Therefore, the observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis H1: [tex]p > 0.5[/tex]is z= 1.8. The correct answer is (e) z=[tex]0.59— 0.5/√0.5(0.5)/100.[/tex]
In this particular question, the observed value of the test statistic for testing the null hypothesis H0: p = 0.5 versus the alternative hypothesis H1: p > 0.5 is to be determined. The question provides a random sample of 100 likely voters in a small city, which produced 59 voters in favor of Candidate A. The formula for calculating the test statistic is as follows:
z= (p- P) / sqrt[P(1-P) / n]
where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.
Substituting the given values into the formula, we get:
z= [tex](0.59- 0.5) / sqrt[0.5(0.5) / 100][/tex]
z= [tex]0.09 / 0.05[/tex]
z=[tex]1.8[/tex]
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What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2)? Express your answer in simplest form
The slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2) is 0.
When determining the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 2) and the midpoint of the segment with endpoints at (5, 0) and (6, 2), there is a specific formula that can be used.
The formula for finding the slope of the line that contains the midpoints of two segments is:
[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
where the points (x1, y1) and (x2, y2) are the midpoints of the two segments.
Steps:1. Determine the midpoint of the first segment by using the midpoint formula:
[tex]\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)[/tex]
The midpoint of the segment with endpoints at (0, 0) and (2, 2) is:
[tex](\frac{0+2}{2}, \frac{0+2}{2}) = (1, 1)2.[/tex]
Determine the midpoint of the second segment by using the midpoint formula:
[tex]\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)[/tex]
The midpoint of the segment with endpoints at (5, 0) and (6, 2) is:
[tex](\frac{5+6}{2}, \frac{0+2}{2}) = (\frac{11}{2}, 1)3.[/tex]
Substitute the midpoints into the slope formula and simplify:
[tex]\frac{y_2-y_1}{x_2-x_1}=\frac{1-1}{\frac{11}{2}-1}=\frac{0}{\frac{9}{2}}=0[/tex]
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A chain fits tightly around two gears as shown. The distance between the centers of the gears is 32 inches. The radius of the larger gear is 19 inches. Find the radius of the smaller gear. Round your answer to the nearest tenth, if necessary. The diagram is not to scale.
Answer:
The answer to your problem is, C.
Step-by-step explanation:
From the given figure it is noticed that the radius of a circle is 11 inches and the centers of two circles are 20 inches apart. The length of the direct common tangent between both circles is 19 inches.
If the centers of two circles of radius r₁ and r₂ are d units apart, then the length of the direct common tangent between them is
[tex]L = \sqrt{d^{2} - (r_{1} - r_{2} )^{2} }[/tex]
[tex]19 = \sqrt{20^{2} - (11-r_{2} )^{2} }[/tex]
Next, Square both sides.
[tex]361 = 400 - ( 11 - r_{2} )[/tex]
[tex]( 11 - r_{2} )^{2} = 400 - 361[/tex]
[tex]( 11 - r_{2} )^{2} = 39[/tex]
Change the square root both sides.
[tex]11-r = \sqrt{39}[/tex]
[tex]11- 6.245 = r[/tex]
[tex]4.775 = r[/tex]
[tex]R = 4.8ish[/tex]
Therefore the radius of second circle is 4.8 inches
Thus the answer to your problem is, C.
I will give brainiest to whatever answered correctly.
Determine the inverse of the matrix
(5 -4)
( -8 6)
A) c^-1 =
(-5 8)
(4 -6)
B) c^-1 =
(6 4)
(8 5)
C) c^-1 =
(2.5 2)
(4 3)
D) c^-1 =
(-3 -2)
(-4 -2.5)
Step-by-step explanation:
that is the correct answer above
Answer:
So correct option is D)
Hope it helps you:)
One - third of a number y is 14.
Answer:
42
Step-by-step explanation:
Answer:
Step-by-step explanation:
If one-third of a number y is 14, we can express this mathematically as:
1/3 y = 14
To solve for y, we can isolate y on one side of the equation by multiplying both sides by 3:
1/3 y * 3 = 14 * 3
Simplifying, we get:
y = 42
Therefore, the number is 42.
a camper attaches a rope from the top of her tent, feet above the ground, to give it more support. if she takes the rope to the ground feet from the middle of her tent, about how long is the rope from the ground to the tent?
4 feet.
The length of the rope from the ground to the top of the tent is 4 feet.
