Find the mean, median, interquartile range and mean absolute deviation of the set of numbers. Round to the nearest tenth, if necessary. 1, 1, 4, 8, 9, 3, 8 please help ​

The probability that the mean weight of 12 fruit will be between 681 and 682 grams is 0.0184.

We can solve this problem by using the central limit theorem, which tells us that the distribution of sample means will be approximately normal if the sample size is sufficiently large.

First, we need to calculate the standard error of the mean:

standard error of the mean = standard deviation / sqrt(sample size)

= 23 / sqrt(12)

= 6.639

Next, we can standardize the sample mean using the formula:

z = (x - mu) / (standard error of the mean)

where x is the sample mean, mu is the population mean, and the standard error of the mean is calculated above.

z1 = (681 - 692) / 6.639 = -1.656

z2 = (682 - 692) / 6.639 = -1.506

Using a standard normal distribution table or calculator, we can find the probabilities corresponding to these z-scores:

P(z < -1.656) = 0.0484

P(z < -1.506) = 0.0668

The probability of the sample mean being between 681 and 682 grams is the difference between these probabilities:

P(-1.656 < z < -1.506) = P(z < -1.506) - P(z < -1.656)

= 0.0668 - 0.0484

= 0.0184

Therefore, the probability that the mean weight of 12 fruit will be between 681 and 682 grams is 0.0184.

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If we choose ε > 0, we can find a δ such that ||f – ft||1 < ε for all t with 0 < |t - 1| < δ, and we have shown that lim||f - ft||1 = 0 as t -> 1.

(a) To prove that lim||f – ft||1 = 0 as t -> 0, we need to show that for any ε > 0, there exists a δ > 0 such that ||f – ft||1 < ε for all t with 0 < |t| < δ.

We have:

||f – ft||1 = ∫|f(x) – f(x – t)| dx

By the continuity of f, we know that for any ε > 0, there exists a δ > 0 such that |f(x) – f(x – t)| < ε whenever |t| < δ. Therefore:

||f – ft||1 = ∫|f(x) – f(x – t)| dx < ε∫dx = ε

This holds for all t with 0 < |t| < δ, so we have shown that lim||f – ft||1 = 0 as t -> 0.

(b) To prove that lim||f - ft||1 = 0 as t -> 1, we need to show that for any ε > 0, there exists a δ > 0 such that ||f – ft||1 < ε for all t with 0 < |t - 1| < δ.

We have:

||f – ft||1 = ∫|f(x) – f(tx)| dx

Using the change of variables y = tx, we can write this as:

||f – ft||1 = (1/t)∫|f(y/t) – f(y)| dy

Since f is integrable, it is also bounded. Let M be a bound on |f|. Then we have:

||f – ft||1 ≤ (1/t)∫|f(y/t) – f(y)| dy ≤ (1/t)∫M|y/t – y| dy = M|1 – t|

This holds for all t with 0 < |t - 1| < δ, where δ = ε/2M. Therefore, if we choose ε > 0, we can find a δ such that ||f – ft||1 < ε for all t with 0 < |t - 1| < δ, and we have shown that lim||f - ft||1 = 0 as t -> 1.

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Answer:

0.6

Step-by-step explanation:

WE KNOW THAT

P(E)+P(F)=1

P(E)=0.4

NOW

P(E)+P(F)=1

0.4+P(F)=1

P(F)=0.6

HENCE THE PROBABILITY OF NOT SPINNING A BLUE COLOUR IS 0.6

Probability of not spinning a blue colour is 0.6

We know that sum of all Probability is 1,

So the probability of not spinning a blue is = 1 - Probability of  spinning a blue colour.

Putting values we get, = 1 - 0.4 = 0.6

Hence the probability of not spinning a blue colour is 0.6

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The probability that the mean weight of 20 randomly selected men is between 170 and 185 lbs is approximately 0.7189 or approximately 72%.

To solve this problem, we need to use the formula for the sampling distribution of the mean, which states that the mean of a sample of size n drawn from a population with mean μ and standard deviation σ is normally distributed with a mean of μ and a standard deviation of σ/sqrt(n).

In this case, we have a population of men with a mean weight of 178 lbs and a standard deviation of 26 lbs. We want to know the probability that 20 randomly selected men will have a mean weight between 170 and 185 lbs.

