Which expression is equivalent to (8×10−2) (2.4×10−3)?

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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Answer:

Area = 12a m²

Step-by-step explanation:

Given information,

→ Length = 4a m

→ Width = 3a m

Now we have to,

→ find the area of rectangular park.

Formula we use,

→ Area = L × W

Then the area of rectangle is,

→ L × W

→ 4a × 3a

→ (4 × 3)a

→ 12a m²

Therefore, the area is 12a m².

The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.

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.

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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Answer

Step-by-step explanation:

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 .

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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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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What is the greatest common factor of 20 and 40?

20

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.

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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]

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.

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.

(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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[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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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.

Answer:

113

Step-by-step explanation:

Answer:

-5/8

Step-by-step explanation:

5.(-2^-3)

-5. 1/2³

-5. 1/8

-5/8

Answer:

4xy ×3xy -2xy

12xy-2xy

10xy

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.

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.

For a linear differential equation, the coefficients of y or the derivatives of y are either:

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:

For the differential equation in item 3, we have that:

Hence statement 3 is correct.

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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

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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The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.

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]

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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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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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.

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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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In the arrangement of letters and numericals, the total number of different paths can one spell AMC8 are 24 in count.

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

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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The amount of sugar, in ounces, that is needed to make 28 ounces of hummingbird nectar is 7 ounces.

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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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.

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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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

Step-by-step explanation:

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.

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.

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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