r1 r2 f1 r3 f2 f3 r3 is a reflection.
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Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0 (see Equation (1) of Section 1.3). Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0. (Use P for P(t).)
dP dt =
What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0?
dP dt =
Answer:
A) Differential equation for population growth in case of individual immigration is:
[tex]\frac{dP}{dt}=kP+r[/tex]
B) Differential equation for population growth in case of individual emigration is:
[tex]\frac{dP}{dt}=kP-r[/tex]
Step-by-step explanation:
Population growth rate in the absence of immigration and emigration is given as:
[tex]\frac{dP}{dt}=kP--(1)[/tex]
A) When individuals are allowed to immigrate:
Let r be the constant rate of individual immigration given that r >0.
Differential equation for population growth in this case is:
[tex]\frac{dP}{dt}=kP+r[/tex]
B) In case of individual emigration:
Let r be the constant rate of individual emigration given that r >0.
Differential equation for population growth in this case is:
[tex]\frac{dP}{dt}=kP-r[/tex]
On one of its routes across Asia, Alpha Airlines flies an aircraft with checked-in luggage capacity of 8500 lbs. There are 121 seats on the flight.
The average (per passenger) weight of checked-in luggage is 68 lbs with a standard deviation of 11 lbs.
What is the probability that on a randomly selected full flight on this route the checked-in luggage capacity will be exceeded?
Answer:
the probability is P=0.012 (1.2%)
Step-by-step explanation:
for the random variable X= weight of checked-in luggage, then if X is approximately normal . then the random variable X₂ = weight of N checked-in luggage = ∑ Xi , distributes normally according to the central limit theorem.
Its expected value will be:
μ₂ = ∑ E(Xi) = N*E(Xi) = 121 seats * 68 lbs/seat = 8228 lbs
for N= 121 seats and E(Xi) = 68 lbs/person* 1 person/seat = 68 lbs/seat
the variance will be
σ₂² = ∑ σ² (Xi)= N*σ²(Xi) → σ₂ = σ *√N = 11 lbs/seat *√121 seats = 121 Lbs
then the standard random variable Z
Z= (X₂- μ₂)/σ₂ =
Zlimit= (8500 Lbs - 8228 lbs)/121 Lbs = 2.248
P(Z > 2.248) = 1- P(Z ≤ 2.248) = 1 - 0.988 = 0.012
P(Z > 2.248)= 0.012
then the probability that on a randomly selected full flight, the checked-in luggage capacity will be exceeded is P(Z > 2.248)= 0.012 (1.2%)
Determine if the collection is not well defined and therefore is not a set. The collection of current NHL players
Answer:
Step-by-step explanation:
The collection of current NHL players cant be defined as some may be on injury so therefore the current NHL players is ill-defined thereby they are described as Not well defined set
Final answer:
A collection of current NHL players is a well-defined set. We can precisely determine what is and is not a member of this set.
Explanation:
A collection of current NHL players is a well-defined set. A set is a collection of distinct objects, and in this case, the objects are the current NHL players. It is possible to precisely determine what is and is not a member of this set. We can provide a clear and concise definition of the set by listing the names of all current NHL players.
For example:
Auston Matthews
Connor McDavid
Alex Ovechkin
Sidney Crosby
Nathan MacKinnon
These are just a few examples, but by listing all the current NHL players, we can determine exactly what is included in the set and what is not.
Find g(-x) when g(x) = 2x^2 -5x + 5
Answer: g(-x) = 2x^2 +5x + 5
Step-by-step explanation:
This Is a topic under functions in mathematics, it deals with substituting "x" for the new entity that has been placed in the new function you're asked to look for.
Hence, to solve this, we simply substitute "-x" for "x" wherever "x" appears in the equation.
By doing so, we go straight to the given function.
2x^2 -5x + 5
And we go ahead with our substitution which goes thus:
2(-x^2) -5(-x) + 5
On simplifying, we then finally have have:
2x^2 +5x + 5.
This means:
g(-x) = 2x^2 +5x + 5
One group, which contains 28 dogs, averages 20.5 inches. Another group that contains 19 dogs, averages 32.1 inches.
What is the average height of poodles in your kennel?
Answer:
The average height of poodles in your kennel is 25.19 inches.
Step-by-step explanation:
This is a weighed average problem.