First, we need to calculate the standard deviation of the sampling distribution of the mean. Since we are taking a sample of size 20, the standard deviation of the sampling distribution is:

σ/sqrt(n) = 26/sqrt(20) = 5.82

Next, we need to standardize the interval between 170 and 185 lbs using the formula:

z = (x - μ) / (σ/sqrt(n))

For x = 170 lbs:

z = (170 - 178) / 5.82 = -1.37

For x = 185 lbs:

z = (185 - 178) / 5.82 = 1.20

Now we can use a standard normal distribution table (or a calculator) to find the probability of the interval between -1.37 and 1.20:

P(-1.37 < z < 1.20) = 0.8042 - 0.0853 = 0.7189

Therefore, the probability that 20 randomly selected men will have a mean weight between 170 and 185 lbs is 0.7189 or approximately 72%.

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Answer:

I assume you trying to find a surface area (tell me if I'm wrong. okay?

Step-by-step explanation:

V = (1/2)bhL

where b is the base of the triangle, h is the height of the triangle, and L is the length of the prism.

The formula for the surface area of a triangular prism is:

SA = bh + 2(L + b)s

where b and h are the same as above, L is the length of the prism, and s is the slant height of the triangle.

To use these formulas, we need to identify the values of b, h, L, and s from the given dimensions. The base of the triangle is 8 units, the height of the triangle is 6 units, and the length of the prism is 15 units. The slant height of the triangle can be found using the Pythagorean theorem:

s^2 = b^2 + h^2 s^2 = 8^2 + 6^2 s^2 = 64 + 36 s^2 = 100 s = sqrt(100) s = 10

Now we can plug these values into the formulas and simplify:

V = (1/2)bhL V = (1/2)(8)(6)(15) V = (1/2)(720) V = 360

SA = bh + 2(L + b)s SA = (8)(6) + 2(15 + 8)(10) SA = 48 + 2(23)(10) SA = 48 + 460 SA = 508

Therefore, the volume of the triangular prism is 360 cubic units and the surface area is 508 square units.

A linear or exponential model would not model the relationship between the number of employees and number of customers

From the question, we have the following parameters that can be used in our computation:

number of employees 1 2 3 4 10

number of customers 8 4 13 17 39

Testing a linear model

To do this, we calculate the difference between the y values

So, we have

13 - 4 = 4 - 8

9 = -4 ---- this is false

So, the function is not a linear function

Testing an exponential model

To do this, we calculate the ratio of the y values

So, we have

13/4 = 4/8

3.25 = 1/2 ---- this is false

So, the function is not an exponential function

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The times it rotates is given by 207 rotations, the weapon that will do the more damage is sword and percent probability of rolling a six on a single six sided die is 17%.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

a) Number of rotation in 1min = 143

No of rotation in 60 seconds = 143

No. of rotation in 1 seconds = 143/60

number of rotation in 87 seconds = 143/60 x 87 = 207 rotations.

b) Sword damage 14 in 1 seconds

Axe damage is 25 in 3 seconds

so in 1 seconds it is 25/3

Sword damage in 1 min = 14 x 60 = 840 units

Axe damage in 1 min = 25/3 x 60 = 500 units

Swords will do more damage in 1 min .

c) Probability = No of favorable outcome / Total number of outcome x 100

= Total outcomes = {1, 2, 3, 4, 5, 6}

= 1/6 = 100

= 17%.

Therefore, percent probability is 17%.

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The quantity q(t) of a radioactive substance left at time t > 0 with a half-life of 3200 years can be found using the formula: q(t) =

[tex]q(0) * 0.5^(t/3200)[/tex]

after 6400 years, only 10 grams of the substance will be left. where q(0) is the initial quantity of the substance.

Given q(0) = 20 g, we can find q(t) for any time t > 0 using the formula above. For example, if we want to find q(6400) - the quantity of the substance left after 6400 years - we can substitute t = 6400 in the formula and get: q(6400) =

[tex]20 * 0.5^(6400/3200)[/tex]

= 10 g.

After 6400 years, only 10 grams of the substance will be left. It is important to note that the half-life of a radioactive substance is the time it takes for half of the substance to decay.

After one half-life (3200 years), the initial quantity of the substance will be reduced to half (10 g). After two half-lives (6400 years), it will be reduced to one-fourth (5 g), and so on.

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the answer to your question is  130

Step-by-step explanation:

the main triangle is an isoceles triangle (both legs are equally long). that means that the height y bergen the 2 legs splits the baseline in half.

therefore,

x = 10/2 = 5

Pythagoras gives us y.

c² = a² + b²

c being the Hypotenuse (the side opposite of the 90° angle). in our case 10.

a and b are the legs. in our case x and y.

