The probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.
What is the probability?To calculate the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election, we can use the binomial probability formula.
The formula is: P(x) = (ⁿCₓ) × pˣ × (1-p)⁽ⁿ⁻ˣ⁾
In this case, n = 328, p = 46.37%, and x = 164 (since half of 328 is 164).
Plugging in the numbers we get:
P(x) = ³²⁸C₁₆₄ × (0.4637)¹⁶⁴ × (0.5363)⁽³²⁸⁻¹⁶⁴⁾ = 0.0532
Therefore, the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.
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What is the missing side on a rectangle 8 8 3 12
The missing side of the rectangle is 21.
In a rectangle, opposite sides are congruent and parallel. Therefore, if we know the length and width of a rectangle, we can find the length of any missing side using the formula for the area or the perimeter of a rectangle.
In the given rectangle 8 8 3 12, the two sides are labeled 8 and 12, which represent the length and the width of the rectangle, respectively. To find the missing side, we can use the formula for the perimeter of a rectangle, which is:
Perimeter = 2 x (length + width)
Substituting the given values, we get:
Perimeter = 2 x (8 + 12)
Perimeter = 2 x 20
Perimeter = 40
Since we have the length and width of the rectangle, we can use the perimeter formula to solve for the missing side. We know that the perimeter is equal to the sum of all four sides of the rectangle, so we can write:
Perimeter = 8 + 8 + 3 + x
where x is the missing side.
Substituting the value of the perimeter (40) and simplifying, we get:
40 = 19 + x
x = 21
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There are 30 students in Mr. Hall class. 1/3 of the students are traveling somewhere this summer. Of those traveling, 1/5 are going out of the country. How many students are traveling out of the country?
The number of students traveling are 2 students out of the 30 who are traveling out of the country this summer. If 1/3 of the students are traveling somewhere this summer, then we can calculate the number of students who are traveling by multiplying the total number of students by 1/3:
Number of students traveling = 30 x 1/3 = 10
Now, we need to find out how many of these 10 students are traveling out of the country. We know that 1/5 of the students who are traveling are going out of the country, so we can find the number of students who are traveling out of the country by multiplying the total number of traveling students by 1/5:
Number of students traveling out of the country = 10 x 1/5 = 2
Therefore, there are 2 students out of the 30 who are traveling out of the country this summer.
It's important to note that fractions can be converted to decimals or percentages to make them easier to work with. For example, 1/3 can be written as 0.33 or 33%, and 1/5 can be written as 0.20 or 20%. This can be particularly useful when dealing with more complex problems or when working with larger numbers.
In summary, by using the information given, we can determine that out of the 30 students in Mr. Hall's class, 10 are traveling somewhere this summer, and out of those 10 students, 2 are going out of the country. This type of problem-solving helps build math skills that are applicable in real-world scenarios, such as budgeting and planning for travel.
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Factor completely.
7b^2-63
Thank you :DDD
Since both terms are perfect squares, factor using the difference of squares formula, [tex]a^2-b^2=(a+b)(a-b)[/tex] where [tex]a=b[/tex] and [tex]b=3[/tex]
Answer:[tex]7(b+3)(b-3)[/tex]Suppose we create a box model for the outcome of a game of darts. The player has a 1/3 chance of throwing a dart in the inner ring, and a 2/3 chance of the dart landing in the outer ring. In our model, we have two unique tickets marked "inner" and "outer." We put in 1 ticket marked "inner." How many tickets do we put in that are marked "outer?"
a. 0
b. 1
c. 2
d. 3
As per the combination method, the number of tickets that we put in that are marked "outer" is 3 (option d).
In this case, we want to choose the number of tickets marked "outer." Let's call this number k. We know that we already put one ticket marked "inner" in the box, so the total number of tickets in the box is 2. Therefore, n = 2.
Now we need to determine k. We want to know how many tickets we need to put in that are marked "outer." We can represent this as a. So we have:
ᵃC₁ = a! / ((1!)(a-1)!) = a
We want to find the value of a that satisfies the condition that the probability of choosing an "inner" ticket is 1/3 and the probability of choosing an "outer" ticket is 3/2.
Since we already put in 1 ticket marked "inner," the probability of choosing an "inner" ticket is 1/2, which means the probability of choosing an "outer" ticket is also 1/2.
