Answer:
The box plot shows information
about the heights, in cm, of a Yr 10
group of Year 10 students.
Yr 7
130
a) The table shows information about
the heights, in cm, of a group
of Year 7 students.
Complete the boxplot for Year 7.
b) The heights distributions of Year 10
and Year 7 are compared.
140
height
(cm)
150
160
Height (cm)
median
170
least
upper
lower
quartile quartile
height
145 158 154 166
range
25
A- The median height is higher for Yr 7 than for Yr 10
Select the statements that apply. B-The interquartile range is higher for Yr 7 than for Yr 10
C - The data range is wider for Yr 7 than for Yr 10
D- The median height is lower for Yr 7 than for Yr 10
180
(3)
(2)
Total marks: 5
Step-by-step explanation:
A - The median height is higher for Yr 7 than for Yr 10. (Incorrect)
B - The interquartile range is higher for Yr 7 than for Yr 10. (Correct)
C - The data range is wider for Yr 7 than for Yr 10. (Correct)
D - The median height is lower for Yr 7 than for Yr 10. (Incorrect)
Therefore, the correct statements are B and C.
how to integrate directly using substitution given the same initial condition does this match your taylor series
The integration of the given function using substitution and matching it with the Taylor series is equal to 4/3.
Step 1: Find the Taylor series of the given function f(x).
f(x) = e^(x^2)cos(x)
Therefore, we can say that the Taylor series of the given function is: f(x) = 1 + x^2 + (1/2)x^4 + (1/6)x^6 + (1/24)x^8 + ...
Step 2: Substitute the required value of x into the Taylor series. Let us substitute x = 1 into the Taylor series we got above.
f(1) = 1 + 1^2 + (1/2)1^4 + (1/6)1^6 + (1/24)1^8 + ...
f(1) = 1 + 1 + (1/2) + (1/6) + (1/24) + ...
Step 3: Simplify the expression to get the answer. We can simplify the above expression by using the formula for the sum of an infinite geometric series. The formula is given as follows:
Sum of infinite geometric series = a / (1 - r), where a is the first term and r is the common ratio. Here, a = 1 and r = 1/4. Sum of the series = a / (1 - r) = 1 / (1 - 1/4) = 4/3.
Therefore, we can say that the integration of the given function using substitution and matching it with the Taylor series is equal to 4/3.
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Find the value of HCF of (Log ²2^18 + log²4)(log²32 + log ²216)
After calculating, the value of the expression HCF(log²(2^18) + log²(4))(log²(32) + log²(216)) is: 3 log²2 (5 - 3 log 3)
We can start by simplifying the logarithmic expressions inside the parentheses:
log²(2^18) + log²(4) = (18 log 2)^2 + (2 log 2)^2 = 324 log²2 + 4 log²2 = 328 log²2
log²(32) + log²(216) = (5 log 2)^2 + (3 log 6)^2 = 25 log²2 + 27 log²3
Now, we can express the original expression as:
HCF(log²(2^18) + log²(4))(log²(32) + log²(216)) = HCF(328 log²2, 25 log²2 + 27 log²3)
To find the highest common factor of these two terms, we can factor out the common factor of log²2:
HCF(328 log²2, 25 log²2 + 27 log²3) = log²2 HCF(328, 25 + 27 log²3)
Now, we need to find the highest common factor of the two integers 328 and 25 + 27 log²3.
We can factor out 3 from 25 and 27, and then use the difference of squares formula to write:
25 + 27 log²3 = (5 + 3 log 3)(5 - 3 log 3)
So, the highest common factor of 328 and 25 + 27 log²3 is the product of the common factors, which is:
HCF(328, 25 + 27 log²3) = 3(5 - 3 log 3)
Therefore, the value of the expression HCF(log²(2^18) + log²(4))(log²(32) + log²(216)) is:
log²2 HCF(328, 25 + 27 log²3) = log²2 * 3(5 - 3 log 3) = 3 log²2 (5 - 3 log 3)
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using the net below find the area of the triangular prism
6 cm
3 cm
4 cm
6 cm
5 cm
2 cm
Answer:153
Step-by-step explanation:
7: find the domain and range of these functions. (10 pts in total) (a) the function that assigns to each pair of nonnegative integers the first integer of the pair. (b) the function that assigns to each positive integer its largest decimal digit. (c) the function that assigns to a bit string the number of ones minus the number of zeroes in the string. (d) the function that assigns to each positive integer the largest integer not exceeding the square root of the integer. (e) the function that assigns to a bit string the longest string of ones in the string. 7
a) The domain of the function is {0,1,2,3,4,...}. The range is also a non-negative integer.
