find the critical points for the function f(x,y)=x3 y3−9x2−3y−8

The numerical expressions are:

(2 + 4) x 7 = 42

(12 ÷ 3) x 5 = 20

A mathematical expression is made up of integers and mathematical operators including addition, multiplication, subtraction, and division.

A number can be expressed in numerous ways, including word form and numerical form.

A numerical expression is a mathematical statement that only contains numbers and one or more operation symbols. Addition, subtraction, multiplication, and division are examples of operation symbols. It can alternatively be expressed using the radical symbol (square root symbol) or the absolute value symbol.

The numerical expressions are:

(2 + 4) x 7 = 42

(12 ÷ 3) x 5 = 20

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Using the Central Limit Theorem, we have that:

a) Since there are at least 10 successes and 10 failures, the condition is met

b) Using the formula  [tex]$SE_{\hat{p}}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$[/tex]with n = 400 and p = 0.11, the standard error is of 0.0156.

(a) In order to apply the normal approximation to the binomial distribution, the sample size must be large enough such that np and n(1-p) are both greater than or equal to 10, where n is the sample size and p is the probability of success.

In this case, we have n=400 and the observed proportion of defective bulbs is [tex]$\hat{p}=44/400=0.11$[/tex].

Thus, np=4000.11=44 and n(1-p)=4000.89=356.6, which are both greater than 10. Therefore, it is reasonable to assume that the condition for obtaining an interval estimate for [tex]$p_x$[/tex]using the normal approximation is met.

(b) The standard error of the sample proportion [tex]$\hat{p}$[/tex]is given by:

[tex]$SE_{\hat{p}}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$[/tex]

Plugging in the values, we get:

[tex]$SE_{\hat{p}}=\sqrt{\frac{0.11(1-0.11)}{400}}\approx 0.0156$[/tex]

Therefore, the standard error of [tex]$p_x$[/tex]is approximately 0.0156.

It states that for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean  and standard deviation , as long as  and .

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Full Question: A large company produces an equal number of brand-mame lightbulbs and generic lightbulbs. The director of quality control sets guidelines that lightbulbs that are defect lightbulbs that are defective is not equal to the director wants to estimate the average of the two proportions. Production will be stopped if there is evidence that the proportion of all ve is greater than 0. 10. The director also believes that the proportion of brand-name proportion of generic lightbulbs that are defective. Therefore, the o estmate the proportion of brand-name lightbulbs that are defective, a simple random sample of 400 brand-name lightbulbs brand-name lightbulbs that are defective in a sample of 400, and let px represent the proportion of all brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.

(a) One condition for obtaining an interval estimate for px is that the distribution of px is approximately number of is taken and 44 are found to be defective. Let X represent the normal Is it reasonable to assume that the condition is met? Justify your answer.

(b) The suandard error of hr is approumately O 0156 Show how the value of the standard error is calculated

Answer: 8 to 40

6:30

2/10

Step-by-step explanation:To check if this is the answer you can multiply the first number of each of these ratios by 5. (We multiply them by five because it is the second number in the ratio 1:5)

8 times 5 equals 40 -so this one is right

6 times 5 equals 30 -yep this one is right too

2 times 5 equals 10 -yeperdoodle

Yeah..so these are the answers! let me know if this helps you out.

The value of n when t = 4 can be found using the given data points and an appropriate mathematical model. Since the problem does not specify the nature of the relationship between n and t, we will assume a linear relationship and use the slope-intercept form of a straight-line equation to find the value of n when t = 4.

First, we need to find the slope of the line. Using the two data points provided, we can calculate:

slope = (change in n) / (change in t) = (400 - 150) / (1 - 0) = 250

Next, we can use the point-slope form of a line equation to find the equation of the line:

n - 150 = 250(t - 0)
n = 250t + 150

Finally, we can substitute t = 4 into the equation to find the value of n:

n = 250(4) + 150 = 1150

Therefore, the value of n when t = 4 is 1150.

In summary, to find the value of n when t = 4, we assumed a linear relationship between n and t and used the two given data points to calculate the slope of the line. We then used the point-slope form of a line equation to find the equation of the line, and substituted t = 4 into the equation to find the value of n.

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(a) The set {(x1,x2)T|x1 + x2 = 0} is a subspace of R2.