We know that the probability of choosing an "outer" ticket is 3/2, so we can set up the following equation:
ᵃC₁ / 2 = 3/2
Solving for a, we get:
a = 3
In conclusion, the answer is (d) 3.
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F(x)=-(x+3)(x+10) pls help
Answer:
Zeros: x = -10 and x = -3
Vertex: [tex](-\frac{13}{2} , \frac{49}{4} )[/tex]
Step-by-step explanation:
Pre-SolvingWe are given the following function:
f(x) = -(x+3)(x+10)
We want to find the zeros and the vertex of the parabola.
SolvingZerosThe zeros are the values of the function where f(x) = 0.
So, in order to find the zeros, we can set f(x) = 0.
0 = -(x+3)(x+10)
We can divide both sides by -1, to get:
0 = (x+3)(x+10)
To solve this, we will use zero product property.
Split and solve:
x+3 = 0
x = -3
x+10=0
x = -10
Vertex
Now, to find the vertex, we first get the average of the zeros.
Add the values of the zeros together, then divide by two:
[tex]\frac{-3-10}{2}[/tex] = [tex]\frac{-13}{2}[/tex]
Now, we plug this in for x to get the y value (found through f(x)) of the vertex.
[tex]f(-\frac{13}{2}) = -(-\frac{13}{2} + 3) (-\frac{13}{2} + 10)[/tex] = [tex]\frac{49}{9}[/tex]
So, the vertex is [tex](-\frac{13}{2} , \frac{49}{4} )[/tex]
13. The diagonals of a trapezium ABCD intersect at O. AB is parallel to DC, AB = 3 cm and DC = 6 cm. If CO = 4 cm and OB = 3 cm, find AO and DO.
Answer:
AO = 2 cmDO = 6 cmStep-by-step explanation:
You want the measures of AO and DO in a trapezium in which AB║CD, the diagonals intersect at O, and AB = 3 cm, CD = 6 cm, CO = 4 cm, OB = 3 cm.
Similar trianglesDiagonal AC is a transversal to parallel lines AB and CD, so alternate interior angles BAO and DCO are congruent. Vertical angles AOB and COD are also congruent, so ∆ABO ~ ∆CDO by the AA similarity postulate.
This means the side lengths are proportional, so ...
AB/CD = AO/CO = BO/DO
3/6 = AO/4 = 3/DO ⇒ AO = 2, DO = 6
The measures of AO and DO are 2 cm and 6 cm, respectively.
__
Additional comment
It can help to draw a diagram.
MR. Swanson wants to buy some mugs as gifts on his trip to California.There are three gifts shops, and each is offering a different deal. Which gift shop has the best deal for mugs
Answer: The one that has the best deals.
Step-by-step explanation:
To celebrate Halloween, Florence's class is making candy necklaces. Florence is helping pass out string from a 50-yard-spool. She gives 30 inches of string to each student. If there are 24 students in her class, how many yards of string will be leftover?
There will be 30 yards of the string that will be leftover.
What are Arithmetic operations?
It is a field of mathematics that deals with the study of numbers and the operations on those numbers that are relevant to all other areas of mathematics. The basic operations included in it are addition, subtraction, multiplication, and division. The term "arithmetic operator" refers to the operator that does the calculation.
Given that,
Total Length of string = 50 yards.
The total number of students = 24.
Total used string = 24 × 30 = 720.
We know that 1 foot = 12 inches,
So, 150 feet = 1800 inches.
Therefore, yards of string leftover = (1800 - 720)/36
= 1080/36
= 30 yards.
Hence, there will be 30 yards of string that will be leftover.
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Let d be a positive integer. Show that among any group of d+1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d. HINT: Use the Pigeon-hole Principle!
The Pigeon-hole Principle states that if there are more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. This proves that among any group of d+1 integers there are two with exactly the same remainder when divided by d.
This concept can be applied to your question: If there are d+1 integers, and we divide each of them by d, there are at most d remainders that can be obtained. This means that two of the integers must have the same remainder when divided by d.
To prove this, let us assume that all d+1 integers have different remainders when divided by d. Then the remainders must range from 0 to d-1. For example, if d=5, then the remainders must be 0, 1, 2, 3 and 4. Let us denote the integers by x0, x1, x2, ... , xd. Now we can apply the Pigeon-hole Principle. We have d+1 pigeons (x0, x1, x2, ... , xd) and d pigeonholes (remainders 0, 1, 2, 3, 4). Since d+1 is greater than d, at least one pigeonhole must contain more than one pigeon. This means that two of the integers must have the same remainder when divided by d.
