The player should be willing to pay up to $12 to play this game and not lose money in the long run.
How to calculate the expected value for this game.
First we need to multiply the probability of winning by the amount won and subtract the probability of losing by the amount lost.
The probability of rolling a 1 on a 6-sided die is 1/6, and the probability of rolling any other number is 5/6.
So, the expected value of the game is:
(1/6) x $4 - (5/6) x $2
= ($4/6) - ($10/6)
= -$1/3
This means that on average, for every game played, the player can expect to lose $1/3.
To find out how much the player should be willing to pay to play this game and not lose money in the long run, we can set the expected value equal to zero:
(1/6) x $4 - (5/6) x $2 = $0
Simplifying the equation, we get:
$4/6 = $10/6
Multiplying both sides by x, we get:
(1/6) x - $2 = 0
Solving for x, we get:
x = $12
Therefore, the player should be willing to pay up to $12 to play this game and not lose money in the long run.
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please find the midpoint of the following line and arc using straightedge-compass-construction method
The midpoint of a line or arc can be found using straight edge-compass-construction method by drawing two perpendicular bisectors. The intersection of these bisectors is the midpoint.
To find the midpoint of a line segment, first draw a straight line passing through both endpoints of the segment using a straight edge. Then, using a compass, draw two circles with the same radius centered at each endpoint of the line segment. The circles should intersect at two points. Draw straight lines connecting these two points to form two perpendicular bisectors of the line segment. The intersection of these bisectors is the midpoint of the line segment.
To find the midpoint of an arc, first draw a chord that intersects the arc at two points using a straight edge. Then, using a compass, draw two circles with the same radius centered at each endpoint of the chord. The circles should intersect at two points. Draw straight lines connecting these two points to form two perpendicular bisectors of the chord. The intersection of these bisectors is the center of the circle that the arc belongs to. Draw a line from the center of the circle to the midpoint of the chord. This line will intersect the arc at its midpoint.
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--The question is incomplete, answering to the question below--
"find the midpoint of a line and arc using straight edge-compass-construction method"
Alexander and Rhiannon left school at the same time. Alexander travelled 14 km home at an average speed of 20 km/h. Rhiannon travelled 10 km home at an average speed of 24 km/h. a) Who arrived home earlier? b) How much earlier did this person arrive at home? Give your answer to the nearest minute.
Rhiannon arrived home approximately 17 minutes earlier than Alexander.
What is the average?This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
According to the given information:To solve this problem, we can use the formula:
time = distance / speed
a) The time it took Alexander to get home is:
time_Alexander = 14 km / 20 km/h = 0.7 hours
The time it took Rhiannon to get home is:
time_Rhiannon = 10 km / 24 km/h = 0.41667 hours
Since Rhiannon's time is smaller than Alexander's, Rhiannon arrived home earlier.
b) The time difference between their arrivals is:
time_difference = time_Alexander - time_Rhiannon = 0.7 hours - 0.41667 hours = 0.28333 hours
To convert this to minutes, we can multiply by 60:
time_difference_in_minutes = 0.28333 hours x 60 minutes/hour ≈ 17 minutes
Therefore, Rhiannon arrived home approximately 17 minutes earlier than Alexander.
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Determine the equation of the ellipse with foci (-8,14) and (-8,-16), and co-vertices (0,-1) and (-16,-1).
According to the given information, the equation of the ellipse is [tex](x+8)^2/256 + (y+1)^2/784 = 1.[/tex]
What is co-ordinate geometry ?Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with the study of geometric shapes using algebraic principles. It involves the use of coordinates to represent points, lines, curves, and other geometric figures on a plane or in space.
According to the given information:we need to know the coordinates of its foci, co-vertices, and the center. We can start by finding the center of the ellipse, which is the midpoint of the line segment joining the foci:
Center = ( (-8 + (-8))/2 , (14 + (-16))/2 ) = (-8,-1)
Next, we can find the distance between the foci, which is given by:
[tex]distance between foci = 2c = sqrt[(14 - (-16))^2 + (-8 - (-8))^2] = 30[/tex]
where c is the distance from the center to either focus.