b) The largest decimal digit is 9, thus the domain of the function is all positive integers and the range is {9}.
c) The domain of the function is all bit strings of length n and the range is all integers between -n and n.
d) The domain of the function is all positive integers and the range is also all positive integers.
e) The domain of the function is all bit strings and the range is a non-negative integer.
Domain and range of the function that assigns to each pair of non-negative integers the first integer of the pair.In the given problem, (a) the function that assigns to each pair of non-negative integers the first integer of the pair. The first integer of the pair is always non-negative. Therefore, the domain of the function is {0,1,2,3,4,...}. The range is also a non-negative integer.
Domain and range of the function that assigns to each positive integer its largest decimal digit. In this problem, (b) the function that assigns to each positive integer its largest decimal digit. For all positive integers, the largest decimal digit is 9, thus the domain of the function is all positive integers and the range is {9}.
Domain and range of the function that assigns to a bit string the number of ones minus the number of zeroes in the string.In the problem, the function that assigns to a bit string the number of ones minus the number of zeroes in the string. The possible bit strings have a length of n. Therefore, the domain of the function is all bit strings of length n and the range is all integers between -n and n.
Domain and range of the function that assigns to each positive integer the largest integer not exceeding the square root of the integer.In the given problem, the function that assigns to each positive integer the largest integer not exceeding the square root of the integer. Let’s say f(n) is the largest integer not exceeding the square root of n, then the domain of the function is all positive integers and the range is also all positive integers.
Domain and range of the function that assigns to a bit string the longest string of ones in the string.In the given problem, the function that assigns to a bit string the longest string of ones in the string. In this case, the domain of the function is all bit strings and the range is a non-negative integer.
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For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse.
a. E = 1 0 3
0 1 0
0 0 1
b. E = 0 0 1
0 1 0
1 0 0
c. E = 1 0 0
0 1/2 0
0 0 1
d. E = 1 0 0
-2 1 0
0 0 1
e. E = 0 1 0
1 0 0
0 0 1
f. E = 1 0 0
0 1 0
0 0 5
In the given elementary matrix, the corresponding elementary row operation is multiplication of the second row of a given matrix with a scalar of 5. The given matrix is an elementary matrix because it can be obtained by multiplying an identity matrix by an elementary matrix. The inverse of the given matrix can be obtained by multiplying a scalar of 1/5 to the second row of the identity matrix.
The given elementary matrix is:
[1 0 0]
[0 0 5]
[0 1 0]
The corresponding elementary row operation for this matrix is multiplication of the second row of a given matrix with a scalar of 5. This means that if we have a matrix A and we multiply the second row of A with a scalar of 5, we get a new matrix B which is represented by the above elementary matrix.
The inverse of the given matrix is:
[1 0 0]
[0 1/5 0]
[0 0 1]
The inverse of a given elementary matrix can be obtained by applying the inverse of the corresponding elementary row operation to an identity matrix. In this case, the inverse of the given elementary matrix can be obtained by multiplying a scalar of 1/5 to the second row of the identity matrix. This gives us the above inverse matrix.
Therefore, the corresponding elementary row operation for the given matrix is multiplication of the second row of a given matrix with a scalar of 5 and the inverse of the given matrix is:
[1 0 0]
[0 1/5 0]
[0 0 1]
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place a checkmark by the situations that indicate area
The area of a triangle is calculated by multiplying the base of the triangle by the height and dividing by 2. Therefore, the number of sides of the triangle is relevant to calculating its area.
