A random sample of size n=200 is to be taken from a uniform population with α=24 and β=48. Based on the central limit theorem, what is the probability that the mean of the sample will be less than 35?

The probability of you and your best friend being paired up is 2/11.

To calculate the probability of being paired up with your best friend, we need to consider the total number of possible pairings and the number of favorable outcomes where you and your best friend are paired up.

First, let's find the total number of possible pairings. Since there are 11 players, we can pair them up in (11 choose 2) ways, which is calculated as:

C(11, 2) = 11! / (2!(11-2)!) = 55

So, there are 55 possible pairings in total.

Now, let's determine the number of favorable outcomes where you and your best friend are paired up. Since your best friend can be paired with any of the remaining 10 players (excluding yourself), there are 10 favorable outcomes.

Therefore, the probability of being paired up with your best friend is given by:

Probability = Favorable outcomes / Total outcomes

Probability = 10 / 55

Probability = 2 / 11

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To write and solve a story problem with 6 divided by 6, we need to come up with a situation in which 6 is divided equally among 6 parts. For example:

There are 6 pieces of candy to be divided equally among 6 children. Solution: To solve this problem, we can simply divide the total number of candies (6) by the number of children (6):6 ÷ 6 = 1Therefore, each child will receive 1 piece of candy. Another way to solve this problem is to use multiplication. Since division is the inverse of multiplication, we can think of this problem as:6 ÷ 6 = x can be rewritten as 6 = x × 6, where x is the number of candies each child receives. Then we can solve for x by dividing both sides by 6:x = 6 ÷ 6x = 1Therefore, each child will receive 1 piece of candy.

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To determine whether the given series converges or diverges, we'll analyze it using the terms you provided. The series is:

Σ [1/(n^5)(n^6)] for n = 1 to ∞

First, simplify the expression:

1/(n^5)(n^6) = 1/n^(5+6) = 1/n^11

Now, we have the series:

Σ [1/n^11] for n = 1 to ∞

This is a p-series with p = 11. A p-series converges if p > 1. In this case, p = 11 > 1, so the series converges. To find the sum of the convergent series, we use the formula for the sum of a convergent p-series:

Sum = 1/(p-1)

In this case, p = 11:

Sum = 1/(11-1) = 1/10

So, the series converges, and the sum is 1/10.

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The hydrated Calcium Chloride compound is CACl₂.0.2998H₂O

To determine the formula of the hydrated Calcium Chloride compound,  to calculate the number of water molecules present in the hydrated salt.

First to calculate the mass of water lost during the heating process. This can be done by subtracting the mass of the anhydrous salt from the mass of the hydrated salt.

Mass of water lost = Mass of hydrated salt - Mass of anhydrous salt

= 5.4769 g - 2.7745 g

= 2.7024 g

To convert the mass of water lost to moles. the molar mass of water, which is approximately 18.015 g/mol.

Number of moles of water lost = Mass of water lost / Molar mass of water

= 2.7024 g / 18.015 g/mol CACl₂x²

= 0.1499 mol

Calcium Chloride (CACl₂x²) has a molar mass of approximately 110.98 g/mol.

Since calcium chloride has a 1:2 ratio with water in the hydrated form,  the following equation:

0.1499 mol water / 1 mol CACl₂ = X mol water / 2 mol CACl₂x²

0.1499 mol water = X mol water / 2

X mol water ≈ 0.1499 mol water × 2

X mol water ≈ 0.2998 mol water

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Therefore, the determinant of matrix B is 13. The determinant of A is the product of the pivots in the upper triangular matrix U is 6/5.