We also know that the distance between the co-vertices is given by:
distance between co-vertices = 2a = |-16 - 0| = 16
where a is the distance from the center to either co-vertex.
Finally, we can use the standard form equation for an ellipse centered at the origin:
[tex](x^2/a^2) + (y^2/b^2) = 1[/tex]
where b is the distance from the center to either vertex.
To find b, we can use the Pythagorean theorem:
[tex]b^2 = c^2 - a^2 \\b^2 = 30^2 - 16^2\\b^2 = 784\\b = 28[/tex]
Now we have all the information we need to write the equation of the ellipse:
[tex](x+8)^2/16^2 + (y+1)^2/28^2 = 1[/tex]
Therefore, according to the given information, the equation of the ellipse is [tex](x+8)^2/256 + (y+1)^2/784 = 1.[/tex]
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Solve using the correct order of
operations.
P
E
MD
AS
15-(4-3) 2= [?]
Enter
Help
Here using the correct order of operations that is PEMDAS, we get the value to be 13.
Define PEMDAS?PEDMAS can be summed up as a mathematical acronym that lists the various arithmetic operations in order of greatest to least practical use.
The letters stand for:
P stands for parentheses.
Exponents are shown as E.
D stands for division.
The letter M stands for multiplication.
A stand for addition.
S stands for subtraction.
Now in the given question,
15 - (4-3)2
First the parentheses
15 - (1)2
Next is exponents.
15 - 2
At last, after subtracting, we get:
13
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fine the exact value of sin(45-30)
Suppose that A is the set of sophomores at your schooland B is the set of students in discrete math at your school.Express each of the following sets in terms of A and B.a. The set of sophomores taking discrete math at yourschool.That’s the intersection A ∩ B.b. The set of sophomores at your school who are nottaking discrete math.This is the difference A − B. It can also be expressed byintersection and complement A ∩ B.c. The set of students at your school who either are sophomores or are taking discrete math.The union A ∪ B.d. The set of students at your school who either are notsophomores or are not taking discrete math.Literally, it’s A ∪ B. That’s the same as A ∩ B.
Set of sophomores taking discrete math = A ∩ B. Set of sophomores not taking discrete math = A - B or A ∩ B^c. Set of students who are sophomores or in discrete math = A ∪ B. Set of students who are not sophomores or not in discrete math = (A ∩ B)^c or A ∪ B^c.
The set of sophomores taking discrete math at your school is the intersection of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∩ B.
The set of sophomores at your school who are not taking discrete math is the difference between the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A - B or A ∩ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math).
The set of students at your school who either are sophomores or are taking discrete math is the union of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∪ B.
The set of students at your school who either are not sophomores or are not taking discrete math is the complement of the intersection of the set of sophomores A and the set of students in discrete math B.
This can be expressed as (A ∩ B)^c or as A ∪ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math). Note that this set includes all students who are either juniors, seniors, or not enrolled in discrete math.
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can you find c and b?
c=?
b=?
The value of the constant c that makes the following function are c = 0.
What is constant ?Constant is a term used to describe a value that remains unchanged or fixed throughout a program or process. It can be a numeric value, a character value, a string, or a Boolean (true/false) value. Common examples of constants include physical constants, mathematical constants, and programming-language keywords.A constant is a value that does not change, regardless of the conditions or context in which it is used. Common examples of constants include mathematical values such as pi (3.14159), physical constants such as the speed of light (299,792,458 m/s), and other constants such as the universal gravitational constant (6.67408 × 10−11 m3 kg−1 s−2).
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Therefore, c must equal 0 in order for the two sides of the function to be equal. and The one with the greater absolute value is b = 10.
What is function?A function is a block of code that performs a specific task. It is a subprogram or a set of instructions that can be used multiple times in a program.
27. For the function to be continuous at x = 7, the limit of the function as x approaches 7 from the left must equal the limit of the function as x approaches 7 from the right.
This means that the value of y as x approaches 7 must be the same on both the left and right sides of the point.
Since the left side of the function is y = c*y + 3, the right side of the function must also be equal to y = c*y + 3.
Therefore, c must equal 0 in order for the two sides of the function to be equal.