To find the weighed average, we sum each value of the set multiplied by it's weight, and then we divide by the sum of the weights.
In this problem, we have that:
Average 20.5 inches has weight 28.
Average 32.1 inches has weight 19.
What is the average height of poodles in your kennel?
[tex]A = \frac{32.1*19 + 20.5*28}{19+28} = \frac{1183.9}{47} = 25.19[/tex]
The average height of poodles in your kennel is 25.19 inches.
To start with, in finding the height of the poodle, we consider both groups. We then add the multiplication of the number of dogs by the average, and finally, we divide them all by the number dogs. Mathematically, i am saying that;
Group 1 = 28 * 20.5 = 574Group 2 = 19 * 32.1 = 609.9Adding the number of dogs together, we have 28 + 19 = 47Finally,
[tex]\frac{574+609.9}{47}[/tex] = 25.19
Therefore, the average height of the dogs in the kennel is 25.19 inches
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Which expression is equivalent to 6 minus (negative 8)?
Answer:
6 - (-8)
is 6 + 8
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
Use De Morgan's law to select the statement that is equivalent to: "It is not true that the employee received a large bonus and has a big office."
a. The employee received a big bonus or has a big office.
b. The employee did not receive a big bonus and does not have a big office.
c. The employee did not receivę a big bonus or does not have a big office.
d. The employee received a big bonus and has a big office
Answer:
c. The employee did not receive a big bonus or does not have a big office.
Step-by-step explanation:
From the original statement, "It is not true that the employee received a large bonus and has a big office.", it can be concluded that the employee either has a big office, received a large bonus or neither. That is, either one, the other, or neither conditions are met.
Applying De Morgan's law:
a. This leaves out the possibility that both conditions are not met
b. This only considers that both conditions are not met
c. This statement means that one of the conditions are not met, which is correct.
d. This assumes both conditions are true, which is incorrect.
The statement equivalent to the negation of both an employee receiving a large bonus and having a big office is that the employee did not receive a big bonus or does not have a big office. Hence the correct
To apply De Morgan's laws to the statement "It is not true that the employee received a large bonus and has a big office," we need to negate the conjunction and change the 'and' to 'or' while negating each of the individual statements within the conjunction. De Morgan's laws tell us that the negation of a conjunction is equivalent to the disjunction of the negations. So the statement becomes:
The employee did not receive a large bonus or does not have a big office.
Therefore, the correct answer is (c) The employee did not receive a big bonus or does not have a big office.
Compute the line integral of the vector field F=⟨3zy−1,4x,−y⟩F=⟨3zy−1,4x,−y⟩ over the path c(t)=(et,et,t)c(t)=(et,et,t) for −9≤t≤9
Answer:
Step-by-step explanation:
Find the appropriate percentile from a t-distribution for constructing the following confidence interval.
99% t-interval with n = 3.
Answer: 9.9248
Step-by-step explanation:
We know that the critical t- value for confidence interval is a two-tailed value from the t-distribution table corresponding to degree of freedom (df = n-1 , where n is the sample size) and the significance level ([tex]\alpha/2[/tex]) .
The given confidence interval = 99%
⇒ Significance level = [tex]\alpha=100\%-99\%=1\%=0.01[/tex]
Sample size : n=3
DEgree of freedom : df = n-1 = 2
Then, the critical t- value for 99% confidence interval will be;
[tex]t_{\alpha/2, df}=t_{0.01/2,\ 2}[/tex]
[tex]t_{0.005 , 2}=\pm9.9248[/tex] [From t-distribution table]
Hence, the appropriate percentile from a t-distribution for constructing the 99% confidence interval with n = 3 is 9.9248.
A top-fuel dragster ran a 1/4-mile (1320 ft) race. It had traveled 1305.48 feet after 3.58 seconds and it traveled the entire 1320 feet in 3.61 seconds. What was its speed, approximately, in feet per second at the end of the race? In miles per hour?
The speed approximately of this top-fuel dragster is given as 330 miles per hour.
How to solve for the speedWe have 1320 feet distance in 3.61 sec
then we have 1305.48 feet in 3.58
Such that 1320-1305.48 = 14.52 feet
we have to do the conversions of feet into miles and seconds to become hours
1 mile = 5280feet
hence we have 14.52 feet as 14.52/5280
= 0.00275miles
then 0.03 seconds to hour = 0.03/3600
= 0.0000083
Speed = distance / time
0.00275miles/ 0.0000083
= 330 miles per hour
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To find the speed of the top-fuel dragster at the end of the race, divide the total distance traveled by the time taken. The speed at the end of the race is approximately 365.098 ft/s or 249.340 mph.