(a) Using the cofactor formula, we have:

|B| = 3 * |3 0 7|

- 2 * |2 0 1|

+ 0 * |-2 0 1|

= 3 * (3*1 - 0*5) - 2 * (2*1 - 0*(-2)) + 0 * (-2*0 - 0*1)

= 9 + 4 + 0

= 13

(b) To find the PA=LU factorization of matrix A, we perform Gaussian elimination with partial pivoting. The first step is to interchange the first and second rows to get a nonzero pivot in the (1,1) position:

| 3 1 2 | | 3 1 2 |

| 0 2 5 | -> | 0 -5 -1 |

| 3 5 5 | | 0 0 5 |

Next, we perform row operations to get zeros below the pivot in the second row:

| 3 1 2 | | 3 1 2 |

| 0 -5 -1 | -> | 0 -5 -1 |

| 0 4 3 | | 0 19 11 |

Finally, we divide the second row by -5 and subtract 3 times the second row from the third row to get zeros below the (3,2) position:

| 3 1 2 | | 3 1 2 |

| 0 1 1/5| -> | 0 1 1/5|

| 0 0 2/5| | 0 0 32/5|

Therefore, we have:

A = LU = | 3 1 2 | | 1 0 0 | | 3 1 2 |

| 0 1 1/5 | * | 0 1 0 | = | 0 1 1/5|

| 0 0 2/5 | | 0 0 32/5| | 0 0 2/5 |

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The answer is:

(a) 47.

We can use the following formula to find the sample size needed:

n = [(t-value * standard deviation) / margin of error]²

where the margin of error is the maximum amount we allow the estimate to be off by, and the t-value is based on the desired level of confidence and the degrees of freedom (n-1).

In this case, we want the margin of error to be $100 and we want to have a 95% level of confidence. Using a z-value of 1.96 for a 95% confidence interval, we can find the corresponding t-value with 39 degrees of freedom (n-1) using a t-table or calculator.

t-value = 2.0227

Substituting the values into the formula, we get:

n = [(2.0227 * 350) / 100]²

n = 47.22

we get a required sample size of 47 (option a).

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The equation that results from solving the formula of the circumference for b is given as b² = [27v / (4π) - 26 / 4]²(1 - e²). The circumference of an ellipse is approximated by C = 27v, where 2a and 26 are the lengths of the axes of the ellipse.

We have to find the equation that results from solving the circumference formula b. Now, the formula for the circumference of an ellipse is given by;

C = π [2a + 2b(1 - e²)½], Where a and b are the semi-major and semi-minor axes of the ellipse, respectively, and e is the ellipse's eccentricity. As given, C = 27v Since 2a = 26, a = 13

Putting this value of 2a in the formula for circumference;

27v = π [2a + 2b(1 - e²)½]

27v = π [2 × 13 + 2b(1 - e²)½]

27v = π [26 + 2b(1 - e²)½]

Now, dividing by π into both sides;

27v / π = 26 + 2b(1 - e²)½

Subtracting 26 from both sides;

27v / π - 26 = 2b(1 - e²)½

Squaring both sides, we get;

[27v / π - 26]² = 4b²(1 - e²)

Multiplying by [1 - e²] on both sides;

[27v / π - 26]²(1 - e²) = 4b²

Multiplying by ¼ on both sides;

[27v / (4π) - 26 / 4]²(1 - e²) = b²

So, the equation that results from solving the formula of the circumference for b is;

b² = [27v / (4π) - 26 / 4]²(1 - e²). Therefore, the correct option is (A) b² = [27v / (4π) - 26 / 4]²(1 - e²).

Thus, the equation that results from solving the formula of the circumference for b is given as :

b² = [27v / (4π) - 26 / 4]²(1 - e²).

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The two measures of central tendency that best describe the typical number of minutes Addison spent reading each day are the mean and the median.

To determine the two measures of central tendency that best describe the typical number of minutes Addison spent reading each day, we need to calculate the mean, median, and mode of the data, and consider their respective strengths and weaknesses as measures of central tendency.

The mean is calculated by adding up all the numbers and dividing by the total number of numbers.

The median is the middle value when the data is arranged in order from smallest to largest.

The mode is the value that appears most frequently in the data

The range is the difference between the largest and smallest values in the data

The two measures of central tendency that best describe the typical number of minutes Addison spent reading each day are the mean and the median.

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The list below shows the number of minutes Addison spent reading on each of six days. 90, 60, 89, 94, 60, 93

Which two measures of these data best describe the typical number of minutes Addison spent reading each day?