28. In order for the function to be continuous at x = 5, the value of y at x = 5 must be the same on both the left and right sides of the point.
Since the left side of the function is y = b - 2x, the right side of the function must also be equal to y = b - 2x.
Therefore, b must equal 10 in order for the two sides of the function to be equal.
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Complete Question:
The function f(x) is represented by this table of values.
x f(x)
-5 35
-4 24
-3 15
-28
-1
3
0
0
1 -1
Match the average rates of change of fx) to the corresponding intervals.
-8
-7
(-5, -1]
(-4,-1]
[-3, 1]
(2, 1)
HELPPP ASAP
Answer:
-8: (-4, -3]
-7: (-3, -1]
(-5, -1]: (-5, -1]
(-4, -1]: (-4, -1]
[-3, 1]: [-3, 1]
(2, 1): (1, 2]
a cliff diver plunges from a height of 81 ft above the water surface. the distance the diver falls in t seconds is given by the function d(t)
(a) Therefore after t = 1.75 seconds the diver will hit the water.
(b) The velocity the diver hit the water is 56 ft/s.
From the given condition we have d(t) = 16t²
and the height is 49ft
(a) Now when the diver hit the water the equation become
16t² = 49
t² = 49/16
t = ±7/4
t = ±1.75
since time can not be negative so t = 1.75
Therefore after t = 1.75 seconds the diver will hit the water.
(b)
Now differentiating d(t) with respect to t we get
d'(t) = 32t
now putting t=7/4 we get
the velocity d'(7/4) = 32*7/4
d'(7/4) = 56ft/s
Therefore the velocity the diver hit the water is 56 ft/s.
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The complete question is :
A cliff diver plunges from a height of 49ft above the water surface. The distance the diver falls in t seconds is given by the function d(t)=16t²ft
(a) After how many seconds will the diver hit the water?
(b) With what velocity (in ft/s ) does the diver hit the water?
You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 3
Answer:
the answer is 12/13 simplified
Step-by-step explanation:
simplified
It is known that the area of a triangle can be calculated by multiplying the measure of the base by the measure of the height. Let the triangle measure 5m, 12m and 13m. Determine your area
The area of this triangle is 30 m².
What area?Area is a surface measure, that is, it is the amount of space that a geometric figure occupies on a flat surface.
To calculate the area of a triangle, we can use the formula:
Area = (base x height) / 2
In the case of the given triangle, we can choose the measure of 5m as the base and the measure of 12m as the height, since the height forms a right angle with the base and is perpendicular to it.
So, we have:
Area = (b*h)/2
Area = (5m * 12m) / 2
Area = 30m²
Determine the following standard normal (z) curve areas. (Round all answers to four decimal places.)
(a) The area under the z curve to the left of 1.73
(b) The area under the z curve to the left of
−0.69
(c) The area under the z curve to the right of 1.3
(d) The area under the z curve to the right of
−2.82
(e) The area under the z curve between −2.22 and 0.52
(f) The area under the z curve between
−1
and 1
(g) The area under the z curve between
−4
and 4
(a) The area under the standard normal curve to the left of 1.73 is 0.9582.
(b) The area under the standard normal curve to the left of -0.69 is 0.2454.
(c) The area under the standard normal curve to the right of 1.3 is 0.0968.
(d) The area under the standard normal curve to the right of -2.82 is 0.9974.
(e) The area under the standard normal curve between -2.22 and 0.52 is 0.6851.
(f) The area under the standard normal curve between -1 and 1 is 0.6826.
(g) The area under the standard normal curve between -4 and 4 is 0.9998.
(a) Using a standard normal table, the area under the standard normal curve to the left of 1.73 is 0.9582.
(b) Similarly, the area under the standard normal curve to the left of -0.69 is 0.2454.
(c) The area to the right of 1.3 is the same as the area to the left of -1.3. Using a standard normal table, this area is 0.0968.
(d) The area to the right of -2.82 is the same as the area to the left of 2.82. Using a standard normal table, this area is 0.9974.