What is triangle?Triangle is a three-sided polygon that is considered one of the most basic shapes in geometry. It is composed of three line segments that intersect at three points called vertices. The angles formed by the three line segments are the angles of the triangle. A triangle has three sides, three angles, and three vertices. Depending on the length of the sides and the angles, triangles can be classified as acute, right, or obtuse. Acute triangles have all three angles measuring less than 90 degrees, right triangles have one angle measuring 90 degrees, and obtuse triangles have one angle measuring more than 90 degrees. Triangles are also classified by their sides, such as equilateral, isosceles, and scalene. An equilateral triangle has three equal sides and three equal angles, an isosceles triangle has two equal sides and two equal angles, and a scalene triangle has three sides and three angles that are all different. Triangles are used in many areas of mathematics and are seen in many engineering structures and designs.
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The area of a triangle is calculated by multiplying the base of the triangle by the height and dividing by 2. Therefore, the number of sides of the triangle is relevant to calculating its area.
What is triangle?Triangle is a three-sided polygon that is considered one of the most basic shapes in geometry. It is composed of three line segments that intersect at three points called vertices. The angles formed by the three line segments are the angles of the triangle. A triangle has three sides, three angles, and three vertices. Depending on the length of the sides and the angles, triangles can be classified as acute, right, or obtuse. Acute triangles have all three angles measuring less than 90 degrees, right triangles have one angle measuring 90 degrees, and obtuse triangles have one angle measuring more than 90 degrees.
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Complete Question:
Place a checkmark by the situations that indicate area1 is a triangle.
you are going to play mini golf. a ball machine that contains 19 green golf balls, 24 red golf balls, 20 blue golf balls, and 20 yellow golf balls, randomly gives you your ball. what is the probability that you end up with a green golf ball? express your answer as a simplified fraction or a decimal rounded to four decimal places.
The probability of getting a green golf ball from the machine is 19/83 or 0.2289. The result is obtained by the number of green golf balls divided by the total number of golf balls.
How to calculate probability?Probability of an event can be calculated by
P(A) = n(A) / n(S)
Where
P(A) is the probability of an event An(A) is the number of favorable outcomesn(S) is the total number of events in the sample spaceYou have 19 green golf balls, 24 red golf balls, 20 blue golf balls, and 20 yellow golf balls.
Find the probability of getting a green golf ball from the machine!
The total number of golf balls in the machine is
n(S) = 19 + 24 + 20 + 20 = 83 balls
You will end up with a green golf ball with the probability of
P(A) = n(A) / n(S)
P(A) = 19/83
P(A) = 0.2289
Hence, the probability that you will end up with a green golf ball is 19/83 or 0.2289.
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dayna writes the integers 1,2,3,4,5,6,7,8,9,10,11,12 on a chalkboard, then she erases the integers from 1 through 6, as well as their multiplicative inverse $\mod{13}$. what is the only integer dayna does not erase?
The integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 have been written on a chalkboard by Dayna. She then erased the integers from 1 through 6, as well as their multiplicative inverse $\mod{13}$.
We can find the multiplicative inverse of an integer { a modulo 13 } by using the extended Euclidean algorithm.The integers from 1 to 6 are 1, 2, 3, 4, 5, and 6.The multiplicative inverse of : 1 modulo 13 is 1, 2 modulo 13 is 7, 3 modulo 13 is 9, 4 modulo 13 is 10, 5 modulo 13 is 8, and 6 modulo 13 is 11.The only integer that Dayna does not erase is 12.
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A random sample of 100 preschool children in Camperdown revealed that only 60 had been vaccinated. Provide an approximate 95% confidence interval for the proportion vaccinated in that suburb.
a) We have 95% confidence that the interval, .5842 ≤ μ ≤ .7903 will contain the population proportion of children who have been vaccinated.
b) We have 95% confidence that the interval, .5504 ≤ μ ≤ .6560 will contain the population proportion of children who have been vaccinated.
c) We have 95% confidence that the interval, .5199 ≤ μ ≤ .6800 will contain the population proportion of children who have been vaccinated.
d) We have 95% confidence that the interval, .5040 ≤ μ ≤ .6960 will contain the population proportion of children who have been vaccinated.
To summarize, the 95% confidence interval for the proportion of vaccinated preschool children in Camperdown is [tex]0.5040 ≤ μ ≤ 0.6960[/tex], indicating that we can be 95% confident that the true proportion of vaccinated preschool children in Camperdown lies between 50.4% and 69.6%.
