Answer:
True
Step-by-step explanation:
A linear system of equations will have a unique solution if and only if the number of variables equal the number of independent equations.
By independent equations we mean the same equation not repeated by multiplying by any constant.
Suppose number of variables are n, we must have the determinant formed by the coefficients non zero to have a unique solutions.
Here the no of equations are less than the number of variables. So we cannot have a unique solution but can have a parametric solution using the number of parameters as n-m where n = number of variables and m = the number of independent equations given.
So the given statement is true.
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
What is the rate of change of a linear function that includes the points (5, 2) and (9, 4)?
A) -2
B) 2
C) 4
D) 1/2
1
2
Answer:
D) 1/2
Step-by-step explanation:
A linear function has the following format:
[tex]y = ax + b[/tex]
In which a is the slope, that is, the rate of change of the function and b is the y-intercept, that is, the value of y when x = 0.
We have that:
Point (5,2), which means that when x = 5, y = 2. So
[tex]y = ax + b[/tex]
[tex]5a + b = 2[/tex]
Point (9,4), which means that when x = 9, y = 4. So
[tex]9a + b = 4[/tex]
I am going to write b as a function of a in the second equation, and replace on the first. So
[tex]b = 4 - 9a[/tex]
Then
[tex]5a + b = 2[/tex]
[tex]5a + 4 - 9a = 2[/tex]
[tex]-4a = -2[/tex]
[tex]4a = 2[/tex]
[tex]a = 1/2[/tex]
This means that the slope, which is the rate of change of the function, is 1/2.
So the correct answer is:
D) 1/2
Consider the following planes. 3x − 2y + z = 1, 2x + y − 3z = 3
Find parametric equations for the line of intersection of the planes.
Solve for any of the variables; for instance, [tex]z[/tex]:
[tex]z=1-3x+2y[/tex]
[tex]z=-\dfrac{3-2x-y}3[/tex]
Then
[tex]1-3x+2y=-\dfrac{3-2x-y}3\implies11x-5y=6\implies\begin{cases}y=\frac{11x-6}5\\z=1-3x+\frac{22x-12}5\end{cases}[/tex]
Let [tex]x=t[/tex]; then the intersection is given by the vector-valued function
[tex]\vec r(t)=\left(t,\dfrac{11t-6}5,1-3t+\dfrac{22t-12}5\right)[/tex]
or
[tex]\vec r(t)=\left(t,\dfrac{11t-6}5,\dfrac{7t-7}5\right)[/tex]
Express the Cartesian coordinates ( - 5,5) in polar coordinates in at least two different ways. Write the point in polar coordinates with an angle in the range 0 < = theta < 2pi (Type an ordered pair. Type an exact answer, using pi as needed.) Write the point in polar coordinates with an angle in the range - 2pi < = theta < 0. [ ] (Type an ordered pair. Type an exact answer. using pi as needed.)
The Cartesian coordinate point (-5,5) can be re-written in polar coordinates in two ways. The polar coordinates equivalent to (-5,5) in the range 0 <= theta < 2pi are (5sqrt(2), 7pi/4) and in the range -2pi <= theta < 0 are (5sqrt(2), -pi/4).
Explanation:The Cartesian coordinate point (-5,5) can be expressed in polar coordinates using the polar coordinates formula: r = sqrt(x^2 + y^2) and theta = arctan(y/x).
For the first part of the question, we are asked to find the polar coordinates where the angle lies in the range 0 <= theta < 2pi. Using the two formulas above, we find r to be sqrt((-5)^2 + 5^2) = sqrt(50) or 5sqrt(2). The angle theta can be calculated using arctan(5/-5) = -pi/4. We add 2pi to this angle because theta should not be negative, and thus obtain a final result of (7pi/4). So, the polar coordinates equivalent to (-5,5) in the range 0 <= theta < 2pi are (5sqrt(2), 7pi/4).
