PLS HELPPP

For how many integers between 1 and 2023 inclusive is the improper fraction [tex]\frac{n^{2} +4}{n+5}[/tex] not in simplest form.

The Poisson distribution is a discrete probability distribution, the probability of getting a value between 5 and infinity is effectively zero.

This problem can be solved using the Poisson distribution, which is a discrete probability distribution that gives the probability of a given number of events occurring in a fixed interval of time or space, if these events occur independently and at a constant average rate.

Let X be the number of births in a minute, then X follows a Poisson distribution with a mean of λ=3.

(a) The probability of exactly 5 births in a minute is:

P(X=5) = (e^(-λ) * λ^5) / 5! = (e^(-3) * 3^5) / 5! = 0.1008 (rounded to four decimal places)

(b) The probability of at least 5 births in a minute is:

P(X>=5) = 1 - P(X<5) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]

To find the probabilities for X=0,1,2,3,4, we can use the Poisson formula:

P(X=k) = (e^(-λ) * λ^k) / k!

P(X=0) = e^(-3) = 0.0498 (rounded to four decimal places)

P(X=1) = e^(-3) * 3 / 1! = 0.1494 (rounded to four decimal places)

P(X=2) = e^(-3) * 3^2 / 2! = 0.2240 (rounded to four decimal places) P(X=3) = e^(-3) * 3^3 / 3! = 0.2240 (rounded to four decimal places) P(X=4) = e^(-3) * 3^4 / 4! = 0.1680 (rounded to four decimal places)

Therefore,

P(X>=5) = 1 - [0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680] = 1 - 0.8152 = 0.1848 (rounded to four decimal places)

(c) The probability of more than 5 births in a minute is:

P(X>5) = 1 - P(X<=5) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)]

To find the probability for X=5, we can use the Poisson formula as in part (a):

P(X=5) = (e^(-λ) * λ^5) / 5! = (e^(-3) * 3^5) / 5! = 0.1008 (rounded to four decimal places)

Therefore,

P(X>5) = 1 - [0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680 + 0.1008] = 1 - 1.0160 = 0 (rounded to four decimal places)

Since the Poisson distribution is a discrete probability distribution, the probability of getting a value between 5 and infinity is effectively zero.

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The experimental probability for the numbers are:

P(3) = 1/12 (in fraction)

P(3) = 0.08 (in decimal)

P(3) ≈ 8% (in percentage)

P(6) = 3/12  

P(6) = 0.25

P(6) = 25%

P(less than 4) = 6/12

P(less than 4) = 0.5

P(less than 4) = 50%

Probability is the likelihood of a desired event happening.

Experimental probability is a probability that relies mainly on a series of experiments.

Blake rolled a die 12 times and 3 appears once (Given in the table). The experimental probability for 3 will be:

P(3) = 1/12 (in fraction)

P(3) = 0.08 (in decimal)

P(3) ≈ 8% (in percentage)

Blake rolled a die 12 times and 6 appears 3 times (Given in the table). The experimental probability for 6 will be:

P(6) = 3/12  

P(6) = 0.25

P(6) = 25%

The numbers less than are 1, 2 and 3. Six of the twelve outcome in the table are less than 4.

P(less than 4) = 6/12

P(less than 4) = 0.5

P(less than 4) = 50%

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Answer:

The discounted price of the ticket was $75 x .80 = $60

The service fee was $60 x .10 = $6

The total price Jeanette paid for the ticket was $60 + $6 = $66

So the answer is 66

Step-by-step explanation:

The  error interval for the depth of the lake would be:

72.5 m ≤ d < 73.5 m

Since the depth of the lake is given as 73 m, truncated to an integer, the actual depth could be anywhere between 72.5 m and 73.5 m. This is because if the depth is between 72.5 m and 73 m, it would be rounded down to 73 m, and if it is between 73 m and 73.5 m, it would be rounded up to 73 m.

This error interval takes into account the rounding that occurred when the depth was truncated to an integer and gives a range of possible depths that the lake could be within.

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The answers are:

a). The required monthly percentage rate of APR of 19% is 1.59%.

b). The Monthly percentage rate is 2.08%

c). Oscar pays $0.96 more than Felix.

a). Felix received a card with an APR of 19%; it is unknown what his monthly percentage rate was.

APR = 19%

Given that APR is increased monthly throughout the year,

Monthly rate = 19% / number of months in year

Monthly rate = 19% / 12

Monthly rate = 1.59%

Therefore, 1.59% is the minimum monthly percentage rate for an APR of 19%.

b) Oscar received a credit card with a 21% APR.