To check whether the given set is a subspace of R2, we need to check whether it is closed under vector addition and scalar multiplication. Let u = (u1,u2)T and v = (v1,v2)T be two arbitrary vectors in the set, and let c be an arbitrary scalar. Then:

u + v = (u1 + v1, u2 + v2)

Since u1 + v1 + u2 + v2 = (u1 + u2) + (v1 + v2) = 0 + 0 = 0 (since u and v are in the set), we see that u + v is also in the set.

c*u = (c*u1, c*u2)

Since c*u1 + c*u2 = c*(u1 + u2) = c*0 = 0 (since u is in the set), we see that c*u is also in the set.

Therefore, the set {(x1,x2)T|x1 + x2 = 0} is a subspace of R2.

(b) It is not a subspace of R2

To check whether the given set is a subspace of R2, we need to check whether it is closed under vector addition and scalar multiplication.

Let u = (u1,u2)T and v = (v1,v2)T be two arbitrary vectors in the set, and let c be an arbitrary scalar. Then:

u + v = (u1 + v1, u2 + v2)

Since u21 = u22 and v21 = v22 (since u and v are in the set), we see that (u1 + v1)2 = (u2 + v2)2. Therefore, u + v is in the set.

c*u = (c*u1, c*u2)

Since u21 = u22 (since u is in the set), we see that (c*u1)2 = (c*u2)2. Therefore, c*u is in the set.

However, the set {(x1,x2)T|x21 = x22} is not a subspace of R2 because it does not contain the zero vector (0,0)T, which is required for any set to be a subspace.

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Answer:

y= -5(x+6)^2 +4

Step-by-step explanation:

The zoo keeper is 5.5 feet tall.

When the corresponding properties of two or more triangles are compared, and there is a common relations among them, then the triangles are said to be similar. Thus their corresponding sides may be compared in form of ratio.

In the question, the giraffe casts a shadow as given. Then by comparison, the height of the zoo keeper (h) can be determined as follows:

4/ 12 = h/ 16.5

12h = 4*16.5

      = 66

h = 66/ 12

  = 5.5

The zoo keeper is 5.5 feet tall.

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1. Affordably presented by objects or environments, such as a ball providing the opportunity to practice grasping, throwing, and catching.
2. Individuals perceive these affordances and decide to engage with them, based on their current motor abilities and developmental stage.
3. Through interacting with affordances, individuals practice and develop their motor skills by attempting, refining, and mastering the actions associated with affordance.

Affordably is a term used to describe the relationship between an individual's perception of their environment and their ability to interact with it. In terms of motor development, affordances refer to the opportunities for movement that the environment presents. These opportunities can be both physical and social and can include objects to manipulate, spaces to explore, and people to interact with.

The concept of affordance is important for motor development because it provides children with opportunities to practice and refine their motor skills. As children explore their environment, they are able to perceive the various affordances that it presents, and they can use these affordances to develop their motor skills.

For example, a child may perceive that a box can be used as a stepping stool, and they may use this affordance to climb up onto a table. In doing so, they are developing their balance, coordination, and strength. Similarly, a child may perceive that a ball can be thrown, caught, and bounced, and they can use these affordances to develop their hand-eye coordination, spatial awareness, and timing.

Overall, affordance plays an important role in motor development by providing children with opportunities to explore and interact with their environment and to develop their motor skills in the process.
Affordance contributes to motor development by providing opportunities for individuals to interact with their environment, which in turn helps them develop and refine their motor skills. Affordance refers to the potential actions or uses that an object or environment provides to an individual. In the context of motor development, affordances can be seen as opportunities for practicing and enhancing motor abilities.


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The statement "if B is a 3x3 matrix, and det(B) = -4, then det([tex]2BB^tB^{-1[/tex]) = -8" is false.

If B is a 3x3 matrix, and det(B) = -4, then det([tex]2BB^tB^{-1[/tex]) = -8. Here are the terms I will include in my answer: matrix, determinant, transpose, and inverse.

