: Exercise 5. The rank r nonnegative matrix factorisation of an m x n matrix, A, may be estimated using the following algorithm • Set w to be any mxr matrix, and h to be any r x n matrix, both non-negative and of full rank. • Iteratively compute h = h - *(w? A). /(w? wh) w = w • *((Aht). /(whht), where here we use MATLABesque notation, and denote the entry-wise matrix multiplication and division operators as * and/ and (a) Give example of a situation where, due to the initial choices of w and h, this algorithm would fail. (b) If the algorithm does not fail, must the entries of h and w aleays be non-negative? Explain your answer. (c) Use the algorithm to compute a nonnegative matrix factorisation of [34] A= 6 8 ]

The area between -2 and 0 is 0.5000 - 0.0228 = 0.4772. Therefore, P(-2 < z ≤ 0) = 0.4772.

(a) To find P(z ≤ -3.0), we can use a standard normal distribution table or technology such as a calculator or statistical software. Looking at a standard normal distribution table, we find that the area to the left of -3.0 is 0.0013 (rounded to four decimal places). Therefore, P(z ≤ -3.0) = 0.0013.

(b) To find P(z ≥ -3), we can use the fact that the standard normal distribution is symmetric about its mean of 0. Therefore, P(z ≥ -3) is the same as the area to the right of 3, which we can find using a standard normal distribution table or technology. Looking at a table, we find that the area to the right of 3 is also 0.0013. Therefore, P(z ≥ -3) = 0.0013.

(c) To find P(z ≥ -1.7), we can use a standard normal distribution table or technology. Looking at a table, we find that the area to the left of -1.7 is 0.0446 (rounded to four decimal places). Therefore, the area to the right of -1.7 (which is the same as P(z ≥ -1.7)) is 1 - 0.0446 = 0.9554. Therefore, P(z ≥ -1.7) = 0.9554.

(d) To find P(-2.6 ≤ z), we can use a standard normal distribution table or technology. Looking at a table, we find that the area to the left of -2.6 is 0.0047 (rounded to four decimal places). Therefore, P(-2.6 ≤ z) is the same as the area to the right of -2.6, which is 1 - 0.0047 = 0.9953. Therefore, P(-2.6 ≤ z) = 0.9953.

(e) To find P(-2 < z ≤ 0), we can use the fact that the standard normal distribution is symmetric about its mean of 0. Therefore, we can find the area to the left of -2 and the area to the left of 0 and subtract them to find the area between -2 and 0. Looking at a standard normal distribution table, we find that the area to the left of -2 is 0.0228 (rounded to four decimal places), and the area to the left of 0 is 0.5000. Therefore, the area between -2 and 0 is 0.5000 - 0.0228 = 0.4772. Therefore, P(-2 < z ≤ 0) = 0.4772.

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The entire surface area must be at least 18.21 square units.

A function connects an input with an output. It is analogous to a machine with an input and an output. And the output is somehow related to the input. The standard manner of writing a function is f(x) "f(x) =... "

Let's start by finding the first derivative of the function A with respect to r:

A(r) = 2πr³ - 6πr² + 71.1r - 10.8

A'(r) = 6πr² - 12πr + 71.1

To find the stationary points of A, we need to solve the equation A'(r) = 0:

6πr² - 12πr + 71.1 = 0

Dividing both sides by 6π, we get:

r² - 2r + 11.85/π = 0

Using the quadratic formula, we can solve for r:

r = [2 ± √(2² - 4(1)(11.85/π))] / 2

r = 1 ± √(1 - 11.85/π)

Since the value under the square root is negative, there are no real solutions to this equation. Therefore, there is no value of r for which the function A has a stationary point.

Next, let's verify that the value of r = 2.1 results in the total surface area being minimized. To do this, we need to find the second derivative of A:

A''(r) = 12πr - 12π

Setting this equal to zero and solving for r, we get:

12πr - 12π = 0

r = 1

Since A''(1) > 0, we know that r = 1 corresponds to a minimum value of A. Therefore, the value of r = 2.1, which is greater than 1, also corresponds to a minimum value of A.

To find the minimum total surface area, we can substitute r = 2.1 into the expression for A:

A(2.1) = 2π(2.1)³ - 6π(2.1)² + 71.1(2.1) - 10.8

A(2.1) ≈ 18.21

Therefore, the minimum total surface area is approximately 18.21 square units.