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Assume there are 4 collector cards with one card inside every box of cereal you buy. Each card has a probability of 1/4 of being inside your box. What's the probability you get the complete set of cards on Box 5? Go from left to right as in accordance to time. 4.3 2 3/ 6 mar
The probability of getting the complete set of cards in Box 5 is 3/64.
To calculate the probability of getting the complete set of cards in Box 5, we have to consider the probability of getting each of the 4 cards in the previous 4 boxes.
For the first box,
The probability of getting any one of the 4 cards is 1,
since there are no previous boxes to consider.
For the second box,
The probability of getting a card that is not already in our possession is 3/4 (since there is one card already collected).
For the third box,
The probability of getting a card that is not already in our possession is 2/4 (since there are now two cards already collected).
For the fourth box,
The probability of getting a card that is not already in our possession is 1/4 (since there are now three cards already collected).
Multiplying these probabilities together, we get,
⇒ 1 x 3/4 x 2/4 x 1/4 = 3/64
So the probability of getting the complete set of cards in Box 5 is 3/64.
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Brainliest if correct
Answer: a > -1
Explanation is in the image.
Answer:
[tex]a > -1[/tex]
Step-by-step explanation:
1) Write the equation
[tex]-2a+14 < 5a+21\\[/tex]
2) Collect like terms on their corresponding side
[tex]-7a < 7[/tex]
3) Divide -7 from both sides and flip the sign
[tex]a > -1[/tex]
josh borrowed $250 from his mother to buy an electric scooter. josh will pay her back in 1 year with 3% simple annual interest. how much interest will josh pay?
The interest which josh will pay on the electric scooter with a simple annual interest of 3% is 7.50.
What is interest rate?Interest rate can be defined as the amount of interest which is due per period, as a proportion of the amount lent, deposited, or borrowed by someone.
The interest rate formula is:
Interest Rate = {(Simple Interest × 100)}/{ (Principal × Time)}
Here,
Josh borrowed 250 from his mother to buy an electric scooter and will pay her back in one year with three simple annual interest.
The amount of interest that Josh will pay is calculated as:
Interest = Principal Amount × Rate of Interest × Time
Interest = 250 × 3
Therefore, Josh will pay his mother $7.50 in interest for the loan.
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In order for a confidence interval based on de Moivre's equation to be valid, which of the following conditions must be true?
a. We must be forming a confidence interval for a coefficient in a multiple regression model.
b. All of these answers are correct.
c. We must be forming a confidence interval for a population mean based on a sample mean.
d. The underlying distribution of the data must be normally distributed
The condition that must be true in order for a confidence interval based on de Moivre's equation to be valid is:
d. The underlying distribution of the data must be normally distributed.
What is a confidence interval?A confidence interval is an interval estimate of a population parameter that specifies a range of values within which the parameter is likely to lie with a certain level of confidence. In other words, it represents the degree of uncertainty associated with the estimate.
De Moivre's equationDe Moivre's equation is a formula for approximating the probability of a specific number of successes in a series of independent Bernoulli trials. This formula is only relevant if the sample size is large enough such that the normal approximation to the binomial distribution is valid. Thus, this formula can be used to calculate confidence intervals for binomial proportions when the sample size is large enough to apply the normal approximation.
Answers to other options:
a. We must be forming a confidence interval for a coefficient in a multiple regression model - This statement is incorrect. De Moivre's equation is not related to multiple regression models.
b. All of these answers are correct - This statement is incorrect because not all of the options are correct. Only one option is correct.
c. We must be forming a confidence interval for a population mean based on a sample mean - This statement is incorrect. De Moivre's equation is not relevant for calculating confidence intervals for population means. The Central Limit Theorem is used instead.
Hence, option "d" only is true.
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a machine that manufactures automobile parts produces defective parts of the time. if parts produced by this machine are randomly selected, what is the probability that at most of the parts are defective? carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. (if necessary, consult a list of formulas.)
The probability that at most of the parts are defective is 0.96.
A machine that manufactures automobile parts produces defective parts of the time. If parts produced by this machine are randomly selected, what is the probability that at most of the parts are defective?
The given probability of producing defective parts is P(defective) = 0.15. Now, we need to find the probability that at most of the parts are defective. This can be done by finding the probability of producing 0, 1, or 2 defective parts.
