true or false: with a well-designed sample survery, the sample percentage is very likely to equl the population percentage

15 different combinations can be used by John to buy the cake and the hammer.

Given,

John buys a cake at a bakery and a hammer at a hardware store

Number of hardware shops = 5

Number of bakeries = 3

We have to find the combinations of stores he can purchase;

Here,

5 hardware shops; H₁, H₂, H₃, H₄, H₅

3 bakeries; B₁, B₂, B₃

Now,

The combinations be like;

B₁ ; (H₁, H₂, H₃, H₄, H₅)

B₂ ; (H₁, H₂, H₃, H₄, H₅)

B₃ ; (H₁, H₂, H₃, H₄, H₅)

That is,

3 bakeries and 5 hardware shops

= 3 x 5

= 15

Therefore,

John can buy the cake and hammer in 15 different combinations.

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Answer:

68 = 2L + 2(9)

Step-by-step explanation:

Just write down exactly what it says!

The perimeter of a rectangle is 68 inches

68

The perimeter equals twice the length of L inches

68 = (2 × L)

Plus twice the width of 9 inches.

68 = (2 × L) + (2 × 9)

Now just clean up the equation.

68 = 2L + (2 × 9)

68 = 2L + 2(9)

The answer is 68 = 2L + 2(9).

If John and Sam insist on sitting next to each other , then 6 people can sit at a round table in 48 ways .

In the question ,

it is given that ,

John and Sam insist to sit together ,

we have to find the number of ways , we can sit 6 people at a round table

the number of people = 6 people

there is no head of the table ,

So, first person is just a place marker so there are really only 5 people whose positions matter.

But since John and Sam are tied together, they are the equivalent of one person.

that means there  are only 4 people.

= 4! × 2 ways   ...because it could be (John , Sam) or (Sam, John) ,

= 48  ways .

Therefore , 6 people can sit in 48 ways .

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The dimensions of the can are radius of can = 4cm and

height of can = 78.6cm

What is the volume of cylinder?

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up.

Thus, the volume of the cylinder can be given by the product of the area of base and height.

Volume of cylinder = πr²h

According to the given question:

We have V cylinder = V = 400cc

We will first find the areas involved

Let x = radius of base then area of the base =  π*x² and this is the area of the top too

For lateral area we need to get h the height of the cylinder as function of x

V = πx²h   ⇒  h= v/πx²    ⇒   h = 400/πx²

Now the total area of the cylinder is:

A(x) = Area of the base + area of the top + lateral area

A(x) = 2*π*x² + 2πx h ⇒  A(x) =  2*π*x² + 2πx (V/πx² )

A(x) =  2*π*x² + 800/x

Taking derivatives:

A´(x)  =  4πx  - 800/x²

A´(x) = 0                4πx  - 800/x² =0     ⇒ πx - 200/x² = 0

( πx³  -  200 )/ x²  = 0      

πx³  -  200 = 0      

x = 4 cm

and h = 400/πx²    ⇒  h = 78.6 cm

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The perimeter of the triangle ABC is 12 units. Hence, option (B) is correct.

A triangle is a geometric figure with three edges, three angles and three vertices. It is a basic figure in geometry.

The sum of the angles of a triangle is always 180°

In the given right angle triangle,

The vertices are A(-3, 2), B(-3, -1) and C(1, -1)

The distance between AB = 3,

And the distance between BC = 4

To find the distance of AC, use Pythagoreans theorem,

AC² = AB² + BC²

       = 3² + 4²

       = 9 + 16

AC² = 25

AC = 5

The perimeter of triangle ABC = AB + BC + AC

= 3 + 4 + 5

= 12

The perimeter of the triangle ABC is 12 unit.

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Answer:

Step-by-step explanation:

kwik trip

Answer:

Let x be the number of pennies, and y  be the number of nickels.

1 penny = 0.01 dollar

1 nickel = 0.05 dollar.

The equations that will describe the given problem will be,

0.01 x + 0.05 y = 0.81

y = x + 9

Plugging the second equation into the first gives us,

0.01 x + 0.05(x+ 9) = 0.81

0.06 x + 0.45 = 0.81

0.06 x = 0.36

y = 15

So, Kennedy has 6 pennies, and 15 nickels.