1. Determine det(B)
Given: det(B) = -4

2. Compute det(2B)
The determinant of a scalar multiple of a matrix is the scalar raised to the power of the matrix's dimension multiplied by the determinant of the matrix. Since the matrix is 3x3, we have:
det(2B) = ([tex]2^3[/tex]) * det(B) = 8 * (-4) = -32

3. Compute det([tex]B^t[/tex])
The determinant of a matrix and its transpose are equal, so:
det([tex]B^t[/tex]) = det(B) = -4

4. Compute det([tex]B^{-1[/tex])
For an invertible matrix, the determinant of its inverse is the reciprocal of the determinant:
det([tex]B^{-1[/tex]) = 1/det(B) = 1/(-4) = -1/4

5. Calculate det([tex]2BB^tB^{-1[/tex])
Using the property of determinants that det(AB) = det(A) * det(B), we have:
det([tex]2BB^tB^{-1[/tex]) = det(2B) * det([tex]B^t[/tex]) * det([tex]B^{-1[/tex]) = -32 * (-4) * (-1/4) = -32

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To find T, N, and k for the space curve r(t) = -7ti - (7a cosh(t/a))j with a > 0.6, follow these steps:

1. Calculate the first derivative, r'(t), to find the tangent vector T:
r'(t) = -7i - (7/a sinh(t/a))j
To find the unit tangent vector T, normalize r'(t):
T = r'(t) / ||r'(t)||

2. Calculate the second derivative, r''(t), to find the normal vector N:
r''(t) = - (7/a² cosh(t/a))j
To find the unit normal vector N, normalize r''(t):
N = r''(t) / ||r''(t)||

3. Calculate the curvature k:
k = ||r'(t) x r''(t)|| / ||r'(t)||³

In summary, find the first and second derivatives of r(t), normalize them to get T and N, and compute the curvature k using the given formula.

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There is not sufficient evidence to conclude that infestation by spider mites induces resistance to wilt disease caused by the Verticillium fungus in cotton plants at the 1% level.

The question being investigated is whether infestation by spider mites can induce resistance to wilt disease caused by the fungus Verticillium in cotton plants. The experiment involved two groups of cotton plants - one group was infested with spider mites, while the other group served as controls. After two weeks, the mites were removed and both groups were inoculated with the Verticillium fungus. The number of plants that developed symptoms of wilt disease was recorded for each group.

To test whether infestation by spider mites can induce resistance to wilt disease, we can use a hypothesis test. The null hypothesis (H0) is that there is no difference in the proportion of plants that develop wilt disease between the mites and no mites groups, while the alternative hypothesis (Ha) is that the mites group has a lower proportion of plants with wilt disease compared to the no mites group.

We can use a chi-square test for independence to determine whether the data provide sufficient evidence to reject the null hypothesis at the 1% level. The test statistic is calculated as follows:

chi-square = (ad - bc)^2 / [(a+b)(c+d)]

where a = number of plants in the mites group with wilt disease, b = number of plants in the mites group without wilt disease, c = number of plants in the no mites group with wilt disease, and d = number of plants in the no mites group without wilt disease.

Using the data from the table, we can calculate the test statistic as follows:

chi-square = (14*28 - 18*26)^2 / [(14+18)(28+26)] = 1.079

The degrees of freedom for the chi-square test is (2-1)*(2-1) = 1. The critical value of chi-square at the 1% level with 1 degree of freedom is 6.635.

Since the calculated chi-square value (1.079) is less than the critical value (6.635), we fail to reject the null hypothesis.

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The solution to the equation for x is given as follows:

x = 2.92.

The equation for x in this problem is solved applying the proportions of the problem.

The equivalent side lengths are given as follows:

Hence the proportional relationship to obtain the value of x is given as follows:

27/21 = (9x - 19)/(-9x + 83)

Applying cross multiplication, we obtain the value of x as follows:

21(9x - 19) = 27(-9x + 83)

432x = 1263

x = 1263/432

x = 2.92.

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(112)3 =336

(211)3 =633

sum of the numbers:

(112)3 +(211)3=

336+633=

969

product of the numbers:

(112)3 × (211)3=

336×633=

212688

If N = 25 and K = 5, the degrees of freedom (df) you would use is C. 5,20.

This is because the formula for degrees of freedom in this case is (K-1)(N-1), which gives us (5-1)(25-1) = 4x24 = 96, and we divide by the total sample size (N) to get 96/25 = 3.84.

Since we cannot use a decimal for degrees of freedom, we round down to the nearest whole number, which gives us 3.