Monthly percentage rate = 21%/12 = 1.75%

c). We are informed that for a particular month, each of them had an average daily amount of $800.

Thus;

Felix's payments in full = 800 * 1.63%

Felix's payments in full  = $13.04

Amount Oscar issued = 800 * 1.75%

Amount Oscar issued = $14

Variation in Payments = $14 - $13.04 = $0.96

Thus, Oscar pays $0.96 more than Felix.

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Answer:

21 cm

Step-by-step explanation:

We can start by using the formula for the area of a triangle:

Area = (1/2) * base * height

Where the base is b = 24 cm and the height by side a is h = 12 root 3 cm.

Area = (1/2) * 24 cm * 12 root 3 cm = 144 cm^2

We can also use the formula for the area of a triangle in terms of its sides and circumradius:

Area = (abc) / (4R)

Where a, b, and c are the sides of the triangle and R is the radius of its circumcircle.

Plugging in the values we know, we get:

144 cm^2 = (24 cm * a * c) / (4 * 7 root 3 cm)

Simplifying, we get:

9 root 3 cm = a * c / 7

Multiplying both sides by 7 and dividing by a, we get:

c = (63 root 3 cm^2) / a

Substituting the value we know for a, we get:

c = (63 root 3 cm^2) / (12 root 3 cm) = 21 cm

Therefore, the length of side c is 21 cm.

Hopes this helps

The length of side a is approximately 24.1 units.

As per the given information in question:

side b = 20side c = 12angle A = [tex]42^{0}[/tex]

Here we have to find the Length of side a.

The Law of Cosines is given by the equation: c² = a² + b² - 2ab cos(C) where a, b, and c are the sides of a triangle, and C is the angle opposite side c.

To find the length of side a, given side b = 20, side c = 12, and angle A = 42 degrees, we can use the Law of Sines.

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides. In this case, we have:
a / sin(A) = b / sin(B) = c / sin(C)
We are given side b, side c, and angle A, so we can use the formula to find the length of side a.

First, let's find angle B using the given information:
a / sin(42) = 20 / sin(B)
Since we have one angle (A) and the sides opposite to it (a) and another side (b), we can find angle B:
sin(B) = 20 x sin(42) / a
Now we can find angle C using the fact that the sum of angles in a triangle is always 180 degrees:
angle C = 180 - angle A - angle B
angle C = 180 - 42 - angle B
Now we have angle C, and we can use the Law of Sines again to find the length of side a:
a / sin(42) = 12 / sin(angle C)
Solve for a:
a = 12 x sin(42) / sin(angle C)
Now, plug in the values of angle B and angle C that we found earlier and solve for a:
a = 12 x sin(42) / sin(180 - 42 - angle B)
a ≈ 24.1 units.

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Step-by-step explanation:

Remember for RIGHT triangle  Tan Φ = Opposite Leg/Adjacent Leg

        tan (40) = 900 / x

              x =  900 / tan(40) = 1072.6 ft

The equation 4b + 1 = 73 is used to represent the relationship between the number of oranges in each box and the total number of oranges received. Solving for "b" gives the answer that each box contains 18 oranges.

Model/diagram:

Let "b" be the number of oranges in each box. Since Mr. Franclemont ordered 4 boxes and received an additional orange, he received a total of 4b + 1 oranges. We are given that this is equal to 73:

4b + 1 = 73

To solve for "b", we can subtract 1 from both sides and then divide by 4:

4b = 72

b = 18

Therefore, there are 18 oranges in each box.

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Answer:

7% interest rate.

Step-by-step explanation:

The highest interest you can earn from a savings account is 8% annual interest. This is because the banks don't want to lose money.

The equation that represents the graph of y = x² + 2x - 3 moved 3 units to the left is: D. y = (x+3)² + 2(x+3)-3

What is graph equation?

To move a function 3 units to the left, we need to replace x with (x + 3) in the equation. This is because, if we plug in x + 3 into the equation, we will get the same value as if we had plugged in x but moved 3 units to the left.

So, starting with the original equation y = x² + 2x - 3, we replace x with (x + 3):

y = (x + 3)² + 2(x + 3) - 3

Now, we can simplify this equation to get it into standard form:

y = x² + 8x + 18

Therefore, the equation that represents the graph of y = x² + 2x - 3 moved 3 units to the left is:

D. y = (x+3)² + 2(x+3)-3

which simplifies to:

y = x² + 8x + 18.