Step-by-step explanation:

1. The expected number of cars is given as follows: 2.87.

2. The standard deviation of the number of cars is given as follows: 0.94.

The discrete distribution is given from the table in this problem, as follows:

The expected value is given by the sum of each outcome multiplied by it's probability, as follows:

E(X) = 0 x 0.005 + 1 x 0.12 + 2 x 0.125 + 3 x 0.5 + 4 x 0.25 = 2.87.

The standard deviation is given by the square root of the sum of the differences squared between each observation and the mean, as follows:

S(X) = sqrt((0-2.87)² x 0.005 + (1-2.87)² x 0.12 + (2-2.87)² x 0.125 + (3-2.87)² x 0.5 + (4-2.87)² x 0.25) = 0.94.

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The average score is 97

What is average ?

In plain English, an average is a single number chosen to represent a group of numbers; it is often the sum of the numbers divided by the number of numbers in the group. The average of the numbers 2, 3, 4, 7, and 9 is, for instance, 5.

Average, mean, median, and norm all refer to something that sits in the middle. The average is the ratio created by dividing the total number of figures in a set by the total number of figures. scored 85 on tests on average. The term "mean" can refer to the simple average or to a value that lies halfway between two extremes.

Let the grade he got on final test be x

The average of all four grades is 91.

So, we have  

(92+89+86+x)/4 = 91

⇒x=364−267

= 97

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1. Write an expression for the total area of the wall: 10x

2. The area of each section: 2x ft²

3. The area of each section of the wall:  2x ft²

A flat (2-D) surface's or an object's shape's overall area is known as its area. On a sheet of paper, draw a square using a pencil. It's a two-dimensional figure. The term "Area" refers to the area that a form occupies on paper. 

We use cm² or m² as our units for measuring area. Area is computed by multiplying a shape's length by its breadth.

Given that,

Todd and his friends are painting 5 equal sections of a wall.

The wall is 10 feet.

1. Total area of the wall this is rectangular size wall which length is 'x' (unknown) and width/height is 10 feet.

So, area = length × height

area = 10x

2. To find the area of each section, there are 5 section (each are equal) so area of each section = total area / 5

= 10x /5

= 2x ft²

we divided total area by 5.

3.  Now area of each section

= 10x/5

= 2x ft²

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The radian measures of the central angle of the sector is 2

Given,

In the question:

Perimeter of sector is 4 times of its radius .

To find the radian measures of the central angle of the sector is?

Now, According to the question:

We know:

The perimeter of the circumference is = 2[tex]\pi[/tex]r

If the perimeter of a sector is 4 times its radius.

then, the arc measure of the sector is (4r-2r) → 2r

So, if 2×[tex]\pi[/tex] radians (full circle) has a length → 2[tex]\pi[/tex]r

X  radians → 2× r  (the sector)

X=2× r ×(2×[tex]\pi[/tex])/(2[tex]\pi[/tex]r)

X = 2

Hence,  The radian measures of the central angle of the sector is 2.

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1. Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.

2.The implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

1. a) check the image attached

r= radius

h=height

b) Volume of conical tank ,V = (1/3)πr²h

= (1/3)πr²(2r)

V = 2/3πr³

c) Rate of change of volume with respect to time is denoted as dV/dt

Rate of change of radius is denoted as dr/dt

d) V = 2/3πr³

dV/dt =  2/3π (3r²) dr/dt

dV/dt = 2πr²  dr/dt

e) Rate of change in volume, dV/dt = 15cm³ / sec  ,  r =75cm

∴ 15 = 2π(75)² .  dr/dt

dr/dt = 15 / (2π(75)²)

dr/dt = 15/35342.92

dr/dt = 0.00042 cm/sec

∴Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.

2. sin(2x² y³) - 3x³ = 1

Differentiating :

cos(2x² y³) [2x² y³dy/dt + 2y³ 2x] - 9x² = 0

6x²y² dy/dx . cos(2x² y³) + 4xy³ cos(2x² y³) - 9x² = 0

dy/dx = 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

Hence, the implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

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Answer:

3.5 Points

Step-by-step explanation:

Change means how many more points did you gain. 6.4 - 2.9= 3.5

The stocks overall change in points over the two days is 3.5 points.

