The standard deviation of the sampling distribution is approximately 0.3292.
What is central limit theorem?The behaviour of the sampling distribution of the mean is described by the central limit theorem, a key conclusion in statistics. It asserts that regardless of how the population distribution is shaped, if a random sample of size n is taken from a population with mean and standard deviation, the distribution of sample means will tend towards a normal distribution as n increases.
Because it enables us to utilise the normal distribution to draw conclusions about the population mean based on sample means, the central limit theorem has significant practical ramifications. Additionally, it offers a foundation for confidence interval estimation and statistical hypothesis testing.
The standard deviation is given by the formula:
SD = σ/√(n)
Now, substituting the value of σ = 40, n = 147,400 we have:
SD = σ/√(n) = 40/√(147400) ≈ 0.3292
Hence, the standard deviation of the sampling distribution is approximately 0.3292.
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ms george baked 2 cakes of equal size each of her 12 guests at the same amount of cake, and no cake was left over.
a What fraction of a cake did each guest eat
b. How do you think ms george cut the cakes into equal pieces? Explain
Taut gets 14% discount. What should he do to compute what his cost is without tax
The 18% he gets over the goods purchased exceeds Rs. 6300.
What is the discount?a decrease from the gross amount or value of something: for example, a(1): a reduction from the usual or list price.
Total Purchase Value = Total discount in rs * % discount
= 1520/16 x 100 = 9500
Discount on upto 3200 = 3200 x 14% = 448
Discount on upto 3201-6300 = (6300-3200) x 14% = 496
Balance discount = 1520 - 448 - 496 = 576
Above 6300 purchase amount = 9500 - 6300 = 3200
Discount % = 576/3200 x 100 = 18%
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Complete question:
A customer gets 14% discount on purchasing goods whose value is up to Rs. 3200 and gets 16% discount whose value is between Rs. 3201- Rs. 6300 and customer gets certain % discount while purchase the goods exceeds Rs. 6300. A customer gets a total discount of Rs. 1520 is 16% of the total purchase value. find the %age he gets over the goods purchased exceeds Rs. 6300 ?
poppy seed muffins 1
cinnamon rolls 3
bran muffins 2
apple fritters 2
croissants 2
Considering this data, how many of the next 20 pastries served should you expect to be bran muffins?
We can expect 4 of the next 20 pastries served to be bran muffins based on the given data.
What is meant by the term data?
Data refers to a collection of facts, statistics, measurements, or observations that are gathered and analyzed to gain insights or information about a particular topic or subject. Data can be both quantitative (numerical) and qualitative (descriptive) in nature, and it can be obtained from various sources, such as surveys, experiments, sensors, or records. The processing and interpretation of data can lead to new discoveries, trends, or predictions in various fields, including science, business, and technology.
According to the given information
The proportion of bran muffins out of the total pastries served can be calculated by dividing the number of bran muffins by the total number of pastries:
The proportion of bran muffins = 2 / (1 + 3 + 2 + 2 + 2) = 2 / 10 = 0.2
This means that 20% of the pastries served are bran muffins. To find out how many of the next 20 pastries served should be expected to be bran muffins, we can multiply 20% by 20:
The number of expected bran muffins = 0.2 * 20 = 4
Therefore, we can expect 4 of the next 20 pastries served to be bran muffins based on the given data.
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The function = 50 − 1.6 can be used to determine the amount, in fluid ounces, of laundry detergent remaining after x loads of laundry have been done. What is the rate of change amount of laundry in fluid ounces with respect to the number of loads of laundry that have been done?
The rate of change of the amount of laundry with respect to the number of loads is a constant value of -1.6 fluid ounces per load.
What is linear equation?Any equation that has terms that are either constants or the product of a constant and a variable of the first degree (exponent 1) is said to be linear. In other words, the equation's greatest degree for any variable is 1. One way to express a linear equation is as follows:
ax + b = 0
If an is not equal to zero, a and b are constants, and x is the variable. This is how a linear equation is typically expressed.
The given function is f(x) = 50 - 1.6x.
Now, the rate of change is the slope of the equation.
Comparing the equation with that of the linear equation y = mx + c we see that the slope is -1.6.
Hence, the rate of change of the amount of laundry with respect to the number of loads is a constant value of -1.6 fluid ounces per load.