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To use the least squares regression line to predict the final grade for a student who missed 15 class periods. It depends on the relationship between the number of missed class periods and the final grade.

If there is a strong linear relationship between these variables, then it would be reasonable to use the least squares regression line to predict the final grade for a student who missed 15 class periods. However, if there is a weak or nonlinear relationship between these variables, then the prediction may not be reliable.

Even if there is a strong linear relationship between the variables, there may be other factors that affect a student's final grade that are not accounted for in the regression model. Therefore, it's always important to interpret the results of a regression analysis with caution and consider other sources of information as well.

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For set A: Mean = 6, Mean absolute deviation = 1 and For set B: Mean = 7, MAD = 1.33 based on deviations.

We must first determine the average or mean of the data points in order to calculate the mean and mean absolute deviation (MAD) of two data sets. The mean is calculated by dividing the total number of data points by their sum. The absolute deviation of each data point from the mean must then be determined, and the average of these deviations is used to get the MAD.

Using set A:

Mean = (4 + 6 + 7 + 8 + 5 + 6) / 6 = 6

MAD = (|4-6| + |6-6| + |7-6| + |8-6| + |5-6| + |6-6|) / 6 = 1

Using set B:

Mean = (5 + 7 + 4 + 8 + 9 + 9) / 6 = 7

MAD = (|5-7| + |7-7| + |4-7| + |8-7| + |9-7| + |9-7|) / 6 = 1.33

It is clear from a comparison of the two sets that set B has a higher mean than set A, indicating that set B's values are generally higher. Moreover, set B's MAD is larger than set A's, suggesting that set B's values are more dispersed or volatile. We can therefore say that set B has larger and more varied values than set A.

It is crucial to keep in mind that the mean and MAD are merely two indicators of central tendency and dispersion, respectively, and that to thoroughly examine and contrast data sets, they should be combined with additional statistical indicators and visualizations.

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The explicit rule for the mentioned graph is, If the involved variables are correlated, the points will fall along a line or curve.

Scatter plots have been described as one of the most useful inventions in statistical graphs. Scatter plots were originally presented by British scientist John Frederick W. Herschel in 1833. Herschel used it when studying the orbits of binary stars. He plotted the angle of the double star against the measured year. Scatter plots were used to understand the underlying relationship between the two measurements. Bar charts and line charts are commonly used, but scatter charts still dominate the world of science and business. It's very easy to see the points on a scale and understand their relationships.

A scatterplot is a way of displaying data in a graphical format. A simple scatterplot uses axes to plot points based on their values.

Since the cost of safari increases with the number of passengers, it shows the direct and positive correlation between the variables and hence the formation of linear scatter plot.

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The measure of angle BCD is 62 degrees.

When a transversal intersects two parallel lines, alternate interior angles are congruent.

This is given by the Alternate Interior Angles Theorem. Likewise, when two parallel lines are cut by a transversal, consecutive interior angles add up to 180 degrees.

This is given by the Consecutive Interior Angles Theorem.

In the following image, AB is parallel to DC, and BC is a transversal intersecting both parallel lines.

The measure of angle ABC is 118°, and the measure of DAB is 132.

Given that AB is parallel to DC, and BC is a transversal intersecting both parallel lines.

The measure of angle ABC is 118°, and the measure of DAB is 132.

We need to find the measure of angle BCD.

By the Consecutive Interior Angles Theorem, we know that angle BCD + angle ABC = 180 degrees

Therefore, angle BCD = 180 - angle ABC angle BCD = 180 - 118angle BCD = 62 degrees

Therefore, the measure of angle BCD is 62 degrees.

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Answer:

The skater has done half of the  potential energy 2750

Step-by-step explanation:

5500 divided by 2 = 2750

Answer:

Step-by-step explanation:

Answer:

a = 1

Step-by-step explanation:

The zeros of a polynomial P(x) are the x-values that make the polynomial equal to zero, P(x) = 0.

The sum of the zeros of a quadratic equation ax² + bx + c = 0 is -b/a.

The product of the zeros of a quadratic equation ax² + bx + c = 0 is c/a.

Given polynomial:

[tex]x^2+ 4x + 2a[/tex]

Therefore, the zeros of the given polynomial are the x-values that make x²+ 4x + 2a = 0.

