Answer:
We can not see the graph, but that does not matter.
For a function y = f(x), the domain is the set of the possible values of x that we can input in the function, and the range is the set of the possible outputs.
First, for the domain, we start assuming that the domain is the set of all real numbers.
Then we look at our function, there is a point that causes a problem?
(a problem can be a zero in a denominator, for example)
Well, we have a quadratic equation, then we can not have any "x" in a denominator, then there are not problems, which means that the domain is the set of all real numbers.
For the range we first need to find the minimum (or maximum value) of f(x).
Here we know that the vertex is (1, - 3)
If the arms of the graph open upwards, then the minimum is at the vertex, which means that the minimum value of y is y = -3
And the arms go upwards infinitely, so there is no upper limit.
Then the range is the set of all numbers such that:
-3 ≤ y
Now, if the arms of the graph open downwards, then the maximum is at the vertex, which means that the maximum of our function is at y = -3
And with similar thinking than in the above case, we can find that the range is:
R: y ≤ -3.
Solve.(Refer to the picture)
Answer:
[tex]y=2.182[/tex]
Step-by-step explanation:
Solve for y.
[tex]3.082=y+\frac{9}{10}[/tex]
Move all terms not containing [tex]y[/tex] to the right side of the equation.
[tex]y=3.082-\frac{9}{10}[/tex]
Write [tex]\frac{9}{10}[/tex] as a decimal.
[tex]y=3.082-0.9[/tex]
[tex]y=2.182[/tex]
What is the y-intercept
6,328 divide by 56 does anyone know
Answer:
113
Step-by-step explanation:
Howard Cunningham has a $90,000.00 whole life policy. What will the cash value of her policy be at the following anniversary dates?
(Include decimal places for cents in each answer, even when cents are not calculated. Use a thousands comma where applicable.)
Answer:
5 years = 2,700.00
25 years = 30,600.00
Step-by-step explanation:
Hey, these are difficult, I know.
I'll explain how you solve them.
In this case, $90,000.00 needs to first be divided by 1,000.
That gives us 90.
You then multiply 90 by 30, which is the 5 yr cash value.
That gives you 2,700.00.
Since you have 90, you next need to multiply that by 340 (25 yr cash value), which gives you 30,600.00.
Hope this helps. Feel free to mark Brainliest.
small regional carrier accepted 22 reservations for a particular flight with 18 seats. 16 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 58% chance, independently of each other.
(Report answers accurate to 4 decimal places.)
Find the probability that overbooking occurs. Find the probability that the flight has empty seats.
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
How to calculate probability?We remove the 16 regular customers and their seats from consideration. Thus, we consider the 22 - 16 remaining customers = 6 and the 18-16 = 2 seats overbooking means 3 or more of the 6 remaining passengers; each has a 58% chance remaining. We can either use the binomial distribution formula P(x) = C(6, x) .58^x .42^(6-x) or use packages that provide the result, including cumulative probabilities.
Calculation:Overbooking = 1 - P(X <= 2) In Excel, we use =1 - binom.dist(2,6,.58,1) ; the 1 is for cumulative.
The solution is 0.7920
Probability of empty seats = P(x <= 1) = binom.dist(1,6,.58,1) = 0.0510
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
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Can somebody please tell me how do i solves these. Thank you.
2. It is false that the differential equation is non-linear.
The statement regarding the differential equation y''y + (y')² = sin(x) is false.
3. Statement 3: fourth order and nonlinear differential equation.
How to classify the differential equations?For a linear differential equation, the coefficients of y or the derivatives of y are either:
Constant terms.Other variables that are not y.Hence the differential equation for question 2 is linear, as the coefficients of the derivatives are x and -1, neither of which is y, hence the statement is false.
For the differential equation y''y + (y')² = sin(x), we have that:
It is non-linear, as y multiplies y''.It is of the second order, as the order of the highest derivative is of 2.For the differential equation in item 3, we have that:
It is of the fourth order, as the derivative of the third order multiplies the derivative of the first order.It is non linear, as the variable y multiplies the derivative of the third order of y.Hence statement 3 is correct.