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regular priced at $85 on sale for 75%
off regular price
Answer: $21.25
Step-by-step explanation:
This can be solved using the following formula:
discount = (discount rate) x (regular price) sale price = regular price - discountPlugging in the values you provided:discount = (0.75) x (85) discount = 63.75sale price = 85 - 63.75 sale price = 21.25
Therefore, the sale price of the item is $21.25.
Jamie and Polly share £24 in the ratio 2:3. Work out how much money Jamie gets.
Answer:
Jamie gets £9.60
Step-by-step explanation:
sum the parts of the ratio, 2 + 3 = 5 parts
divide the amount by 5 to find the value of one part of the ratio.
£24 ÷ 5 = £4.80 ← value of 1 part of the ratio , then
2 parts = 2 × £4.80 = £9.60 ← amount Jamie gets
Which is the most precise description of quadrilateral ABCD?
The most precise description of quadrilateral ABCD is a square.
We have,
Quadrilateral ABCD.
We see that,
AB is parallel to CD
AB = CD _____(1)
AD is parallel to BC.
AD = BC _____(2)
Now,
AB = AD ______(30
From (1), (2), and (3).
AB = BC = CD = AD
This means,
The Quadrilateral ABCD is a square.
Thus,
The most precise description of quadrilateral ABCD is a square.
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Consider the information below for Postman Builders Inc. Suppose that the expected inflation rate and thus the inflation premium increases by 2.0 percentage points, and Postman acquires risky assets that increase its beta by the indicated percentage. What is the firm's new required rate of return? Beta 1.5, Required Return 10.2, RPm 6%, Percentage increase in beta 65
Therefore, Postman Builders Inc’s new required rate of return is 23.7%1.
SET A RATE?Rate might signify two different things. As a noun, it denotes an amount, frequency, or measure that is typically compared to another quantity or measure. Or, it can be used as a verb to indicate "assign a standard or value to" using a specific scale.
Postman Builders Inc. has a necessary return of 10.2% and a beta of 1.5, according to the information given.
We may get the new necessary rate of return as follows if Postman purchases risky assets that increase its beta by 65% and the predicted inflation rate and inflation premium increase by 2 percentage points.
New Required Rate of Return
= Risk-free rate + Beta * (Expected Market Return - Risk-free rate)
= 2% + (1.5 * (6% + 2% + (65% * 6%))) = 23.7%
New Required Rate of Return equals 8% plus 2.47 x (-2%) to 3.85%.
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What amount of a 80% acid solution must be mixed with a 55% solution to produce 600 mL of a 60% solution?
Answer:The amount of a 80% acid solution that must be mixed with a 55% solution to produce 600 mL of a 60% solution can be calculated using a simple algebraic equation. Let x represent the amount of the 80% acid solution in mL that needs to be mixed with the 55% solution.
The equation can be set up as follows:
0.80x + 0.55(600 - x) = 0.60(600)
Step-by-step explanation: The left-hand side of the equation represents the total amount of acid in the mixture. The first term, 0.80x, represents the amount of acid from the 80% solution, and the second term, 0.55(600 - x), represents the amount of acid from the 55% solution. The right-hand side of the equation represents the total amount of acid in the final 60% solution, which is 60% of 600 mL.
By solving this equation for x, we can determine the amount of the 80% acid solution that needs to be mixed with the 55% solution. Once we have the value of x, we can then calculate the amount of the 55% solution by subtracting x from 600 mL.
Internal Link: For a verified answer on a similar topic, check out this link: https://brainly.com/question/1234567 intext:Answer Expert Verified.
Let x=amount of 80% solution needed
Then 600-x=amount of 30% solution needed
Now we know that the pure acid in the 80% solution (0.80x) plus the amount of pure acid in the 30% solution ((0.30)(600-x)) has to equal the amount of pure acid in the final mixture(0.40*600), so our equation to solve is:
0.80x+0.30(600-x)=0.40*600 get rid of parens
0.80x+180-0.30x=240 subtract 180 from each side
0.80x+180-180-0.30x=240-180 collect like terms
0.50x=60 divide each side by 0.50
x=120 ml---------------------amount of 80% solution needed
600-x=600-120=480 ml---------------amount of 30% solution needed
CK
120*0.80+480*0.30=0.40*600
96+144=240
240=240
Which equation best shows that 48 is a multiple of 3?
Choose 1 answer:
A
B
C
144= 3 x 48
3 = 48 ÷ 16
52 = 48 + 3
D3=48-45
You roll a 6-sided die.
What is P(less than 4)?
Write your answer as a fraction or whole number.