10² = 5² + y²

100 = 25 + y²

75 = y²

y = sqrt(75) = 8.660254038...

a. Yes. Nonlinear Solver will be guaranteed to identify the level of training that maximizes productivity b. If training is set to 5, the resulting level of productivity is 62.

a. Yes, Nonlinear Solver will be guaranteed to identify the level of training that maximizes productivity.

This is because the formula given is a quadratic equation with a positive coefficient for the x-squared term (2x2), indicating a concave upward curve. The maximum point of a concave upward curve is always at the vertex, which can be found using the Nonlinear Solver.

b. If training is set to 5, the resulting level of productivity can be found by substituting x=5 into the equation:

y = -3x + 2x^2 + 27
y = -3(5) + 2(5)^2 + 27
y = -15 + 50 + 27
y = 62

Therefore, the resulting level of productivity when training is set to 5 is 62 (rounded to the nearest whole number).

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The inverse function of the function f(x) = -3x/8 is f⁻¹(x) = -8x/3

To find the inverse of a function, we need to switch the roles of x and y and then solve for y.

Let's begin by rewriting the function f(x) in terms of y:

y = f(x) = -3x/8

Now, let's switch x and y:

x = -3y/8

Next, we'll solve for y:

x = -3y/8

8x = -3y

y = -8x/3

So the inverse function of f(x) = -3x/8 is f⁻¹(x) = -8x/3

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Answer:

44/35

Step-by-step explanation:

this answer cannot be further simplified*

The equation of the graph is determined as y - 1 = 5x/3.

The equation of the graph is calculated by applying the general equation of a line form.

y = mx + c

where;

The slope of the graph is calculated as follows;

m = Δy/Δx

m = (y₂ - y₁ ) / (x₂ - x₁ )

m = ( -4 - 1 ) / (3 - 0)

m = -5/3

y = -5x/3 + 1

y - 1 = 5x/3

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Step-by-step explanation:

which of the principles and the question is not clear i saw something different before i clicked on it

Answer:

[tex] \dfrac{x^2 + 2x}{(x + 1)^2} [/tex]

Step-by-step explanation:

[tex] f(x) = \dfrac{x^2}{x + 1} [/tex]

[tex] \dfrac{d}{dx} \dfrac{x^2}{x + 1} = [/tex]

[tex] = \dfrac{d}{dx} [(x^2)(x + 1)^{-1}] [/tex]

[tex]= (x^2)(-1)(x + 1)^{-2} + (x + 1)^{-1}(2x)[/tex]

[tex] = \dfrac{-x^2}{(x + 1)^{2}} + \dfrac{2x}{x + 1} [/tex]

[tex] = \dfrac{-x^2}{(x + 1)^{2}} + \dfrac{2x^2 + 2x}{(x + 1)^2} [/tex]

[tex] = \dfrac{x^2 + 2x}{(x + 1)^2} [/tex]

The probability that a random point on AK will be on CH is 1/2

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event is 1 and it is equal to 100% in percentage.

probability = sample space /total outcome

The total outcome is the range of AK.

range = highest - lowest

= 10-(-10) = 10+10 = 20

sample space of CH = 4-(-6)

= 4+6 = 10

Therefore probability that a random point on AK will be on CH is 10/20

= 1/2

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Answer:

1. Based on the grades, it is likely that Class 1 was the lecture class and Class 2 was the small group class. This is because the grades in Class 1 have a wider range (70-90) and a larger variance, while the grades in Class 2 are more tightly clustered together (82-90) and have a smaller variance.

2. Histograms or box plots could be drawn to compare the shapes of the distributions, but we cannot do this through text.

3. The most appropriate measure of center for these data sets is the mean, since the distributions are approximately symmetric. The most appropriate measure of spread for these data sets is the standard deviation, since the distributions are not strongly skewed and there are no extreme outliers.

Step-by-step explanation:

The correct values are,

                     Q1            Q2       IQR   Mean      Median         MAD

Class 1            76.25         81.75    5.5      79.35         79.50       3.12

Class 2             84             88        4           84.60         86           3.85

Subtraction in mathematics means that is taking something away from a group or number of objects. When you subtract, what is left in the group becomes less.

Now, The first step is to arrange the grades in the classes in ascending order.