Explanation:To find the speed of the top-fuel dragster at the end of the race, we can divide the total distance traveled (1320 ft) by the time it took to cover that distance (3.61 s). This will give us the average velocity of the dragster.
So, the speed at the end of the race is approximately 365.098 ft/s.
To convert this speed to miles per hour, we can multiply it by the conversion factor 0.6818. This gives us the speed of the dragster at the end of the race in miles per hour, which is approximately 249.340 mph.
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 ¯ x x¯ of these residents is to be computed. We know the random variable ¯ x x¯ has approximately a Normal distribution because____________.
a. of the central limit theorem.
b. of the 68‑95‑99.7 rule.
c. the population from which we’re sampling has a Normal distribution.
d. of the law of large numbers.
Answer:
a. of the central limit theorem.
Step-by-step explanation:
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, larger than 30, can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex]
In this problem, the sample size is 100, so it is sufficiently large to use the Central Limit Theorem. The mean of the sample in 69 and the standard deviation of the sample is 0.58.
So the correct answer is:
a. of the central limit theorem.
An aptitude test known as the Gesell adaptive score test is given to children to measure their level of cognitive development. It is of interest to know whether or not a relationship exists between this test score and the age (in months) at which a child speaks his/her first word. To examine this, the following data were collected for 21 children: (a) Treating the Gesell score as the response variable (y) and the age at first word as the explanatory variable (x), make a scatterplot of these data.
Does there appear to be a linear relationship among these variables?
To assess if there's a linear relationship between the Gesell score and the age at first word, one would create a scatterplot with the Gesell score as (y) and age as (x). The presence of a linear trend could be indicated by a clustering of points near a line, while the strength of the relationship would be further analyzed using the least-squares regression line and correlation coefficient.
Explanation:To determine whether there is a linear relationship between the Gesell adaptive score test (Gesell score) and the age at which children speak their first word, you would start by plotting a scatterplot with the Gesell score as the response variable (y) and the age at first word as the explanatory variable (x). In the scatterplot, each point represents one child's data with their corresponding age at first word on the x-axis and Gesell score on the y-axis.
After plotting the data, you would examine the scatterplot to see if the points suggest a linear trend. If the points cluster around a line that slopes upwards or downwards, this could indicate a positive or negative linear relationship respectively. Conversely, if the points are widely scattered without any discernible pattern, it might suggest that there is no significant relationship between the variables.
If there seems to be a potential linear relationship, you might proceed to calculate the least-squares regression line to find the best-fitting line through the data and the correlation coefficient to measure the strength and direction of the relationship between the variables. Significant correlation coefficients (typically those near -1 or 1) would support the presence of a linear relationship, while coefficients near zero would suggest little to no linear relationship.
Final answer:
To analyze the relationship between the Gesell adaptive score test and the age of first word spoken by children, one would plot a scatterplot with Gesell score on the y-axis and age at first word on the x-axis. This visualization aids in identifying any potential linear or non-linear patterns.
Explanation:
To investigate if there is a relationship between the Gesell adaptive score test and the age at which a child speaks their first word, we would create a scatterplot with the age at first word (in months) as the x-axis (explanatory variable) and the Gesell score as the y-axis (response variable). By analyzing the pattern of the dots on the scatterplot, we could determine if there is any apparent linear relationship or if the data suggests a more complex relationship, such as an inverted U-shaped relationship.
It's important to note that an absence of linear correlation from a statistical test doesn't necessarily deny the existence of any relationship between two measures; the relationship could be non-linear or might change over time. An example of this would be a change in problem-solving strategies in children causing an inverted U-shaped relationship in cognitive ability over different stages. Therefore, a scatterplot is a crucial tool for visually identifying patterns and potential relationships in data.
Identify the type of sampling used. Thirty-five math majors, 52 music majors and 38 history majors are randomly selected from 447 math majors, 480 music majors and 451 history majors at the state university. What sampling technique is used?
Answer:
Stratified
Step-by-step explanation:
The stratified sampling is defined as the type of random sampling when the population is divided into non-overlapping groups known as strata and then sample is selected from each of the stratum.
