there are 3 seniors and 15 juniors in mrs. gillis’s math class. three students are chosen at random from the class. a. what is the probability that the group consists of a senior and two juniors? b. if the group consists of a senior and two juniors, what is the probability that stephanie, a senior, and jan, a junior, are chosen?

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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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:

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.

Answer: Nancy ran, 14080 yards.

Step-by-step explanation:

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 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 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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The ancient stone pyramids in Egypt, which have a square base with a width of 148 meters and a height of 126 meters, have a volume of 9394 square metres.

Mathematics uses cubic units to quantify volume, including cubic meters, cubic centimeters, cubic millimeters, etc. Typically, liters and gallons are used to calculate liquid volume. The loudness of a sound is measured by its volume. A container's capacity, or how much it can hold, would be determined by its volume. Cubic units established by the International System of Units are frequently used to express volume. Learn how to easily measure volume.

Here,

Side of the pyramid=148 m

Height of pyramid=126 m

Volume,

=1/2*b*h

=1/2*148*126

=9394 m²

The volume of the ancient stone pyramids in Egypt that has a square base of side 148 m and height 126m is 9394 m².

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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 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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There are 55 sophomores taking French II and Chemistry I.

In mathematics, Multiplication operations perform Multiplying values on either side of the operator.

For example 4×2 = 8

According to the given information, the required solution would be as:

First, we need to find the number of sophomores taking French II, Chemistry  I, or both:

⇒ 1/3 × 465

Apply the multiplication operation, and we get

⇒ 155

Of the 155 sophomores, 40 are taking only French II and 60 are taking only Chemistry I:

⇒ 155 - 40 - 60

⇒ 155 - 100

Apply the subtraction operation, and we get

⇒ 55

Thus, there are 55 sophomores taking French II and Chemistry I.

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When the standard deviation is $1200 and the mean is $8000, the test statistic's result will be -1.804.

A statistic employed in statistical hypothesis testing is known as a test statistic. A test statistic, which can be thought of as a numerical summary of a data set that condenses the data into a single number that can be used to conduct the hypothesis test, is how a hypothesis test is commonly expressed. A number derived from a statistical test of a hypothesis is the test statistic. It displays how closely your actual data fit the distribution predicted by the statistical test's null hypothesis.

Here,

The claimed mean=$8250

The actual mean=$8000

The sample size=75

The standard deviation=$1200

The test statistic,

=(Actual mean-claimed mean)/Standard deviation/√(sample size)

=(8000-8250)/1200/√75

=-250/1200*1/√75

=-1.804

The value of test statistic will be -1.804 when the mean is $8000 and standard deviation is $1200.

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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

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 probability that you have seen both the red and green marbles at least once is 0.3104

Given,

1/5 chance of winning a marble of any color in a drawing.

Every draw is independent and has the same probability because the marbles are replaced.

Calculating = 1- the likelihood of not receiving both red and green marbles in any of the four draws yields the same result as P, the probability of receiving both red and green marbles at least once (Q).

The union of the following three examples represents ways to not receive both red and green in all four draws:

(1)

Probability of no red and no green in a draw = 3/5.

Probability of no red and no green in 4 draws,

Q1 = 3/5 * 3/5 * 3/5* 3/5 = 81/54

The probabilities are multiplied as each draw is independent and has the same probabilities.

(2)

The probability of no red and 1 green in 4 draws is given by binomial probability.

= (4 3) (1/5)(3/5)³ = (4 * 3³)/5¹ = 108/5⁴

The probability of no red and 2 green in 4 draws is given by binomial probability.

= (4 2) (1/5)²(3/5)² == (6 * 3²)/54 = 54/5⁴

The probability of no red and 3 green in 4 draws is given by binomial probability.

= (4 3) (1/5)³(3/5) == (4 * 3)/5¹ = 12/5⁴

The probability of no red and 4 green in 4 draws is given by binomial probability.

= (4 4) = 1 * 1/5¹ = 1/5⁴⁴

Q2= sum of all 4 above probabilities.

Q2 (108 +54 + 12+1)/5 = 175/5⁴

(3)

This probability is the same as in Q2. Q3 = 175/5⁴

Therefore the probability of not seeing both red and green at all,

Q = Q1 + Q2 + Q3.

   = 81 +175 + 175/5⁴ = 413/5⁴

Finally, the probability of getting both red and green at least once,

P = 1 - Q

  = 1-413/5⁴

  = (625 431)/625

  = 194/625

  = 0.3104.

The answer is 0.3104.

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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).