For the second part of the question, we need the polar coordinates where the angle theta lies in the range -2pi <= theta < 0. In this case, the value for r remains the same as before. However, the angle is -pi/4, because it needs to be negative. Hence, the polar coordinates equivalent to (-5, 5) in the range -2pi <= theta < 0 are (5sqrt(2), -pi/4).
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two cards are chosen at random from a deck of 52 playing cards. what is the probability that they are both aces?
Answer:
The probability that they are both aces is 0.00452.
Step-by-step explanation:
Consider the provided information.
Out of 52 playing card we need to select only 2.
Thus, the sample space is: [tex]^{52}C_2=1326[/tex]
Two cards are Ace.
The number of Ace in a pack of playing card are 4 and we need to select two of them.
This can be written as: [tex]^4C_2=6[/tex]
Thus, the probability that they are both aces is:
[tex]P(\text{Both are Ace})=\dfrac{6}{1326} \approx0.00452[/tex]
Hence, the probability that they are both aces is 0.00452.
The correlation between a car’s engine size and its fuel economy (in mpg) is r = - 0.774.
What fraction of the variability in fuel economy is accounted for by the engine size?
A. 27.5%
B. 63.8%
C. 71.8%
D. 84.1%
Answer: 59.91%.
Step-by-step explanation:
We know that the fraction of the variability in data values accounted by a model is given by [tex]r^2[/tex] , where r is the coefficient of correlation.We are given , that the correlation between a car’s engine size and its fuel economy (in mpg) is r = - 0.774.
Then, the fraction of the variability in fuel economy is accounted for by the engine size would be [tex]r^2=( - 0.774)^2=0.599076\approx59.91\%[/tex]
[Multiply 100 to convert a decimal into percent]
Hence, the fraction of the variability in fuel economy is accounted for by the engine size is 59.91%.
None of the options are correct.
To solve such problems we must know about the fraction of the variability in data values or R-squared.
R- SquaredThe fraction by which the variance of the dependent variable is greater than the variance of the errors is known as R-squared. It is called so because it is the square of the correlation between the dependent and independent variables, which is commonly denoted by “r” in a simple regression model.fraction of the variability in data values = (coefficient of correlation)²= r²
The fraction of the variability in fuel economy is accounted for by the engine size is 59.91%.
Given to usThe correlation between a car’s engine size and its fuel economy (in mpg) is r = - 0.774.SolutionVariability in fuel economyAs it is given that the correlation between a car’s engine size and its fuel economy (in mpg) is r = - 0.774.
the variability in fuel economy = r²
= (-0.774)²
= 0.599076
= 59.91%
Hence, the fraction of the variability in fuel economy is accounted for by the engine size is 59.91%.
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Answer the following question using the appropriate counting technique, which may be either arrangement with repetition, permutations, or combinations. Be sure to explain why this counting technique applies to the problem.
How many possible birth orders with respect to gender are possible in a family with five children? (For example, BBBGG and BGBGGBBBGG and BGBGG are different orders.)
A. Permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.
B. Arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated.
C. Combinations because the selections come from a single group of items, no item can be selected more than once, and the order of the arrangement does not matter.
D. Arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
Final answer:
The total number of possible birth orders with respect to gender in a family with five children is 32.
Explanation:
The appropriate counting technique to answer this question is A. permutations. Permutations are used when the selections come from a single group of items, no item can be selected more than once, and the order of the arrangement matters. In this case, the birth orders with respect to gender can be represented by arranging the genders of the children in different orders.
Since there are 5 children, there are 5 positions to fill. The first position can be filled with either a boy or a girl, so there are 2 options. The second position can also be filled with either a boy or a girl, so there are 2 options again. This goes on for all 5 positions.
Therefore, the total number of possible birth orders with respect to gender is 2 * 2 * 2 * 2 * 2 = 32.