True, the sample statistics may have a different value for each observation that is randomly chosen since they are a numerical explanatory measure of a sample.

Using sample data, a statistic is a quantitative descriptive measure. A parameter is a quantitative way to describe a population. Since whole populations (N) aren't always possible, we must rely on samples (n) as well as the statistics that can be calculated from them.

The mean, median, & mode, which are utilized at practically all math and statistics levels, are the most well-known forms of descriptive statistics. By summing together all the data set's figures & then dividing the total number of figures, the mean or average can be derived.

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The commission amount is $60 with the rate at 8% of earned sales amount $750.

Percentage means one part of every cent or hundred. And it is represented by symbol '%'.

Given that sales = $750 ,

and the commission rate is 8% .

Now to find the commission , we find 8% of $790

    = 750 x 8 / 100

    = 6000 / 100

    = $60

Therefore, the commission amount is $60 with the rate at 8% of earned sales amount $750.

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The correct answer is:

Reject null hypothesis H0, we cannot conclude the proportion of customer complaints is more for the old dough.

As given in the question,

Let us consider proportion of customer complaints with old dough is p₁

And proportion of complaints with new dough is p₂

Difference in proportion of complaints in old and new dough

= p₁ - p₂

Null hypothesis : H0 : p1 ≥ p2

p₁ - p₂  ≥ 0

Alternative hypothesis : Ha : p1 < p2

p1 - p2 < 0

Let 'x' represents the number of complaints

Sample size = n

Sample proportion = x/n

Old dough:

Number of complains with old dough 'x₁' = 6

Number of pizzas with old dough 'n₁' = 385

p₁ = 6/385

   = 0.01558

   = 0.016

New dough:

Number of complains with new dough 'x₂' = 16

Number of pizzas with new dough 'n₂' = 340

p₂ = 16/340

   = 0.04705

   = 0.047

Pooled proportion test:

[tex]\bar{p}[/tex] = (x₁+ x₂)/(n₁ + n₂)

  = (6 + 16)/(385 + 340)

  = 0.0303

  = 0.03

1 -  [tex]\bar{p}[/tex] = 1 - 0.03

      = 0.97

z = (p₁ - p₂)/√[tex]\bar{p}[/tex](1 - [tex]\bar{p}[/tex])(1/n₁ + 1/n₂)

 = (0.016 - 0.047)/√(0.03)(0.97)(1/385 + 1/340)

 = (-0.031)/ √(0.03)(0.97)(0.0026 + 0.0029)

 = - 0.031/√0.00016005

 = -0.031 / .01265

 = -2.45

It is left tailed test.

p value for the area to the left of the z score using normal distribution table:

p value = 0.007143

Significant level 'α' for  95% confidence level:

= 1 - 0.95

= 0.05

Since 0.05 > 0.0.007143

Reject null hypothesis.

Therefore, reject H0, we cannot conclude the proportion of customer complaints is more for the old dough.

The complete question is :

Paul owns a mobile wood-fired pizza oven operation. A couple of his clients complained about his dough at a recent catering, so he changed his dough to a newer product. Using the old dough, there were 6 complaints out of 385 pizzas. With the new dough, there were 16 complaints out of 340 pizzas. Let p1 be the proportion of customer complaints with the old dough and p2 be the proportion of customer complains with the new dough. Based on a 95% confidence for the difference of the proportions, what can be concluded?

Multiple Choice

A. Reject H0, we can conclude the proportion of customer complaints is more for the old dough

B. Do not reject H0, we cannot conclude the proportion of customer complaints is more for the old dough

C. Do not reject H0, we can conclude the proportion of customer complaints is more for the old dough

D. Reject H0, we cannot conclude the proportion of customer complaints is more for the old dough

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The HAACP principle addressed by cooking the duck breast to 165 degrees F is a 'critical limit'.

What is the critical limit?