The value of the probability of rolling a number less than 4 is 1/2
Calculating the value of P(less than 4)?The probability of rolling a number less than 4 on a fair 6-sided die can be calculated by counting the number of favorable outcomes and dividing it by the total number of possible outcomes.
There are three favorable outcomes: 1, 2, and 3.There are six possible outcomes: 1, 2, 3, 4, 5, and 6.Therefore, the probability of rolling a number less than 4 is:
P(less than 4) = favorable outcomes / total outcomes = 3/6 = 1/2
So the probability of rolling a number less than 4 is 1/2, or 50%.
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Making observations often involves:
A. forming a hypothesis, finding answers, and planning experiments.
OB. forming a hypothesis, asking questions, and writing articles.
C. asking questions, identifying problems, and gaining new
information.
D. finding ansers, making predictions, and identifying problems.
The correct statement is C. asking questions, identifying problems, and gaining new information is the most accurate option.
What is observation?The definition of "observation" given by Merriam-Webster is "an act of recognizing and noting a fact or occurrence often involving measurement with instruments," or "a record or description so obtained."
C. asking questions, identifying problems, and gaining new information is the most accurate option.
Making observations often involves the process of asking questions, identifying problems, and gaining new information. Observations are made to gather data and information about a phenomenon or event, which can then be used to answer questions or develop hypotheses for further research.
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15th term of 2, 10, 18, 26, 34
Step-by-step explanation:
a1 = 2
a2 = 10
d = a2 - a1 = 10 - 2 = 8
a15 = a + 14d = 2 + 14(8) = 2 + 112 = 114
Answer:
2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114
Step-by-step explanation:
all you do is add 8 each time
The average high temperatures in degrees for a city are listed.
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
If a value of 120.5° is added to the data, how does the mean change and by how much?
Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers.
For the given logarithmic expression the simplified result is given by option b.) [tex]2 \log_b({x})-\frac{7}{3} \log_b({y) -\frac{8}{3} \log_b (z) \\\end{align*}[/tex].
How to solve logarithmic expression?Check for the type of logarithmic expression.Make use of logarithmic properties to simplify the expression.Further simplify using algebric methods.Check for more answers or further simplify the statement if necessary.[tex]\begin\text{Given \;expression:} \quad & \log_b \sqrt[3]{\frac{x^6}{y^7 \cdot z^8}} \\[/tex]
[tex]\text{1: Simplify the expression inside the cube root:} \quad & \frac{(x^6)^{\frac{1}{3}}}{((y^7 \cdot z^8)^{\frac{1}{3}})} \\\\[/tex][tex]\text{2: Simplify the exponents:} \quad & x^{6 \cdot \frac{1}{3}} = x^2, \quad (y^7 \cdot z^8)^{\frac{1}{3}} = y^{\frac{7}{3}} \cdot z^{\frac{8}{3}} \\\\[/tex]
[tex]\text{3: Substitute the simplified expression back:} \quad & \log_b\frac{x^2}{y^{\frac{7}{3}} \cdot z^{\frac{8}{3}}} \\\end{align*}[/tex]
Now Using ,[tex]\log_c\frac{a}{b}} = \log_c a -\log_cb[/tex]
[tex]\text{4: Equation becomes:} \quad & \log_b({x^2})-\log_b({y^{\frac{7}{3}} \cdot z^{\frac{8}{3})}} \\\end{align*}[/tex]
Also, [tex]\log_c{b^n}} = n\log_c b[/tex]
[tex]\text{5: Using property stated above, we get:} \quad &2 \log_b({x})-\frac{1}{3} \log_b({y^7 \cdot z^{{8}}) \\\end{align*}[/tex]
Finally,[tex]\log_c{a}\cdot{b}} = \log_c a +\log_cb[/tex]
[tex]\text{6:Making the final result:} \quad &2 \log_b({x})-\frac{7}{3} \log_b({y) -\frac{8}{3} \log_b (z) \\\end{align*}[/tex]
Thus, option b is the required result.
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* (3) Find the area of the shaded area 2 of each rectangle. 7 cm 8 cm 12 cm
The area of the shaded region is 73.65 cm².
We have,
The shaded region is a triangle.
Base = x
Height = y
Now,
Using the Pythagorean theorem,
y² = 12² + 10²
y² = 144 + 100
y² = 244
y = √244
And,
x² = 8² + 5²
x² = 64 + 25
x² = 89
x = √89
Now,
Area of the shaded region.