Suppose a manufacturer of light bulbs produces a 75-watt bulb that burns a mean of 7500 hours before it burns out. It has a standard deviation of 220 hours. What percentage of their bulbs would they have to replace under warranty if they offered a warranty of 7200 hours?
Answer:
8.69% of their bulbs would they have to replace under warranty if they offered a warranty of 7200 hours.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 7500, \sigma = 220[/tex]
What percentage of their bulbs would they have to replace under warranty if they offered a warranty of 7200 hours?
This is the probability that X is lower or equal than 7200 hours. So this is the pvalue of Z when X = 7200.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{7200 - 7500}{220}[/tex]
[tex]Z = -1.36[/tex]
[tex]Z = -1.36[/tex] has a pvalue of 0.0869
So 8.69% of their bulbs would they have to replace under warranty if they offered a warranty of 7200 hours.
Final answer:
The manufacturer would need to replace approximately 8.69% of their 75-watt light bulbs under a warranty of 7200 hours, based on the given mean lifetime and standard deviation.
Explanation:
To calculate the percentage of 75-Watt light bulbs that a manufacturer would have to replace under warranty if they offered a warranty of 7200 hours, we need to use the normal distribution. The mean lifetime of the bulbs is 7500 hours with a standard deviation of 220 hours. To find the percentage of bulbs that burn out before 7200 hours, we first calculate the z-score, which is the number of standard deviations an observation is from the mean.
The z-score is calculated as:
Z = (X - μ) / σ
Where μ is the mean, σ is the standard deviation, and X is the value we are interested in (7200 hours in this case).
Plugging in the numbers:
Z = (7200 - 7500) / 220 = -300 / 220 = -1.36
Using a Z-table or a statistical calculator, we find that the probability corresponding to a Z-score of -1.36 is approximately 0.0869 or 8.69%. This means that the manufacturer would have to replace about 8.69% of their bulbs under this warranty.
Which equation can be used to find the total number of contestants, C that audition in H hours?
Answer:
The correct answer is C. 32h = C
Step-by-step explanation:
For finding out the equation that can be used to find the total number of contestants, C, that audition in h hours. we will use these variables:
Number of contestants = C
Number of hours of auditions = h
Number of contestants that can audition per hour = 32
Now, we can affirm that the equation that can be used to find the total number of contestants, C, that audition in h hours is:
C = 32 * h
The correct answer is C. 32h = C
Let X be the time in minutes between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with λ = 1, compute the following: (If necessary, round your answer to three decimal places.)
(a) The expected time between two successive arrivals is minutes.
(b) The standard deviation of the time between successive arrivals is minutes.
Answer:
a) 1
b) 1
Step-by-step explanation:
Data provided in the question:
X = The time in minutes between two successive arrivals
X has an exponential distribution with λ = 1
Now,
a) The expected time between two successive arrivals is minutes i.e
E(X) = [tex]\frac{1}{\lambda}[/tex]
or
E(X) = [tex]\frac{1}{1}[/tex]
or
E(X) = 1
b) The standard deviation of the time between successive arrivals is minutes
i.e
σₓ = [tex]\sqrt{\frac{1}{\lambda^2}}[/tex]
or
σₓ = [tex]\sqrt{\frac{1}{1^2}}[/tex]
or
σₓ = 1
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.
The mean height of a basketball team is 6 feet with a standard deviation of 0.2 feet. The team's center is 6.7 feet tall. Find the center's z score. Is his score unusual?
Answer:
z-score = 3.5, Yes, z-score is unusual.
Step-by-step explanation:
Given information:
Population mean ; μ = 6 feet
Standard deviation ; σ = 0.2 feet
Sample mean = 6.7 feet
The formula for z-score is
[tex]z=\dfrac{\overline{X}-\mu}{\sigma}[/tex]
where, [tex]\overline{X}[/tex] is sample mean, μ is population mean and σ is standard deviation.
Substitute the given values in the above formula.