To calculate this, subtract the distance from the tent to the ground (2 feet) from the height of the tent (6 feet), and you will get the length of the rope (4 feet).
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I tried and it did not make sense help
Answer: D) -20.99
Step-by-step explanation:
-4.97-2.36+-5.19-8.47 = -20.99
Solve this and show work
Answer:
see explanation below
Step-by-step explanation:
(secθ - cosθ)/secθ = (sinθ)^2
(secθ - cosθ)/secθ = (sinθ)^2
since secθ = 1/cosθ, you substitute
(secθ - cosθ)/(secθ)
= (1/cosθ - cosθ)/(1/cosθ)
= cosθ(1/cosθ - cosθ)
= 1 - cosθ^2
= sinθ^2
grandma forgot how many raisins she put in a batch; you sample one loaf and count 40 raisins. how many do you estimate are in a batch? what is the the uncertainty in your estimate?
Using the sample of 40 raisins, you can estimate that there are 40 raisins in a batch. However, due to the uncertainty of the sample size, there is an element of uncertainty as to how many raisins there truly are in a batch.
Thus, the uncertainty of your estimate can range from slightly below 40 raisins to slightly above 40 raisins.
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the top of an l-shaped desk has the dimensions shown. what volume of wood is needed to make the top of the desk? complete the explanation of how you solved this problem.
The volume of wood needed to make the top of the l-shaped desk is 0.728 cubic feet
The top of an l-shaped desk has the dimensions shown in the diagram. To determine the volume of wood needed to make the top of the desk, we must use the formula for volume. The formula for volume is length times width times height, or V = lwh. In this case, the length of the top is 28 inches, the width is 45 inches, and the height is 0.75 inches.
Plugging these values into the volume formula gives us V = 28 x 45 x 0.75, which equals 1260 cubic inches. Since wood is typically sold in cubic feet, we must convert this volume to cubic feet by dividing it by 1728, the number of cubic inches in one cubic foot. 1260/1728 = 0.728, so the volume of wood needed to make the top of the desk is 0.728 cubic feet.
To summarize, the volume of wood needed to make the top of the l-shaped desk is 0.728 cubic feet. This was determined by using the volume formula, which is V = lwh, where l is the length, w is the width, and h is the height. The length, width, and height of the desk top were 28 inches, 45 inches, and 0.75 inches, respectively. By plugging these values into the volume formula and converting to cubic feet, we determined the volume of wood needed is 0.728 cubic feet.
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What is 102.5% of 80?
Answer:
Step-by-step explanation:
102.5/80=
82
Answer: 82
Step-by-step explanation:
FirstMove the percent 2 times to the right to get a decimal.
102.5% ——> 1.025
Secondly
You’re dividing the part with the whole always to get the percent. So if we have the precent (converted into a decimal) and the whole. So we are finding the part. So to find the part we multiple 1.025 and 80.
Third
You solve now. So 1.025 x 80 = 82
Solve the following quadratic function by utilizing the square root method.
Answer:
x = ±9
Step-by-step explanation:
If x² = k, then x = ±√k.
x² - 81 = 0
x² = 81
x = ±√81
x = ±9
PLEASE HELP WITH EXPLANATION PLEASE PLEASE PLEASE NO SCAMMING FOR THE COINS OR I WILL REPORT YOU
The correct solutions for the given equation is given by table in option 3.
Explain about the solution of linear equation?A straight line is produced when a linear equation, which is a two-variable equation, is graphed. The intersection of corresponding values which satisfy the equation results in the line.
The variables x and y are most frequently used in linear equations, hence the associated values can be shown as (x,y) on the coordinate plane, pairs. A collection of two maybe more linear equations is known as a system of linear equations.
The given equation is-
y = x - 6
Put the value of 'x' from the options and check for 'y', if its correct.
The option are the solution of the equation.
option 1:
y = x - 6
x = --5
y = -5 - 6 = -11 (incorrect)
option 2:
y = x - 6
Put x = -8
y = -8 - 6 = -14 (incorrect)
option 3:
y = x - 6
Put x = -7
y = -7 - 6 = -13 (correct)
Thus, the correct solutions for the given linear equation is given by table in option 3.
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Find the area of the following
rhombus:
10 cm
6 cm
8 cm
A = [?] cm²
Answer:96
Step-by-step explanation:
[tex]Area=\frac{d1 d2}{2}[/tex]
[tex]Area=\frac{(6+6)(8+8)}{2}[/tex]
[tex]Area=96[/tex]
Find the area of each of the regular polygon below.