A 95% confidence interval for the proportion of preschool children in Camperdown who have been vaccinated is [tex].5040 ≤ μ ≤ .6960.[/tex] This indicates that we can be 95% confident that the true proportion of vaccinated preschool children in Camperdown lies between 50.4% and 69.6%.
To calculate this interval, we first need to calculate the sample proportion of vaccinated preschool children. To do this, we divide the number of vaccinated children (60) by the total number of children in the sample (100). This yields a sample proportion of 0.6.
Next, we need to calculate the standard error of the proportion, which is calculated by taking the square root of (sample proportion * (1 - sample proportion) / sample size). Plugging in our values yields a standard error of 0.0435.
We then use the standard error to calculate the 95% confidence interval, which is equal to (sample proportion +/- (1.96 * standard error)). Plugging in our values yields an interval of 0.5040 ≤ μ ≤ 0.6960. This indicates that we can be 95% confident that the true proportion of vaccinated preschool children in Camperdown lies between 50.4% and 69.6%.
To summarize, the 95% confidence interval for the proportion of vaccinated preschool children in Camperdown is 0.5040 ≤ μ ≤ 0.6960, indicating that we can be 95% confident that the true proportion of vaccinated preschool children in Camperdown lies between 50.4% and 69.6%.
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In which condition vector a.b has the minimum value? Write it.
Answer:
if it is perpendicular to eacha other I e 0
14. The perimeter of the table shown is 18 feet. Write an equation
in the form px + q = r to solve for x.
6
Answer:
its 3
Step-by-step explanation:
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Write the model giving the number of s of subscribers in t years after 1970
Model: S(t) = 5000 + 3000t. With a starting value of 5000 subscribers in 1970 and a steady rise of 3000 subscribers each year, this model predicts a linear increase in the number of subscribers through time.
With a starting value of 5000 subscribers in 1970 and a steady rise of 3000 subscribers each year, this model predicts a linear increase in the number of subscribers through time. Simply enter the value of t into the equation to get the number of subscribers after t years. For example, if you want to know the number of subscribers in 1990, which is 20 years after 1970, you would enter in t=20 into the equation and obtain S(20) = 5000 + 3000(20) = 65000. As a result, our model predicts that there would be 65,000 members in 1990. It's crucial to note that this is a simplified model and may not adequately reflect the genuine behaviour of subscriber growth over time.
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what is the weighted mean, mean, and median overall rate of return on this investment portfolio?
The weighted mean takes into account the weight of each investment and can provide a more accurate measure of the overall rate of return on the portfolio than the mean or median.
The weighted mean, mean, and median overall rate of return on this investment portfolio can be calculated using the following formulas:
Weighted Mean: Weighted mean = (R1 x W1) + (R2 x W2) + (R3 x W3) + ...
Mean: Mean = (R1 + R2 + R3 + ...) / N
Median: Median = (N + 1) / 2
where R1, R2, etc. are the returns on individual investments, W1, W2, etc. are the weights assigned to each investment, and N is the total number of investments in the portfolio.
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Solve each equation and verify the solution.Please heeelp
We have established that the answer to the equation, [tex]x = 24/7[/tex] , is that the left side equals the right side.
What is the equation in algebra?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation [tex]3x + 5 = 14[/tex] consists of the two equations [tex]3x + 5[/tex] and 14, which are separated by the 'equal' sign.
The equation is as follows:
[tex]\frac{1}{2}\times (x-2)+ 1+2/3x-3[/tex]
This equation can be made simpler by first merging like terms:
[tex]\frac{1}{2}\times (x-2)+ 1+2/3x-3[/tex]
[tex]= 1/2x + 2/3x - 4[/tex]
[tex]= 7/6*x - 4[/tex]
As a result, the equation becomes [tex]7/6*x - 4 = 0.[/tex]
We can now get the value of x by multiplying both sides by 6/7 after adding 4 on both sides: [tex]7/6*x = 4 x = 24/7.[/tex]
We rewrite the original equation with x = 24/7 to confirm the solution:
[tex]1/2*(24/7-2)+1+2/3(24/7)-3[/tex]
[tex]= 1/2*(10/7)+1+16/21-3[/tex]
[tex]= 5/7+1-2/3[/tex]
[tex]= 6/7[/tex]
Therefore, 6/7 demonstrates the equality.