[tex]z=\dfrac{6.7-6}{0.2}[/tex]
[tex]z=\dfrac{0.7}{0.2}[/tex]
[tex]z=3.5[/tex]
The z-score is 3.5.
If a z-score is less than -2 or greater than 2, then it is known as unusual score.
3.5 > 2
It means z-score is unusual.
Since the z score is greater than 3, hence his score is unusual.]
Z scoreZ score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
[tex]z=\frac{x-\mu}{\sigma} \\ \\ where\ x=raw\ score,\mu=mean\ and\ \sigma=standard\ deviation[/tex]
Given that mean = 6, standard deviation = 0.2.
For x = 6.7:
[tex]z=\frac{6.7-6}{0.2}=3.5 [/tex]
Since the z score is greater than 3, hence his score is unusual.
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The population of a region is growing exponentially. There were 40 million people in 1980 (when ????=0) and 80 million people in 1990. Find an exponential model for the population (in millions of people) at any time ????, in years after 1980.
Answer:
[tex]P(t) = 40e^{0.06931t}[/tex]
Step-by-step explanation:
The population exponential equation is as follows.
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(t) is the population in t years from now, P(0) is the population in the current year and r(decimal) is the growth rate.e = 2.71 is the Euler number.
Find an exponential model for the population (in millions of people) at any time ????, in years after 1980.
There were 40 million people in 1980 (when ????=0).
This means tht P(0) = 40.
So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]P(t) = 40e^{rt}[/tex]
80 million people in 1990.
1990 is 10 years after 1980. So P(10) = 80. We use this to find the value of r.
So
[tex]P(t) = 40e^{rt}[/tex]
[tex]80 = 40e^{10r}[/tex]
[tex]e^{10r} = 2[/tex]
Applying ln to both sides, since [tex]\ln{e^{a}} = a[/tex]
[tex]\ln{e^{10r}} = \ln{2}[/tex]
[tex]10r = 0.6931[/tex]
[tex]r = \frac{0.6931}{10}[/tex]
[tex]r = 0.06931[/tex]
So the exponential model for the population is:
[tex]P(t) = 40e^{0.06931t}[/tex]
A music school has budgeted to purchase 3 musical instruments. They plan to purchase a piano costing $4,000, a guitar costing $600, and a drum set costing $750. The mean cost for a piano is $4,500 with a standard deviation of $2,500. The mean cost for a guitar is $500 with a standard deviation of $200. The mean cost for drums is $850 with a standard deviation of $100. (Enter your answers to two decimal places.)
1.) How many standard deviations above or below the average piano cost is the piano?
2.) How many standard deviations above or below the average guitar cost is the guitar?
3.) How many standard deviations above or below the average drum set cost is the drum set?
4.) Which cost is the lowest, when compared to other instruments of the same type?
5.) Which cost is the highest when compared to other instruments of the same type?
The piano is 0.2 standard deviations below the average piano cost, while the guitar is 0.5 standard deviations above the average guitar cost. The drum set is 1 standard deviation below the average drum set cost. The drums have the lowest cost compared to other instruments of the same type, and the piano has the highest cost.
Explanation:1.) To find how many standard deviations above or below the average piano cost is the piano, we need to calculate the z-score. The formula for calculating the z-score is
z = (x - mean) / standard deviation
Let's plug in the values for the piano:
z = ($4,000 - $4,500) / $2,500
z = -0.2
So, the piano is 0.2 standard deviations below the average piano cost.
2.) Following the same formula, let's calculate the z-score for the guitar:
z = ($600 - $500) / $200
z = 0.5
Thus, the guitar is 0.5 standard deviations above the average guitar cost.
3.) Using the formula for the z-score again, let's calculate the z-score for the drum set:
z = ($750 - $850) / $100
z = -1
The drum set is 1 standard deviation below the average drum set cost.
4.) The cost of the drums is the lowest when compared to other instruments of the same type.