Class 1: 70, 72, 74, 75, 76, 77, 77,77, 78, 79, 80, 80, 80, 81, 81, 82, 84, 86, 88, 90

Class 2: 70, 77, 79, 82, 84, 84, 84, 84, 86, 86, 86, 86, 86, 87, 88, 88, 88, 88, 89, 90

Hence, We get;

Q1 for class 1= 1/4(n + 1) = 21/4 = 5.25 = 76.25

Q2 for class 2 = 1/4(n + 1) = 5.25 = 84

Q3 for class 1= 3/4(n + 1) = 15.75 = 81.75

Q3 for class 2 = 3/4(n + 1) = 15.75 = 88

And,

IQR for class 1 = Q3 - Q1 = 81.75 - 76.25 = 5.50

IQR for class 2 = Q3 - Q1 = 88 - 84 = 4

Mean for class 1 = sum of grades / total number of grades = 1587 / 20 = 79.35

Mean for class 2 = sum of grades / total number of grades= 1692 / 20 = 84.6

Median for class 1 = (n + 1) / 2 = 21/2 = 10.5 = 79.50

Median for class 1 = (n + 1) / 2 = 21/2 = 10.5 = 86

Since, We know that;

MAD = 1/n ∑ l x - m(x) l

Where: n = number of observations

x = number in the data set

m = mean

Hence,

Mean absolute deviation for class 1 = 62. 3/ 20 = 3.12

Mean absolute deviation for class 2. = 77/ 20 = 3.85

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The equation that represents the length of the completed tunnel based on the number of days is y = 45x + 140.

Option A is the correct answer.

We have,

From the table,

We take two ordered pairs:

(15, 815) and (20, 1040)

Now,

The equation can be written as y = mx + c.

And,

m = (1040 - 815) / (20 - 15)

m = 225/5

m = 45

And,

(15, 815) = (x, y)

815 = 15 x 45 + c

c = 815 - 675

c = 140

Now,

y = mx + c

y = 45x + 140

Thus,

The equation that represents the length of the completed tunnel based on the number of days is y = 45x + 140.

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a. The probability of getting a z-score less than -2.67 is 0.0038

b. The range to capture the middle 95% of averages is 66.56 mph to 69.44 mph.

c. The range to capture the middle 90% of averages is 66.77 mph to 69.23 mph.

d. the probability of having an average exceeding 67 mph is 0.9082.

a. The sampling distribution of the mean of a sample of 16 cars is normally distributed with a mean of 68 mph and a standard deviation of 3/√16 = 0.75 mph. The shape of the distribution is normal, the center is 68 mph, and the standard deviation is 0.75 mph. To find the probability that the mean speed of a random sample of 16 cars is less than 66 mph, we need to calculate the z-score:

z = (66 - 68) / 0.75 = -2.67

Using a z-table, we find that the probability of getting a z-score less than -2.67 is 0.0038.

b. To capture the middle 95% of averages, we need to find the z-scores that correspond to the 2.5th and 97.5th percentiles of the normal distribution. Using a z-table, we find that these z-scores are -1.96 and 1.96, respectively. Then we can use the formula:

68 + (-1.96)(0.75) < μ < 68 + (1.96)(0.75)

which gives us the range of 66.56 mph to 69.44 mph.

c. To capture the middle 90% of averages, we need to find the z-scores that correspond to the 5th and 95th percentiles of the normal distribution. Using a z-table, we find that these z-scores are -1.645 and 1.645, respectively. Then we can use the formula:

68 + (-1.645)(0.75) < μ < 68 + (1.645)(0.75)

which gives us the range of 66.77 mph to 69.23 mph.

d. To find the probability of having an average exceed 67 mph, we need to find the z-score that corresponds to 67 mph:

z = (67 - 68) / 0.75 = -1.33

Using a z-table, we find that the probability of getting a z-score less than -1.33 is 0.0918. Therefore, the probability of having an average exceed 67 mph is 1 - 0.0918 = 0.9082.

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Answer:

Sure, I can help you with that! First, let's simplify the expression inside the parentheses:

5 3/2 * 2 - 1/2 = 5 * 3 - 1/2 = 14.5

Now we can substitute this value back into the original expression and simplify:

(14.5)^2 = 210.25

Therefore, the simplified expression is 210.25.

Step-by-step explanation:

To find the orthogonal trajectories of the given family of curves, we first need to understand what the term "orthogonal" means. In simple terms, two lines or curves are said to be orthogonal if they intersect at a right angle. Now, coming back to the problem, the given family of curves can be written as x^2 - 2y^2 = c, where c is a constant.