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The sale price of the cell phone that costs $600, after a 25% discount from the cost is $450

A discount is a reduction or deduction from the actual price of goods and or services.

The cost of the cell phone = $600

The percentage discount on the of the cell phone = 25%

The sale price after the discount can therefore be calculated as follows;

Sale price = ((100 - 25)/100) × $600 = $450

The sale price after the discount = $450

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The interval contains 0, we cannot reject the null hypothesis that the two proportions are equal at the 0.10 level of significance.

Let p1 be the proportion of male teachers in the first school district, and p2 be the proportion of male teachers in the neighboring school district. We want to construct a 90% confidence interval for the difference in the proportions, p1 - p2.

The point estimate for the difference in proportions is:

[tex]p1-p2=\frac{19}{97}-\frac{19}{114} = 0.049[/tex]

Under the assumption of independence between the two samples, the standard error can be estimated using the following formula:

[tex]SE=\sqrt{\frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2} }[/tex]

[tex]SE=\sqrt{\frac{0.196(1-0.196)}{97} + \frac{0.167(1-0.167)}{114} }= 0.056[/tex]

To find the endpoints of the confidence interval, we can use the following formula:

(p1 - p2) ± z(SE)

Substituting the values, we get: 0.049 ± 1.645(0.056)

The lower endpoint of the interval is:

0.049 - 1.645(0.056) = -0.006

The upper endpoint of the interval is:

0.049 + 1.645(0.056) = 0.104

Since the interval contains 0, we cannot reject the null hypothesis that the two proportions are equal at the 0.10 level of significance.

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The value of the vector [tex]proj_{vu}[/tex] is <-972/169, -405/169>. (option d).

The projection of one vector onto another vector can be thought of as the shadow of one vector onto another in the direction of the second vector. Mathematically, the projection of vector v onto vector u can be calculated as follows:

[tex]proj_{vu}[/tex] = (v · u / ||u||²) x u

Here, · denotes the dot product of two vectors, and ||u|| denotes the magnitude or length of vector u.

Now, let's apply this formula to the given vectors u and v:

u = <-12,-5>

v = <3,9>

To calculate [tex]proj_{vu}[/tex], we first need to find the dot product of vectors u and v:

u · v = (-12 x 3) + (-5 x 9) = -36 - 45 = -81

Next, we need to find the magnitude of vector u:

||u|| = √((-12)² + (-5)²) = √(144 + 25) = √169 = 13

Now, we can substitute the values we have found into the formula for [tex]proj_{vu}[/tex]:

[tex]proj_{vu}[/tex] = (-81 / (13²)) x <-12,-5> = <-81/13, -405/169>

Therefore, the answer to the given question is option (d): [tex]proj_{vu}[/tex] = <-972/169, -405/169>.

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The value of the surface integral using the Divergence Theorem is approximately 6.702.

(g) To use Stokes's Theorem, we need to find the curl of F:

curl(F) = ∂Fz/∂y i + (∂Fx/∂z - ∂Fz/∂x)j + ∂Fy/∂x k

= (-xzsin(y) + 1) i + xcos(y) j + xcos(y) k

Now, we need to find the boundary of S above the plane z = 0, which is the curve C.

Parameterizing C, we have:

r(t) = ti + (t²-t)j, 0 ≤ t ≤ 1

Now, we can use Stokes's Theorem:

∫ ∫s+, F. ds = ∫ ∫C curl(F) . dr

= ∫₀¹ (-t²sin(t) + 1) dt + ∫₀¹ tcos(t) dt + ∫₀¹ tcos(t) dt

= [-t³cos(t) + 3t²sin(t) + t]₀¹ + sin(1) - 1

≈ -0.732

(h) To use the Divergence Theorem, we need to find the divergence of F:

div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

= zcos(y) + cos(y)z - xzsin(y) + 1

Now, we can use the Divergence Theorem:

∫ ∫s F.ds = ∭v div(F) dv

= ∫₀¹ ∫₀²π ∫₀^t (zcos(y) + cos(y)z - xzsin(y) + 1) r dz dy dt

≈ 6.702

Therefore, the value of the surface integral using the Divergence Theorem is approximately 6.702.

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Answer:

6.3cm

Step-by-step explanation:

The sides can be found by taking the square root of the area.

(Area)1/2=s, where s = side.

(40)1/2=6.3.

So the length of a side is 6.3 cm.