A critical limit is a value to which a biological, chemical or physical factor must be controlled to prevent food safety hazards. These are the conditions under which potential hazards can be controlled, reduced, or eliminated. They are monitored by observation or measurement, such as measuring the temperature of food storage to avoid spoilage.

Critical limits are based on factors such as:

Temperature, Humidity, Moisture level, pH, Water activity (Aw), Time, Salt concentration, and Preservatives.

Production processes that fail to stay within the bounds of the critical limits must be stopped or the products must be discarded.

Hence the answer is critical limit.

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Initial value that take the LaPlace transform of both sides of the given differential equation to create the corresponding algebraic equation is [tex]y(t) = 3t + sin4t + 3cos4t[/tex]

In terms of its usefulness in resolving physical issues, the Laplace transform is an important transform that is probably second only to the Fourier transform. The Laplace transform is especially helpful in solving linear ordinary differential equations, such as those that arise in the analysis of electronic circuits.

The initial value-

[tex]y" + 16y = 48t\\y(0)=3,y'(0)=7\\L( f^{n}(t) ) = s^{n}L(f(t)) - s^{n - 1}(f(0)) - s^{n - 2}(f(0)).........................................f^{n - 1} (0)\\[/tex]

By the formula,

[tex]s^{2}Y(s) - sy(0) - y'(0) + 16Y(s) = 48L(t)\\L(t^{n}) = \frac{n!}{s^{n+1} } \\\\On substitutinf the values,\\Y(s)(s^{2} + 16) - 3s - 7 = \frac{48}{s^{2} }\\Y(s)(s^{2} + 16) = \frac{48}{s^{2} } +3s + 7\\Y(s) = \frac{48 + s^{3} -7s^{2} }{s^{2}((s^{2} + 16) }\\Y(s) = \frac{3}{s^{2} } + \frac{4}{(s^{2} + 16} + \frac{3s}{(s^{2} + 16}[/tex]

On taking LaPlae inverse both side,

[tex]Y(t) = L( \frac{3}{s^{2} } ) + L( \frac{4}{(s^{2} + 16)} ) + L( \frac{3s}{(s^{2} + 16)} )\\Y(t) = 3t + sin4t + 3cos4t\\[/tex]

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Answer:

12

Step-by-step explanation:

(5x-8)/6x

The variable expression are

cost at each agency to rent a car for 280 miles

The company that has a lower cost for 400 miles

The required expression is a linear function

A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.

y = mx + c

Definition of variable to suit the problem

y = output variable (total cost/)

m = slope (cost  per mile driven)

x = input variable (number of miles)

c = y intercept (Car Rental charges)

The equation as described in the problem is represented using linear function as as follows

y = $0.18x + $27.60 - Blue Car Rental

y = $0.14x + $40.40 - Red Car Rental

the total cost at each agency to rent a car for one day and drive it 280 miles

y = $0.18x + $27.60 - Blue Car Rental

y = 0.18 * 280 + 27.60

y = $78

y = $0.14x + $40.40 - Red Car Rental

y = 0.14 * 280 + 40.40

y = $79.60

Write and solve an inequality

$0.18x + $27.60 < $0.14x + $40.40

0.18x + 27.60 < 0.14x + 40.40

0.18x - 0.14x < 40.40 - 27.60

0.04x < 12.80

x < 320 miles

Mr. Kao rents a car for one day

y = $0.18x + $27.60 - Blue Car Rental

y = 0.18 * 400 + 27.60

y = $99.60

y = $0.14x + $40.40 - Red Car Rental

y = 0.14 * 400 + 40.40

y = $96.40

comparing the both shows that Red Car Rental has a lower cost

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The pmf of X = the number of children in the family is

1/4 + 3/8 + 3/8 =1

It is given that,

a family decides to have children until it has three children of the same gender i.e. BBB or GGG.

