Suppose that we have a bulbs box containing 60 bulbs, of which 13 are defective. 2 bulbs are slected at random, with replacement from the box (Round your answer to three decimals) A) Find the probability that both bulbs are defective. B) Find the probability that atleast one of them is defective.

A.  P(x > 19) is also approximately 0.0228.

B. P(x < 13) is also approximately 0.1587.

C. P(15.58 < x < 19.58) is also approximately 0.4893.

D. P(10.28 ≤ x ≤ 17.94) is also approximately 0.8226.

a. P(x>19):

We need to standardize the variable x using the z-score formula:

z = (x - µ) / σ

Substituting the values we get,

z = (19 - 15) / 2 = 2

Using a standard normal distribution table or calculator, we find that P(z > 2) is approximately 0.0228. Therefore, P(x > 19) is also approximately 0.0228.

b. P(x < 13):

Again, we use the z-score formula:

z = (x - µ) / σ

Substituting the values we get,

z = (13 - 15) / 2 = -1

Using a standard normal distribution table or calculator, we find that P(z < -1) is approximately 0.1587. Therefore, P(x < 13) is also approximately 0.1587.

c. P(15.58 < x < 19.58):

We need to standardize both values of x using the z-score formula:

z1 = (15.58 - 15) / 2 = 0.29

z2 = (19.58 - 15) / 2 = 2.29

Using a standard normal distribution table or calculator, we find that P(0 < z < 2.29) is approximately 0.9893 - 0.5 = 0.4893. Therefore, P(15.58 < x < 19.58) is also approximately 0.4893.

d. P(10.28 ≤ x ≤ 17.94):

We standardize both values of x using the z-score formula:

z1 = (10.28 - 15) / 2 = -2.36

z2 = (17.94 - 15) / 2 = 0.97

Using a standard normal distribution table or calculator, we find that P(-2.36 ≤ z ≤ 0.97) is approximately 0.8325 - 0.0099 = 0.8226. Therefore, P(10.28 ≤ x ≤ 17.94) is also approximately 0.8226.

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We cannot determine the percentage of pegilar grade gasoline sales between 3.27 and 53.63 per gallon or the percentage of regular gasoline sale price > 3.33/gallon as the total sales for both are not provided.

Given data:Pegilar grade gasoline sales between 3.27 and 53.63 per gallon.

Percentage of pegilar grade gasoline sale between 3.27 and 53.63 per gallon can be calculated as:X %.

Therefore,X% = (Sale between 3.27 and 53.63 per gallon) / Total sales * 100.

However, the total sales are not provided so we cannot calculate the percentage.

Further information is required.Similarly, for the second part, given data is:Regular gasoline sale price > 3.33/gallon.

Percentage of regular gasoline sale price > $3.33/gallon can be calculated as:Y %.

Therefore,Y % = (Regular sale price > $3.33/gallon) / Total sales * 100.

However, the total sales are not provided so we cannot calculate the percentage. Further information is required.

To summarize, we cannot determine the percentage of pegilar grade gasoline sales between 3.27 and 53.63 per gallon or the percentage of regular gasoline sale price > 3.33/gallon as the total sales for both are not provided.

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The probability that exactly 11 of 37 randomly selected college students major in STEM can be calculated using the binomial probability formula, which is:

P(X = k) = (n choose k) * p^k * q^(n-k)Where:

P(X = k) is the probability of k successesn is the total number of trials (37 in this case)k is the number of successes (11 in this case)

p is the probability of success (30%, or 0.3, in this case)q is the probability of failure (100% - p, or 0.7, in this case)(n choose k) is the binomial coefficient, which can be calculated using the formula

:(n choose k) = n! / (k! * (n-k)!)where n! is the factorial of n, or the product of all positive integers from 1 to n.

The calculation of the probability of exactly 11 students majoring in STEM is therefore:P(X = 11)

= (37 choose 11) * (0.3)^11 * (0.7)^(37-11)P(X = 11) ≈ 0.200

So the probability that exactly 11 of the 37 randomly selected college students major in STEM is approximately 0.200 or 20%.

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The graph G can be represented by the shorthand notation G = (V, E), where V is the set of vertices and E is the set of edges.

To write the mathematical description of G4 in terms of (V, E), we need to consider the graph G with four iterations. It can be denoted as G4 = (V4, E4), where V4 is the set of vertices in the fourth iteration and E4 is the set of edges in the fourth iteration.