Using a significance level of α = 0.05, the critical value is -1.645, and we should reject H0, indicating that there is evidence to support the claim that p < 0.45.

We have,

You are testing the claim that p < 0.45 using a test statistic of z = -3.19 and a significance level of α = 0.05.

Step 1:

Determine the critical value(s) using the significance level:
Since this is a left-tailed test (claim is p < 0.45), you'll need to find the critical value for a one-tailed test at α = 0.05. Using a standard normal distribution table or a calculator, you find that the critical value is z = -1.645.

Step 2:

Compare the test statistic to the critical value:
The test statistic z = -3.19 is more negative (to the left) than the critical value z = -1.645.

Step 3:

Make a decision regarding H0:
Since the test statistic is more negative than the critical value, we reject H0. This means there is sufficient evidence to support the claim that p < 0.45 at a significance level of 0.05.

Thus,
Using a significance level of α = 0.05, the critical value is -1.645, and we should reject H0, indicating that there is evidence to support the claim that p < 0.45.

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Answer:

BC = 30

Step-by-step explanation:

We know that opposite sides of a parallelogram are congruent. Because of this, we can equate their lengths and solve for the variable z:

15z = 19z - 8

↓ adding 8 to both sides

15z + 8 = 19z

↓ subtracting 15z from both sides

8 = 4z

↓ dividing both sides by 4

2 = z

z = 2

Now, we can plug this z-value into the length of side BC and simplify:

BC = 19z - 8

BC = 19(2) - 8

BC = 30

The general solution of the differential equation is:

xy^2/2 + x^2y + (x^2/2)y - (x^2/4)y^2 + h(x) = C

where C is the constant of integration.

To solve this differential equation, we can use the method of exact differential equations.

First, we need to check if the equation is exact by verifying if the following condition is satisfied:

∂(y^2 + xy)/∂y = ∂(x^2)/∂x

Differentiating y^2 + xy with respect to y, we get:

∂(y^2 + xy)/∂y = 2y + x

Differentiating x^2 with respect to x, we get:

∂(x^2)/∂x = 2x

Since these two expressions are equal, the equation is exact.

To find the general solution, we need to find a function f(x,y) such that:

∂f/∂x = y^2 + xy

∂f/∂y = x^2

Integrating the first equation with respect to x, we get:

f(x,y) = xy^2/2 + x^2y + g(y)

where g(y) is a constant of integration that depends only on y.

Taking the partial derivative of f(x,y) with respect to y and equating it to x^2, we get:

∂f/∂y = x^2 = xy + 2xg'(y)

where g'(y) is the derivative of g(y) with respect to y.

Solving for g'(y), we get:

g'(y) = (x^2 - xy)/2x

Integrating both sides with respect to y, we get:

g(y) = (x^2/2)y - (x^2/4)y^2 + h(x)

where h(x) is a constant of integration that depends only on x.

Therefore, the general solution of the differential equation is:

xy^2/2 + x^2y + (x^2/2)y - (x^2/4)y^2 + h(x) = C

where C is the constant of integration.

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A 5% risk of being wrong, as accepted by convention in psychology, is an acceptable level of risk as it balances decisions making and potential for error well. The costs of lowering the risk to 1% requires more time and efforts.

In my opinion, a 5% risk of being wrong is a generally acceptable level of risk in many situations in psychology. This is because it balances the need for making decisions with the potential for error.

If we were to lower the risk to 1%, it might require more time, resources, and effort to gather additional evidence and conduct more in-depth analysis. This could slow down the decision-making process, which might not be desirable in certain situations.

Ultimately, the acceptable level of risk depends on the context and the potential consequences of the decision being made. If the consequences of a wrong decision are severe, it may be worthwhile to invest in reducing the risk to 1% or lower.

However, for most everyday decisions, a 5% risk of being wrong is a reasonable compromise between accuracy and efficiency.

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Answer:

Its the first one

Step-by-step explanation:

correct answer

If there are 48 tulips, the number of daisies present would be 56.

If there are 6 tulips for every 7 daisies, we can express the ratio of tulips to daisies as 6/7.

Let's use the information that there are 48 tulips to find out how many daisies there are:

If 6 tulips correspond to 7 daisies, then we can set up the proportion:

6/7 = 48/x

where x is the number of daisies.

To solve for x:

Therefore, there are 56 daisies in the flower garden.

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The flower garden contains 56 daisies.

Asteraceae, a family of flowering plants with approximately 32,000 recognized species, contains daisies among its members.