Therefore, the degrees of freedom for this scenario is 5-1 = 4 for the numerator and 25-1 = 24 for the denominator, which gives us a final answer of C. 5,20.

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Ans: B (=20)

p/s: sorry i use my calculator :')))) bc it's too long. you can do it by substituting the x values of each one according to the answer into the given equation.If there are any mistakes, please forgive me :'))))

Ok done. Thank to me >:333

Answer:

One number is 6 and the other number is 4

Step-by-step explanation:

Helping in the name of Jesus.

The statement is true. If B is obtained from A by row operations, the rows of B are linear combinations of the rows of A and vice-versa. (D)

When two matrices A and B are row equivalent, it means they can be obtained from each other through a series of elementary row operations. These row operations include row swapping, row multiplication by a nonzero scalar, and adding/subtracting multiples of one row to another row.

Since row operations preserve the row space of a matrix, the row spaces of A and B remain the same throughout these operations. In other words, the rows of B are linear combinations of the rows of A and vice-versa.

Thus, if A and B are row equivalent, their row spaces are the same, as the span of the rows in both matrices is identical. This supports the statement's truth that if A and B are row equivalent, their row spaces are the same.(D)

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Question: what do negative exponents do?

Answer: (B)

Step-by-step explanation: I believe

We use integration by calculating the area of the elliptical cross-section and the height of the wedge. Setting up the integral, solving it using u-substitution, and simplifying it, the volume of the wedge is 5000*pi/243 cubic units.

To find the volume of the wedge cut from the elliptical cylinder x^2 + 9y^2 = 25 by the planes z = 0 and z = 3x that is above the xy-plane, we first need to visualize the shape. The elliptical cylinder is a three-dimensional shape that looks like a stretched-out cylinder, with an elliptical cross-section. The plane z = 0 is the xy-plane, which is the flat surface at the bottom of the cylinder. The plane z = 3x is a diagonal plane that intersects the cylinder at an angle. The wedge that we need to find the volume of is the portion of the cylinder that is above the xy-plane and below the plane z = 3x.

To find the volume of this wedge, we need to use integration. We will integrate over the x and y dimensions to find the volume of the shape. We start by finding the limits of integration. The elliptical cylinder has a horizontal axis of length 5 (the square root of 25) and a vertical axis of length 5/3 (the square root of 25/9). We can use these dimensions to set the limits of integration. We will integrate over the x dimension from -5/3 to 5/3 (the limits of the elliptical cross-section) and over the y dimension from -sqrt((25-x^2)/9) to sqrt((25-x^2)/9) (the limits of the elliptical cross-section at each value of x).

Now we need to set up the integral to find the volume. The volume of a wedge can be calculated using the formula V = (1/3)Bh, where B is the area of the base and h is the height of the wedge. In this case, the base is the elliptical cross-section and the height is the distance between the planes z = 0 and z = 3x.

The area of the elliptical cross-section at each value of x and y is given by A = pi * x * 3y. The height of the wedge at each value of x and y is given by h = 3x. So we can set up the integral as follows:

V = integral from -5/3 to 5/3 (integral from -sqrt((25-x^2)/9) to sqrt((25-x^2)/9) of (1/3) * pi * x * 3y * 3x dy) dx

Simplifying this integral, we get:

V = (pi/3) * integral from -5/3 to 5/3 (integral from -sqrt((25-x^2)/9) to sqrt((25-x^2)/9) of 9x^2y dy) dx

Integrating over y, we get:

V = (pi/3) * integral from -5/3 to 5/3 of 9x^2 * [(sqrt((25-x^2)/9))^2 - (-sqrt((25-x^2)/9))^2] dx

Simplifying this integral, we get:

V = (10*pi/9) * integral from -5/3 to 5/3 of x^2 * (25-x^2)^(1/2) dx

This integral can be solved using a u-substitution, where u = 25-x^2 and du/dx = -2x. We get:

V = (10*pi/27) * integral from 0 to 25 of u^(1/2) du

Simplifying this integral, we get:

V = (100*pi/81) * (u^(3/2)/3)| from 0 to 25

V = (100*pi/81) * (125/3)

V = 5000*pi/243

Therefore, the volume of the wedge cut from the elliptical cylinder x^2 + 9y^2 = 25 by the planes z = 0 and z = 3x that is above the xy-plane is 5000*pi/243 cubic units.