A. Mean and mode

B. Mean and median

C. Mode and range

D. Median and range

Please explain and justify your answe

The diameter of the tree trunk is 6.5 feet (to the nearest half-foot).

Given: Length of the rope wrapped around the tree trunk = 21 feet 8 inches.How the group of students can use this length to estimate the diameter of the tree trunk to the nearest half-foot is described below.Using this length, the students can estimate the diameter of the tree trunk by finding the circumference of the tree trunk. For this, they will use the formula of the circumference of a circle i.e.,Circumference of the circle = 2πr,where π (pi) = 22/7 (a mathematical constant) and r is the radius of the circle.In this question, we are given the length of the rope wrapped around the tree trunk. We know that when the rope is wrapped around the tree trunk, it will go around the circle formed by the tree trunk. So, the length of the rope will be equal to the circumference of the circle (formed by the tree trunk).

So, the formula can be modified asCircumference of the circle = Length of the rope around the tree trunkHence, from the given length of rope (21 feet 8 inches), we can calculate the circumference of the circle formed by the tree trunk as follows:21 feet and 8 inches = 21 + (8/12) feet= 21.67 feetCircumference of the circle = Length of the rope around the tree trunk= 21.67 feetTherefore,2πr = 21.67 feet⇒ r = (21.67 / 2π) feet= (21.67 / (2 x 22/7)) feet= (21.67 x 7 / 44) feet= 3.45 feetTherefore, the radius of the circle (formed by the tree trunk) is 3.45 feet. Now, we know that diameter is equal to two times the radius of the circle.Diameter of the circle = 2 x radius= 2 x 3.45 feet= 6.9 feet= 6.5 feet (nearest half-foot)Therefore, the diameter of the tree trunk is 6.5 feet (to the nearest half-foot).

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The value of the definite integral ∫3−3(3x3−2x2 x 1) dx is 0.

The given question asks us to find the definite integral of a polynomial function of degree 3 over the interval [-3, 3]. When we integrate a polynomial function, we get a polynomial function of one degree higher. In this case, we get a degree 4 polynomial function, which we can evaluate at the upper and lower limits of the interval and take the difference to get the definite integral.

After simplifying the expression, we get the definite integral to be 0. This result suggests that the area under the curve of the given polynomial function over the interval [-3, 3] is zero. Definite integrals have many applications in calculus, physics, engineering, and economics, and understanding their properties is crucial in these fields.

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The profit function is 9x - 7000 and the revenue function is 12x.

Given that the cost function for a company to produce a lunch box is c(x)= 3x+7000 where x is the number of lunch boxes and the company sells the lunch boxes for $12 each.

To write a profit function, the revenue function is required to calculate the profit earned by the company.

The revenue function is given as:

Revenue = Selling Price × Quantity Sold

Price is $12 for each lunch box, therefore

Revenue = $12 × Quantity sold

Quantity sold is represented as x, therefore,

Revenue = 12x

The profit function is given as:

Profit = Revenue - Cost

The cost function is given as c(x)= 3x+7000

Therefore,

Profit = 12x - (3x + 7000)

Profit = 9x - 7000

Hence, the profit function is 9x - 7000 and the revenue function is 12x.

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The evaluated value of the given expression when j = 0.5 and k = 0.25 is 6.5.

The given expression is 7j+5−8k7j+5−8k7, j, plus, 5, minus, 8, k.

We need to evaluate the given expression when j=0.5j=0.5j, equals, 0, point, 5 and k=0.25k=0.25k, equals, 0, point, 25.

Now we substitute the values of j and k in the given expression.

7(0.5)+5−8(0.25)7(0.5)+5−8(0.25)7, times, 0, point, 5, plus, 5, minus, 8, times, 0, point, 25=3.5+5-2=6.5

The value of the expression when j=0.5j=0.5j, equals, 0, point, 5 and k=0.25k=0.25k, equals, 0, point, 25 is 6.5, which is the final answer.

Therefore, the evaluated value of the given expression when j = 0.5 and k = 0.25 is 6.5.

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The smallest number of pigeons in the pigeonhole that contains the most number of pigeons is 2341.