In the floral garden, if there are 6 tulips for every 7 daisies, we may apply a ratio to determine how many daisies there are:

6 tulips / 7 daisies = 48 tulips / x daisies

Cross-multiplying, we get:

6 tulips * x daisies = 48 tulips * 7 daisies

To put it simply, we have:

6x = 336

x = 56 is obtained by multiplying both sides by 6.

Therefore, the flower garden contains 56 daisies.

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The two events are rolling a 1 on fair die and rolling even number .

Rolling a 1 face up does not represents even number both the mutually exclusive events.

Rolling a fair die twice represents independent events.

Main difference is both will not occur at the same time.

Mutually exclusive events represents the events are disjoint set.

Suppose from the field of interest like to studying,

The events of rolling a '1' on a fair six-sided die and rolling an 'even number' on the same die.

A scenario in which these events are mutually exclusive is,

When the die shows a '1' face-up after rolling.

The event of rolling an even number did not occur since '1' is an odd number.

Thus, these events cannot occur at the same time, and they are mutually exclusive.

A scenario in which these events are independent is when we roll the die twice.

The first roll may result in an odd or even number.

But it does not affect the probability of getting an even number on the second roll.

The events of rolling a '1' on the first roll and rolling an even number on the second roll are independent.

As the occurrence of one event does not affect the probability of the other event happening.

The main difference between mutually exclusive and independent events is,

That mutually exclusive events cannot occur at the same time.

While independent events can occur simultaneously without affecting each other's probability of occurring.

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Answer: ==8

Step-by-step explanation:

move the letter terms left number terms right. But like balencing a scale what you do to one side you have to do to the other for the equation to be correct. So

Rounding up, we need a sample size of n = 443 to be 99% confident that our estimate of the population mean is within 4 of the true population mean.

The formula to calculate the sample size needed to estimate a population mean with a specified margin of error is:

n = (z^2 * σ^2) / E^2

Where:

z = the z-score corresponding to the desired confidence level (in this case 99%, which gives z = 2.576)

σ = the population standard deviation

E = the desired margin of error

Plugging in the values given in the problem, we get:

n = (2.576^2 * 33.9^2) / 4^2

n = 442.74

Rounding up, we need a sample size of n = 443 to be 99% confident that our estimate of the population mean is within 4 of the true population mean.

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We have shown that A is a subset of B and that B is not a subset of A, we can conclude that A and B are not equal, i.e., A ≠ B.

a. To prove that A is a subset of B, we need to show that every element in A is also in B. Let n be an arbitrary element in A. Then, we have to show that n is also in B, i.e., n = 10r + 2 for some integer r. Since n is in A, we know that n = 5q + 2 for some integer q. We can rewrite this as:

n = 10q + 2q + 2

= 10q + 2(q + 1)

= 10r + 2

where r = q + 1. Since r is an integer, we have shown that n is in B. Therefore, A is a subset of B.

b. To disprove that B is a subset of A, we need to find at least one element in B that is not in A. Let m = 5s + 3 for some integer s be an arbitrary element in B. We need to show that m is not in A, i.e., m ≠ 10r + 2 for any integer r. Suppose for the sake of contradiction that m = 10r + 2 for some integer r. Then we have:

5s + 3 = 10r + 2

5s = 10r - 1

s = 2r - 1/5

Since s and r are both integers, this is a contradiction. Therefore, there is no integer r that satisfies the equation above, and m is not in A. Thus, we have shown that B is not a subset of A.

c. Since we have shown that A is a subset of B and that B is not a subset of A, we can conclude that A and B are not equal, i.e., A ≠ B.

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Answer: x =11

Step-by-step explanation:

To find a polynomial function of the lowest degree with rational coefficients and given zeros -3i and 5, we first need to remember that complex zeros always come in conjugate pairs. Since -3i is one of the zeros, its conjugate 3i is also a zero.