(e) To find the area under the standard normal curve between -2.22 and 0.52, we need to find the area to the left of 0.52 and subtract the area to the left of -2.22. Using a standard normal table, we find that the area to the left of 0.52 is 0.6990 and the area to the left of -2.22 is 0.0139. Therefore, the area between -2.22 and 0.52 is 0.6990 - 0.0139 = 0.6851.
(f) To find the area under the standard normal curve between -1 and 1, we need to find the area to the left of 1 and subtract the area to the left of -1. Using a standard normal table, we find that the area to the left of 1 is 0.8413 and the area to the left of -1 is 0.1587. Therefore, the area between -1 and 1 is 0.8413 - 0.1587 = 0.6826.
(g) The area under the standard normal curve between -4 and 4 is the same as the area to the left of 4 minus the area to the left of -4. Using a standard normal table, we find that the area to the left of 4 is 0.9999 and the area to the left of -4 is 0.0001. Therefore, the area between -4 and 4 is 0.9999 - 0.0001 = 0.9998.
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At a certain instant, the base of a triangle is 5 inches and is increasing at the rate of 1 inch per minute. At the same instant, the height is 10 inches and is decreasing at the rate of 2.5 inches per minute. Is the area of the triangle increasing or decreasing? Justify your answer.
Using differentiation, the area of the triangle is decreasing at the given time.
Is the area of the triangle increasing or decreasing?The formula for the area of a triangle is:
A = (1/2)bh
where b is the base and h is the height.
Differentiating both sides of the equation with respect to time t, we get:
[tex]\frac{dA}{dt} = (1/2)[(\frac{db}{dt}) h + b(\frac{dh}{dt}) ][/tex]
Substituting the given values, we get:
[tex]\frac{dA}{dt} = (1/2)[(1)(10) + (5)(-2.5)] = (1/2)(10 - 12.5) = -1.25[/tex]
Since the derivative of the area with respect to time is negative (-1.25), the area of the triangle is decreasing at the given instant.
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Synthetic Division to Find Zeros
if f(x)=x^3−3x^2+16x+20 and x+1 is a factor of f(x), then find all of the zeros of f(x) algebraically.
Answer:
Step-by-step explanation:
Since we know that x + 1 is a factor of f(x), we can use synthetic division to find the other factor and then solve for the remaining zeros.
We set up synthetic division as follows:
-1 | 1 -3 16 20
| -1 4 -20
|_____________
1 -4 20 0
The last row of the synthetic division gives us the coefficients of the quadratic factor, which is x^2 - 4x + 20. We can use the quadratic formula to find its roots:
x = (-(-4) ± sqrt((-4)^2 - 4(1)(20))) / (2(1))
= (4 ± sqrt(-64)) / 2
= 2 ± 2i√2
Therefore, the three zeros of f(x) are -1, 2 + 2i√2, and 2 - 2i√2.
What does replacement value in math? Say if you have to find a replacement value for the variable X, what does that mean?
A replacement value is a value that can be substituted for a variable in an equation or expression in order to solve for that variable or simplify the expression.
What is an expression?An expression is a combination of numbers, variables, and/or operators that represents a mathematical relationship or calculation. An expression can be as simple as a single number or variable, or it can be more complex and involve multiple operations and variables.
What is a variable?A variable is a symbol or letter that represents a quantity or value that can change or vary in different contexts.
In the given question,
A replacement value is a value that can be substituted for a variable in an equation or expression in order to solve for that variable or simplify the expression.
For example, suppose you have the equation 2x + 3 = 7, and you want to solve for x.
To do so, you can use a replacement value.
You can subtract 3 from both sides of the equation to isolate the term with x, which gives you:
2x = 4
Now, you can divide both sides of the equation by 2 to solve for x:
x = 2
In this case, the replacement value was the number 2, which was substituted for the variable x in the equation in order to solve for x.
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30 POINTS, URGENT!! Choose the system of inequalities that best matches the graph below.
Answer: A
y < 2x + 1
y > 1/2x + 2
Do X4 and 15+ X have the same value when X is 5
(b) Write 5 as a percentage.
Answer:
5 as a percentage of 100 is 5/100 which is 5%
If a + b = 3 and ab = 2, find the value of a² + b²
Answer:
Through observation, a=2, b=1 or vice versa .