To determine the smallest number of pigeons in the pigeonhole that contains the most number of pigeons, we can use the pigeonhole principle.

The pigeonhole principle states that if you distribute more than m objects into m pigeonholes, then at least one pigeonhole must contain more than one object.

In this case, we have 32761 pigeons and 14 pigeonholes. To minimize the number of pigeons in the pigeonhole that contains the most, we want to distribute the pigeons as evenly as possible.

Dividing 32761 by 14, we get:

32761 / 14 = 2340 remainder 1

This means we can evenly distribute 2340 pigeons to each of the 14 pigeonholes, leaving 1 pigeon remaining.

To minimize the number of pigeons in the pigeonhole that contains the most, we distribute the remaining 1 pigeon to one of the pigeonholes, resulting in the exact integer is 2341.

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Based on the inequality y > 2x + 5, Lisa is allowed to attend schools D, E, and F.

Part A: To create a system of inequalities that only contain points B and C in the overlapping shaded regions, we need to identify the boundaries of those regions and set up appropriate inequalities.

Looking at the graph, we can see that the shaded region where points B and C overlap is bounded by two lines: one vertical line passing through x = 3, and one horizontal line passing through y = -3.

The vertical line passing through x = 3 divides the coordinate plane into two regions: one to the left of x = 3 and one to the right. To include point B in the overlapping shaded region, we need to consider the left side of the line, so we set up the inequality x < 3.

The horizontal line passing through y = -3 also divides the coordinate plane into two regions: one above y = -3 and one below. To include point C in the overlapping shaded region, we need to consider the region below the line, so we set up the inequality y < -3.

Therefore, the system of inequalities that only contains points B and C in the overlapping shaded region is:

x < 3

y < -3

To graph these inequalities, you would draw a dotted vertical line at x = 3 and shade the region to the left of the line. Then, draw a dotted horizontal line at y = -3 and shade the region below the line. The overlapping shaded region represents the area where both inequalities are satisfied, and that's where points B and C lie.

Part B: To verify that points B and C are solutions to the system of inequalities created in Part A, we substitute the coordinates of each point into the inequalities and check if the resulting statements are true.

For point B (3, 1):

x < 3 becomes 3 < 3, which is false.

y < -3 becomes 1 < -3, which is false.

Since both inequalities are false when substituting point B, it means that point B is not a solution to the system of inequalities. Therefore, it does not lie in the overlapping shaded region.

For point C (3, -3):

x < 3 becomes 3 < 3, which is false.

y < -3 becomes -3 < -3, which is also false.

Similar to point B, both inequalities are false when substituting point C. Hence, point C is not a solution to the system of inequalities and does not lie in the overlapping shaded region.

Part C: To identify the schools Lisa is allowed to attend based on her designated zone defined by y > 2x + 5, we need to check which schools satisfy this inequality.

Let's evaluate the inequality for each school's coordinates:

Point A (1, 3):

3 > 2(1) + 5

3 > 2 + 5

3 > 7

The inequality is false, so Lisa cannot attend school A.

Point B (3, 1):

1 > 2(3) + 5

1 > 6 + 5

1 > 11

The inequality is false, so Lisa cannot attend school B.

Point C (3, -3):

-3 > 2(3) + 5

-3 > 6 + 5

-3 > 11

The inequality is false, so Lisa cannot attend school C.

Point D (-4, 2):

2 > 2(-4) + 5

2 > -8 + 5

2 > -3

The inequality is true, so Lisa can attend school D.

Point E (-1, 5):

5 > 2(-1) + 5

5 > -2 + 5

5 > 3

The inequality is true,

so Lisa can attend school E.

Point F (-3, -3):

-3 > 2(-3) + 5

-3 > -6 + 5

-3 > -1

The inequality is true, so Lisa can attend school F.

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The ordered pair is (3, 12)Hence, the two ordered pairs are (2, 8) and (3, 12).

Given that Rachel the Eagle flies at a rate of 1 mile per hour and is modeled by the equation y = x. She increases her rate by 3 miles per hour and we are to plot two ordered pairs showing the distances she will fly at 2 hours and 3 hours, respectively, at her new rate.