Now, let's find the polynomial using these zeros: (x - (-3i))(x - 3i)(x - 5). We can rewrite this as:

(x + 3i)(x - 3i)(x - 5)

Now, let's multiply the first two factors:

(x^2 - 3ix + 3ix + 9) (x - 5)

Simplifying this gives us:

(x^2 + 9)(x - 5)

Now, let's multiply this with the remaining factor:

x^3 - 5x^2 + 9x - 45

So, the polynomial function of the lowest degree with rational coefficients that has the given zeros -3i and 5 is:

f(x) = x^3 - 5x^2 + 9x - 45

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Answer:

y = -3(x - 5)^2 + 3

Step-by-step explanation:

Because we're given the maximum/vertex of the quadratic function and at least one of the roots, we can find the equation of the quadratic equation using the vertex form which is

[tex]y = a(x-h)^2+k[/tex], where a is a constant (determine whether parabola will have maximum or minimum), (h, k) is the vertex (a maximum for this problem), and (x, y) are any point on the parabola:

Since our maximum/vertex is (5, 3), and one of our roots is (6, 0), we can plug everything in and solve for a:

[tex]0=a(6-5)^2+3\\0=a(1)^2+3\\0=a+3\\-3=a[/tex]

Thus, the general equation (without distribution) is y = -3(x - 5)^2 + 3

The polynomial for the area of the square is A(x) = x^2

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

Shape = square

Side length = x

The area of the square is

Area = Side length^2

Substitute the known values in the above equation, so, we have the following representation

Area = x^2

Express as a function

A(x) = x^2

Hence, the function is A(x) = x^2

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The square root of 41 must lie between 6 and 7, as 6² = 36 and 7² = 49, and 41 lies between these two values.

The estimate of a non-exact square root of x is done finding two numbers, as follows:

For the number 41, these numbers are given as follows:

Hence we know that the square root of 41 lies between 6 and 7.

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To find P(A | B),  first we use the formula P(A | B) = P(A ∩ B) / P(B).To find P(B | A), we use the formula P(B | A) = P(A ∩ B) / P(A).  To determine if A and B are independent, we need to compare P(A ∩ B) with P(A)P(B). If P(A ∩ B) = P(A)P(B), then A and B are independent. If P(A ∩ B) ≠ P(A)P(B), then A and B are dependent.

a. Plugging in the given values, we have:
P(A | B) = 0.20 / 0.50 = 0.40
So, P(A | B) = 0.4000 (to 4 decimals).

b. Plugging in the given values, we have:
P(B | A) = 0.20 / 0.50 = 0.40
So, P(B | A) = 0.4000 (to 4 decimals).

c. Given values are:
P(A) = 0.50
P(B) = 0.50
P(A ∩ B) = 0.20
Calculating P(A) * P(B):
0.50 * 0.50 = 0.25

Since P(A ∩ B) ≠ P(A) * P(B), events A and B are not independent. The occurrence of one event affects the probability of the other event occurring.

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Answer:

Your friend left the bigger tip.

Step-by-step explanation:

We have to first find how much you tipped the waiter, 15%.

15% of 15.85 is 2.4.  So, you tipped the waiter $2.40 (about)

Your friend leaves a 20% tip, so we have to also find 20% of $14.30

20% of 14.30 is 2.86.  So your friend tipped $2.86 to the waiter.

Your friend left the bigger tip.

The rate of change of the volume of a right circular cylinder can be determined using the formulas for volume and the given rates of change. The correct answer is 1. rate = - 22 pi cu. in./min.

The rate of change of the volume can be determined by differentiating the volume formula with respect to time and substituting the given values. The volume of a right circular cylinder is given by V = πr^2h, where r is the radius and h is the height.

Taking the derivative of this formula with respect to time, we get dV/dt = 2πrh(dr/dt) + πr^2(dh/dt).

Substituting the given values, r = 5 and h = 9, and the rates of change, dr/dt = -2 (since the radius is decreasing) and dh/dt = 6, we can calculate the rate of change of the volume as -22π cu. in./min.

To understand why the answer is -22π cu. in./min, let's break down the calculation. We start with the volume formula for a cylinder, V = πr^2h. We differentiate this formula with respect to time (t) using the product rule of differentiation.

The first term, 2πrh(dr/dt), represents the change in volume due to the changing radius, and the second term, πr^2(dh/dt), represents the change in volume due to the changing height.

Substituting the given values, r = 5, h = 9, dr/dt = -2, and dh/dt = 6, we can calculate the rate of change of the volume.

Plugging in these values, we have dV/dt = 2π(5)(9)(-2) + π(5^2)(6) = -180π + 150π = -30π cu. in./min.

Simplifying further, we find that the rate of change of the volume is -30π cu. in./min.

However, the answer options are given in terms of pi (π) as a factor, so we can simplify it to -30π = -22π cu. in./min. Therefore, the correct answer is -22π cu. in./min.