= 1/2 x √89 x √244
= 1/2 x 9.43 x 15.62
= 73.65 cm²
Thus,
The area of the shaded region is 73.65 cm².
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A diagram shows a garden bed. The area of the garden bed is 21ft to the power of 2. What is the Height of the garden bed. Need help ASAP
Answer: 1.75
Step-by-step explanation:
Answer:
6
Step-by-step explanation:
Since the garden bed looks like the shape of a trapezoid the equation to find the area of a trapezoid is A = 1/2h (base 1 + base 2)
The area is already given so you can just plug it in. Also you can plug in the bases.
21 = 1/2h (3 + 4)
21 = 1/2h (7) rewrite
21 = 1/2(7)h
21 = 3.5h Divide both sides by 3.5
6 = h
Someone please help I have no idea what I’m doing
The compositions are:
f * f(x) = x⁴ - 2x²
g * g(x) = 81/64x
How did we get the values?To find the composition f * f, we start by substituting f(x) into f(x) wherever we see x:
f * f (x) = f(f(x)) = f(x² - 1) = (x² - 1)² - 1
Expanding the expression, we get:
f * f (x) = x⁴ - 2x² + 1 - 1 = x⁴ - 2x²
So, f * f(x) = x⁴ - 2x².
To find the composition g * g, we substitute g(x) into g(x) wherever we see x:
g * g (x) = g(g(x)) = g(9/8x) = 9/8(9/8x) = 81/64x
So, g * g(x) = 81/64x.
Therefore, the compositions are:
f * f(x) = x⁴ - 2x²
g * g(x) = 81/64x
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The text format of the question is:
Suppose that the functions f and g are defined as follows.
f(x) = x² - 1
g(x) = 9/8x, x ≠
Find the compositions f * f and g * g
Simplify your answers as much as possible.
(Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain)
An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, s?
2 < s < 18
6 < s < 12.8
s < 2 or s > 18
s < 6 or s > 12.8
We can see here that the best representation of the possible range of values for the third side, s is: 2 < s < 18.
What is an acute triangle?An acute triangle is a type of triangle where all three angles are acute angles, meaning they are less than 90 degrees. In other words, an acute triangle is a triangle with all three angles being sharp or narrow.
The best representation of the possible range of values for the third side, s, can be determined using the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have two sides measuring 8 cm and 10 cm. Let's call the third side "s". Using the Triangle Inequality Theorem, we can write:
8 + 10 > s
18 > s and
8 + s > 10
s > 2
and
10 + s > 8
s > -2 (since a length cannot be negative)
Therefore, we know that s must be greater than 2 and less than 18.
The best representation of the possible range of values for s is: 2 < s < 18
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Find the circumference of a child swimming pool that has a radius of 3 feet
The circumference of a child swimming pool is 6π.
What is circumference?
The circumference of a circle or ellipse in geometry is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. The curve length around any closed figure is more often referred to as the perimeter.
Here, we have
Given: A child swimming pool that has a radius of 3 feet.
We have to find the circumference.
The circumference is diameter x π
The diameter is twice the radius
c = 2πr
c = 2π(3)
= 6π
Hence, the circumference of a child swimming pool is 6π.
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any help? i dont seem to understand which radius of circle is.
First of all, radius is half the diameter. Radius is from the center-most point of a circle straight to the edge.
Answer:
The area of the whole object is 1,237.68 m^2 (meters squared).
Step-by-step explanation:
The circles are equal to:
π * d = a
3.14 * 12 = a
37.68 = a.
Now, since the circles are half in the rectangle, it would be easier to calculate the rectangle using these halves, so the total of the half-circles outside the rectangle is equivalent to a whole circle, or 37.68. Now, let's calculate the rectangle:
20 * (30 * 2) = a
20 * 60 = a
1200 = a.
The whole area is equivalent to the area of the two half-circles and the area of the rectangle:
37.68 + 1200 = a
1237.68 = a
A restaurant owner orders new plates and spoons based on the information below. • plates are sold in spoons are sold in packages of 9 packages of 12 The restaurant owner orders an equal number of plates and spoons. What is the least number of packages of plates and packages of spoons she should order to have an equal number of plates and spoons?
The least number of packages of plates and spoons will be 12 and 9 respectively so she should order to have an equal number of plates and spoons.
What is inequality?
It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.
Given that, a restaurant owner will use the data below to place new orders for plates and spoons. Packages of nine plates and twelve spoons are available for purchase. Equal quantities of plates and spoons are ordered by the restaurant owner.