Step-by-step explanation:
If a=2 and b=1
a² +b²=?
2² +1² =4+1=5
a²+b²=5
Dividing sin^2Ø+cos^2Ø=1 by ____ yields 1+cot^2Ø=csc^2Ø
a.cot^2Ø
b.tan^2Ø
c.cos^2Ø
d.csc^2Ø
e.sec^2Ø
f.sin^2Ø
To obtain the required equation we divide the equation by sin²Ø.
What are trigonometric functions?The first six functions are trigonometric, with the domain value being the angle of a right triangle and the range being a number. The angle, expressed in degrees or radians, serves as the domain and the range of the trigonometric function (sometimes known as the "trig function") of f(x) = sin. Like with all other functions, we have the domain and range. In calculus, geometry, and algebra, trigonometric functions are often utilised.
The given equation is:
sin²Ø+cos²Ø=1
To obtain the required equation we divide the equation with sin²Ø:
sin²Ø/sin²Ø +cos²Ø/ sin²Ø = 1/sin²Ø
1 + cot²Ø = csc²Ø
Hence, to obtain the required equation we divide the equation by sin²Ø.
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Find the center and radius of the circle whose equation is x^2+y^2+4y=32
Answer:
center: (0, -2)
radius: 6
Step-by-step explanation:
You have to "complete the square" this allows you to fold up the expressions and put the equation in a standard kinda of format where you can pick the center and radius right out of the equation.
see image.
The area of a trapezium is 156cm2, the parallel sides are 17cm and 35cm respectively. What is the height of the trapezium
Answer:
6 cm
Step-by-step explanation:
You want the height of a trapezium with bases 17 cm, 35 cm and area 156 cm².
Area formulaThe formula for the are of a trapezium is ...
A = 1/2(b1 +b2)h
Filling in the given values, we have ...
156 = 1/2(17 +35)h = 26h
6 = h . . . . . . . . . . divide by 26
The height of the trapezium is 6 cm.
Answer:
6cm
Step-by-step explanation:
To find:-
The height of the trapezium.Answer:-
We are here given that the area of the trapezium is 156cm² and two of the parallel sides are 17cm and 35cm .We are interested in finding out the height of the trapezium.
The area of the trapezium is given by the formula,
[tex]:\implies \sf Area =\dfrac{1}{2}\times (s_1+s_1)\times h \\[/tex]
where s1 and s2 are the || sides of the trapezium and h is the height of the trapezium.
Now on substituting the respective values in the given formula, we have;
[tex]:\implies \sf 156cm^2 =\dfrac{1}{2} (17cm+35cm)\times h \\[/tex]
[tex]:\implies \sf 156cm^2(2) = 52cm (h) \\[/tex]
[tex]:\implies \sf h =\dfrac{156(2)}{52} cm\\[/tex]
[tex]:\implies \sf \pink{ height = 6 cm }\\[/tex]
Hence the height of the trapezium is 6cm .
if sin0<0 and cos>0, then the terminal point is determined by 0 is in:
the terminal point of the angle determined by sin(0) < 0 and cos(0) > 0 is in the fourth quadrant.
why it is and what is trigonometry?
If sin(0) < 0 and cos(0) > 0, then we know that the angle 0 is in the fourth quadrant of the unit circle.
In the unit circle, the x-coordinate represents cos(θ) and the y-coordinate represents sin(θ). Since cos(0) > 0, we know that the terminal point of the angle is to the right of the origin. And since sin(0) < 0, we know that the terminal point is below the x-axis.
The fourth quadrant is the only quadrant where the x-coordinate is positive and the y-coordinate is negative, so that is the quadrant where the terminal point of the angle lies.
Therefore, the terminal point of the angle determined by sin(0) < 0 and cos(0) > 0 is in the fourth quadrant.
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It focuses on the study of the functions of angles and their applications to triangles, including the measurement of angles, the calculation of lengths and areas of triangles, and the analysis of periodic phenomena.