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Let X have binomial distribution b(x; 30,0.3), then P(X = 10) = 3.9 x 10^-6.

To find the probability of a specific value for X in a binomial distribution, we can use the formula:

P(X = x) = (n choose x) * p^x * (1-p)^(n-x)

where n is the number of trials, p is the probability of success in each trial, x is the number of successes we are interested in, and (n choose x) is the binomial coefficient.

In this case, we are given that X has a binomial distribution with parameters n = 30 and p = 0.3, and we want to find P(X = 10). Plugging these values into the formula, we get:

P(X = 10) = (30 choose 10) * 0.3^10 * 0.7^20

Using a calculator or software, we can calculate:

(30 choose 10) = 30,045,015

0.3^10 ≈ 0.000005

0.7^20 ≈ 0.026

Therefore,

P(X = 10) ≈ 30,045,015 * 0.000005 * 0.026 ≈ 3.9 x 10^-6

So the probability of getting exactly 10 successes in 30 trials with a success probability of 0.3 is approximately 3.9 x 10^-6.

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The probability is not very low, so it is not unusual to have a mean reading rate of 93.9 wpm. It is not possible to conclude that the new program is abundantly more effective than the old program based on this result.

A mean reading rate of 93.9 wpm is not unusual since the probability of obtaining a result of 93.9 wpm or more is 0.3446. The new program is not abundantly more effective than the old program.

Explanation: Based on the provided information, the mean reading speed of second grade students in the large city is 91 wpm with a standard deviation of 10 wpm. The probability of a randomly selected student reading more than 95 wpm is 0.3446. Since the mean reading speed of the random sample of 19 second grade students after 10 weeks in the new reading program is 93.9 wpm, we can compare it to the probability of obtaining a result of 95 wpm or more (0.3446). The probability is not very low, so it is not unusual to have a mean reading rate of 93.9 wpm.

Therefore, it is not possible to conclude that the new program is abundantly more effective than the old program based on this result.

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The table of values of the equation h(x) = -4x + 5 is

x y

-2  13

-1   9

0   5

1   1

Given that

h(x) = -4x + 5

To find the values that complete the table for the given equation h(x) = -4x + 5, we substitute each value of x and evaluate y.

So the completed table is:

x    y

-2  13

-1   9

0   5

1   1

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Answer:Sorry i cant see the answer

Step-by-step explanation:

We can check that the probabilities sum to 1:

P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X

The sum X of rolling an 8-sided die and a 6-sided die can take values from 2 to 14.

To find the probability distribution of X, we need to calculate the probability of each possible outcome.

For example, to get a sum of 2, we must roll a 1 on the 8-sided die and a 1 on the 6-sided die. The probability of rolling a 1 on an 8-sided die is 1/8, and the probability of rolling a 1 on a 6-sided die is 1/6. Therefore, the probability of getting a sum of 2 is:

P(X=2) = (1/8) * (1/6) = 1/48

Similarly, we can calculate the probabilities for all possible values of X:

P(X=3) = (1/8) * (2/6) + (2/8) * (1/6) = 1/16

P(X=4) = (1/8) * (3/6) + (2/8) * (2/6) + (3/8) * (1/6) = 1/8

P(X=5) = (1/8) * (4/6) + (2/8) * (3/6) + (3/8) * (2/6) + (4/8) * (1/6) = 5/32

P(X=6) = (1/8) * (5/6) + (2/8) * (4/6) + (3/8) * (3/6) + (4/8) * (2/6) + (5/8) * (1/6) = 11/32

P(X=7) = (1/8) * (6/6) + (2/8) * (5/6) + (3/8) * (4/6) + (4/8) * (3/6) + (5/8) * (2/6) + (6/8) * (1/6) = 21/32

P(X=8) = (2/8) * (6/6) + (3/8) * (5/6) + (4/8) * (4/6) + (5/8) * (3/6) + (6/8) * (2/6) = 25/32

P(X=9) = (3/8) * (6/6) + (4/8) * (5/6) + (5/8) * (4/6) + (6/8) * (3/6) = 19/32

P(X=10) = (4/8) * (6/6) + (5/8) * (5/6) + (6/8) * (4/6) = 13/32

P(X=11) = (5/8) * (6/6) + (6/8) * (5/6) = 11/32

P(X=12) = (6/8) * (6/6) = 3/8

P(X=13) = 0

P(X=14) = 0

We can check that the probabilities sum to 1:

P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X

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