Comparing the given polynomial with ax² + bx + c = 0:

If the zeros of the polynomial x²+ 4x + 2a are α and α/2, then using the zeros product formula:

[tex]\begin{aligned}\alpha \cdot \dfrac{2}{\alpha} &= \dfrac{c}{a} \\\\\implies 2&= \dfrac{2a}{1}\\\\\implies 2&=2a\\\\\implies \dfrac{2}{2}&=a\\\\\implies a&=1\end{aligned}[/tex]

Therefore, the value of a is 1.

The value of angle C in the  cyclic quadrilateral is determined as 61⁰.

In a cyclic quadrilateral, opposite angles are the pair of angles that are opposite to each other, meaning they are located at the ends of a diagonal that divides the quadrilateral into two triangles.

The key property of a cyclic quadrilateral is that its four vertices lie on a circle. This means that opposite angles of a cyclic quadrilateral are supplementary, meaning they add up to 180 degrees.

The value of angle C is calculated as follows;

m ∠ C = 180 - m ∠A  ( opposite angles of cylic quadrilateral )

m ∠ C = 180 - 119

m ∠ C = 61⁰

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Answer:

$512.50

Step-by-step explanation:

500 + (.025 x 500)  2.5% move the decimal two places to the left.

500 + 12.50 = 512.50

Helping in the name of Jesus.

The probability of getting a full house is approximately 0.0014

To calculate the probability of getting a full house, we need to first count the total number of possible 5-card hands that can be dealt from a deck of 52 cards.

There are 52 choices for the first card, 51 for the second, 50 for the third, 49 for the fourth, and 48 for the fifth (since we are drawing without replacement). However, the order in which the cards are dealt does not matter, so we must divide by the number of ways in which the same 5 cards can be ordered. Therefore, the total number of possible 5-card hands is

C(52,5) = 52! / (5! × 47!) = 2,598,960

Now we need to count the number of full house hands we can get. We can choose the value of the triple in C(13,1) ways (since there are 13 possible face values), and we can choose the three cards of that value in C(4,3) ways. We can then choose the value of the pair in C(12,1) ways (since there are now only 12 possible face values left), and we can choose the two cards of that value in C(4,2) ways. Therefore, the number of full house hands is

C(13,1) × C(4,3) × C(12,1) × C(4,2) = 13 × 4 × 12 × 6 = 3,744

Finally, the probability of getting a full house is

P(full house) = number of full house hands / total number of hands

= 3,744 / 2,598,960

= 0.0014

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The weights of 1,000 men in a certain town follow a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds.

Approximately 841 men weigh more than 165 pounds, and 841 men weigh less than 135 pounds. This is due to the fact that the area under the normal curve between the z-scores of -1 and 1 is 0.8413, or 84.13%.

Standard deviation is a measure of how spread out a set of data is. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. Standard deviation can be used to measure the variability of a set of data points or to compare different sets of data. It is a useful tool for understanding how a set of data is distributed, as it gives an indication of how much of the data is close to the mean and how much is far away from it.

To calculate these values, we can use the z-score formula to calculate the standard deviation from the mean weight. The z-score formula is z = (x - μ)/σ, where x is a data point, μ is the mean, and σ is the standard deviation. Thus, for men weighing more than 165 pounds, z = (165 - 150)/15 = 1, and for men weighing less than 135 pounds, z = (135 - 150)/15 = -1.

We can then use the z-score to calculate the probability of finding a man with a weight less than or equal to 165 pounds and greater than or equal to 135 pounds. This probability is equal to the area under the normal curve between the z-scores of -1 and 1. From a normal table, we can find that the area under the normal curve between -1 and 1 is 0.8413, which is equal to 84.13%.

Therefore, the number of men weighing more than 165 pounds is equal to 0.8413 * 1,000, which is approximately 841 men, and the number of men weighing less than 135 pounds is equal to 0.8413 * 1,000, which is also approximately 841 men.

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Complete questions as follows-

The weights of 1,000 men in a certain town follow a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds.

From the data, we can conclude that the number of men weighing more than 165 pounds is about , and the number of men weighing less than 135 pounds is about .

The system of equations are:

A = 80x + 70    

A = 980 - 50x  

Weeks: x = 7 weeks

Amount of money: $630

Let x represent the number of weeks it will take for both girls to have the same amount of money.

Let A represent the amount both will have equally after x number of weeks.