The adjacency matrix A of graph G represents the connections between vertices. It is a square matrix where the entry A[i][j] is 1 if there is an edge between vertices i and j, and 0 otherwise.

To calculate  [tex]A^2[/tex], we need to multiply the adjacency matrix A with itself. The resulting matrix represents the number of paths of length 2 between vertices.

To find the number of paths of length 2 from vertex 0 to vertex 1, we can look at the entry  [tex]A^2[/tex][0][1]. The value of this entry indicates the number of paths of length 2 from vertex 0 to vertex 1. Similarly, we can determine the number of paths of length 2 from vertex 0 to vertex 2 by examining the entry  [tex]A^2[/tex][0][2].

In summary, the shorthand notation for the graph G is G = (V, E). The mathematical description of G4 is G4 = (V4, E4). The adjacency matrix A represents the connections between vertices in G. To calculate [tex]A^2[/tex], we multiply A with itself. The number of paths of length 2 from vertex 0 to vertex 1 is determined by the entry  [tex]A^2[/tex][0][1], and the number of paths of length 2 from vertex 0 to vertex 2 is determined by the entry [tex]A^2[/tex][0][2].

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The equation of the tangent plane to the surface z = e^(3x/17)ln(4y) at the point (1, 3, 2.96449) is:  z - 2.96449 = (3/17)e^(3/17)(x - 1)ln(4)(y - 3).

To find the equation of the tangent plane, we need to compute the partial derivatives of the given surface with respect to x and y. Let's denote the given surface as f(x, y) = e^(3x/17)ln(4y). The partial derivatives are:

∂f/∂x = (3/17)e^(3x/17)ln(4y), and

∂f/∂y = e^(3x/17)(1/y).

Evaluating these partial derivatives at the point (1, 3), we get:

∂f/∂x (1, 3) = (3/17)e^(3/17)ln(12),

∂f/∂y (1, 3) = e^(3/17)(1/3).

Using these values, we can construct the equation of the tangent plane using the point-normal form:

z - 2.96449 = [(3/17)e^(3/17)ln(12)](x - 1) + [e^(3/17)(1/3)](y - 3).

Simplifying this equation further will yield the final equation of the tangent plane.

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The equation of the plane passing through the point (0,0,0) with normal vector n=i+j+k is x + y + z = 0

The equation of a plane can be determined when the normal vector and a point on the plane are known. Given that the point (0,0,0) lies on the plane and its normal vector is n = i + j + k, the equation of the plane can be determined as follows:

Step-by-step solution:

Let the equation of the plane be Ax + By + Cz + D = 0

where A, B, C, and D are constants to be determined and (x, y, z) is a point on the plane.

The normal vector of the plane is given as n = i + j + k. This vector is perpendicular to every vector lying on the plane.

Now let's take a point on the plane, say (0, 0, 0).

This vector is parallel to the plane, so its dot product with the normal vector of the plane should be zero.i.e.

0 + 0 + 0 = (0)(1) + (0)(1) + (0)(1)

This gives us: 0 = 0. Hence, the point (0,0,0) satisfies the equation of the plane.

Substituting these values into the equation of the plane, we get:

A(0) + B(0) + C(0) + D = 0

Simplifying, we obtain:

D = 0

Therefore, the equation of the plane is Ax + By + Cz = 0, where A, B, and C are constants to be determined and (x, y, z) is a point on the plane.

Now let's find the values of A, B, and C. To do so, we need to find another point on the plane.

Since the normal vector of the plane is i + j + k, we can choose another point with coordinates that are multiples of the coefficients of this vector. Let's choose the point (1,1,1).

Substituting (1,1,1) into the equation of the plane, we get:

A(1) + B(1) + C(1) = 0

Simplifying, we get:

A + B + C = 0

Therefore, the equation of the plane passing through the point (0,0,0) with normal vector n=i+j+k is x + y + z = 0

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The margin of error (E) can be calculated using the formula [tex]E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}}$[/tex], where [tex]z_{\frac{\alpha}{2}}$[/tex] is the z-value with a cumulative probability of -2.33. Using the standard normal distribution table, the z-value corresponding to 0.01 is -2.33. The margin of error (E) is 0.0736, allowing for a 95% confidence interval for the true population proportion (p) using the normal approximation to the binomial distribution.