We know that Rachel’s new rate is 1 + 3 = 4 miles per hour.We are to find the distance she will fly at 2 hours and 3 hours at her new rate.Using the formula for distance, d = rt (distance = rate x time)We have the following;For 2 hours,d = rt= 4 x 2 = 8 miles∴ Ordered pair = (2, 8)For 3 hours,d = rt= 4 x 3 = 12 miles

∴ Ordered pair = (3, 12)Therefore, the two ordered pairs are (2, 8) and (3, 12).Hence, our solution is complete. We can present this solution in about 150 words as follows;Rachel the Eagle is known to fly at a rate of 1 mile per hour. This is modeled by the equation y = x.

If she increases her rate by 3 miles per hour, we can calculate the new rate as follows:New rate = 1 + 3 = 4 miles per hour.

To determine the distance Rachel will fly at 2 hours and 3 hours, we can use the formula for distance, d = rt. By substitution of the new rate and given time, we obtain the following:For 2 hours,d = rt= 4 x 2 = 8 miles

Therefore, the ordered pair is (2, 8)For 3 hours,d = rt= 4 x 3 = 12 milesTherefore, the ordered pair is (3, 12)Hence, the two ordered pairs are (2, 8) and (3, 12).

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To determine the fraction that Lee writes on the first mark to the right of 17, we need to understand the numbering pattern and the position of the marks.

If Lee marks sixths on the number line, it means that the interval between each mark is 1/6.

Starting from 0, the first mark to the right of 17 would be located at 18.

To find the fraction written on this mark, we can calculate the difference between 18 and 17 and express it as a fraction of the interval between each mark (1/6).

18 - 17 = 1

Therefore, the fraction that Lee writes on the first mark to the right of 17 is 1/6.

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5.880 can be expressed as the ratio of integers 127/25.

To express 5.880 as a ratio of integers, we can write it as follows:

5.880 = 5 + 0.880

To convert the decimal part (0.880) into a fraction, we can write it as a repeating decimal by observing the repeating pattern:

0.880880880...

The repeating part is "880", which has three digits.

Now, we can express 5.880 as a ratio of integers:

5.880 = 5 + 0.880 = 5 + 880/1000

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 10:

5.880 = 5 + 880/1000 = 5 + (880 ÷ 10)/(1000 ÷ 10) = 5 + 88/100

Finally, we can simplify the fraction further:

5.880 = 5 + 88/100 = 5 + 22/25

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The point that splits the segment AB into a ratio of 2:5 is (-6, 3).

To find the point that splits segment AB into a ratio of 2:5, we can use the concept of a weighted average.

The x-coordinate of the point is found by taking 2 parts of B's x-coordinate and 5 parts of A's x-coordinate and summing them, then dividing by the total parts (2+5=7).

Similarly, the y-coordinate is found by taking 2 parts of B's y-coordinate and 5 parts of A's y-coordinate, then dividing by the total parts.

For point A (-10, 1) and B (4, 8), the calculations would be as follows:

x-coordinate: (2 * 4 + 5 * -10) / 7 = (8 + -50) / 7 = -42 / 7 = -6

y-coordinate: (2 * 8 + 5 * 1) / 7 = (16 + 5) / 7 = 21 / 7 = 3

Among the given points, only (-6, 3) matches the calculated coordinates. Therefore, (-6, 3) is the point that splits segment AB into a ratio of 2:5.

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a.  chi-square value is greater than the critical chi-square value for the given level of significance and degrees of freedom. b. there are 2 degrees of freedom.

(a) The decision rule for this hypothesis test is to reject the null hypothesis if the calculated chi-square value is greater than the critical chi-square value for the given level of significance and degrees of freedom.

(b) To determine the degrees of freedom, we use the formula df = (r - 1)(c - 1), where r is the number of rows and c is the number of columns in the contingency table. In this case, there are 3 rows and 1 column, so df = (3 - 1)(1 - 1) = 2. Therefore, there are 2 degrees of freedom.