Round non-terminating decimals to the nearest hundredth.
nonagon (9 sided figure)
apothem = 16.5
side = 12
Rοunding tο the nearest hundredth, the area οf the nοnagοn is 891.00 square units.
What is the regular pοlygοn?A regular pοlygοn is a pοlygοn that has all sides οf equal length and all angles οf equal measure.
Tο find the area οf a regular pοlygοn, we use the fοrmula:
Area = (1/2) × Perimeter × Apοthem
The perimeter οf a nοnagοn (9-sided figure) with a side length οf 12 is:
Perimeter = 9 × 12 = 108
Therefοre, the area οf the nοnagοn is:
Area = (1/2) × 108 × 16.5
Area = 891
Hence, Rοunding tο the nearest hundredth, the area οf the nοnagοn is 891.00 square units.
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seven students go on vacations and decide that each one will send postcards to three others. is it possible that every student receives postcards from exactly the same three students to whom he/she sent postcards? show all of your work.
It is not possible for every student to receive postcards from exactly the same three students to whom they sent postcards.
Let's assume it is possible that every student receives postcards from exactly the same three students to whom he/she sent postcards.
Each student sends postcards to 3 other students. So the total number of postcards sent is 7 x 3 = 21.
If we count the number of postcards received by all 7 students, it should be equal to 21. However, this is not the case if each student receives postcards from exactly the same three students to whom he/she sent postcards.
If a student A sends postcards to B, C, and D, then B, C, and D should send postcards back to A. This means that each of B, C, and D will send postcards to two other students, and each of those two students will send postcards to two others as well.
So if we draw a diagram, it will look like this:
A
/ | \
B C D
/ \ | / \
E F G H
In the above diagram, A sends postcards to B, C, and D, and receives postcards back from them. B sends postcards to A, E, and F. C sends postcards to A, G, and H. D sends postcards to A, E, and H. E sends postcards to B, D, and G. F sends postcards to B, C, and H. G sends postcards to C, E, and F. H sends postcards to D, C, and F.
If we count the number of postcards received by each student, we get:
A: 3
B: 3
C: 3
D: 3
E: 3
F: 3
G: 3
H: 3
Therefore, it is possible that every student receives postcards from exactly the same three students to whom he/she sent postcards.
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in january , the temperature in parts of minnesota fell from f to f over a -hour period. what was the average temperature change per hour?
The average temperature change per hour was -2.708 degrees Fahrenheit.
To calculate the average temperature change per hour, we need to find the total temperature change over the 24-hour period and then divide by the number of hours.
The total temperature change is the difference between the starting temperature (41 degrees Fahrenheit) and the ending temperature (-13 degrees Fahrenheit), which is -54 degrees Fahrenheit.
Dividing the total temperature change by the number of hours, we get:
-54 degrees Fahrenheit / 24 hours = -2.25 degrees Fahrenheit per hour.
Rounding to three decimal places, we get an average temperature change per hour of -2.708 degrees Fahrenheit per hour.
The complete question is: In January 2008, the temperature in parts of Minnesota fell from 41∘41∘F to −13∘-13∘F over a 24-hour period. What was the average temperature change per hour?
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What is the value of angle h?
How do you know (what type of angle pairs are they)?
The value of h is 34.
What is the angle formed when two lines intersect?
Intersecting lines are two or more lines that have precisely one point in common. The point of junction is this central location that connects all of these lines. It should be observed that the intersecting lines only come together at one point, regardless of the angle at which they do so.The vertical angles are the opposing angle pairs created by two intersecting lines. Non-adjacent angles created by two crossing lines are known as opposite angles. Angles at opposite sides are equal. (equal in measure)According to the Vertically opposite angle theorem, opposite angles are equal.
So, h = 34.
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two angles are complementary. the measure of the larger angle is 26 degrees more than three times the measure of the smaller angle. what is the measure of each angle?
the measure of the smaller angle is 16 degrees, and the measure of the larger angle is 74 degrees.
The two angles are complementary, which means that their sum equals 90 degrees. Let x be the smaller angle, and y be the larger angle.