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The best approximation, in cubic inches, to the increase in volume of a sphere when the radius is increased from 3 to 3.1 inches, is Use V = Tr3 (a) 0.04n/3 (b) 3.61 n'
(c) 0.04 n'
(d) 1.2 n'
(e)36 n'
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
How is volume of a sphere calculated?In simple words, when a sphere is placed in the space or three-dimension space, then the total space acquired by the sphere generally shows the volume of the sphere. The mathematical formula needed to determine the volume of a sphere is shown below,
[tex]V = \frac{4}{3} \pi r^3[/tex]
Calculation:Given- The initial radius of the sphere is r1 = 3 inch and the final radius of the sphere is r2 = 3.1 inch.
Now , the change in the radius of the sphere is [tex]dR =[/tex][tex]r2 - r1 =\\[/tex] 3.1 - 3 = 0.1 inch.
The mathematical expression of the volume of a sphere is shown below,
[tex]V = \frac{4}{3}\pi r^3[/tex] (here, [tex]r[/tex] is the radius of sphere)
Differentiate the above shown expression with respect to radius ([tex]r[/tex]) as shown below,
[tex]\frac{d}{dr} (V) = \frac{d}{dr} (\frac{4}{3} ) \pi r^3[/tex]
∴
dV/dR = 4/3 π × d/dR(r^3)
= 4/3 π × 3r^2
dV/dR = 4π r^2
∴ dV = 4π r^2 dR ................(1)
Substitute all values in equation (1) and calculate the approximation as shown below,
dV = 4π r^2 dR
= 4π (3)^2 × (0.1)
∴ dV = 3.6π cubic inches.
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
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Kate Drew has been hand-painting wooden Christmas ornaments for several years. Recently, she has hired some friends to help her increase the volume of her business. In checking the quality of the work, she notices that some slight blemishes occasionally are apparent. A sample of 24 pieces of work resulted in the following number of blemishes on each piece: 1,1,2,3,1,0,2,0,0,1,3,1,0,1,2,0,0,2,1,2,3,2,1,1. Develop upper and lower control limits (99.73%) for the number of blemishes on each piece. Is the process in control? Why? **Please do this in excel*
A sample of 24 pieces of work is considered for the checking the quality of the work,
a) Upper control limit of 99.73% confidence interval is 4
Lower control limit of 99.73% confidence interval is Zero.
The process is in control because all the values of the blemishes per unit are lie within the control limits .
What is Control Limits?The Control limits consist of two limits lower control limit and upper control limit within which the value of the statistical observations of a product are expected to lie. If the values lie outside these limits, then the process is called out of statistical control. We have,
Kate Drew has been hand-painting wooden Christmas ornaments for several years.
Sample size = 24 pieces
confidence level = 99.73% = 0.9973
The number of blemishes per unit is given as:
1,1,2,3,1,0,2,0,0,1,3,1,0,1,2,0,0,2,1,2,3,2,1,1.
The control limits are to be computed for the number of defects per piece . Let us consider the number of defects per piece be denoted by c. In that case, the control limits for c chart are used. The upper control limit (UCL) and lower control limit (LCL) for c chart is given by,
UCL = c-bar + 3 s.d
LCL = c-bar - 3 s.d
where c-bar = 1/n(€c) average number of defects
So, c-bar is computed as,
c-bar = (1+1+2+3+1+0+2+0+0+1+3+1+0+ 1+2+0 +0+2 + 1+2 +3+ 2+1+1) /24 = 30/24 = 1.25
put the data in Excel sheet and calculate required limits using Excel Commands.
Excel Command for Average, c-bar
=Average( Range of numbers who's average you want to calculate )
In my excel sheet, = Average ( H2 : H25)
= 1.25
Excel Command for Standard deviations,
=STDEV( numbers who's standard deviations you want to calculate)
here, =STD(H2:H25) , output= 0.98907
UCL = 1.25 + 3( 0.98907)
= 4.217
LCL = 1.25 - 3( 0.98907)
= - 1.717
but defective item can never be negative so, Lower control limit is Zero.
Hence, for the given data, the upper control limit for the number of defects is 4 and the lower control limit is 0.
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DUE SOON! REAL ANSWERS ONLY!!!
Answer:
A.) The sum will be an even integer.
Step-by-step explanation:
The conjecture that can be made about the sum of two even integers and one odd integer is that the sum will be an even integer. This is because even integers are numbers that are divisible by 2, and odd integers are numbers that are not divisible by 2.
For example, consider the sum of the even integer 6 and the odd integer 7. The sum of these two numbers is [tex]6 + 7 = 13[/tex], which is an odd integer. However, if we add another even integer to the sum, such as 4, the resulting sum is [tex]13 + 4 = 17[/tex], which is an even integer. This shows that the sum of two even integers and one odd integer is always an even integer.