For the inequality, we have to apply the arithmetic operation in which we do the multiplication of x and apply the inequality for the given data.
If the plates are sold in packages of 9 and there are 12 packages the number of plates will be 108.
If the plates are sold in packages of 12 and there are 9 packages the number of plates will be 108.
Thus, the least number of packages of plates and spoons will be 12 and 9 respectively so she should order to have an equal number of plates and spoons.
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Determine if the function is linear or exponential
Answer:
linear
Step-by-step explanation:
function declines at a constant rate of 1/2x with a y-intercept of 20 giving the equation y= 1/2x + 20
Cuantos subconjuntos tiene A={a, r, i, t, m, e, t, i, c, a}
The number of subsets that the set has are 128 subsets.
How to find the number of subsets?First, let's remove the duplicates from the set A. The distinct elements in set A are:
A = {a, r, i, t, m, e, c}
Now, we can find the number of subsets. For each element in the set, we have two choices: either it is included in the subset, or it is not. Therefore, there are 2^n subsets for a set with n distinct elements.
In our case, n = 7, so there are 2^7 = 128 subsets for the given set A.
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Select all the correct answers. If the measure of angle is , which statements are true? The measure of the reference angle is . The measure of the reference angle is . The measure of the reference angle is . 0 =9/2
The statement cos(theta) = √3/2 is correct.
Which statement is correct?Angles are the angles formed by 2 bisecting lines or surfaces at or near their intersection.
The following information has been provided:
The angle (theta) = [tex]\frac{2\pi }{3}[/tex] measurement has been given to us.
We must determine which propositions are true by determining the measure of angle (theta) = [tex]\frac{2\pi }{3}[/tex]
cos(theta) = [tex]\frac{\sqrt{3} }{2}[/tex]
= cos30
theta = 30
tan(theta)= [tex]-\sqrt{3} = tan(90 + 30)[/tex]
tan(theta) = tan120
theta = 120
sin(theta) = [tex]-\frac{1}{2}[/tex] = [tex]cos(90 + 30)[/tex]
sin(theta) = sin120
theta = 120
Therefore the correct statement is cos(theta) = √3/2
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Complete question:
The measure of angle(theta) Is 2pi/3, which statements are true?
cos(theta) = √3/2
The measure of the reference angle is 30°.
The measure of the reference angle is 45°.
The measure of the reference angle is 60°.
tan(theta) = -√3
sin(theta)=-1/2
Determine the interval(s) for which the function shown below is decreasing.
The interval(s) for which the function is decreasing are (3, 1) and (4, ∝)
Determining the interval(s) for which the function is decreasing.From the question, we have the following parameters that can be used in our computation:
The graph
The interval(s) for which the function is decreasing is when
As x increases, the function values decreases
Using the above as a guide and examining the graph, we have the following:
The function is decreasing at (3, 1) and (4, ∝)
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In need of assistance! If possible, I'd appreciate it!
Vector a: start at (1, 3) and end at (-4, -2) in blue.
Vector b: start at (-4, -2) and end at (1, 3) in red.
Vector a+b: start at (1, 3) and end at (2, 7) in green.
How do we calculate?To represent a+b using the parallelogram method,
we must draw vectors representing a and b.
The initial point of vector a is (1, 3), and its terminal point is (-4, -2). The initial point of vector b is (-4, -2), and its terminal point is (1, 3).
Using the vector tool, we then draw the vectors a and b. We start at the initial point of each vector and select the terminal point.
Vector a: start at (1, 3) and end at (-4, -2)
Vector b: start at (-4, -2) and end at (1, 3)
We the diagonal, we start at the initial point of vector a, (1, 3), and draw a line parallel to vector b that passes through the initial point of vector b, (-4, -2). This line intersects the line parallel to vector a that passes through the initial point of vector b at point (2, 7). This point is the terminal point of the diagonal vector, which is a+b.
We use the vector tool, we can draw vector a in blue, vector b in red, and vector a+b in green.
Vector a: start at (1, 3) and end at (-4, -2) in blue.
Vector b: start at (-4, -2) and end at (1, 3) in red.
Vector a+b: start at (1, 3) and end at (2, 7) in green.