To find the orthogonal trajectories, we need to differentiate this equation with respect to y, treating x as a constant. This gives us:
-4xy = dy/dx

Now, we need to find the equation of the curves that intersect the given family of curves at a right angle, i.e., the slopes of the curves must be negative reciprocals of each other. Therefore, we can write:
dy/dx = 4xy/k

where k is a constant. To solve this differential equation, we can separate the variables and integrate:

∫dy/4xy = ∫dx/k

ln|y| - ln|x^2| = ln|c| + ln|k|

ln|y/x^2| = ln|ck|

y/x^2 = ±ck

Therefore, the orthogonal trajectories of the given family of curves are given by y = ±kx^2/c, where k is a constant. These curves intersect the original family of curves at right angles.

1. Identify the family of curves: The given equation is x^2 + 2y^2 = c, where c is a constant. This represents a family of ellipses with different sizes depending on the value of c.

2. Calculate the derivative: To find the orthogonal trajectories, we first need to find the derivative of the given equation with respect to x. Differentiate both sides with respect to x:

d/dx(x^2) + d/dx(2y^2) = d/dx(c)
2x + 4yy' = 0

3. Find the orthogonal slope: The slope of the orthogonal trajectory is the negative reciprocal of the original slope. Since the original slope is y', the orthogonal slope is -1/y':

Orthogonal slope = -1/y'

4. Replace the original slope with the orthogonal slope:

2x + 4y(-1/y') = 0

5. Solve for y':

y' = -2x/(4y)

6. Solve the differential equation: Now we have a first-order differential equation to find the equation of the orthogonal trajectories:

dy/dx = -2x/(4y)

Separate variables and integrate both sides:

∫(1/y) dy = ∫(-2x/4) dx

ln|y| = -x^2/4 + k

7. Solve for y:

y = e^(-x^2/4 + k) = C * e^(-x^2/4), where C is a new constant.

The orthogonal trajectories of the given family of curves are represented by the equation y = C * e^(-x^2/4).

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The approximate height of the cliff, given the angle of elevation would be 480. 09 meters.

One approach to determining the cliff's height involves applying trigonometry's tangent function . In this instance, a given angle of elevation (58°) and 300 meters' distance from its base are known.

The tangent function is defined as:

tan (angle) = opposite side / adjacent side

tan ( 58 ° ) = h / 300

h = 300 x tan(58°)

h = 300 x 1. 6003

=  480. 09 meters

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The HR administrator must collect a sample size of 108 employees to be accurate within 0.09 at the 95% confidence interval.

To determine the necessary sample size, we need to use the formula:

[tex]n = \frac{(z^2 )(p) (1-p)}{E^2}[/tex]

Where:
- n = sample size
- z = the z-score for the desired level of confidence (in this case, 1.96 for 95%)
- p = the estimated proportion of employees using the benefit (since we have no preliminary notion, we will use 0.5 as the most conservative estimate)
- E = the desired margin of error (0.09)

Plugging in these values, we get:

[tex]n = \frac{(1.96^2 )(0.5) (1-0.5)}{0.09^2}[/tex]
n = 107.92 = 108

We round up to the next whole number since we can't have a fraction of a person in our sample.

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Answer:

24 is the answer

Step-by-step explanation:

add the numbers and divide them by 7

42 inches divided by 1 gives the measurement 3 feet and 6 inches.

We have to find what number divides the number 42 inches to 3 feet and 6 inches.

We know that the conversion of measurement units,

1 foot = 12 inches

3 feet = 3 × 12 inches = 36 inches

3 feet 6 inch = 36 + 6 = 42 inches

So the required number divides 42 inches in to 42 inches itself.

Any number divided by 1 gives the same number.

So the required number is 1.

Hence the unknown number which divide 42 inches to 3 feet and 6 inches is 1.

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The short-run supply curve shows the quantity of output a firm is willing to supply at different market prices in the short run. It is typically upward sloping, meaning that as the price of the product increases, the firm is willing to produce and supply more units. This is because higher prices will allow the firm to cover its variable costs and potentially earn a profit.

When used in conjunction with the average variable cost (AVC) curve, the supply curve can give a firm valuable information about its profits. The AVC curve represents the average variable cost per unit of output, which includes the costs that vary with the level of production (such as labor and materials).

If the market price is above the AVC curve, the firm is covering all of its variable costs and may earn a profit. If the market price is below the AVC curve but still above the average total cost (ATC) curve, the firm is not covering all of its costs but is still producing because it is covering its variable costs. If the market price falls below the ATC curve, the firm is not covering all of its costs and is likely to shut down production in the short run.