Let X denotes the number of defective parts. So, we have to calculate the probabilities for P(X = 0), P(X = 1), and P(X = 2). To calculate these probabilities, we will use the binomial probability formula:
P(X = x) ={n}C{x} p^x (1 - p)^{n - x}, Here, n = number of parts produced, p = probability of producing defective parts
x = number of defective parts
First, we need to find P(X = 0),
P(X = 0) = (0.85)^5 = 0.4437
P(X = 1) = {5}C{1} (0.15)^1 (0.85)^4 = 0.3672
P(X = 2) = {5}C{2} (0.15)^2 (0.85)^3 = 0.1459
Now, we can find the probability that at most of the parts are defective as follows:
P{at most 2 defective parts}) = P(X = 0) + P(X = 1) + P(X = 2) = 0.957
Therefore, the probability that at most of the parts are defective is 0.96.
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(3x+1)^2=3(x+1). Solve for X
Answer:
Step-by-step explanation:
(3x+1)^2 = 3x+3
9x^2 +6x +1=3x+3
9x^2+3x-2=0
finally we got a trinomial quadratic equation solve by factorizing
9x^2 -6x+3x-2=0
3x(3x-2)+(3x-2)=0
3x-2 = 0 or 3x+1=0
x= 2/3 or x= -1/3
If two triangles have two pairs or corresponding angles and three pairs are corresponding sides that are congruent then the two triangles are congruent
AAS conguerence rule states that if two congruent triangles have any two pairs of corresponding angles that are equal and one pair of corresponding sides that are equal.
Triangles that are congruent have the same area since all of their matching angles and side lengths are the same.
You are aware that a triangle's three angles add up to 180 degrees. As a result, the third pair of angles is also equal if the first two pairs are (180° - sum of equal angles). Hence, if any two pairs of angles and one pair of corresponding sides are equal, two triangles are said to be congruent. It could be referred to as the AAS Congruence Rule.
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how do you use TAN in equations and what is it?
Answer:
TAN is a mathematical function in trigonometry that stands for tangent. It is used to calculate the tangent of an angle in a right triangle, which is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In equations, you can use TAN to find the value of the tangent of an angle. For example, if you have an angle of 30 degrees in a right triangle and you want to find the value of the tangent of that angle, you can use the TAN function in your calculator or programming language.
The syntax of the TAN function is usually "tan(x)", where x is the angle in radians. If your calculator or programming language uses degrees instead of radians, you may need to convert the angle to radians first using the conversion formula: radians = degrees * (pi/180).
For example, to find the value of the tangent of 30 degrees, you can use the TAN function as follows:
In degrees mode: TAN(30) = 0.57735027
In radians mode: TAN(30*pi/180) = 0.57735027
TAN can be used in various trigonometric equations and identities to solve for unknown sides or angles of a right triangle.
Step-by-step explanation:
In the given truss bridge, parallelograms ABCD and PQRS are congruent. If AB = 24 feet, what is PQ?
Answer:
D. 24 ft
Step-by-step explanation:
Congruent means equal. Since ABCD and PQRS are congruent they are the same.
Answer:
D
Step-by-step explanation:
Graph the equation y = 2.
Graph Linear Equations-Quiz-Level H
0
X
Answer:
See below picture.
Explanation:
y=2 has a y-intercept of 2, so we start graphing there. with most equations, we would follow the slope starting from that point, but y=2 doesn't have a slope. The 2 stays going all the way "across" for all x-values. I used demos to graph.
3p^2 +7p=0 solve by factoring
Answer:
p = 0, p = -7/3
Step-by-step explanation:
Pre-SolvingWe are given the following equation:
3p² + 7p = 0
We want to solve the equation by factoring.
Solving
To factor, we want to look for a common term that we can pull out.
You may notice that both terms have 'p' in common, so we can pull out p from both terms.
This will then make the equation:
p(3p + 7) = 0
Now, we can use zero product property to solve the equation.
p = 0
3p + 7 = 0
Subtract.
3p = -7
Divide.
p = -7/3
Our answers are p = 0 and p = -7/3
jarrod wants to find out the most popular football game between the home team and the visiting team. which of the following would give him the most accurate results? * 1 point a.surveying the cheerleaders b.surveying the people waiting in line for tickets c.surveying the people with the visiting team's hat on d.surveying the people wearing the home team's jersey
Out of the given options, Jarrod can get the most accurate results by surveying the people wearing the home team's jersey. Therefore, the correct option is D.