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Use this image to find
sin 0
Answer:
[tex]sin(theta) = \frac{5}{13} [/tex]
EASY ALGEBRA 2 !! PLEASE HELP
according to the question the balance in the account earning compound interest after 6 years is $4,156.22.
what is interest ?Multiply the principal with the rate of interest, the time period, and various other factors to determine simple interest. Simple returns = money + interests + hours is the marketing formula. This approach makes calculating interest the simplest. The approach to calculating interest that is most usually employed is as a percentage over the principal sum. For instance, he will just pay his share of the interest at 100% when he borrows $100 with a friend and agrees to shell out it back with 5% interest. $100 has been reduced to $5 (0.05). Every time you borrow money, you must pay interest, which is applied to whatever loans you make. On the loan balance, interest is typically calculated using an annual percentage.
given,
To calculate the balance in the account earning compound interest after 6 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the sum of money at the end of the period of time
P stands for the principle (beginning sum).
the decimal representation of the annual interest rate (r).
n represents how many times the interest rate is compounded annually.
t is the duration (in years).
In this case, we have:
P = $3500 (the principal)
r = 2.16% = 0.0216 (the annual interest rate)
Since interest is compounded four times a year (quarterly), n is equal to four.
t = 6 (the time period is 6 years)
With these values entered into the equation, we obtain:
A = 3500(1 + 0.0216/4)^(4*6)
A = $4,156.22
Therefore, the balance in the account earning compound interest after 6 years is $4,156.22.
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The balance in the account earning compound interest after 6 years when the principal is $3500 at an annual interest rate of 2.16% compounded quarterly is approximately $4,029.53.
What is compound interest?Compound interest is the term used to describe the interest that is calculated on both the initial investment and any accumulated interest. It is interest on top of interest, to put it another way.
We can use the formula for compound interest to find the balance in the account after 6 years:
A = P(1 + r/n)^(nt)
where:
A = balance after 6 years
P = principal = $3500
r = annual interest rate = 2.16%
n = number of compounded times per year = 4 (quarterly)
t = time in years = 6
Plugging in the values, we get:
A = 3500(1 + 0.0216/4)^(4×6)
A ≈ $4,029.53
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I need help understanding it and it being broken down
Therefore, the expected value of the event P(2k) for 2 ≤ k ≤ 6, given the event E, is 96/1 or simply 96.
What is probability?Probability theory is widely used in many fields, including mathematics, statistics, physics, engineering, economics, and social sciences. It provides a framework for analyzing and understanding uncertain events and making predictions based on available information. Probability theory is also used in decision-making, risk management, and game theory, among other applications.
Here,
We can start by calculating the probability of each event in the sample space:
P(1) = 1/32
P(4) = 1/8
P(8) = 1/4
P(16) = 1/2
P(32) = 1
P(64) = 1
Next, we can calculate the probability of the event E by summing the probabilities of the individual outcomes in E:
P(E) = P(1) + P(8) + P(32) + P(64)
= 1/32 + 1/4 + 1 + 1
= 33/32
Now, we can calculate the expected value of the event P(2k) for 2 ≤ k ≤ 6:
E[P(2k)] = Σ[2 ≤ k ≤ 6] P(2k) * ΘS(2k)
= P(4)*ΘS(4) + P(8)*ΘS(8) + P(16)*ΘS(16) + P(32)*ΘS(32) + P(64)*ΘS(64)
= (1/8)*4 + (1/4)*8 + (1/2)*16 + (1)*32 + (1)*64
= 4/8 + 8/4 + 16/2 + 32 + 64
= 1/2 + 2 + 8 + 32 + 64
= 106.5
Therefore, the expected value of the event P(2k) for 2 ≤ k ≤ 6, given the event E, is 106.5. However, since the event E has a probability greater than 1, this result does not make sense. So, we need to normalize the probabilities of the events by dividing them by P(E) to obtain the correct expected value:
P'(1) = P(1)/P(E) = (1/32)/(33/32) = 1/33
P'(8) = P(8)/P(E) = (1/4)/(33/32) = 8/33
P'(32) = P(32)/P(E) = 1/(33/32) = 32/33
P'(64) = P(64)/P(E) = 1/(33/32) = 32/33
Now, we can calculate the normalized expected value:
E[P(2k)] = Σ[2 ≤ k ≤ 6] P'(2k) * ΘS(2k)
= P'(8)*ΘS(8) + P'(32)*ΘS(32) + P'(64)*ΘS(64)
=(8/33)*8 + (32/33)*32 + (32/33)*64
= (64/33) + (1024/33) + (2048/33)
= 3136/33
= 96/1
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