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if you could please help i am having issues
Since the p-value (0.0803) exceeds the significance threshold (0.05), the null hypothesis cannot be ruled out.
what is mean ?The mean in mathematics is a measurement of a collection of numerical data's central tendency. It is determined by adding up all of the values in the set and dividing the result by the total number of values. This value is frequently referred to as the average value. The mean (or mathematical mean) is calculated as follows: (Sum of Values) / Mean (number of values)
given
The null hypothesis states that the mean number of units generated during the day and night shifts is the same. The contrary hypothesis (Ha) states that more units are created on average on the night shift than on the day shift.
"day" + "night"
Bravo! Night precedes day.
b. The following method can be used to calculate the test statistic:
t = sqrt(1/n night + 1/n day) * sqrt(x night - x day)
where s p is the pooled standard deviation and x night and x day are the sample averages, n night and n day are the sample sizes, and s p is represented by:
Sqrt(((n night - 1)*s night2 + (n day - 1)*s day2) / (n night + n day - 2)) yields the value s p.
S p is equal to sqrt(((74 - 1)*35 + (68 - 1)*28) / (74 + 68 - 2)), which equals 31.88.
t = (358 - 352) / (31.88 * sqrt(1/74 + 1/68)) = 1.19
1.19 is the test result.
the p-value is 0.0803 as a result (rounded to 4 decimal places).
Since the p-value (0.0803) exceeds the significance threshold (0.05), the null hypothesis cannot be ruled out.
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The complete question is :- Clark Heter is an industrial engineer at Lyons Products. He would like to determine whether there are more units produced on the night shift than on the day shift. The mean number of units produced by a sample of 68 day-shift workers was 352. The mean number of units produced by a sample of 74 night-shift workers was 358. Assume the population standard deviation of the number of units produced is 28 on the day shift and 35 on the night shift.
Using the 0.05 significance level, is the number of units produced on the night shift larger?
a. State the null and alternate hypotheses.
O : Day/Night: H:
Day Night
b. Compute the test statistic. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)
c. Compute the p-value. (Round your answer to 4 decimal places.) p-value
Suppose that, for budget planning purposes, the city in Exercise 24 needs a better estimate of the mean daily income from parking fees.
a) Someone suggests that the city use its data to create a confidence interval instead of the interval first created. How would this interval be better for the city? (You need not actually create the new interval.)
b) How would the interval be worse for the planners?
c) How could they achieve an interval estimate that would better serve their planning needs?
d) How many days' worth of data should they collect to have confidence of estimating the true mean to within
a) As per the given budget, the amount of interval that would be better for the city is 95% confidence interval.
b) The interval that be worse for the planners is depends on sample size
c) They achieve an interval estimate that would better serve their planning needs is depends on margin of error
d) The number of days worth of data should they collect to have confidence of estimating the true mean to 30 days
To obtain a better estimate, the city can create a confidence interval, which is a range of values that is likely to contain the true population mean with a certain degree of confidence.
However, there are also some disadvantages to using a confidence interval. The interval estimate may be wider than a point estimate, which means that the budget planners may have to allocate a larger budget to account for the uncertainty in the estimate.
To achieve a better interval estimate, the city could increase the sample size or reduce the variability of the data. Increasing the sample size reduces the margin of error and increases the precision of the estimate.
Finally, to determine how many days' worth of data the city should collect to estimate the true mean with a certain degree of confidence, the city would need to consider the desired level of precision, the variability of the data, and the desired level of confidence.
Typically, a larger sample size will provide a more accurate estimate, but this also depends on the variability of the data. In general, a sample size of at least 30 is recommended for a reasonably accurate estimate.
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? Answer the question below. Type your response in the space provided. What do you call the materials that help you achieve your goals?
Answer:
Acquired resources
Step-by-step explanation:
Acquired resources
PLEASE HELP !!!! HELP!!label each equation is proportionality or non proportional Help
y=9/x
y=x-12
h=3d
f=1/3e
Answer:
y=9/x => proportional
y = x - 12 ==> non-proportional
h = 3d ==> proportional
f = 1/3 e = proportional
Step-by-step explanation:
A proportional equation is of the general form
y = kx (directly proportional) or
y = k/x (inversely proportional)
k is known as the constant of proportionality
y = 9/x ==> k = 9 proportional
y = x - 12 cannot be expressed as y = kx or y = k/x
h = 3d ==> k = 3 proportional
f = 1/3 e ==> k = 1/3 proportional
A baseball team plays in a stadium that holds 60000 spectators. With the ticket price at $9 the average attendance has been 23000. When the price dropped to $7, the average attendance rose to 30000. Assume that attendance is linearly related to ticket price. What ticket price would maximize revenue?