A survey is an organized process of collecting and recording information from a particular group of people for the purpose of understanding and discovering the state of affairs in a particular area or issue. It is one of the most prevalent research methods used by social scientists and market researchers, among others. Surveys are used to obtain accurate data on a variety of topics, such as people's opinions and behaviors, demographic data, socioeconomic data, and much more.
Accurate surveying is critical because it assists in collecting precise data, which is critical in making sound decisions. An accurate survey provides the data needed to establish reliable conclusions, identify trends, and draw meaningful insights.
The method of conducting a survey can influence the outcome; therefore, it is critical to use the most effective approach to get accurate data.
To get the most accurate results from a survey, a researcher must be sure to construct the questionnaire correctly and analyze the data effectively.
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plain why the statistic is misleading.
Wilson was 42 inches tall on Jan 1, 2000, and 51 inches tall on Jan 1, 2002. William was 5 feet tall on Jan 1, 2000, and 6 feet tall on Jan 1, 2002.
Conclusion: The difference between 42 and 51 is greater than the difference between 5 and 6, so Wilson grew more during one year.
Wilson and William were measured at various times, therefore drawing the inference that Wilson grew more over the course of a year than William is incorrect and the statistics is misleading.
What is descriptive statistics?Inferential statistics and descriptive statistics are two disciplines of statistics with distinct applications.
Summarizing and characterizing gathered data is the focus of descriptive statistics. It uses techniques including graphical presentations, measurements of variability, and measures of central tendency, such as mean, median, and mode (e.g., histograms, box plots). The purpose of descriptive statistics is to shed light on a sample's or population's properties, such as its distribution, dispersion, and shape.
Wilson and William were measured at various times, therefore drawing the inference that Wilson grew more over the course of a year than William is incorrect.
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PLEASEEE HELP I NEED THIS ASAPPP
Answer:
perimeter=42
Area=90
Step-by-step explanation:
find the length of DEFG:
perimeter=a=b+a+b
28=10+b+10+b
28=20+2b
8=2b
4=b
finding the scale factor:
15/10=3/2
to find the perimeter of WXYZ:
the missing side:
=4x3/2
=6
perimeter=a+b+a+b
perimeter=15+6+15+6
perimeter=42
Area of WXYZ:
Area=bh
Area=15x6
Area=90
Unit 7 polygons and quadrilaterals
Homework 7 trapezoids
** this is a 2-page document **
Directions: if each quadrilateral below is a trapezoid, find the missing measures
Angle L can be calculated as follows:
angle L = angle - 180 LMO stands for angle. MNO \sangle L = 180 - 150 - 30 degree angle L = 0
As a result, angle L is 0 degrees.
1. We know that sides AB and CD of trapezoid ABCD are parallel. We can use the tangent function to find the length of side AD because angle B is a right angle and angle ABD is 45 degrees:
AD/AB = tan(45)
AD=AB * tan(45) AD=AB
As a result, AD = 10.
2. We know that the sides PQ and RS of the trapezoid PQRS are parallel. We can use the sine function to find the length of side PS because angle Q is a right angle and angle PSQ is 60 degrees:
PS/QS sin(60) =
PS = sin * QS (60)
5 * sqrt = PS (3)
As a result, PS = 5*sqrt (3).
3. We know that the sides UV and WX of a trapezoid UVWX are parallel. We can use the cosine function to find the length of side WU because angle V is a right angle and angle WVU is 30 degrees:
WU/UV cos(30) =
UV * cos WU (30)
5 * sqrt(3) / 2 = WU
As a result, WU = (5/2)*sqrt (3).
4. We know that the sides LM and NO of the trapezoid LMNO are parallel. We can use the sine function to find the length of side MO because angle L is a right angle and angle MNO is 30 degrees:
MO/NO sin(30) =
MO = 4 / 2 MO = NO * sin(30)
As a result, MO = 2.
Because angles MNO and LMO add up to 180 degrees, we can calculate angle LMO as follows:
LMO angle = 180 - angle LMO MNO angle = 150 degrees
Finally, because angle N is a right angle, we can calculate angle L as follows:
angle L = angle - 180 LMO stands for angle. MNO\sangle L = 180 - 150 - 30 degree angle L = 0
As a result, angle L is 0 degrees.