Therefore, the supply curve in conjunction with the AVC curve allows a firm to determine whether it should produce and supply output in the short run based on the prevailing market price. If the market price is high enough to cover variable costs and potentially earn a profit, the firm will continue to produce. However, if the market price falls below the AVC curve, the firm will likely reduce or cease production to minimize losses.

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The correct mean adjusts in the records for all 230 children is  96 pounds

The given issue includes finding the right cruel weight of the 230 children after adjusting for the scale blunder. The first mean weight is given as 98 pounds, but we got to alter for the scale blunder of 2 pounds that the scale was perusing over the genuine weight.

To correct the scale mistake, we ought to subtract 2 pounds from each child's recorded weight. This will shift the complete conveyance of weights by 2 pounds to the cleared out, so the unused cruel weight will be lower than the first cruel weight.

The first cruel weight is given as 98 pounds, but we got to alter for the scale blunder:

Rectified mean weight = Original cruel weight - Scale blunder

Rectified mean weight = 98 - 2

Rectified mean weight = 96 pounds

Subsequently, the proper cruel weight of the children after altering the scale blunder is 96 pounds. This implies that on normal, the children weighed 96 pounds rather than 98 pounds as initially recorded. 

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Answer:

A, C, E

Step-by-step explanation:

To determine which expressions have a value greater than 1, evaluate the expressions following the order of operations (PEMDAS) and remembering the following:

[tex]\;\;\;\:-\frac{1}{3} \div (-2)+4\\\\= -\frac{1}{3} \cdot \left(-\frac{1}{2}\right)+4\\\\=\frac{(-1) \cdot (-1)}{3 \cdot 2}+4\\\\= \frac{1}{6}+4\\\\= 4\frac{1}{6}[/tex]

[tex]\;\;\;-\frac{1}{3} \cdot (-2)-4\\\\= \frac{2}{3} -4\\\\= \frac{2}{3} -\frac{12}{3}\\\\= \frac{2-12}{3}\\\\= -\frac{10}{3}\\\\=-3\frac{1}{3}[/tex]

[tex]\;\;\:\:-\frac{1}{3} \cdot (-2-4)\\\\= -\frac{1}{3} \cdot (-6)\\\\=\frac{(-1)\cdot (-6)}3{}\\\\= \frac{6}{3} \\\\= 2[/tex]

[tex]\;\;\:\:-\frac{1}{3} \cdot (-2)(-4)\\\\= -\frac{1}{3} \cdot (8)\\\\=\frac{(-1) \cdot 8}{3}\\\\= -\frac{8}{3} \\\\= -2\frac{2}{3}[/tex]

[tex]\;\;\:\:-\frac{1}{3} + (-2)-(-4)\\\\= -\frac{1}{3} -2+ 4\\\\= -2\frac{1}{3} + 4\\\\=4 -2\frac{1}{3}\\\\= 1\frac{2}{3}[/tex]

Therefore, the expressions that have a value greater than 1 are:

Answer:

To solve this problem, we need to use the concept of hypergeometric distribution, which gives the probability of selecting a certain number of objects with a specific characteristic from a population of known size without replacement. We will use the formula:

P(X = k) = [ C(M, k) * C(N - M, n - k) ] / C(N, n)

where:

P(X = k) is the probability of selecting k objects with the desired characteristic;

C(M, k) is the number of ways to select k objects with the desired characteristic from a population of M objects;

C(N - M, n - k) is the number of ways to select n - k objects without the desired characteristic from a population of N - M objects;

C(N, n) is the total number of ways to select n objects from a population of N objects.

In our case, we want to select 5 rats out of a population of 80, and we want exactly 3 of them to have 2 genetic mutations. We can calculate this probability as follows:

P(3 rats have exactly 2 mutations) = [ C(12, 3) * C(68, 2) ] / C(80, 5)

where:

M is the number of rats that have exactly 2 mutations, which is the sum of the rats that have mutations AB, AC, AD, BC, BD, and CD, or M = 5 + 6 + 4 + 3 + 3 + 1 = 22;

N - M is the number of rats that do not have exactly 2 mutations, which is the remaining population of 80 - 22 = 58 rats;

n is the number of rats we want to select, which is 5.

We can simplify this expression as follows:

P(3 rats have exactly 2 mutations) = [ C(12, 3) * C(68, 2) ] / C(80, 5)

= [ (12! / (3! * 9!)) * (68! / (2! * 66!)) ] / (80! / (5! * 75!))

= 0.03617

Therefore, the probability that if 5 rats are selected at random, 3 will have exactly 2 genetic mutations is 0.03617 (rounded to five decimal places).