5.) The cost of the piano is the highest when compared to other instruments of the same type.
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
a) For 30 randomly selected Rolling Stones concerts, the mean gross earnings is 2.79 million dollars. Assuming a population standard deviation gross earnings of 0.47 million dollars, obtain a 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions). Confidence interval: ( __________________ , __________________ )
Answer:
The 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions) is (2.5690, 3.0110).
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex]
Now, find M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 2.575*\frac{0.47}{\sqrt{30}} = 0.2210[/tex]
The lower end of the interval is the mean subtracted by M. So it is 2.79 - 0.2210 = 2.5690 million dollars.
The upper end of the interval is the mean added to M. So it is So it is 2.79 + 0.2210 = 3.0110 million dollars.
The 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions) is (2.5690, 3.0110).
Cody and Chelsea are 270 feet apart when they start walking toward one another. They are walking at the same speed, so whenever Cody travels some number of feet, Chelsea travels the same number of feet. Let x represent the number of feet Cody has traveled since he started walking toward Chelsea.
Write an expression in terms of xx that represents the number of feet Cody has walked toward Chelsea since they started walking.
The expression in terms of [tex]x[/tex] that represents the number of feet Cody has walked toward Chelsea since they started walking is [tex]x_{Co} = x[/tex].
Procedure - Uniform motionIn this question we assume that both Cody and Chelsea have a uniform motion, in opposite sides and to their encounter. If both travel the same number of feet, the kinematic formulas for each walker are described below:
Cody[tex]x_{Co} = x[/tex] (1)
Chelsea[tex]x_{Ch} = 270 - x[/tex] (2)
Encounter[tex]x_{Co} = x_{Ch}[/tex] (3)
Where:
[tex]x[/tex] - Number of feet walked by each walker, in feet.[tex]x_{Co}[/tex], [tex]x_{Ch}[/tex] - Position of each walker, in feet.By (3), (2) and (1) we have the following equation:
[tex]x = 270 - x[/tex]
[tex]2\cdot x = 270[/tex]
[tex]x = 135\,ft[/tex]
The expression in terms of [tex]x[/tex] that represents the number of feet Cody has walked toward Chelsea since they started walking is [tex]x_{Co} = x[/tex]. [tex]\blacksquare[/tex]
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The expression to represent the number of feet Cody has walked toward Chelsea since they started walking is '270 - 2x'. Where 'x' represents the distance Cody has traveled and '2x' reflecting that Chelsea has traveled the same distance. The total is subtracted from 270, the initial distance, to find the remaining distance.
Explanation:Given that Cody and Chelsea start walking towards each other from a distance of 270 feet and that they are walking at the same speed. If x represents the number of feet Cody has traveled since he started walking, given that Chelsea travels the same distance, the total distance they have both traveled together is represented by 2x. Since they started 270 feet apart, to find out how much more they still have to walk until they meet, you would subtract the total feet they've walked from the original distance. So this can be represented by the expression: 270 - 2x. This expression will give us the remaining distance after Cody has walked 'x' feet towards Chelsea.
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Find the area of the region bounded by the hyperbola 25x2 − 4y2 = 100 and the line x = 3. (Using trigonometric substitution)
The question involves calculating the area between a hyperbola and a line using trigonometric substitution, yet it's more directly related to understanding hyperbola properties and standard integration techniques of hyperbolic functions rather than a straightforward trigonometric substitution problem.
Explanation:The problem involves finding the area bounded by the hyperbola 25x² - 4y² = 100 and the line x = 3 using trigonometric substitution. This specific problem requires a solid understanding of calculus, particularly integration, and knowledge of hyperbolas and trigonometric identities. To solve this, we first rewrite the hyperbola in its standard form, which is √(25x²/100 - 4y²/100) = 1, simplifying to x²/4 - y²/25 = 1, indicating that a = 2 and b = 5. Using trigonometric substitution, we set x = a sec(θ) and y = b tan(θ) where a and b are the semi-major and semi-minor axes of the hyperbola, respectively. However, the given question simplifies to the computation of an area for a given range of x, particularly up to x = 3, which doesn't directly imply a need for integral computation but rather an understanding of the geometry of hyperbolas.