According to the problem statement, the larger angle is 26 degrees more than three times the measure of the smaller angle. In mathematical terms, this means:
y = 3x + 26We also know that the sum of the two angles is 90 degrees. In mathematical terms, this means:
x + y = 90 Substituting the expression for y from the first equation into the second equation,
we get: x + (3x + 26) = 90Simplifying this equation,
we get: 4x + 26 = 90 Subtracting 26 from both sides of the equation,
we get: 4x = 64Dividing both sides of the equation by 4,
we get: x = 16Substituting this value of x into the equation for y,
we get: y = 3(16) + 26 = 74
Therefore, the measure of the smaller angle is 16 degrees, and the measure of the larger angle is 74 degrees.
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need help to this what would the variance be?
The variance of the set of data is 10. Option A
What is variance?Variance can be defined as a statistical measurement of the spread of numbers in a data set.
It is used to measure how far each number in the set is distant from the mean and from every other number in the set.
It is sometimes described as the measure of variability.
It is denoted with the mathematical symbol,'σ²'
The formula for calculating the variance of a set of data is;
Variance = ∑(x-x⁻)²/n - 1
Substitute the values, we have;
Variance = 16 + 4 + 4 + 16 + 0/5 - 1
Add the values
Variance = 40/4
Divide the values
Variance = 10
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in general, which point size would you use if you wanted each character to be approximately one inch in size? a.) 1 pt b.) 72 pts c.) 24 pts d.) 36 pts
If you want each character to be approximately one inch in size, you would use a point size of 72 pts.
Point size is a unit of measurement used to determine the size of typefaces. It represents the height of the characters in a font. One point is equal to 1/72 of an inch, which means there are 72 points in one inch.
Therefore, if you want each character to be approximately one inch in size, you would need to use a font size of 72 points. This would ensure that the characters are roughly one inch tall, assuming that the font is designed to be proportional and not condensed or expanded.
Choosing a smaller point size, such as 1 pt or 24 pts, would result in characters that are much smaller than one inch. Choosing a larger point size, such as 36 pts, would result in characters that are larger than one inch.
Therefore the correct answer is option b.) 72 pts.
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A cylindrical glass with a radius of 5 cm and a height of 20 cm is half full of milk. How
many milliliters of milk are in the glass? (1cm3
= 1ml)
There is 78.54 mI of miIk in the gIass.
What is VoIume of a cyIinder ?The voIume of a cyIinder means the space inside the cyIinder that can hoId a specific amount of materiaI quantity. In simpIer words, the capacity of a cyIinder to hoId a thing is its voIume. Inside the space of a cyIinder, you can hoId either of the three types of matter – soIid, Iiquid, or gas. This capacity can onIy be witnessed in a three-dimensionaI cyIinder, i.e., you cannot hoId any Iiquid, soIid, or gas in a two-dimensionaI cyIinder.
The equation for the voIume of a cyIinder is V = Ah,
where A is the area of the base and h is the height.
Thus, the voIume can aIso be expressed as V = πr2h.
Find capacity of the gIass :
VoIume of the gIass = πr²h
Given radius = 5 ; Height = 2
VoIume of the gIass = π(5)²(2)
VoIume of the gIass = 157.08cm³
Find the voIume of the miIk :
If the gIass is haIf fuII,
VoIume of the miIk = 177.08 ÷ 2
VoIume of the miIk = 78.54 cm³
Convert cm³ to mI :
1 mI = 1000 cm³
78.54cm³ = 78.54 mI
There is 78.54 mI of miIk in the gIass.
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Write and solve an inequality Diego family tank
Answer:
please make me brainalist and keep smiling dude I hope you will
Step-by-step explanation:
The inequality represents the number of days Diego's father can drive the car without the warning light coming on is, The remaining fuel is greater than 1.5 gallons Diego's father can drive the car without the warning light coming on.
A random group of adults was asked to complete a survey regarding the number of pets in their households. No two adults sur came from the same household. The number of households, , with no pets is one fourth of the number of households with multip Which of the following equations represents this situation if of the households have a single pet?
The equation representing the situation is 4x = y - 1, where x is the number of households with one pet and y is the total number of households. This can be answered by the concept of Simple equation.
Let's assume that there are x households with a single pet. The number of households with no pets is given as one-fourth of the number of households with multiple pets, which means there are 4 times as many households with multiple pets as there are households with no pets. Let's represent the total number of households as y.
Therefore, we can say that the number of households with no pets is (y - x)/4, and the number of households with multiple pets is 3(y - x)/4 since there are 4 times as many households with multiple pets as there are households with no pets.