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Please help its gone overdue
80 points
Hence, AA"B"Ccoordinates "'s in green are: A" (0, 0), B" (-4, -5), and C" (1, -4).
what is graph ?An illustration of data or mathematical functions is a graph. It is a method for presenting numerical data in a way that is simple to comprehend and interpret. A set of axes, one for each variable being represented, and one or more plots or lines that reflect the data or function being graphed are the main components of a graph. There are many distinct graph kinds, each of which is appropriate for displaying various sorts of data, including line graphs, bar graphs, scatter plots, and pie charts.
given
The axes cannot be identified because they are not specified in the query.
The blue ABC bar graph is as follows:
Using the equation (x, y) (y, -x), rotate AABC 90 degrees clockwise by applying the following transformation to each point:
A(-3, 0) → A' (0, 3) (0, 3)
B(-2, 4) → B' (-4, -2) (-4, -2)
C(1, -1) → C' (1, -1) (1, -1)
The red A'B'C' coordinates are as follows: A' (0, 3), B' (-4, -2), and C' (1, -1).
We take the y-coordinate of each point and subtract 3 to translate AA'B'C' three units down:
A' (0, 3) → A" (0, 0) (0, 0)
B' (-4, -2) → B" (-4, -5) (-4, -5)
C' (1, -1) → C" (1, -4) (1, -4)
Hence, AA"B"Ccoordinates "'s in green are: A" (0, 0), B" (-4, -5), and C" (1, -4).
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The complete question is:-
1. Label the axes.
2. Graph: A(-3, 0) B(-2, 4) C(1,-1) Draw ABC in blue.
3. Rotate AABC 90° clockwise to
create ▲A'B'C' in red. List the coordinates below:
Formula: (x, y) → (y, -x)
A(-3, 0) A' ()
B(-2, 4)→ B' ()
C(1, -1) C' ()
4. Translate AA'B'C' three units down to create ▲A"B"C" in green. What are the coordinates of AA"B"C"?
can you find another way to solve this problem without using the original one.
The area of the isosceles trapezoid is 37.25 unit².
What is isοsceles trapezοid?An isοsceles trapezοid is a trapezοid with cοngruent base angles and cοngruent nοn-parallel sides. A trapezοid is a quadrilateral with οnly οne side parallel.
An isοsceles trapezοid can be defined as a trapezοid whοse nοn-parallel sides and base angles have the same measure. That is, if the twο οppοsite sides (bases) οf a trapezοid are parallel and the twο nοn-parallel sides are οf equal length, it is an isοsceles trapezοid. See the diagram belοw.
Area of trapezium = 1/2(h)(a + b)
where h = height
a = smaller base = 10
b = larger base = 15
Now to find height, we see a triangle at the end of trapezium, draw the imaginary altitude
Now, the base = 5/2 ⇒(15 - 10= 5)/2
= 2.5
Using trigonometry ratio
tan(θ) = Height/ Base
tan(40°) = Height/ 2.5
0.8391 = Height/ 2.5
Height = 2.5/ 0.8391
Height ≈ 2.98
Now, The area = 1/2(2.98)(10 + 15)
= 1/2(2.98)(10 + 15)
= 37.25 unit²
Thus, The area of the isosceles trapezoid is 37.25 unit².
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Which of the following represents vector vector t equals vector PQ in trigonometric form, where P (–13, 11) and Q (–18, 2)?
t = 10.296 sin 60.945°i + 10.296 cos 60.945°j
t = 10.296 sin 240.945°i + 10.296 cos 240.945°j
t = 10.296 cos 60.945°i + 10.296 sin 60.945°j
t = 10.296 cos 240.945°i + 10.296 sin 240.945°j
The correct answer is option (C).
What are the fundamental forms of trigonometry?Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent are the six functions (cot).
The equation t = Q - P, where Q and P are the specified locations, can be used to determine the components of the vector t. Therefore:
t = (–18, 2) – (–13, 11) = (–18 + 13, 2 – 11) = (–5, –9) (–5, –9)
The vector's magnitude is given by:
|t| = √(–5)^2 + (–9)^2 = √106 ≈ 10.296
The formula = tan1 (y/x), where x and y are the vector's components, can be used to determine the direction of the vector t. The direction must be expressed in terms of sine and cosine functions because we are required to represent the vector in trigonometric form.