The straight line x = 3 intersects the hyperbola at specific points, and the area in question could be typically found using trigonometric integrals if it required finding an area under a curve or between two curves. In this context, without further details on the exact method of trigonometric substitution to be used for computing the area directly, we often rely on the integrals involving the hyperbolic trigonometric functions, using limits set by the intersections of the curve and the line. Since a direct solution involving trigonometric substitution for area calculation under these conditions is complex and not standard, it's crucial to review the application within this specific problem's context further.
For what value of x does 3^4 x = 27^x - 3? a.–9 b.–3 c.3 d.9
Answer:
A
Step-by-step explanation:
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.
In a recent poll of 1100 randomly selected home delivery truck drivers, 26% said they had encountered an aggressive dog on the job at least once. What is the standard error for the estimate of the proportion of all home delivery truck drivers who have encountered an aggressive dog on the job at least once? Round to the nearest ten-thousandth a) 0.0141 b) 0.0002 c) 0.1322 d) 0.0132
Answer:
d) 0.0132
Step-by-step explanation:
In a sample of size n, with proportion p, the standard error of the proportion is:
[tex]SE_{p} = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem, we have that:
[tex]p = 0.26, n = 1100[/tex]
So
[tex]SE_{p} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.26*0.74}{1100}} = 0.0132[/tex]
So the correct answer is:
d) 0.0132
Final answer:
Explaining the calculation of the standard error for a proportion estimate based on a poll of truck drivers encountering aggressive dogs.
Explanation:
Question: In a recent poll of 1100 randomly selected home delivery truck drivers, 26% said they had encountered an aggressive dog on the job at least once. What is the standard error for the estimate of the proportion of all home delivery truck drivers who have encountered an aggressive dog on the job at least once?
Answer:
Calculate the standard error using the formula: SE = sqrt((p*(1-p))/n)
Substitute the values: p = 0.26 (proportion), n = 1100 (sample size)
SE = sqrt((0.26*(1-0.26))/1100) = sqrt((0.26*0.74)/1100) = sqrt(0.1924/1100) = sqrt(0.000175) ≈ 0.0132 (option d)
Let f(x)=x+2 and g(x)=2x+1 .
Graph the functions on the same coordinate plane.
What are the solutions to the equation f(x)=g(x) ?
Enter your answers in the boxes.
Step-by-step explanation:
Given g(x) = x+2 and f(x) = [tex]2^x+1[/tex]
If f(x) = g(x)
⇔[tex]2^x+1[/tex] = x+2
⇔[tex]2^x= x +1[/tex]
Only x= 1 and x=0 will be satisfy the above equation.
If x= 1, it gives and x=0 gives
[tex]2^1= 1+1[/tex] [tex]2^0=0+1[/tex]
⇔2=2 ⇔1=1
Answer:
1 & 0
Step-by-step explanation:
Inferential statistics involves using population data to make inferences about a sample. Group of answer choices True False
Answer:
False. Is the inverse process.
See explanation below.
Step-by-step explanation:
We need to remember two important concepts:
A parameter, is a quantity or value who describe a population desired, for example the population mean [tex]\mu[/tex] or the population standard deviation [tex]\sigma[/tex]
A statistic, is a quantity or value who represent the information of the sample data, for example the sample mean [tex] \bar X[/tex] or the sample deviation [tex] s[/tex]
Based on this we can analyze the statement:
"Inferential statistics involves using population data to make inferences about a sample"
False. Is the inverse process.
If we know the population data then we indeed have parameters and we don't need to do any type of inference in order to estimate these parameters with the statistics.