This conjecture can be proven using mathematical induction. We can start by showing that the sum of any two even integers and one odd integer is always an even integer. Then, we can assume that the sum of any n even integers and one odd integer is also an even integer. Finally, we can show that the sum of n+1 even integers and one odd integer is also an even integer, using the assumption that the sum of n even integers and one odd integer is even. This will prove that the conjecture is always true.
Two fair dice are rolled. Find the joint probability mass function of x
and y
, when x
is the largest value obtained on any die and y
is the sum of the values.
a) P(X = x₀, Y = 2x₀) = 1/36
P(X = x₀, Y = k) = 1/18 for k between x₀+1 and 2x₀-1 inclusive
Every other event has probability 0. x₀ is any number between 1 and 6 inclusive.
b) P(X = x₀, Y = x₀) = x₀/36
P(X = x₀, Y = k) = 1/36 for k between x₀+1 and 6 inclusive.
x₀ is between 1 and 6 inclusive. Every other event has probability 0.
c) P(X = x₀, Y = x₀) = 1/36
P(X = x₀, Y = k) = 1/18 with k between x₀+1 and 6 inclusive
x₀ between 1 and 6 inclusive. Any other event has probability 0.
What is a joint probability?Joint probability is the probability of two events happening simultaneously.
Let P(A) = probability that event A takes place and P(B) = probability that event B takes place. Then, probability of event A and B happening together i.e. P(A and B) is called joint probability.
What is the Calculation of joint probability?(1) When A and B are independent events: When event A has no impact on event B or vice versa, the two events are said to be independent. For independent events, formula for joint probability is:
P(A and B) = P(A) × P(B)
Example: I am performing two acts at a time (i) tossing a coin and (ii) throwing a dice. What is the probability that the coin shows up “head” and the dice shows up “3”?
There is no doubt that getting “head” on coin and getting “3” on dice are independent events.
Probability of the coin showing up “head” = 1/2
Probability of the dice showing up “3” = 1/6
Therefore, probability of getting “head” on coin and “3” on dice = (1/2)×(1/6) = 1/12
(2) When A and B are dependent events: If event A and event B cannot be considered independent, calculation of joint probability requires knowledge of conditional probability. For dependent events, formula for joint probability is:
P(A and B) = P(A) × P(B, given A) = P(B) × P(A, given B).
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Can I get help with this quick check
The common difference of the sequence is 70, the slope of the 1st and 2nd lines is 70, the slope of the 3rd and 4th points is 70, and the slope is constant.
What is the slope?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. Y and x are the net changes in the y and x coordinates, respectively.
Given:
Time in minutes and jumping jack table, coordinates of lines (1, 70) and (2, 140)
(a)
Consider 70, 140, 210, and 280 as an arithmetic sequence then the common difference will be,
Common difference = 140 - 70 = 70
(b)
Calculate the slope of the line as shown below,
[tex]m = (y_2-y_1) / (x_2 - x_1)[/tex]
m = (140 - 70) / (2 - 1)
m = 70 / 1
m = 70
(c)
Calculate the slope for 3rd and 4th points,
m = (280 - 210) / (4 - 3)
m = 70 / 1
m = 70
(d)
As you can, the slope is constant, and the common difference is also constant.
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How do I answer this assignment?
I do not understand
Answer:
number of caps is the ratio
blue + white + green = 96
5x + 4x + 7x = 96
16x = 96
x = 6
Blue caps : 5x = 5(6) = 30
white caps : 4x = 4(6) = 24
green caps : 7x = 7(6) = 42
total of : 96 caps
Therefore, a caps in the picture represent 6 capsStep-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.8 C 88 C M C 8C M A M C8 C M C 88 C 8(A) 8(B) 9(C) 12(D) 24(E) 36
In the arrangement of letters and numericals, the total number of different paths can one spell AMC8 are 24 in count.
What is counting principle?A basic counting principle is that if there are p ways to do one thing and q ways to do another, then there are p × q ways to do both. Possible results of experiments. The counting principle can be extended to situations where there are two or more alternatives.
We have, the arrangement of letters and numericals in above figure such that
Beginning at the A in middle M a path allows only moves from one letter to an adjacent either above, below, left, or right, but not diagonal letter.There are three different kinds of paths that are on the above diagram to get the required pattern. The first kind is when you directly count A, M, C in a straight line. The second is when you count A, turn left or right to get M, then go up or down to count 8 and C. The third is the one where you start with A, move up or down to count M, turn left or right to count C, then move straight again to get 8.