θ = tan⁻¹ (–9/–5) ≈ 60.945°
In trigonometric form, the vector t is thus represented as follows:
t = [t|cos|i] + [t|sin|j]
We get the following by altering the values of |t| and:
t = 10.296 cos I + 10.296 sin j of angle 60.945
As a result, the following is the proper trigonometric representation of the vector t:
t = 10.296 cos I + 10.296 sin j of angle 60.945
Thus, alternative is the right response (C).
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The sail of a boat is in the shape of a right triangle. which expression shows the height, in meters, of the sail? the sail of a boat is a right triangle with an acute angle equal to 40 degrees and the side adjacent to the 40 degrees angle is 2 meters long. 2(sin 40°) sine 40 degrees over 2 2(tan 40°) tangent 40 degrees over 2
The expression shows the height, in meters, of the sail is 2(sin 40°) (option A)
One way to approach this problem is to use the sine function.
We know that sin(40 degrees) is equal to the length of the side opposite angle 40 degrees divided by the length of the hypotenuse. In other words, sin(40 degrees) = b/c.
We want to find the length of the segment that drops from the top of the sail to side a, which we will call h.
This segment is the side adjacent to angle 40 degrees, so we can use the cosine function to relate it to the length of side b. Specifically, cos(40 degrees) = h/b. Solving for h, we get
h = 2 x sin(40 degrees) = 2(sin 40°)
Hence the option (A) is correct.
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57
Exhibit 18-2
Students in statistics classes were asked whether they preferred a 10-minute break or to get out of class 10 minutes early. In a sample of 150 students, 40 preferred a 10-minute break, 80 preferred to get out 10 minutes early, and 30 had no preference. We want to determine if there is a difference in students' preferences.
Refer to Exhibit 18-2. The test statistic based on the number of students who preferred to get out early equals
Group of answer choices
1.825
0.67
0.82
3.65
The test statistic based on the number of students who preferred to get out early equals 3.65.
What is a test statistic?A test statistic is a numerical summary of the sample data that is employed to determine the strength of the evidence for a hypothesis. It is a quantitative measure that compares the observed data to the hypothesis. The t-value, Z-score, F-value, and chi-squared value are examples of common test statistics. In statistical hypothesis testing, a test statistic is used to make decisions about whether or not to reject the null hypothesis.
Based on the given problem, the number of students who preferred to get out of class early is 80, while the number of students who preferred a 10-minute break is 40. Hence, the number of students who had no preference is 30.
Now, we will find the test statistic based on the number of students who preferred to get out of class early as follows:
Test Statistic = (Observed Value - Expected Value) / Standard Deviation
To find the observed value, we need to calculate the proportion of students who preferred to get out of class early in the sample as follows:
Proportion of students who preferred to get out of class early = 80 / (150-30)= 80 / 120= 0.67
Therefore, the observed value is 0.67.
To find the expected value, we need to assume that there is no difference in students' preferences, and the proportion of students who preferred to get out of class early is the same as the proportion of students who preferred a 10-minute break.
Expected value = Proportion of students who preferred to get out of class early * Total number of students= 0.5 * 150= 75
Therefore, the expected value is 75.
To find the standard deviation, we need to use the formula for the standard deviation of a proportion as follows:
Standard Deviation = √[(p*q) / n]
where p is the proportion of students who preferred to get out of class early, q is the proportion of students who preferred a 10-minute break, and n is the total number of students.
Standard Deviation = √[(80/120)*(40/120) / 150]= √(0.1778 / 150)= 0.048
Therefore, the standard deviation is 0.048.
Now, we will substitute the values in the formula for the test statistic as follows:
Test Statistic = (0.67 - 0.5) / 0.048= 3.65
Therefore, the test statistic based on the number of students who preferred to get out early equals 3.65.
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What is the distance from the point (15,-21) to the line for which f(4)=-8 and f(8)=-18
The distance from the point (15,-21) to the line is 5.39 units.
What is point slope form?The equation of a line in point-slope form is:
y - y1 = m(x - x1) (x - x1)
where (x1, y1) is a point on the line and m is the slope of the line. When we are unsure of the y-intercept but are aware of the line's slope and a point on the line, we can utilise this form of the equation.