What we do generally is use the information from the sample in order to obtain statistics representative of the population with the aim to estimate the parameters unknown of the population
Harry church borrowed 17,500.00 at 6.5% exact interest. He had to pay back a maturity value of 17,873.97 to pay off the loan. What was the term of the loan
Answer:the term of the loan is approximately 4 months
Step-by-step explanation:
The term of the loan means the period for which the loan was given.
We would apply the formula for simple interest which is expressed as
I = PRT/100
Where
P represents the principal
R represents interest rate
T represents time in years
I = interest after t years
From the information given
P = 17500
R = 6.5%
I = total amount paid - principal
I = 17,873.97 - 17,500.00 = 373.97
Therefore
373.97 = (17500 × 6.5 × T)/100
373.97 = 1137.5T
T = 373.97/1137.5
T = 0.32 years
Converting to months, it becomes
0.32 × 12 = 3.84
Approximately 4 months.
a farmer looks over a field and sees 28 heads and 78 feet. some are goats, some are ducks. how many of each animal are there?
Answer:there are 11 goats and 17 ducks.
Step-by-step explanation:
Let x represent the number of goats in the field.
Let y represent the number of ducks in the field.
A farmer looks over a field and sees 28 heads and 78 feet. some are goats, some are ducks. A goat has one head and a duck also has one head. It means that
x + y = 28
A goat has 4 feets and a duck has two feets. It means that
4x + 2y = 78 - - - - - - - - - - - - - 1
Substituting x = 28 - y into equation 1, it becomes
4(28 - y) + 2y = 78
112 - 4y + 2y = 78
- 4y + 2y = 78 - 112
- 2y = - 34
y = - 34/- 2
y = 17
x = 28 - y = 28 - 17
x = 11
To find the number of goats and ducks in the field, set up a system of equations using the given information. Solve the system of equations using elimination or substitution method. After solving, you will find that there are 11 goats and 17 ducks in the field.
Explanation:Let's solve this problem by setting up a system of equations. Let's assume that the number of goats is G and the number of ducks is D.
From the given information, we can set up two equations:
1) G + D = 28 (equation 1) - since the total number of heads is 28
2) 4G + 2D = 78 (equation 2) - since each goat has 4 feet and each duck has 2 feet
We can solve these equations using substitution or elimination. Let's solve by elimination method:
Multiplying equation 1 by 2, we get:
2G + 2D = 56 (equation 3)
Subtracting equation 3 from equation 2, we get:
2G = 22
G = 11
Substituting the value of G into equation 1, we get:
11 + D = 28
D = 17
Therefore, there are 11 goats and 17 ducks in the field.
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Geologists have an interest in the structure and the history of the earth. A geologist can go back in time by drilling deep into the ground, retrieving a core sample, estimating the ages of the various layers, and examining the composition. A timeline can be built of the entire area from where the core sample was drilled. A geologist may retrieve several core samples to confirm the history of the earth's structure in that sampled area. Mountains, lakes, and unstable ground can easily impede a simple random sampling of a desired geographical area, therefore what is the most realistic sampling method that represents the actual drillings, comparisons, and scientific examinations of several core samples within the same geographical area? Stratified sampling Systematic sampling Cluster sampling Simple random sampling
Answer:
The answer is cluster sampling
Step-by-step explanation:
Going back to the basic types of sampling and how they are acconplished, cluster sampling is accomplished by dividing the population into groups (usually geographical) which in the above question is referred to as the geographical area. These groups are called clusters or blocks. The clusters are randomly selected and each element in the selected clusters are used. Therefore the most realistic sampling method that represents the actual drillings, comparisons, and scientific examinations of several core samples within the same geographical area is cluster sampling.
Final answer:
Stratified sampling is the most realistic sampling method for geologists to represent the actual drillings, comparisons, and scientific examinations of several core samples within the same geographical area.