Now, we can use the given information that the number of households with no pets is one-fourth of the number of households with multiple pets to write the equation:
(y - x)/4 = x/3
Solving for y, we get:
y = 4x + 3x/4 = 16x/4 + 3x/4 = 19x/4
But we are given that the total number of households y includes the households with one pet, so we can write:
y = x + (y - x)/4 + 3(y - x)/4 = x + y/4
Substituting the value of y from the previous equation, we get:
19x/4 = x + 19x/16
Solving for x, we get:
x = 16/3
Therefore, the equation representing the situation is 4x = y - 1, where x = 16/3 and y = 19x/4, which simplifies to:
4(16/3) = y - 1
y = 21
Therefore, there are 16 households with one pet and 5 households without any pets.
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how do you find the margin of error given the number of standard deviations and the confidence interval
To find the margin of error given the number of standard deviations and the confidence interval, use the formula Margin of Error = Z * (Standard Deviation / √Sample Size).
The margin of error is a measure of the precision of an estimate or a statistic. It represents the range of values above and below the estimate or statistic that is likely to contain the true population parameter with a certain degree of confidence.
To calculate the margin of error, you need to know the number of standard deviations (Z) from the mean corresponding to the desired level of confidence, the standard deviation of the population, and the sample size.
Once you have the Z-value, standard deviation, and sample size, you can plug these values into the formula and solve for the margin of error. The formula tells us that the margin of error is proportional to the Z-value and the standard deviation, and inversely proportional to the square root of the sample size.
Therefore, a larger Z-value or a larger standard deviation will result in a larger margin of error, while a larger sample size will result in a smaller margin of error.
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there are 20 rows of seats on a concert hall: 25 seats are in the 1st row, 27 seats on the 2nd row, 29 seats on the 3rd row, and so on. if the price per ticket is $2,300, how much will be the total sales for a one-night concert if all seats are taken?
The total sales for a one-night concert with all seats taken will be $2,024,000.
To solve this problem, we need to find the total number of seats in the concert hall. We know that the first row has 25 seats, the second row has 27 seats, and the third row has 29 seats.
We can represent the number of seats in each row using an arithmetic sequence with a first term of 25 and a common difference of 2. The nth term of this sequence can be found using the formula:
a_n = a_1 + (n - 1)d
where a_1 is the first term (25), d is the common difference (2), and n is the number of the row.
Using this formula, we can find the number of seats in the 20th row
a_20 = 25 + (20 - 1)2
a_20 = 25 + 38
a_20 = 63
Therefore, the total number of seats in the concert hall is the sum of the seats in all 20 rows:
total seats = 25 + 27 + 29 + ... + 63
To find this sum, we can use the formula for the sum of an arithmetic sequence
S_n = (n/2)(a_1 + a_n)
where S_n is the sum of the first n terms of the sequence. In this case, n = 20, a_1 = 25, and a_n = 63.
S_20 = (20/2)(25 + 63)
S_20 = 10(88)
S_20 = 880
Therefore, there are a total of 880 seats in the concert hall. If all seats are taken, the total sales for the one-night concert will be
total sales = number of seats x price per ticket
total sales = 880 x $2,300
total sales = $2,024,000
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define "area "in the context
Therefore, the area of the object or surface in this context would be 2 square meters.
What is length?Length is a measure of the distance between two points. It is a physical quantity that describes the extent of an object or space in one dimension, usually measured in units such as meters, centimeters, feet, inches, etc. Length can also refer to the size or duration of something, such as the length of a book, the length of a movie, or the length of a period of time. In mathematics, length is also used to describe the size of a curve or a line segment, as well as the perimeter of a polygon or the circumference of a circle.
by the question.
In the context of length 2m and breadth 1m, the area refers to the measurement of the total amount of space that is covered by a two-dimensional object or surface within those dimensions.
The formula for calculating the area is:
Area = length x breadth
Substituting the given values, we get:
Area = 2m x 1m = 2 square meters
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Name 2 figures for which all cross sections taken at a particular orientation
are congruent.
Answer: Rectangular prism & Cylinder
Step-by-step explanation: In geometry, a rectangular prism is a polyhedron with two congruent and parallel bases. It is also called a cuboid. A rectangular prism has six faces, and all the faces are in a rectangle shape and have twelve edges. Because of its cross-section along the length, it is said to be a prism.
A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder. The line segment joining the two centers is the axis, that denotes the height of the cylinder.