There are 8 paths for each kind of path, making for 8×3= 24 paths.
Hence, the required paths are 24 in count.
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An airline that wants to assess customer satisfaction chooses a random sample of 10 of its flights during a single month and asks all of the passengers on those flights to fill out a survey. This is an example of a
A) Multistage sample.
B) Stratified sample.
C) Cluster sample.
D) Simple random sample.
E) Convenience sample
An airline that wishes to measure customer satisfaction chooses ten of its flights at random over the course of a single month and requests that each passenger fill out a survey. A Cluster sample looks like this.
Why do we utilize cluster sampling?Cluster sampling should be used to study large, dispersed populations because interviewing every person would be costly, time-consuming, and occasionally impossible.Smaller, more comparable groups that represent the population under study can be produced via cluster sampling.You divide a population into groups, such as districts or schools, and then randomly select a sample from among these groupings when using the probability sampling approach known as cluster sampling. Each cluster should ideally be a miniature representation of the total population.An airline that wants to gauge customer happiness selects a haphazard sample of 10 of its flights over the course of a single month and asks each passenger to complete a survey. An illustration of a Cluster sample is this.To learn more about Cluster sample refer to:
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on Monday Therease went to the doctor and got an antibiotic for strep throat. The doctor told her take a dose of 4.8 ml every 12 hours for 7 days. If Thereas took her first dose at 9:00 AM on Monday, what day and time should she take her 7th dose?
Therease should take her 7th dose on Friday at 9:00AM
What is time ?
Time can be described in mathematics as an ongoing and continuous series of events that take place one after another, from the past through the present, and into the future. The duration of events or the gaps between them can be measured, compared, or even ordered using time.
Time is the ongoing progression of existence and things that happen in what seems to be an irrevocable order from the past, present, and forward into the future.
Time is defined by physicists as the flow of events from the past through the present and into the future. In essence, a system is timeless if it is unchanging. When describing events that take place in three-dimensional space, time can be thought of as the fourth dimension of reality.
The doctor told her to take a dose of 4.8 ml every 12 hours for 7 days.
If Theresa took her first dose at 9:00 AM on Monday,
Simply add 12 hours every time up to the 7th dose,
First dose ----> 9:00 AM on Monday,
Second dose ----> 9:00 PM on Monday,
Third dose ----> 9:00 AM on Tuesday,
Fourth dose ----> 9:00 PM on Tuesday,
Fifth dose ----> 9:00 AM on Thursday,
Sixth dose ----> 9:00 PM on Thursday,
Seventh dose ----> 9:00 AM on Friday,
Hence, Theresa should take her 7th dose at 9:00 AM on Friday.
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determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The fifth percentile of the standard normal distribution is -1.64.
In the given question, we have to determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The standard normal distribution, commonly known as the z-distribution, is a unique type of normal distribution in which the mean and standard deviation are both equal to 1. Any normal distribution's values can be transformed into z scores to normalise it. Z score shows the amount of standard deviations from the mean that each consists of numbers.
Since, X is normally distributed.
P(X< c) = 0.05
Using Z table
c = -1.645
Fifth percentile = -1.64
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Write the following expression without negative exponents and without parentheses. (7x)^{-2} (7x) −2
[tex]\frac{1}{(7x)^2}[/tex] is the expression without negative exponents.
What are exponents and indices?
Exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself.
An index, or power, is the small floating number that appears after a number or letter. The plural of index is indices.
According to the given question:
(7x)^{-2} is the given indices expression
To expression without negative exponent, the expression
can be written as
[tex](7x)^{-2} = \frac{1}{(7x)^2}[/tex]
Since the exponent -2 is on the whole 7x term. So, the whole term moves to denominator to make the exponent positive.
Therefore [tex]\frac{1}{(7x)^2}[/tex] is the expression without negative exponents
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What is the greatest common factor of 20 and 40?
What is the greatest common factor of 20 and 40?
20
Hummingbird feeders are filled with hummingbird nectar. Hummingbird
nectar is made using a ratio of 1 part sugar to 3 parts water.
How much sugar, in ounces, is needed to make 28 ounces of
hummingbird nectar? Show your work in the provided space.
The amount of sugar, in ounces, that is needed to make 28 ounces of hummingbird nectar is 7 ounces.