The only option that represents a proportional relationship is;

d) The cost of repairing a computer at $54.75 per hour

Proportional relationships are defined as relationships between two variables where their ratios are equivalent.

a) The cost of purchasing boxes of paper is $25.39 per box plus a shipping fee of $3.25. If the number of boxes are x, then;

y = 25.39x + 3.25

This does not represent a proportional relationship as their ratios are not equivalent.

b) The cost of repairing a copy machine is is $35.75 per hour with a service charge of $80.50. The equation is;

y = 35.75x + 80.50

This does not represent a proportional relationship as their ratios are not equivalent.

c) The cost of purchasing printer cartridges at $34.85 per cartridge including delivery. This is not a proportional relationship as the ratio is not a straightforward one based on the delivery.

d) This represents a proportional relationship because the equation is;

y = 54.75x

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The complete options are;

a) The cost of purchasing boxes of paper for $25.39 per box plus a shipping fee of $3.25 per box .

b) The cost of repairing a copy machine at $35.75 per hour with a service charge of $80.50

c) The cost of purchasing printer cartridges at $34.85 per cartridge including delivery

d) The cost of repairing a computer at $54.75 per hour

Answer:

Step-by-step explanation: y = -3x + 6

The required Interest Rate Compounded Daily​ (365 Days/Year) is 7.227%.

Interest on interest, or compound interest, is the adding of interest to the principle sum of a loan or deposit. It's the outcome of reinvesting interest rather than paying it out, so that interest is received on the principal plus previously collected interest in the next quarter.,

[tex]A = P(1+ \dfrac{r}{n})^{nt}[/tex]

where A is the final amount

P is the principal amount

r is the rate of interest

n is the number of times interest is charged in a year

t is the number of years

Given that interest Rate Compounded Daily​ (365 Days/Year) is required for a ​$23,000 Investment to grow To $35,500 In 6 ​Years. Therefore, we can write,

[tex]A = P(1+\dfrac rn)^{n \times t}[/tex]

[tex]35,500 = 23,000(1+\dfrac{r}{365})^{365 \times 6}\\\\\dfrac{35,500}{23,000} = (1+\dfrac{r}{365})^{2190}\\\\1.543478 = (1+\dfrac{r}{365})^{2190}\\\\1.000198 = 1+\dfrac{r}{365}\\\\\dfrac{r}{365} = 1.000198 - 1\\\\\dfrac{r}{365} = 0.000198[/tex]

r = 1.000198 x 365

r  = 0.07227

r = 7.227%

Hence, the required Interest Rate Compounded Daily​ (365 Days/Year) is 7.227%.

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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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Answer:

[tex]-\frac{4}{51}\approx-0.07843[/tex]

Explanation:

[tex]\frac{4}{15}(-\frac{1}{3.4})\\\frac{4}{15}(-\frac{5}{17})\\-\frac{20}{255}\\-\frac{4}{51}\approx-0.07843[/tex]

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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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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Using the z-score a randomly selected Endure All battery has a 10.56% chance of lasting more than 550 hours.

How closely a value relates to the mean of a set of values is measured using a Z-score.

Z-score is calculated using standard deviations from the mean.

When a data point's Z-score is 0, it indicates that its entire score is equal to the mean score.

So, how far the raw score deviates from the mean is determined using the Z score.

These steps are used to determine the Z score:

z = (x - μ)/σ

We know that:  x > 550 and μ = 500, σ = 40

z = 550 - 500 / 40

z = 1.25

Using the normal distribution graph as a guide:

P(z > 1.25) = 1 - P(z < 1.25)

P(z > 1.25) = 1 - 0.8944

P(z > 1.25) = 0.1056

Therefore, using the z-score a randomly selected Endure All battery has a 10.56% chance of lasting more than 550 hours.

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Step-by-step explanation:

It is not a scalene triangle

because the third angle is 69⁰

Explanation: 180-(69+42)

180-111

69

The volume of the vanilla ice cream in the cone is 4.18 cubic inches

The diameter of the cone = 2 inches

The radius of the cone = 2/1

= 1 inches

The height of the cone = 6 inches

The volume of the cone = (1/3) × π × r^2 × h

Substitute the values in the equation

= (1/3)π × 1^2 × 6

= 6.28 cubic inches

The ratio of  vanilla and chocolate ice cream = 2:1

2x + x = 6.28

3x = 6.28

x = 6.28 / 3

x = 2.09

The volume of vanilla ice cream = 2x

= 2×2.09

= 4.18 cubic inches

Therefore, the volume of vanilla ice cream is 4.18 cubic inches

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