Thus:

For Kendra, the equation is:

A = 80x + 70    -----(1)

For Mallory, the equation is:

A = 980 - 50x   ------(2)

Thus, for both to have the same amount of money:

80x + 70 = 980 - 50x

80x + 50x = 980 - 70

130x = 910

x = 910/130

x = 7 weeks

A = 80(7) + 70

A = $630

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The Student Senate at a large university has 60 members, of which 20% are minorities. The probability of selecting only two minority students is 10.74%, making it surprising to have only two on the Senate. The expected number of minority students is 12, and having only two may indicate under-representation. The standard deviation of minority students is 1.94. If there were nine minority students, it would not necessarily indicate bias as it falls within the usual range of values.

Let n, the number of trials, is 60 since there are 60 students chosen to represent the entire student body in the Senate. Let p, the probability of success (i.e., selecting a minority student) on a single trial, is 0.20 since 20% of the student body is minority.

The probability of selecting only two minority students in the Senate can be calculated as P(x≤2) = P(x=0) + P(x=1) + P(x=2) = 0.1074. Therefore, there is a 10.74% chance of having only two minority students on the Senate purely by chance.

Based on the low probability in part c), it would be surprising to have only two minority members on the Senate. However, it's important to note that statistical probability alone cannot determine whether bias exists, as there may be other factors at play.

The Minority Caucus may have a reasonable complaint of under-representation, as the expected number of minority students on the Senate can be calculated as u = np = 60 * 0.20 = 12. Therefore, having only two minority students on the Senate is significantly lower than the expected number.

The standard deviation of the number of minority students selected can be calculated as σ = sqrt(np(1-p)) = 1.94.

If the Senate had nine minority students, the Minority Caucus would not have a strong case for racial bias. Assuming that the distribution of minority students selected follows a binomial distribution, the expected number of minority students would be u = np = 60 * 0.20 = 12, with a standard deviation of σ = 1.94.

Using the Empirical Rule, the usual range of values would be from u - 2σ to u + 2σ, which in this case would be from 8.12 to 15.88. Therefore, having nine minority students falls within the usual range of values, indicating that it is not necessarily a result of bias.

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--The given question is incomplete, the complete question is given

" Using the binomial distribution. A Student Senate at a large university is made up of 60 students chosen to represent the entire student body. 20% of the students at this university are minorities. When the members of the senate are selected, only two members are minority students. The Minority Caucus claims that this Senate is racially biased. They decide to use statistics to support their claim. Let x be the number of minority students who are selected for the senate. Assume the binomial probability can be used as the university has a large student population (large enough to assume independence when sampling students) What is n? What is p? Calculate the probability of the Senate being made up of so few minority students purely by chance. That is, find P(xS2) for this binomial problem: Based on your answer to part b, would you be surprised if the Senate had only 2 minority members? Why or why not? Do you think the Minority Caucus is reasonable in their complaint of under- representation? Explain. Assuming that the senate represents the entire student body, how many minority students would you expect to be selected for the senate? You will need to find u for this. There is a shortcut to find the mean for a binomial distribution shown in the text. ur Find the standard deviation of the number of minority students selected. Again, use the shortcut for finding the standard deviation for a binomial distribution, 0= Now suppose that the Student Senate has 9 minority students. Could the Minority Caucus make a case for racial bias in this case? Explain your answer using the Empirical Rul bunu need to find +20 the usual range of values and include the range of values Find the standard deviation of the number of minority students selected. Again, use the shortcut for finding the standard deviation for a binomial distribution, 0 = Now suppose that the Student Senate has 9 minority students. Could the Minority Caucus make a case for racial bias in this case? Explain your answer using the Empirical Rule (you need to find to, the "usual range" of values and include the range of values in your answer:"--

When a group of 1919 beavers gets together to build a dam and decides to divide into teams with 44 beavers on each full team, the number of full teams of beavers will be 43. The number of beavers that will be in the smaller team is 7.

You can define a number as a count, like in a race, when we say, 3, 2, 1, GO!” or a measurement such as John Cena

weighs 275 lbs. There are also fractions such as 22/7 and decimals such as 3.14.

There are 1919 beavers, and each team has 44 beavers.

1919 divided by 44 gives 43 with a remainder of 47.

Since there cannot be a partial team, the number of full teams of beavers is 43.

There are 47 beavers remaining after the formation of the full teams.

Because it's impossible to divide 47 beavers into teams of 44 beavers, a smaller team will be created with the remaining beavers.

As a result, the number of beavers in the smaller team is 7.

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The missing angle in the triangle is 70 degrees.