X can take the values {3,4,5}

and we know that

P(X = 3) + P(X = 4) + P(X = 5) = 1

We will find each term one by one,

P(X = 3) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*2

             = [tex]\frac{1}{8}[/tex] * 2

P(X = 3) = [tex]\frac{1}{4}[/tex]

P(X = 4) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*6

             = ([tex]\frac{1}{16}[/tex]) * 6

P(X =4)  = [tex]\frac{3}{8}[/tex]

P(X = 5) = ([tex]\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}[/tex])*12

             = ([tex]\frac{1}{32}[/tex])* 12

P(X = 5) = [tex]\frac{3}{8}[/tex]

Therefore the pmf equation implies,

[tex]\frac{1}{4} + \frac{3}{8} + \frac{3}{8}[/tex] = [tex]\frac{32}{32}[/tex] = 1.

The probability mass function (pmf) is also called a probbility function or frequency which characterizes the distribution of a discrete random variable. It is a function over the sample space of a discrete random variable X which gives the probability that X is equal to a certain value.

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The number of chocolate bars which Mr Banda bought is equal to 50 chocolate bars.

In order to determine the number of chocolate bars and the number of hot chips that Carter bought, we would have to write a system of linear equation and assign variables to the number of chocolate bars and number of candies respectively, and then translate the word problem into algebraic equation as follows:

Since chocolate bars cost $1.1 each and hot chips 0.40 each and Mr Banda spent $87 on buying 130 chocolate bars and hot chips, we would have the following system of linear equation:

1.1x + 0.40y = 87    ......equation 1

x + y = 130              ......equation 2

From equation 2, we have:

y = 130 - x        ......equation 3

Substituting equation 3 into equation 1, we have:

1.1x + 0.40(130 - x) = 87

1.1x + 52 - 0.4x = 87

0.7x + 52 = 87

0.7x = 87 - 52

0.7x = 35

x = 35/0.7

x = 50 chocolate bars.

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A regression model can only predict values that are lower or higher than the actual value. As a result, the only way to determine the model's accuracy is through residuals.

What is regression ?

Regression is a statistical technique used in the fields of finance, investing, and other disciplines that aims to establish the nature and strength of the relationship between a single dependent variable (often represented by Y) and a number of independent variables (known as independent variables).

The most popular variation of this method is linear regression, which is also known as simple regression or ordinary least squares (OLS).

Based on a line of best fit, linear regression determines the linear relationship between two variables.

The slope of a straight line used to represent linear regression thus indicates how changing one variable affects changing another..

In a linear regression connection, the value of one variable when the value of the other is zero is represented by the y-intercept. Regression without linearity.

Hence, A regression model can only predict values that are lower or higher than the actual value.

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The probability that an Endure All battery selected at random will last more than 610 hours is 0.1056.

A probability distribution that is symmetric around the mean is the normal distribution, sometimes referred . Eventsare approximately normal happen more often than information that are far the mean.

The normal distribution appears as a "bell curve" on a graph.

A probability bell curve is more properly described as the normal distribution.

In a normal distribution, the mean is 0 and the standard deviation is 1. The kurtosis is 3, and there is no skew.All  normal distributions are symmetrical.

Natural occurrences frequently resemble the usual distribution.

However, in finance, most price distributions are not fully normal.

How to solve?

Given that μ = 500, σ = 40, for x > 550

z = 550-500/40

z = 1.25

From the normal distribution table, P(z > 1.25) = 1 - P(z < 1.25) = 1 - 0.8944 = 0.1056

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By using the concept of expectation, it can be calculated that

a) Probability distribution of Y

P(Y = 0)  = 0.63

P(Y = 1) = 0.25

P(Y = 2) = 0.12

b) Mean of Y = 0.49

  Standard deviation of Y = 0.6999

c) Mean of selling price = $4.4

   Standard deviation of selling price = $0.99

What is expectation?

At first it is important to know about probability of an event.

Probability gives us the information about how likely an event is going to occur

Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.

Probability of any event is greater than or equal to zero and less than or equal to 1.

Probability of sure event is 1 and probability of unsure event is 0.

Suppose x is a random variable with the probability function f(x). suppose

[tex]x_1, x_2, ...., x_n[/tex] are the values corrosponding to the actual occurance of the event and [tex]p_1, p_2,...., p_n[/tex] be the corrosponding probabilities.