How to find the ounces of sugar needed ?Hummingbird nectar is said to be made by using a ratio of 1 part sugar to 3 parts water. For every 1 ounce of sugar therefore, there would be 3 ounces of water.
Put together, this would give 4 ounces. And the percentage of sugar in this 4 ounces would be:
= 1 / 4
= 25 %
If 28 ounces of hummingbird nectar needed to be made, the sugar needed is :
= Percentage of sugar x Ounces of hummingbird nectar
= 25 % x 28
= 7 ounces of sugar
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A. A random sample of 50 interns has a mean earnings of $6520 with a population standard deviation of $927 over the summer months. Determine whether a normal distribution or a t-distribution should be used or whether neither of these can be used to construct a confidence interval.
B. In a random sample of 50 designer outfits, the average cost was $350 with a standard deviation of $50. Determine whether a normal distribution or a t-distribution should be used or whether neither of these can be used to construct a confidence interval. Assume the distribution of weekly food expenses is normally shaped.
We determined that normal distribution should be used to construct a confidence interval for these two statistical data.
What is t distribution and normal distribution?In statistics, the t-distribution is a system of describing a set of data where most of the data fall close to the mean value and the remaining data adjust the tail on both sides.
The normal distribution is one of the systems of describing a set of data that implies data close to the mean value are more probable to occurrence than the other data.
When does t-distribution and normal distribution used in statistics?In the statistics, we use t- distribution when the number of samples is less than 30 and the population standard deviation is unknown to us.
we use normal distribution when the sample size is greater than 30 and the population standard deviation is known.
We are given two set of statistical data and for both case population standard deviation is given, and the size of sample is greater than 30.
hence, we will use normal distribution for both cases.
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two tangents $\overline{pa}$ and $\overline{pb}$ are drawn to a circle, where $p$ lies outside the circle, and $a$ and $b$ lie on the circle. the length of $\overline{pa}$ is $12,$ and the circle has a radius of $9.$ find the length $ab.$
The value of tangent AB as calculated from the given data is 14.4.
The tangents to the circle as given are PA and PB.
Let the point of intersection of PO and AB be X.
PA = 12
OA = 9
As given that the triangle is right angled we can use hypotenuse theorem to calculate PO,
PO = √ PA² + OA²
= √ 144 + 81
= 15
Now , the given triangles are similar because,
AX / AO = PA / PO
AX / 9 = 12 / 15
Therefore, the value of X is,
X = 7.2
Now we know that AB's midpoint is X,
Hence , AB = 2X = 14.4
Tangent is known as a point which passes through a circle only at one single point A circle can have infinite tangents.
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Perform the following mathematical operation, and report the answer to the appropriate number of significant figures.
1204.2 + 4.72613 = [?]
The answer is not 1208.92613
Answer
Step-by-step explanation:
Realiza las siguientes potencias
Answer:
-5/8
Step-by-step explanation:
5.(-2^-3)
-5. 1/2³
-5. 1/8
-5/8
What is the product of the 4 x and the 3 x cubed y squared minus 2 x y cubed?
Answer:
4xy ×3xy -2xy
12xy-2xy
10xy
Find the y-intercept and the slope of the line.
y=-1/2-1/4x
y-intercept: ?
slope: ?
Answer: slope = -1/4 y-intercept = -1/2
Step-by-step explanation:
Given an equation in the form y = mx + b, m = slope and b = y-intercept.
We can rearrange y= -1/2 - 1/4x into that form.
y = -1/4x - 1/2
We can now obviously see that -1/4x is the slope, and -1/2 is the y-intercept.
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.72.
(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 23 specimens from the seam was 4.85. (Round your answers to two decimal places.)
(b) Compute a 98% CI for true average porosity of another seam based on 14 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.)
(c) How large a sample size is necessary if the width of the 95% interval is to be 0.42? (Round your answer up to the nearest whole number.)
(d) What sample size is necessary to estimate true average porosity to within 0.24 with 99% confidence? (Round your answer up to the nearest whole number.)
For the normal distribution of helium porosity (in percentage) of coal samples,
a) 95% CI for the true average porosity of a certain seam is equals to ( 4.56 , 5.14 ).
b) A 98% CI for true average porosity of another seam based on 14 specimens is equals to (4.11 , 5.01).
c) Sample size with width of confidence interval 0.42 is 45.
d) Necessary Sample size to estimate true average porosity to within 0.24 is 60.
We have Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed.