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The answer is: it depends. If {c} is equal to {a} or {b}, then it is necessarily in the sigma-algebra, since {a} and {b} are already in the sigma-algebra and sigma algebras are closed under subsets.

In order to answer this question, we need to understand what a sigma-algebra is and what properties it has.

A sigma algebra is a collection of subsets of a set that has three properties:
1. It contains the empty set.
2. It is closed under complementation (i.e., if A is in the sigma-algebra, then A^c is also in the sigma-algebra).
3. It is closed under countable unions (i.e., if A1, A2, A3, ... are in the sigma-algebra, then their union is also in the sigma-algebra).

The answer is: it depends. If {c} is equal to {a} or {b}, then it is necessarily in the sigma-algebra, since {a} and {b} are already in the sigma-algebra and sigma algebras are closed under subsets.

However, if {c} is not equal to {a} or {b}, then we cannot say for sure whether it is in the sigma-algebra or not.

To see why, consider the following example. Let X = {a, b, c, d} and let the sigma-algebra be the power set of X (i.e., the collection of all subsets of X).

Then {a} and {b} are in the sigma-algebra, but {c} is not. Therefore, we cannot say that {c} is necessarily in the sigma-algebra.

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Given: An angle of measure 115 degrees We know that: The supplement of an angle is equal to 180 degrees minus the angle, and the complement of an angle is equal to 90 degrees minus the angle

Now, we need to find the complement of the supplement of an angle measuring 115 degrees.So, let's first find the supplement of the given angle:

Supplement of 115 degrees = 180 - 115= 65 degrees

Now, we need to find the complement of the above angle which is:

Complement of 65 degrees = 90 - 65= 25 degrees Therefore, the complement of the supplement of an angle measuring 115º is 25 degrees.

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the correct answer is (c) The ratio of circle A is equal to the ratio of circle B, as the ratio of the circumference to the diameter is the same for both circles.

In a circle, the ratio of the circumference to the diameter is constant and is denoted by the mathematical constant π (pi), which is approximately equal to 3.14159. This means that for any circle, regardless of its size or radius, the ratio of the circumference to the diameter will always be the same.

Since circle A has twice the radius of circle B, it means that the circumference of circle A will be twice the circumference of circle B. Similarly, the diameter of circle A will also be twice the diameter of circle B. Therefore, when we calculate the ratio of the circumference to the diameter for both circles, we will obtain the same value, which is π.

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When government spending increases by $5 billion and the MPC = .8, in the first round of the spending multiplier process, spending increases by $20 billion.


The spending multiplier is the amount by which GDP will increase for each unit increase in government spending. It is calculated as 1/(1-MPC), where MPC is the marginal propensity to consume. In this case, MPC = .8, so the spending multiplier is 1/(1-.8) = 5.

Therefore, when government spending increases by $5 billion, the total increase in spending in the economy will be $5 billion multiplied by the spending multiplier of 5, which equals $25 billion. However, the initial increase in spending is only $5 billion, hence the increase in the first round of the spending multiplier process is $20 billion.

In summary, when government spending increases by $5 billion and the MPC = .8, the initial increase in spending is $5 billion, but the total increase in the first round of the spending multiplier process is $20 billion.

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1.) There are 122 Sour Patch Kids in the bag, 2.) the relative frequency of selecting a Green Sour Patch Kid is approximately 0.3607 (3.)the approximate probability of selecting a Green Sour Patch Kid from a bag of 500 pieces is 180.35.

1.)To determine the total number of outcomes, we sum up the frequencies of all the colors:

Total number of outcomes = Frequency of Red + Frequency of Yellow + Frequency of Green + Frequency of Blue

= 26 + 15 + 44 + 37

= 122

So, there are 122 Sour Patch Kids in the bag.

2.)To determine the relative frequency of selecting a Green Sour Patch Kid, we divide the frequency of Green by the total number of outcomes:

Relative Frequency = Frequency of Green / Total number of outcomes

= 44 / 122

≈ 0.3607 (rounded to four decimal places)

So, the relative frequency of selecting a Green Sour Patch Kid is approximately 0.3607.