The missing angle in the quadrilateral is 120 degrees.

For the first diagram, since all the sides of the hexagon are equal, all the interior angles must also be equal. Therefore, each angle measures 120 degrees.
For the second diagram, we can start by finding the missing angle in the triangle.

Since the sum of the angles in a triangle is 180 degrees,

we can subtract the known angles from 180 to find the missing angle:
180 - 60 - 50 = 70
To find the missing angle in the quadrilateral, we can start by noticing that opposite angles in a parallelogram are equal.

Therefore, we can use the fact that the angle marked 60 degrees is opposite the missing angle:
Missing angle = 180 - 60 = 120
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Option C : The minimum amount of paper needed to wrap 6 cylindrical mini muffins is approximately 214.6 square inches, which is closest to option C, 47 and 1 over 7 in2, when pi is approximated as 22/7.

The surface area of a cylindrical mini muffin can be calculated as follows:

The side of the mini muffin is a rectangle that can be wrapped around the circumference of the circular base. The circumference of a circle with a diameter of 2 inches is pid = pi2 inches. Therefore, the length of the side of the mini muffin is pi2 inches, and the height is 1 and 1/4 inches. The area of the side is the product of the height and the length, which is (5/4) * (2pi) square inches.

Therefore, the total surface area of one mini muffin is:

2pi + (5/4) * (2pi) = (13/4) * pi square inches.

To wrap 6 mini muffins, we need to multiply this area by 6:

6 * (13/4) * pi = 39/2 * pi.

Approximating pi as 22/7, we get:

(39/2) * (22/7) = 429/2 square inches.

This is equivalent to 214.5 square inches, which rounds up to approximately 214.6 square inches.

Therefore, the answer is closest to option C, 47 and 1 over 7 in2.

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A bakery makes cylindrical mini muffins that measure 2 inches in diameter and one and one fourth inches in height. If each mini muffin is completely wrapped in paper, then at least how much paper is needed to wrap 6 mini muffins? Approximate using pi equals 22 over 7.

A. 14 and 1 over 7 in2

B. 23 and 4 over 7 in2

C. 47 and 1 over 7 in2

D. 84 and 6 over 7 in2

Therefore, we can be 99% confident that the true difference in mean amount of food Piper eats when she is offered Pick-a-Pair and when she is offered Pickled Peppers is between 0.7 and 5.1 grams.

A confidence interval is a statistical measure used to estimate the range of values within which a true population parameter is likely to fall with a certain level of confidence. It is often used in inferential statistics to assess the precision and reliability of a sample estimate of a population parameter, such as the mean, proportion, or standard deviation.

Here,

To construct a confidence interval for the difference in mean amount of food Piper eats when she is offered Pick-a-Pair and when she is offered Pickled Peppers, we can use a two-sample t-test. The null hypothesis is that there is no difference in mean amount of food Piper eats between the two brands, and the alternative hypothesis is that there is a difference. We can use a 99% confidence level, so our significance level is α = 0.01. First, we can calculate the sample means, sample standard deviations, and sample sizes for each brand:

For Pick-a-Pair:

Sample mean: x1 = 85.2

Sample standard deviation: s1 = 3.45

Sample size: n1 = 31

For Pickled Peppers:

Sample mean: x2 = 82.1

Sample standard deviation: s2 = 4.62

Sample size: n2 = 30

Next, we can calculate the pooled standard deviation, sp, using the formula:

sp = √(((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2))

sp = sqrt(((31 - 1)(3.45)² + (30 - 1)(4.62)²) / (31 + 30 - 2))

sp = 4.019

Then, we can calculate the t-statistic using the formula:

t = (x1 - x2) / (sp * √(1/n1 + 1/n2))

t = (85.2 - 82.1) / (4.019 * √(1/31 + 1/30))

t = 1.864

Using a t-table with degrees of freedom (df) = n1 + n2 - 2 = 59 and a significance level of 0.01, we find the critical values to be ±2.660. Since our t-statistic of 1.864 is within this range, we fail to reject the null hypothesis.

Finally, we can construct the confidence interval using the formula:

CI = (x1 - x2) ± t*(sp * √(1/n1 + 1/n2))

CI = (85.2 - 82.1) ± 2.660*(4.019 * √(1/31 + 1/30))

CI = (0.7, 5.1)

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[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{40\% of 84}}{\left( \cfrac{40}{100} \right)84}\implies \text{\LARGE 33.6}[/tex]