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Answer:

It is 252

Step-by-step explanation:

Just multiply the base and the height ;-;

The number of ways that endpoint P that is perpendicular to line n is One way.

In a plane, two lines are parallel if they lie in the same plane and never intersect, no matter how far they are extended. If a point is between two parallel lines, it lies on a line that is perpendicular to both of the parallel lines.

We can draw a line that passes through endpoint P and is perpendicular to line n. This line will intersect line m at a right angle. Since there is only one line that passes through a point and is perpendicular to another line, there is only one line that can be drawn from endpoint P that is perpendicular to line n.

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Taking the difference between the ages, we can see that  her grandmother is 56 years older than her.

To find how many years older his her grandmother, we just need to take the difference between both of their ages. (remember that a difference is just a subtraction)

Then we will take the age of the grandmother and we will subtract the age of Anita.

We will get the difference:

D = 68 - 12

D = 56

We can see that her grandmother is 56 years older than her.

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In mathematics, these extreme probabilities are expressed as 0 (impossible) and 1 (certain). This means a probability number is always a number from 0 to 1.

Probability can also be written as a percentage, which is a number from 0 to 100 percent.

a) The null space of A is a subspace of the 7-dimensional vector space R^7, and its dimension is 3.

b) The column space of A is a subspace of the 5-dimensional vector space R^5.

The null space of a matrix is the subspace of the vector space in which the matrix operates. In this case, since A is a 5x7 matrix, its null space is a subspace of the 7-dimensional vector space R^7.

a) The dimension of the null space can be found using the rank-nullity theorem, which states that the dimension of the null space plus the rank of the matrix equals the number of columns. Since the rank of A is 4 and it has 7 columns, we have:

dim(null space) + rank(A) = number of columns

dim(null space) + 4 = 7

dim(null space) = 3

Therefore, the null space of A is a subspace of R^7 with dimension 3.

b) The column space of a matrix is the subspace of the vector space generated by the columns of the matrix. In this case, since A is a 5x7 matrix, its column space is a subspace of the 5-dimensional vector space R^5. This is because the columns of A are vectors in R^5. Therefore, the column space of A is a subspace of R^5.

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The package weighs 6.86 102 N.

We must apply the formula for the mechanical advantage of a lever in order to get the weight of the crate:

Input force minus output force is the mechanical advantage.

When the weight of the crate acts as the output force and the force supplied to the lever acts as the input force.

If we rearrange the formula, we obtain:

Mechanical advantage times input force equals output force.

Inputting the values provided yields:

Output Force is equal to 3.43 x 2.0 102 N.

Force at output: 6.86 102 N.

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We need to check the success/failure condition to ensure that the sampling distribution is approximately normal.

For the sampling distribution of S², we need to assume that the population follows a normal distribution in order to convert S² to a chi-square random variable.

To determine how many people we need to sample to reduce the standard deviation of the mean to 2.6, we found n=217.7515.

To use the t-distribution when finding P(Xˉ < some number), we need to assume that the population is normally distributed or approximately normal.

(c) In the Central Limit Theorem for 1 Proportion, we need to check the success/failure condition to ensure that the sampling distribution is approximately normal. This is because the theorem states that as the sample size increases, the sampling distribution of the proportion approaches a normal distribution, provided that the success/failure condition (np ≥ 10 and n(1-p) ≥ 10) is met. This allows us to make valid inferences about the population proportion.

(d) For the sampling distribution of S², we need to assume that the population follows a normal distribution in order to convert S² to a chi-square random variable. This is because the chi-square distribution is derived from the normal distribution, and using it assumes that the underlying population is normally distributed.

(e) To determine how many people we need to sample to reduce the standard deviation of the mean to 2.6 from a sample size of 92 with a standard deviation of 4, we found n=217.7515. Since we cannot sample a partial person, we need to round up to the nearest whole number, which is 218 people.

(f) To use the t-distribution when finding P(Xˉ < some number), we need to assume that the population is normally distributed or approximately normal. This is important because the t-distribution is derived from the normal distribution and is used when estimating population parameters, especially when the sample size is small and the population standard deviation is unknown.

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Answer: 33 units²

Step-by-step explanation:

      We have four triangles that we will find the area for. Since we have two sets of equivalent triangles, we will use A = BH for both different sizes, but we won't divide by two (since we have two).