Standard deviations, σ = 0.72
a)Average porosity of specimens , X-bar = 4.85
Sample size, n = 23
Confidence level , alpha = 95% = 0.95
From Normal Distribution table
Z for 95% Confidence Interval = 1.96
so,95% Confidence interval = (x-bar - Z× σ/sqrt(n) , x-bar + Z×σ/sqrt(n) )
plugging all known values in above formula,
= ( 4.85-1.96×0.72/sqrt(23) , 4.85+1.96×0.72/sqrt(23) )
So, 95% Confidence interval = ( 4.56 , 5.14 )
b)Now, Sample size, n = 14
sample mean, X-bar = 4.56
from normal distribution Z- table
Z-score for 98% CI is equal to 2.33
so,98% CI = (x-bar-Z×σ/sqrt(n) , x-bar+Z×σ/sqrt(n) )
= ( 4.56-2.33×0.72/√14 , 4.56+2.33×0.72/√14 )
98% CI = ( 4.11 , 5.01 )
c)Using the normal Distribution Z- table
Z for 95% CI = 1.96
Width of confidence interval = 0.42
we have to determine the sample size, n .
Using formula, Width = 2×Z×σ/sqrt(n)
=> n = (2×Z×sd/width)²
=> n= (2×1.96×0.72/0.42)²
=> n = 45.15
=> n = 45 ( whole number)
d) From normal distribution table
Z for 99% CI = 2.58
Margin of error, ME = 0.24
margin of error = ME = Z×σ/sqrt(n)
=> n = (Z× σ/ME)²
=> n= (2.58×0.72/0.24)²
=> n = 59.9
=> n = 60
Hence, required sample size is 60.
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find the area of shaded region
Between which values does lie?
Answer:
lies between 10 and 11.
Step-by-step explanation:
We are asked to find the values between which lies.
We know that square root of 100 is 10 and square root of 121 is 11.
Since is between and , therefore, lies between 10 and 11.
Kadeesha bought snacks for her team's practice. She bought a bag of popcorn for $ 1.59 and a 5-pack of juice bottles. The total cost before tax was $13.34. Which equation or tape diagram could be used to represent the context if j represents each bottle of juice costs?
A
5 j=1.59-13.34
C
1.59 j=13.34-5
The equation or tape diagram could be used to represent the context if j represents each bottle of juice costs is; Option D tape diagram
How to Represent Algebraic equations?We are told that;
1 bag of popcorn costs $1.59
She bought a 5-pack of juice bottles
Total cost before tax = $13.34
Now, we are told that j represents each bottle of juice costs. Thus;
Cost of the 5 packs of juice = 5j
Thus, the equation of the total cost before tax is;
1.59 + 5j = 13.34
Looking at the options, only the tape diagram in option D correctly represents the required equation.
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Q. An arithmetic series has first term a and common difference d, where d is a prime number. The sum of the first n terms of the series is S, and Sm=39 S2m = 320 Find the value of d and the value of m Show clear algebraic working. (Total for question = 5 marks)
Finding the values of d and the value of m with the algebraic working will give us the the value of d to be 7 and the value of m to be 3.
How do we calculate the values using the algebraic expression?Finding the value of d, we know that the sum of the first n terms of an arithmetic series is given by:
S = n/2 * (2a + (n-1)d)
Since the sum of the first n terms is S and the sum of the first 2m terms is S2m, we can set up the following equation:
S = m/2 * (2a + (m-1)d)
S2m = 2m/2 * (2a + (2m-1)d)
Substituting the given values for S and S2m into these equations, we get:
39 = m/2 * (2a + (m-1)d)
320 = 2m/2 * (2a + (2m-1)d)
Solving for d in each equation, we find that d = -7 in the first equation and d = 7 in the second equation. Since d must be a prime number, the only possible value for d is 7.
Now that we know the value of d, we can solve for m. Substituting the value of d back into one of the equations and solving for m, we get:
39 = m/2 * (2a + (m-1)7)
78 = m * (2a + (m-1)7)
78 = m * 2a + 7m^2 - 7m
7m^2 - m - 78 = 0
We can solve for m using the quadratic formula:
m = (-1 +/- sqrt(1^2 - 4*7*(-78)))/(2*7)
= (-1 +/- sqrt(2521))/14
= (-1 + 49)/14 = 3
= (-1 - 49)/14 = -7
Since m must be a positive integer, the only possible value for m is 3.
Therefore, the value of d is 7 and the value of m is 3.
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