To calculate the equation of a line using the point-slope method, we must first determine the slope of the line using the following formula:
m = (y2 - y1)/(x2 - x1) (x2 - x1)
where the two points on the line are (x1, y1) and (x2, y2). We may enter the slope, along with one of the line's points, into the point-slope form to obtain the equation of the line.
Given that, the line has the following values f(4)=-8 and f(8)=-18.
The coordinates of the line are (4, -8) and (8, -18)
Thus, the slope of the line is:
m = y2 - y1/ x2 - x1
m = -18 + 8 / 8 - 4
m = -10/4 = -5/2
Now the slope intercept form is given as:
y - y1 = m (x - x1)
Substitute the values:
y + 8 = -5/2(x - 4)
2y + 16 = -5x + 20
2y = -5x + 20 - 16
2y = -5x + 4
The distance from the line to point is given as:
Distance = |ax + by + c| / √(a² + b²)
Substituting the values:
Distance = |-5(15) + -2(-21) + 4| √(-5² + -2²)
Distance = |-29|/ 5.38
Distance = 5.39
Hence, the distance from the point (15,-21) to the line is 5.39 units.
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The coordinates of the vertices of quadrilateral HIJK are H(1,4), I(3,2), J(-1,-4), and K(-3,-2). If quadrilateral HIJK is rotated 270 about the origin, what are the vertices of the resulting image, quadrilateral H’ I’ J’ K’
The vertices of the resulting image, quadrilateral H’ I’ J’ K’ include the following:
H' (4, -1).
I' (2, -3).
J' (-4, 1).
K' (-2, 3).
What is a rotation?In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.
In Geometry, rotating a point 270° about the origin would produce a point that has the coordinates (y, -x).
By applying a rotation of 270° about the origin to quadrilateral HIJK, the location of its vertices is given by:
(x, y) → (y, -x)
Ordered pair H (1, 4) → Ordered pair H' (4, -(1)) = (4, -1).
Ordered pair I (3, 2) → Ordered pair I' (2, -(3)) = (2, -3).
Ordered pair J (-1, -4) → Ordered pair J' (-4, -(-1)) = (-4, 1).
Ordered pair K (-3, -2) → Ordered pair K' (-2, -(-3)) = (-2, 3).
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an aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. the aquarium is filled with water to a depth of 37 cm. a rock with volume $1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. by how many centimeters does the water level rise? express your answer as a decimal to the nearest 100th.
The water level in the aquarium rises by (A) 0.25.
The volume of water in the aquarium before the rock is added is:
100 cm x 40 cm x 37 cm = 148000 cm^3
When the rock is added, its volume is 1000 cm^3, so the total volume of water and the rock is:
148000 cm^3 + 1000 cm^3 = 149000 cm^3
To find the new water level, we need to divide the total volume by the base area of the aquarium:
149000 cm^3 ÷ (100 cm x 40 cm) = 37.25 cm
Therefore, the water level rises by:
37.25 cm - 37 cm = 0.25 cm
So the answer is (A) 0.25.
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Your question is incomplete, but probably the complete question is :
An aquarium has a rectangular base that measures 100 cm by 40cm and has a height of 50cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume 1000cm^3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
(A) 0.25
(B) 0.5
(C)
(D)1.25
(E) 2.5
a rectangle is drawn around a sector of a circle as shown. if the angle of the sector is 1 radian and the area of the sector is , find the dimensions of the rectangle, giving your answers to the nearest millimetre
The dimensions of the rectangle is 1mm x 0.5mm.
The area of a sector of a circle can be calculated using the formula A = (1/2)*r2*θ,
where r is the radius of the circle and
θ is the angle of the sector.
Therefore, the area of the sector given in the question is A = (1/2)*r²*1, where r = 1.
Since the rectangle has the same area as the sector,
the area of the rectangle can be calculated as A = l*w,
where l is the length and
w is the width.
This equation can be rearranged to give l = A/w,
where A = (1/2) and w is the width.
Substituting the values for A and w into the equation gives l = (1/2) / w.
Since the width of the rectangle is the same as the radius of the circle,
w = 1.
Therefore, the length of the rectangle is l = (1/2), which gives the dimensions of the rectangle as 1mm x 0.5mm, rounded to the nearest millimeter.