A = BH

A = (3)(9)

A = 27 units²

A = BH

A = (3)(2)

A = 6 units²

      Now, we will add these two sets of triangles together.

A = 27 units² + 6 units²

A = 33 units²

(a) To be a probability density function, f(x) must satisfy two conditions: f(x) ≥ 0 for all x. The total area under the curve of f(x) must be equal to 1.

We have:[tex]f(x) = c/(1 + x^2)[/tex]

For f(x) to be non-negative, we need c > 0. To find the value of c such that the total area under the density function of f(x) is equal to 1, we integrate f(x) from −∞ to +∞ and set the result equal to 1:

∫(−∞ to +∞) f(x) dx = ∫(−∞ to +∞) c/(1 + x^2) dx = cπ = 1

Therefore, c = 1/π, and f(x) = 1/(π(1 + x^2)) is a probability density function.

(b) We want to find [tex]P(−9 < X < 9) for X ~ f(x) = 1/(π(1 + x^2))[/tex]

Using the cumulative distribution function (CDF), we have:

[tex]F(x) = P(X ≤ x) = ∫(−∞ to x) f(t) dt = ∫(−∞ to x) 1/(π(1 + t^2)) dt[/tex]

[tex]= (1/π) tan^−1(x) + (1/2)[/tex]

So, using the CDF, we have:

[tex]P(−9 < X < 9) = F(9) − F(−9) =[/tex] [tex][tan^−1(9)/π + 1/2] − [tan^−1(−9)/π + 1/2][/tex]

=[tex][tan^−1(9) − tan^−1(−9)]/π[/tex]

=[tex](1/π) tan^−1(9/−1)[/tex]

= 0.499 (rounded to three decimal places)

Therefore, P[tex](−9 < X < 9) ≈ 0.499.[/tex]

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Answer:

The plane should enter the clouds in about 3.3 hours; the plane should exit the cloud in about 6.0 hours

Step-by-step explanation:

I also made a table using your data in Desmos Graphic Calculator and the quadratic regression equation it gave me was

[tex]y=-15.3604x^2+141.912x+0.56701[/tex]

If your Desmos looks like mine, your table should be in box 1 (y1), and something like

[tex]y_{1}[/tex] ~ [tex]ax^2_{1}+bx_{1}+c[/tex] and it says STATISTICS, RESIDUALS, PARAMETERS (all this should be in the y2 box and it should graph the parabola itself)

First, under this equation, type y = 300 in the fourth box (y4)

Second, click on the wrench on the right and for your x axis, use -2.585 < 12.415 and for your y axis

Third, hover your mouse over the first spot where you parabola and your y = 300 line intersect.  You should see that the intersection coordinate is (3.261, 300)

Because the parabola points up at this first intersection, we know the plane is travelling upward, so this first coordinate is when the plane enters the clouds.  And rounding 3.261 to the nearest tenth gives us 3.3 hours

Fourth, hover your mouse over the second spot where your parabola and your y = 300 line intersect.  You should see that the intersection coordinates are (5.978, 300)

Because the parabola points down at this second intersection point, we know that plane is travelling downward, so this second coordinate is when the plane exits the clouds.  And rounding to the nearest tenth gives us 6.0 hours

If you can't figure out how to made the quadratic regression equation, I attached a picture of the Desmos graph I used for your question.

The margin of error at a 90% confidence level is approximately 0.053.

To find the margin of error at a 90% confidence level, we first need to calculate the standard error, which is the standard deviation of the sampling distribution of the proportion:

SE = [tex]\sqrt{[p(1-p)/n]}[/tex]
where p is the estimated proportion (p-hat) and n is the sample size.

Plugging in the values, we get:

SE = sqrt[0.24(1-0.24)/340] ≈ 0.032

Next, we use the formula for margin of error at a 90% confidence level:

ME = z*SE
where z is the z-score corresponding to the desired confidence level.

Since we are looking for a 90% confidence level, we use z = 1.645:

ME = 1.645*0.032 ≈ 0.053

Therefore, the margin of error at a 90% confidence level is approximately 0.053. This means that if we were to repeat the sampling process many times, 90% of the intervals we construct would contain the true proportion of the population within 0.053 of our estimated proportion.

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