The sampling technique used is stratified sampling because the population is divided into 3 strata math, music and history and then sample of 35,52 and 38 is selected from these 3 strata respectively.
One coin in a collection of 65 coins has two heads; the rest of the coins are fair. If a coin, chosen at random from the lot and then tossed, turns up heads six times in a row, what is the probability that it is the two-headed coin?
There is only 1 two-headed coin in the collection of 65 coins.
The probability of selecting the two headed coin is 1/65.
The outcome achieved when any of the other coins is tossed a number of times is based purely on chance.
Although if the 2-headed coin is selected, the only possible outcome is having a head, but It also possible to have 6 heads in 6 tosses with a coin that is not 2-headed.
What we're concerned with, is the probability that the 2-headed coin was selected from the lot of 65 coins, which is 1/65.
Answer:
There is only 1 two-headed coin in the collection of 65 coins.
The probability of selecting the two-headed coin is 1/65.
The outcome achieved when any of the other coins is tossed a number of times is based purely on chance.
Although if the 2-headed coin is selected, the only possible outcome is having a head, but It is also possible to have 6 heads in 6 tosses with a coin that is not 2-headed.
What we're concerned with, is the probability that the 2-headed coin was selected from the lot of 65 coins, which is 1/65.
Step-by-step explanation:
Let Y1 and Y2 have the joint probability density function given by:
f (y1, y2) = k(1 − y2), 0 ≤ y1 ≤ y2 ≤ 1, 0, elsewhere.
(a) Find the value of k that makes this a probability density function.
(b) Find P(Y1 ≤ 3/4, Y2 ≥ 1/2).
The question involves computing parameters of a joint probability density function given a specific function and ranges. The process involves setting up and evaluating appropriate double integrals over the given ranges.
Explanation:The subject of this problem is related to joint probability density functions (pdfs) and probability theory which comes under mathematics, specifically statistics.
(a) To find the value of k that makes this a valid probability density function, we use the property that the integral of a pdf over its range should equal 1:
So, we integrate the function f(y1, y2) = k(1 – y2) over the range 0 <= y1 <= y2 <= 1. This gives us a double integral: We first integrate with respect to y1, from 0 to y2. Then we integrate with respect to y2, from 0 to 1. Finally, we set this equal to 1 and solve for k.
(b) To find P(Y1 ≤ 3/4, Y2 ≥ 1/2), you integrate the joint pdf over the given intervals:
Integrate from 0 to 3/4 with respect to y1, and from 1/2 to 1 with respect to y2. Learn more about Joint Probability Density Functions here:
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A number is equal to 3 times a smaller number. Also, the sum of the smaller number and 4 is the larger number. The situation is graphed on the coordinate plane below, where x represents the smaller number and y represents the larger number. On a coordinate plane, a line goes through (0, 4) and (2, 6) and another line goes through (1, 3) and (2, 6). Which two equations represent the situation? y = one-third x and y = x minus 4 y = one-third x and y = x + 4 y = 3 x and y = x + 4 y = 3 x and y = x minus 4.
Answer: two equations represent the situation are
y = 3x and y = x + 4
Step-by-step explanation:
The smaller number was represented by x and its values on the x coordinate.
The larger number was represented by y and its values on the y coordinate.
The larger number is equal to 3 times a smaller number. This means that
y = 3x
Also, the sum of the smaller number and 4 is the larger number. This means that
y = x + 4
Answer:
two equations represent the situation are
y = 3x and y = x + 4
Step-by-step explanation:
Derive the equation of motion of the spring-mass system given below. Please show and MARK your derivation step by step. Missing steps will result in losing points. Use the assumptions sin(θ) = θ and cos(θ) = 1.
Answer:
Ö + θ ( (k/m) + (g/l)) = 0
Step-by-step explanation:
Use the FBD attached:
Apply Newtons 2 nd Law in tangential direction:
Sum ( Ft ) = m*a
Sum of all tangential forces is:
m*g*sin(θ) + k*l*sin(θ)*cos(θ) = - m*l*Ö
Using small angle approximations:
sin (θ) = θ
cos (θ) = 1
Ö = angular acceleration.
m*g*θ + k*l*θ = -m*l*Ö
Ö + θ ( (k/m) + (g/l)) = 0