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20. Assertion(A): The sides of a triangle are 5cm, 12cm and 13cm and its area is 30 cm². Reason(R): Area of a triangle is base x height. (a) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion, (b) Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion. (c) Assertion is true but the Reason is false. (d) Assertion is false but the Reason is true.
What is the probability of
drawing a face card, then
drawing a heart with
replacement
Answer:
n(s) =52. n(f) = 12 n(h) = 13
p (f)= 13/52. p(f)= 12/52
The prices of consumer goods do not always exactly follow the CPI. The following chart shows several consumer items, along with their respective prices in 1983 and today.ItemPrice in 1983 ($)Current Price ($)Pair of boots67.45126.85Hair dryer15.2529.95Box of tissues1.152.25Toaster22.8544.25Using the prices shown on this chart, which of the following is a reasonable estimate of the current CPI?a. 208b. 195c. 180d. 173
None of the given options (i.e. a, b, c, or d) is correct.
Given the prices of consumer items, we can find the estimate of the current CPI.
The percentage change in the price of the consumer item over time can be given as:
Percentage change = (Current Price − Price in 1983)/Price in 1983 × 100%
Now, using the percentage change we can find the average percentage change (increase or decrease) in prices. We calculate the average percentage change by finding the sum of the percentage change in prices of all items and then dividing it by the number of items.
Thus, we have:
Average percentage change = [(126.85−67.45)/67.45 + (29.95−15.25)/15.25 + (2.25−1.15)/1.15 + (44.25−22.85)/22.85]/4Average percentage change = 4.4188%
Now, the current CPI can be found using the following formula:
Current CPI = CPI in 1983 × [1 + (Average percentage change)/100]Current CPI = 100 × [1 + 4.4188/100] ≈ 104.42 ≈ 104
Therefore, the reasonable estimate of the current CPI is 104.
None of the given options (i.e. a, b, c, or d) is correct.
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A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 1) is 0.2. When a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2.
Find the probability that a 0 is received. (Enter the value of the probability in decimal format and round the final answer to one decimal place.)
P(0 received correctly) = P(0 sent) × P(0 received correctly | 0 sent)= [tex](2/3) × 0.8= 0.5333[/tex] (rounded to 1 decimal place)Thus, the probability that a 0 is received is 0.5333 (rounded to 1 decimal place).
0.5333
A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 1) is 0.2. When a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2.The probability that a 0 is received correctly is given in the problem as 0.8, and the probability that a 0 is sent is 2/3. Therefore, the probability that a 0 is received correctly
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A new business is hiring managers and employees. The company needs at least 20 managers and no more than 500 employees. It must hire a total of at least 400 people. Write a system of linear inequalities to represent the constraints of this situation. Let x represent the number of managers, and let y represent the number of employees.
Answer:
We can represent the constraints of this situation with the following system of linear inequalities:
x ≥ 20 (the company needs at least 20 managers)
y ≤ 500 (the company needs no more than 500 employees)
x + y ≥ 400 (the company must hire a total of at least 400 people)
So the system of linear inequalities is:
x ≥ 20
y ≤ 500
x + y ≥ 400
Suppose the number of water people drink in a week is normally distributed with a mean of 50 and a standard deviation of 5 glasses of water. Find the value 1 standard deviation below the mean
Answer:
Step-by-step explanation:
.
Alfonso wants to purchase a pool membership for the summer. He has no more than y dollars to
spend. The Aquatics Club charges an initial fee
of $75 plus $20 per month. The Swimming Hole
charges an initial fee of $15 plus $65 per month.
Write a system of inequalities that you can use to
determine which company offers the better deal.
Let x represent the number of months.
Answer:
Let A represent the cost of purchasing a pool membership from the Aquatics Club and let S represent the cost of purchasing a pool membership from the Swimming Hole. Then, we can write the following system of inequalities:
A ≤ y
A = 75 + 20x
S ≤ y
S = 15 + 65x
The first two inequalities represent the cost of purchasing a membership from the Aquatics Club, while the last two represent the cost of purchasing a membership from the Swimming Hole. The inequalities ensure that the cost of purchasing a membership from either company does not exceed Alfonso's budget of y dollars.