The formula to calculate the margin of error in this case is given by the formula below: [tex]$E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}}$[/tex],

where [tex]$z_{\frac{\alpha}{2}}$[/tex] is the z-value with a cumulative probability of [tex]$\frac{\alpha}{2}$, $p^*$[/tex]

is the sample proportion, and n is the sample size. Now, given that p^ = 0.8, n = 120 and α = 0.02, we can calculate the margin of error (E) as follows:

Firstly, we need to find the z-value with a cumulative probability of

[tex]$\frac{\alpha}{2}$ or $\frac{0.02}{2}[/tex] = 0.01

in the standard normal distribution table. The z-value corresponding to 0.01 is -2.33. Then, substituting these values into the formula above we get:

[tex]$$E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}} = -2.33\sqrt{\frac{0.8(1-0.8)}{120}}$$ $$E = 0.0736$$[/tex]

Therefore, the margin of error (E) is 0.0736. This means that we can construct a confidence interval for the true population proportion (p) with 95% confidence using the formula below[tex]:$$CI = \left(p^ - E, p^ + E \right)$$[/tex] Where p^ is the sample proportion. Now substituting the values given above we get:[tex]$$CI = \left(0.8 - 0.0736, 0.8 + 0.0736 \right)$$ $$CI = (0.7264, 0.8736)$$[/tex]

Hence, the 95% confidence interval for the true population proportion (p) is (0.7264, 0.8736). We used the normal approximation to the binomial distribution since the sample size is large enough.

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h(x, y) is a Lipschitzian function with Lipschitz constant K = K1 * K2.

a) To prove that h(x, y) = f(x, y) + g(x, y) is a Lipschitzian function, we need to show that there exists a constant K such that for any two points (x1, y1) and (x2, y2), the following inequality holds:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

where || (x1, y1) - (x2, y2) || represents the Euclidean distance between the points (x1, y1) and (x2, y2).

Since f(x, y) and g(x, y) are Lipschitzian functions, we know that there exist constants K1 and K2 such that:

| f(x1, y1) - f(x2, y2) | ≤ K1 * || (x1, y1) - (x2, y2) ||  ... (1)

| g(x1, y1) - g(x2, y2) | ≤ K2 * || (x1, y1) - (x2, y2) ||  ... (2)

Now, let's consider the difference h(x1, y1) - h(x2, y2):

h(x1, y1) - h(x2, y2) = [f(x1, y1) + g(x1, y1)] - [f(x2, y2) + g(x2, y2)]

                     = [f(x1, y1) - f(x2, y2)] + [g(x1, y1) - g(x2, y2)]

Using the triangle inequality, we have:

| h(x1, y1) - h(x2, y2) | ≤ | f(x1, y1) - f(x2, y2) | + | g(x1, y1) - g(x2, y2) |

Applying inequalities (1) and (2), we get:

| h(x1, y1) - h(x2, y2) | ≤ K1 * || (x1, y1) - (x2, y2) || + K2 * || (x1, y1) - (x2, y2) ||

Since K = K1 + K2, we can rewrite the above inequality as:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

Therefore, h(x, y) is a Lipschitzian function with Lipschitz constant K.

b) To prove that h(x, y) = f(x, g(x, y)) is a Lipschitzian function, we need to show that there exists a constant K such that for any two points (x1, y1) and (x2, y2), the following inequality holds:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

Let's consider the difference h(x1, y1) - h(x2, y2):

h(x1, y1) - h(x2, y2) = f(x1, g(x1, y1)) - f(x2, g(x2, y2))

Since f(x, y) is a Lipschitzian function, we know that there exists a constant K1 such that:

|

f(x1, g(x1, y1)) - f(x2, g(x2, y2)) | ≤ K1 * || (x1, g(x1, y1)) - (x2, g(x2, y2)) ||

Now, let's consider the distance || (x1, y1) - (x2, y2) ||:

|| (x1, y1) - (x2, y2) || = || (x1, g(x1, y1)) - (x2, g(x2, y2)) ||

Since g(x, y) is a Lipschitzian function, we know that there exists a constant K2 such that:

|| (x1, g(x1, y1)) - (x2, g(x2, y2)) || ≤ K2 * || (x1, y1) - (x2, y2) ||

Combining these inequalities, we have:

| h(x1, y1) - h(x2, y2) | ≤ K1 * || (x1, g(x1, y1)) - (x2, g(x2, y2)) || ≤ K1 * K2 * || (x1, y1) - (x2, y2) ||

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