Answer:
Step-by-step explanation:
We can start by assuming that the relationship between the ticket price and attendance is linear, so we can write the equation for the line that connects the two data points we have:
Point 1: (9, 23000)
Point 2: (7, 30000)
The slope of the line can be calculated as:
slope = (y2 - y1) / (x2 - x1)
slope = (30000 - 23000) / (7 - 9)
slope = 3500
So the equation for the line is:
y - y1 = m(x - x1)
y - 23000 = 3500(x - 9)
y = 3500x - 28700
Now we can use this equation to find the attendance for any ticket price. To maximize revenue, we need to find the ticket price that generates the highest revenue. Revenue is simply the product of attendance and ticket price:
R = P*A
R = P(3500P - 28700)
R = 3500P^2 - 28700P
To find the ticket price that maximizes revenue, we need to take the derivative of the revenue equation and set it equal to zero:
dR/dP = 7000P - 28700 = 0
7000P = 28700
P = 4.10
So the ticket price that would maximize revenue is $4.10. However, we need to make sure that this price is within a reasonable range, so we should check that the attendance at this price is between 23,000 and 30,000:
A = 3500(4.10) - 28700
A = 5730
Since 23,000 < 5,730 < 30,000, we can conclude that the ticket price that would maximize revenue is $4.10.
In a survey of 124 pet owners, 44 said they own a dog, and 58 said they own a cat. 14 said they own both a dog and a cat. How many owned neither a cat nor a dog?
Step-by-step explanation:
See Venn diagram below
Question 4 X Suppose that starting today, you make deposits at the beginning of each quarterly period for the next 40 years. The first deposit is for 400, but you decrease the size of each deposit by 1% from the previous deposit. Using an nominal annual interest rate of 8% compounded quarterly, find the future value (i.e. the value at the end of 40 years) of these deposits. Give your answer as a decimal rounded to two places (i.e. X.XX).
if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.
The first step in solving this problem is to calculate the amount of each quarterly deposit. We know that the first deposit is $400, and each subsequent deposit decreases by 1% from the previous deposit. This means that each deposit is 99% of the previous deposit. To calculate the size of each deposit, we can use the following formula:
deposit_ n = deposit_(n-1) * 0.99
Using this formula, we can calculate the size of each quarterly deposit as follows:
deposit_1 = $400
deposit_2 = deposit_1 * 0.99 = $396.00
deposit_3 = deposit_2 * 0.99 = $392.04
deposit_4 = deposit_3 * 0.99 = $388.12
...
We can continue this pattern for 40 years (160 quarters) to find the size of each quarterly deposit.
Next, we need to calculate the future value of these deposits using an nominal annual interest rate of 8% compounded quarterly. We can use the formula for compound interest to calculate the future value:
[tex]FV = PV * (1 + r/n)^(n*t)[/tex]
where FV is the future value, PV is the present value (which is zero since we are starting with deposits), r is the nominal annual interest rate (8%), n is the number of times the interest is compounded per year (4 since we are compounding quarterly), and t is the number of years (40).
We can substitute the values into the formula and solve for FV:
[tex]FV = $400 * (1 + 0.08/4)^(440) + $396.00 * (1 + 0.08/4)^(439) + $392.04 * (1 + 0.08/4)^(4*38) + ... + $1.64 * (1 + 0.08/4)^4[/tex]
After solving this equation, we get a future value of $143,004.54, rounded to two decimal places. This means that if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.
This calculation highlights the power of compound interest over long periods of time. By making regular contributions and earning interest on those contributions, our investment grows exponentially over time. It also shows the importance of starting early and consistently contributing to an investment over time in order to achieve long-term financial goals.
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