3.)Using the relative frequency determined in #2, we can approximate the probability of selecting a Green Sour Patch Kid from a bag of 500 pieces. Since the relative frequency represents the proportion of Green Sour Patch Kids in the bag, we can multiply it by the total number of pieces in the bag:

Probability = Relative Frequency * Total number of pieces

= 0.3607 * 500

= 180.35

Therefore, the approximate probability of selecting a Green Sour Patch Kid from a bag of 500 pieces is 180.35 out of 500, or approximately 0.3617 (or 36.17%) when expressed as a decimal or percentage, respectively.

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Answer:

Step-by-step explanation:

sqrt{-2x^{2}-2x+11 }=\sqrt{-x^{2} +3}

Square both sides:

-2x^2 - 2x + 11 = -x^2 + 3

0 = x^2 + 2x - 8

( x + 4)(x - 2) = 0

x = -4, 2.

As the original equation contains square roots some of these roots might be extraneous.

Checking:

x = -4

sqrt(-2(-4)^2 - 2(-4) + 11 =  sqrt(-13)

sqrt (-(-4)^2 + 3) = sqrt(-13)

x = 2:

sqrt(-2(4) - 2(2) + 11) = sqrt(-8 - 4 + 11) = sqrt(-1)

sqrt(-(2)^2 + 3) = sqrt(-1)

So both are roots

The probability that exactly k tosses are required such that to get exactly two heads is given by P(k) =   [tex]\frac{1}{2}^{k}[/tex] for k = 2, 3, 4, ...

The sample space S consists of all possible sequences of tosses of a fair coin until exactly two heads are obtained.

Represent a head with H and a tail with T.

For example, one possible sequence in S is,

HTTTHH

This represents 6 tosses, with the first two being a head and a tail, the next three being tails, and the final two being heads.

Another example in S is.

HH

This represents 2 tosses, with both being heads.

The sample space S is infinite, since we could continue tossing the coin indefinitely until we get exactly two heads.

To find the probability that exactly k tosses are required, use the following reasoning.

For exactly k tosses to be required,

Need to get exactly one head in the first k-1 tosses, followed by a head in the kth toss.

The probability of getting exactly one head in the first k-1 tosses is [tex]\frac{1}{2} ^{k-1}[/tex].

Since each toss is independent and has a probability of 1/2 of resulting in a head.

The probability of getting a head on the kth toss is also 1/2.

P(k) =  [tex]\frac{1}{2} ^{k-1}[/tex]x (1/2)

       = [tex]\frac{1}{2}^{k}[/tex]

for k = 2, 3, 4, ...

This is a geometric probability distribution with parameter p = 1/2.

Therefore, the probability that exactly k tosses are required to obtain exactly two heads is P(k) =   [tex]\frac{1}{2}^{k}[/tex] for k = 2, 3, 4, ...

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The formal notation for the distribution of the continuous random variable Z in this case is Z ~ N(15, 49).

In formal notation, the distribution of the continuous random variable Z can be written as Z ~ N(μ, σ^2), where N represents the normal distribution, μ represents the mean, and σ^2 represents the variance.

Given that Z has a mean of 15 and a standard deviation of 7, we know that μ = 15 and σ = 7. The variance can be calculated as σ^2 = 49.

Thus, the formal notation for the distribution of the continuous random variable Z in this case is Z ~ N(15, 49).

This means that the values of Z are normally distributed around the mean of 15, with the spread of the distribution determined by the standard deviation of 7. This notation is commonly used in probability theory and statistics to represent the properties of a given random variable.

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The distribution of the continuous random variable z with a mean of 15 and a standard deviation of 7 can be written as:
z ~ N(15, 49)
where N represents the normal distribution, 15 represents the mean, and 49 represents the variance (which is equal to the square of the standard deviation).
In this case, the mean (µ) is 15 and the standard deviation (σ) is 7. Therefore, the formal notation for this distribution is:

z ∼ N(µ, σ²)

where N represents a normal distribution. Plugging in the given values, we get:

z ∼ N(15, 7²)

So the distribution can be written as:

z ∼ N(15, 49)

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Janelle buys 1 pound of raspberries, she can buy a maximum of 4.625 pounds of blackberries.