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Andre wrote the inequality 3x + 10 <= 30 to plan his time. Describe what x , 3x , 10 , and 30 represent in this inequality
Andre can make 6 small cranes. X is the number of small cranes, 3x is the minutes, 10 is the minute for large cranes and 30 is the total time.
3x + 10 is less than or equal to 30
3 is the minutes for the small cranes
X is the number of small cranes
10 is the minutes for the large crane
30 is the total time limit
first, subtract 10 from 30, ( 30-10) which gives you 20 so
3x is less than or equal to 20.
To figure this out, divide 20 by 3, which gives you 6 as a quotient with a remainder of two minutes.
Andre can make 6 small cranes.
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The Complete question is
Andre is making paper cranes to decorate for a party. He plans to make one large paper crane for a centrepiece and several smaller paper cranes to put around the table. It takes Andre 10 minutes to make the centrepiece and 3 minutes to make each small crane. He will only have 30 minutes to make the paper cranes once he gets home.
Andre wrote the inequality 3x + 10 ≤ 30 to plan his time. Describe what x, 3x, 10, and 30 represent in this inequality.
Solve Andre’s inequality and explain what the solution means.
The student council at a large high school is wondering if Juniors or Seniors are more likely to attend Prom. They take a random sample of 50 Juniors and find that 28 are planning on attending Prom. They select a random sample of 45 Seniors and 29 are planning on attending. Do the data provide convincing evidence that a higher proportion of Seniors are going to prom than Juniors? Use a 5% significance level. 1. STATE: (Parameter, Statistic, Hypotheses, Significance level) ______ 2. PLAN: (Name of procedure, Check conditions) _______
3. DO: Mean: Picture: Standard deviation: General Formula: Specific Formula: Work: Test statistic: P-value:
The insufficient evidence to suggest that a higher proportion of Seniors are attending Prom than Juniors.
1. STATE:Parameter: proportion of Juniors attending Prom, proportion of Seniors attending PromStatistic: the sample proportions of Juniors and Seniors who are planning on attending PromHypotheses:H0: pJ = pS (the proportion of Juniors attending Prom is equal to the proportion of Seniors attending Prom)Ha: pJ < pS (the proportion of Juniors attending Prom is less than the proportion of Seniors attending Prom)Significance level: α = 0.05 (5%)2. PLAN:Procedure: two-sample z-test for proportionsConditions:Randomization: The students were randomly selected from each grade level.Independence: The samples are independent as they come from different grade levels.Large Sample: npJ ≥ 10, n(1 − pJ) ≥ 10, npS ≥ 10, n(1 − pS) ≥ 103. DO:Mean:pJ − pSPicture:Standard deviation:Formula:General formula: Z = (pJ − pS) / SEpJ = number of Juniors who are planning to attend Prom / sample size of JuniorsnJ = sample size of JuniorspS = number of Seniors who are planning to attend Prom / sample size of SeniorsnS = sample size of SeniorsSE = sqrt[(pJ(1 - pJ))/nJ + (pS(1 - pS))/nS]Specific formula: Z = (0.56 - 0.64) / 0.092 = -0.87Test statistic: Z = -0.87P-value:P(Z < -0.87) = 0.1935Because the p-value (0.1935) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is insufficient evidence to suggest that a higher proportion of Seniors are attending Prom than Juniors.
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Design a cylindrical can (with a lid) to contain 1 liter (= 1000 cm3) of water, using the minimum amount of metal.
To design a cylindrical can (with a lid) to contain 1 liter (= 1000 cm3) of water, using the minimum amount of metal, we need to consider the following parameters:Height and Diameter of the canThickness of the metalMaterial used for making the canLet's assume we use Aluminium as a material. Now, let's start designing the can:Height of the can:
Volume of water = 1000 cm3Volume of cylinder = πr²hVolume of cylinder = π (d/2)² hVolume of cylinder = π (d²/4) hVolume of cylinder = 1000 cm³π (d²/4) h = 1000 cm³d²h = 4000 cm³h = (4000 cm³) / (π d²) h = (4000 cm³) / (3.14 * d²) h = (1273.24) / d²Diameter of the can:
Volume of cylinder = πr²hVolume of cylinder = π (d/2)² hVolume of cylinder = π (d²/4) hVolume of cylinder = 1000 cm³π (d²/4) h = 1000 cm³d²h = 4000 cm³d² = (4000 cm³) / h d² = (4000 cm³) / (1273.24/d²) d² = 3.1425d = 17.8 cmThickness of the metal:We can assume the thickness to be 0.5 mm.Material used for making the can:AluminiumTotal Surface Area of the can:Total Surface Area of cylinder = 2πrhTotal Surface Area of cylinder = 2π(d/2)(1273.24/d²)Total Surface Area of cylinder = 1273.24/d Total Surface Area of lid = πr²Total Surface Area of lid = π (d/2)²Total Surface Area of lid = π (17.8/2)²Total Surface Area of lid = 248.5Total Surface Area of the Can = 1273.24/d + 248.5Now, we can calculate the minimum amount of Aluminium required to make the can by minimizing the Total Surface Area of the can.Total Surface Area of the can = 1273.24/d + 248.5d (in cm)Total Surface Area of the can = 1273.24/7.09 + 248.5(7.09)Total Surface Area of the can = 584.24Therefore, the minimum amount of Aluminium required to make the can is 584.24 cm².