Explanation:
Stratified sampling is the most realistic sampling method for geologists to represent the actual drillings, comparisons, and scientific examinations of several core samples within the same geographical area. This method involves dividing the population into distinct subgroups based on certain characteristics and then randomly selecting samples from each subgroup.
For example, in geology, geologists may stratify the geographical area based on different rock types or layers to ensure a representative sample of the earth's structure. By using stratified sampling, geologists can account for the diverse composition and ages of various layers within the sampled area, leading to more accurate interpretations of the history and structure of the earth.
Stratified sampling allows geologists to focus their analyses on specific layers or formations within the earth's structure, providing a comprehensive understanding of the geological history of a particular area. This method enhances the reliability and validity of the findings obtained from multiple core samples taken within the same geographical region.
In late June 2012, Survey USA published results of a survey stating that 56% of the 600 randomly sampled Kansas residents planned to set off fireworks on July 4th. Determine the margin of error for the 56% point estimate using a 95% confidence level.
Answer:
The margin of error is of 3.97 percentage points.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
The confidence interval has the following margin of error
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
95% confidence interval
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]z = 1.96[/tex].
We also have that
[tex]n = 600, \pi = 0.56[/tex]
So, the margin of error is:
[tex]M = 1.96*\sqrt{\frac{0.56*0.44}{600}} = 0.0397[/tex]
So the margin of error is of 3.97 percentage points.
The margin of error for the 56% point estimate using a 95% confidence level is; 3.97%
Formula for margin of error here is;
M = z√(p(1 - p)/n)
Where;
z is critical value at given confidence level
p is sample proportion
n is sample size
We are given;
p = 56% = 0.56
n = 600
Confidence level = 95%
Now,from tables, the critical value at a confidence level of 95% is; z = 1.96
Thus;
M = 1.96√(0.56(1 - 0.56)/600)
M = 0.0397 or 3.97%
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The average age of students in a statistics class is 28 years. The 28 years would be considered an example of:
Based on the information given, it should be noted that the 28 years would be considered an example of descriptive statistics.
It should be noted that descriptive statistics gives a summary description of a particular data quantitatively.It is given to an analysis of data which is vital in describing or showing the summary of data.Therefore, 28 years would be considered an example of descriptive statistics.
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The average age of 28 years in the given class is an example of a statistic, which is a numerical value calculated from a sample of a larger population.
In the context of this question from the subject of mathematics, specifically statistics, the average age of 28 years for students in a class is considered an example of a statistic. A statistic is a value calculated from a sample drawn from a larger population. In this case, the sample consists of students in this particular class, and the population could be all students taking a statistics class in the country or the world. The average (or mean) is a basic type of statistic used to summarise a set of values. It's important to note that the average is just one type of statistic - there are many others, such as median, mode, range, and standard deviation that provide different ways to summarise and represent data.
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there is a herd of cattle out on the range. fifty-two percent of the cattle are male. if the herd consists of 1175 animals, how many are female
Answer:
564 of the cattle are female
Step-by-step explanation:
if 52% of them are male, then 48% are female.
.48*1175=564
Answer:
1227
Step-by-step explanation:
Express the confidence interval (0.403 comma 0.487 )(0.403,0.487) in the form of ModifyingAbove p with caretpplus or minus±E. ModifyingAbove p with caretpplus or minus±Eequals=nothingplus or minus±nothing (Type an integer or a decimal.)
L = 0.403 is the lower end of the confidence interval
U = 0.487 is the upper end of the confidence interval
The margin of error E is
E = (U-L)/2
E = (0.487-0.403)/2
E = 0.042
Which is half the distance from the lower to upper end
---------------------------
The midpoint of the confidence interval is the value of [tex]\hat{p}[/tex] (read out as "p hat")
phat = (U+L)/2
phat = (0.487+0.403)/2
phat = 0.445
---------------------------
Answer: The confidence interval (0.403, 0.487) can be rewritten into the form [tex]0.445 \pm 0.042[/tex]