Let's assume Janelle wants to buy blackberries and raspberries and has a maximum budget of $40. We need to find the maximum amount of fruit she can purchase while staying within her budget.

Let's denote the pounds of blackberries as "b" and the pounds of raspberries as "r." The cost of blackberries is $8 per pound, and the cost of raspberries is $9 per pound.

Based on this information, we can set up the following equations:

8b + 9r ≤ 40 (Total cost of blackberries and raspberries should be less than or equal to $40)

b, r ≥ 0 (Pounds of blackberries and raspberries should be non-negative)

To find the maximum amount of fruit Janelle can buy, we need to find the values of b and r that satisfy the given conditions.

There are various methods to solve this problem, such as graphing, substitution, or elimination. Let's use the substitution method:

We can rearrange the first equation as:

8b ≤ 40 - 9r

b ≤ (40 - 9r)/8

Since b and r should be non-negative, we can consider different values of r and substitute them into the equation to find the corresponding maximum values of b.

For example, if we assume r = 0, the equation becomes:

b ≤ (40 - 9(0))/8

b ≤ 5

So, if Janelle buys 0 pounds of raspberries, she can buy a maximum of 5 pounds of blackberries.

Similarly, for r = 1:

b ≤ (40 - 9(1))/8

b ≤ 4.625

Therefore, if Janelle buys 1 pound of raspberries, she can buy a maximum of 4.625 pounds of blackberries.

By exploring different values of r within the given constraints, we can determine various combinations of blackberries and raspberries that Janelle can purchase while staying within her $40 budget.

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The equations that are true regarding the given triangle are: A) w + x + y = 180; B) y + z = w + x + y; E) w + x = z.

Recall the following facts in order to determine the equations that are true:

Therefore, the following equations would be true:

y + z = 180

w + x + y = 180

Therefore, y + x = w + x + y

w + x = z

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To estimate ln(1.35) to within 0.01 using a Taylor polynomial centered at x=0, we can use the formula for the Taylor series expansion of ln(x+1):

ln(x+1) = x - x^2/2 + x^3/3 - x^4/4 + ...

Plugging in x=0.35, we get:

ln(1.35) = 0.35 - 0.35^2/2 + 0.35^3/3 - 0.35^4/4 + ...

To determine how many terms we need to include to get an estimate within 0.01, we can use the remainder term of the Taylor series expansion, which is given by:

Rn(x) = f^(n+1)(c) * (x-a)^(n+1) / (n+1)!

where f^(n+1)(c) is the (n+1)th derivative of f evaluated at some point c between a and x.

For ln(x+1), the (n+1)th derivative is given by:

f^(n+1)(x) = (-1)^n * n! / (x+1)^(n+1)

Using this formula, we can find an upper bound on the remainder term for n=4 (since we need to include up to the x^4 term in the Taylor series) and x=0.35:

|R4(0.35)| <= 4! * 0.35^5 / 5! = 0.000091125

This means that if we include the x^4 term in our estimate, the error will be no larger than 0.000091125. To ensure that our estimate is within 0.01, we need to include enough terms so that the x^5 term and higher are negligible compared to the error bound. Since the terms are decreasing in magnitude, we can stop adding terms once the next term is smaller than the error bound.

Calculating the terms of the Taylor series up to x^4, we get:

ln(1.35) ≈ 0.35 - 0.35^2/2 + 0.35^3/3 - 0.35^4/4

= 0.3228020833

The next term, 0.35^5/5, is approximately 0.004697917, which is larger than our error bound of 0.000091125. Therefore, we need to include the next term, which is -0.35^6/6, to get a more accurate estimate.

Adding this term, we get:

ln(1.35) ≈ 0.35 - 0.35^2/2 + 0.35^3/3 - 0.35^4/4 - 0.35^6/6

= 0.3229268394

This estimate is within 0.01 of the true value of ln(1.35), so we can be confident that it is accurate.

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