Borachio eats at the same fast-food restaurant every day. Suppose the time X between the moment Borachio enters the restaurant and the moment he is served his food is normally distributed with mean 4. 2 minutes and standard deviation 1. 3 minutes. A. Find the probability that when he enters the restaurant today it will be at least 5 minutes until he is served. B. Find the probability that average time until he is served in eight randomly selected visits to the restaurant will be at least 5 minutes
The probability that when he enters the restaurant today it will be at least 5 minutes until he is served is 0.2676 and probability that average time until he is served in eight randomly selected visits is 0.0409.
The square root of the variance is used to calculate the standard deviation, a statistic that expresses how widely distributed a dataset is in relation to its mean. By calculating the deviation of each data point from the mean, the standard deviation can be calculated as the square root of variance.
Given that mean μ = 4.2 , standard deviation σ = 1.3
1. P(X >= 5) = P((X - μ)/σ >
= (5 - 4.2) /1.3
= P(Z ≥ 0.6154)
= 1 - P(Z < 0.6154)
= 1 - 0.7324
= 0.2676
The required probability is 0.2676.
2.Given that n = 8 then [tex]\bar x[/tex] = σ/[tex]\sqrt{(n)[/tex] = 1.3/√(8) = 0.4596
P(x-bar ≥ 5) = P(([tex]\bar x[/tex] - μ)/σx-bar ≥ (5 - 4.2)/0.4596)
= P(Z ≥ 1.7406)
= 1 - P(Z < 1.7406)
= 1 - 0.9591
= 0.0409
The required probability is 0.0409.
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i have no idea abt any of this
Answer:
201.06 [tex]in^{2}[/tex]
Step-by-step explanation:
A = [tex]\frac{1}{4} \pi d^{2}[/tex]
A = [tex]\frac{1}{4} \pi (16)^{2}[/tex]
A = 201.06 [tex]in^{2}[/tex]
Answer:
divide 16 and use pi
pi is this π
π always equally 3.14
100 points
Michael has $16 and wants to buy a mixture of cupcakes and fudge to feed at least 4 siblings. Each cupcake costs $4, and each piece of fudge costs $2.
This system of inequalities models the scenario:
4x + 2y ≤ 16
x + y ≥ 4
Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)
Part B: Is the point (2, 3) included in the solution area for the system? Justify your answer mathematically. (3 points)
Part C: Choose a different point in the solution set and interpret what it means in terms of the real-world context. (3 points)
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Answer:
Step-by-step explanation:
Part A:
The system of inequalities 4x + 2y ≤ 16 and x + y ≥ 4 can be graphed on a coordinate plane. To graph 4x + 2y ≤ 16, we can first graph the line 4x + 2y = 16. We can do this by finding the intercepts:
When x = 0, 4(0) + 2y = 16, so y = 8.
When y = 0, 4x + 2(0) = 16, so x = 4.
So, the intercepts are (0, 8) and (4, 0). We can connect these two points to graph the line.
To determine which side of the line to shade, we can test a point that is not on the line. For example, we can test the point (0, 0):
4(0) + 2(0) = 0 ≤ 16, so (0, 0) is in the shaded region.
Next, we can graph the line x + y = 4. This line passes through the points (0, 4) and (4, 0). To determine which side of the line to shade, we can test a point that is not on the line, such as (0, 0):
0 + 0 = 0 < 4, so (0, 0) is not in the shaded region. Therefore, we shade the region above the line.
The solution set for the system is the region that is shaded by both lines, which is the triangular region in the upper-left corner of the graph.
Part B:
To determine if the point (2, 3) is included in the solution area for the system, we can substitute x = 2 and y = 3 into both inequalities:
4(2) + 2(3) = 14 ≤ 16, so (2, 3) satisfies 4x + 2y ≤ 16.
2 + 3 = 5 ≥ 4, so (2, 3) satisfies x + y ≥ 4.
Therefore, the point (2, 3) is included in the solution area for the system.
Part C:
Let's choose the point (1, 3) as another point in the solution set. This means that Michael can buy 1 cupcake and 3 pieces of fudge, which would cost him:
1 cupcake * $4/cupcake + 3 pieces of fudge * $2/piece of fudge = $10
Since $10 is less than the $16 he has, he can afford to buy this combination of cupcakes and fudge. Therefore, the point (1, 3) represents a valid solution in which Michael buys 1 cupcake and 3 pieces of fudge to feed his siblings.