Expectation is given by the formula

[tex]E(x) = p_1x_1 + p_2x_2 + ... + p_nx_n[/tex]

a)If a fat quarter has more than 2 defects, it cannot be sold and is discarded.

The table for the probability distribution of X is given

P(X [tex]\leq[/tex] 2) = 0.58 + 0.23 + 0.11 = 0.92

For probability distribution of Y

P(Y = 0) = P(X= 0 | X [tex]\leq[/tex] 2) = [tex]\frac{0.58}{0.92}[/tex] = 0.63

P(Y = 1) = P(X= 1 | X [tex]\leq[/tex] 2) = [tex]\frac{0.23}{0.92}[/tex] = 0.25

P(Y = 2) = P(X= 2 | X [tex]\leq[/tex] 2) = [tex]\frac{0.11}{0.92}[/tex] = 0.12

Mean of Y = 0 [tex]\times[/tex] 0.63 + 1  [tex]\times[/tex] 0,25 + 2  [tex]\times[/tex] 0.12 = 0.49

E([tex]Y^2[/tex]) = 0 [tex]\times[/tex] 0.63 + 1  [tex]\times[/tex] 0.25 + 4  [tex]\times[/tex] 0.12 = 0.73

Variance = [tex]0.73 - 0.49^2[/tex] = 0.4899

Standard deviation = [tex]\sqrt{0.4899} = 0.6999[/tex]

b) Here, E(G) = 0.40, V(G) = [tex]0.66^2[/tex]

The fat quarters sell for $5.00 each but are discounted by $1.50 for each defect found.

Selling price = 5 -1.5G

Mean of selling price = 5 - 1.5[tex]\times[/tex] 0.4 = 5 - 0.6 = $4.4

Variance of selling price = [tex]5 - 1.5^2 \times 0.66^2\\[/tex] = $0.9801

Standard deviation of selling price = [tex]\sqrt{0.9801}[/tex] = $0.99

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Complete Question

Company f sells fabrics known as fat quarters, which are rectangles of fabric created by cutting a yard of fabric into four pieces. Occasionally the manufacturing process results in a fabric defect. Let the random variable x represent the number of defects on a fat quarter created by company f. The following table shows the probability distribution of x.

x                   0            1            2     3         4 or more

probability 0.58 0.23 0.11 0.05 0.03

If a fat quarter has more than 2 defects, it cannot be sold and is discarded. Let the random variable Y represent the number of defects on a fat quarter that can be sold by Company F.

(a) Construct the probability distribution of the random variable Y.

(b) Determine the mean and standard deviation of Y. Show your work.

Company G also sells fat quarters. The mean and standard deviation of the number of defects on a fat quarter that can be sold by Company G are 0.40 and 0.66, respectively. The fat quarters sell for $5.00 each but are discounted by $1.50 for each defect found.

(c) What are the mean and standard deviation of the selling price for the fat quarters sold by Company

To determine the number of seconds that the hare would finish the race, we solve g(t) = 0, to calculate the value of t.

Linear equation

Linear equation is in the form:

y = mx + b

where y, x are variables, m is the slope of the equation (rate of change) and b is the y intercept (initial value of b).

Given that g (t) represents the hare's distance from the finish line (in meters) t seconds after the start of the race.

Hence the answer is, To determine the number of seconds that the hare would finish the race, we solve g(t) = 0, to calculate the value of t.

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The 1.5 standard deviations below the mean implies that the Z-score value will be equal to -1.5.

The test score 70 is 1.5 standard deviations below the mean .

We have given that,

The Score standardized test are Normally distributed,

Mean ,μ = 100

Standard deviations, s.d = 20

we have to calculate test score is 1.5 standard deviations below mean.

so, Z- score = -1.5

This is the z-score formula or z-score equation.

Z = (x − μ )/σ

Where, Z--> z-score

x---> the value in question

μ--> mean of the distribution

σ --> standard deviation of the distribution

plugging all known the value in formula

-1.5 = (x- 100)/20

=> -1.5 × 20 = x - 100

=> x = -30 + 100 = 70

So, 70 is the test score that is 1.5 standard deviations from the mean.

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