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y=2x+1
2x-y=3
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Find the tangential and normal components of the acceleration vector for the curve → r ( t ) = 〈 − 3 t , − 5 t ^ 2 , − 2 t ^ 4 〉 at the point t = 1
The tangential component of the acceleration vector at point t = 1 is aT(1) = 233/3 and The normal component of the acceleration vector at point t = 1 is aN(1) = (1/3)√10459
How do we calculate the tangential component?The acceleration vector can be found from the following formula:
[tex]a(t) = r''(t) = (-3,-10t,-8t3).[/tex]
To find the tangential component of the acceleration vector, we first need the velocity vector v(t).
[tex]v(t) = r'(t) = (-3,-10t,-8t3) .[/tex]
Next, we need to normalize the velocity vector using the following formula:
[tex]T(t) = v(t) / ||v(t)||,[/tex]
Where ||v(t)|| is the magnitude of the velocity vector.
[tex](1) = (-3,-10,-8) / \sqrt{(3^2 + 10^2 + 8^2)} = (-3/3, -10/3, -8/3) = (-1 , -10/3, -8/3) .[/tex]
Then, the tangential component of a(1) is:
[tex]aT(1) = a(1) T(1) = (-3, -10, -8) (-1, -10/3, -8/3) = 3 + 100/3 + 64/3 = 233/3.[/tex]
How do we calculate the normal component?To find the normal component of a(1), we simply need to find the magnitude of the tangential component and subtract it from the magnitude of the acceleration vector.
[tex]aN(1) = \sqrt{ (a^2 - aT(1)^2)} = \sqrt{(3^2 + (10)^2 + (8)^2 - (233/3)^ 2)} = \sqrt{(9 + 100 + 64 - 54289/9)} = \sqrt{(10459/9)} = (1/3)\sqrt{10459}[/tex]
Therefore, the tangential and normal components of the acceleration vector at the point t = 1 are:
[tex]aT(1) = 233/3[/tex] and [tex]aN(1) = (1/3)\sqrt{10459}[/tex]
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Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let p be the probability that all four slips bear the same number. Let q be the probability that two of the slips bear a number a and the other two bear a number b≠ab≠a. What is the value of q/p?(A) 162(B) 180(C) 324(D) 360(E) 720
We have that, if they put 40 chips in a hat, each one with the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10, and each number is put on four chips. Four tokens are drawn from the hat at random and without replacement, the value of q/p, q,p as probabilities, will be given by 360, therefore, the correct option is (D) 360
How do we calculate the probability?The probability that all four tokens have the same number (p) is equal to the total number of possible outcomes that meet that criterion divided by the total number of possible outcomes.
In this case, there are 10 possible numbers that could come up (1-10). Therefore, there are 10 possible outcomes for the four slips of paper that have the same number. Each outcome has the same probability of 1/10, so p = (1/10)^4 = 1/10000.
The probability that two of the slips have a number a and the other two have a number b (q) is equal to the total number of possible outcomes meeting that criterion divided by the total number of possible outcomes.
In this case, there are 10 possible numbers that could be drawn (1-10) and 2 ways to choose 2 different numbers out of 10, so there are 20 possible outcomes for two of the slips bearing a number and the other two bearing a number b. Each outcome has the same probability of 1/20, so q = (1/20)^4 = 1/3200000.
The ratio of q to p is q/p = 3200000/10000 = 360. Therefore, the value of q/p is 360.
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