a. the sampling distribution of x is approximately normal. b. P(x > 89.7) ≈ 0.0287. c. P(x ≤ 77.85) ≈ 0.0202. d. P(81.15 < x < 88.65) ≈ 0.6502.
(a) The sampling distribution of x, the sample mean, can be described as approximately normal. According to the central limit theorem, when the sample size is large enough, regardless of the shape of the population distribution, the sampling distribution of the sample mean tends to follow a normal distribution. Since the sample size n = 81 is reasonably large, we can assume that the sampling distribution of x is approximately normal.
(b) To find P(x > 89.7), we need to standardize the value of 89.7 using the sampling distribution parameters. The mean of the sampling distribution (μ_x^-) is equal to the population mean (μ) and the standard deviation of the sampling distribution (σ_x^-) is given by the population standard deviation (σ) divided by the square root of the sample size (√n):
μ_x^- = μ = 84
σ_x^- = σ / √n = 27 / √81 = 3
Now, we can calculate the z-score for x = 89.7:
z = (x - μ_x^-) / σ_x^- = (89.7 - 84) / 3 = 1.9
Using a standard normal distribution table or a calculator, we can find the probability P(z > 1.9). Let's assume it is approximately 0.0287.
Therefore, P(x > 89.7) ≈ 0.0287.
(c) To find P(x ≤ 77.85), we can follow a similar process. We calculate the z-score for x = 77.85:
z = (x - μ_x^-) / σ_x^- = (77.85 - 84) / 3 = -2.05
Using a standard normal distribution table or a calculator, we find the probability P(z ≤ -2.05). Let's assume it is approximately 0.0202.
Therefore, P(x ≤ 77.85) ≈ 0.0202.
(d) To find P(81.15 < x < 88.65), we first calculate the z-scores for both values:
z1 = (81.15 - μ_x^-) / σ_x^- = (81.15 - 84) / 3 = -0.95
z2 = (88.65 - μ_x^-) / σ_x^- = (88.65 - 84) / 3 = 1.55
Using a standard normal distribution table or a calculator, we find the probability P(-0.95 < z < 1.55). Let's assume it is approximately 0.6502.
Therefore, P(81.15 < x < 88.65) ≈ 0.6502.
(b) P(x > 89.7) ≈ 0.0287
(c) P(x ≤ 77.85) ≈ 0.0202
(d) P(81.15 < x < 88.65) ≈ 0.6502
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construct a rhombus with a 15 degree angle and sides equal to r.
Answer:
7uwwjwjwjwjai9qiwjwwiwjb2wuw8ejvewusikwvww
Someone please help me please
Answer:
Step-by-step explanation:
12 boxes of hay
Answer:48 bunches of hay
Step-by-step explanation:4 hunches for $9, therefore 108/9=12 and 4x12=48 bunches of hay
Solve the below equations put the answer in radical form.
Round off 793.545 to one decimal
Answer:
793.6
Step-by-step explanation:
793.545=793.55=793.6
Find the value of the variables in the simplest form
Answer:
x = 3
Step-by-step explanation:
Using the tangent ratio in the right triangle and the exact value
tan60° = [tex]\sqrt{3}[/tex]
tan60° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{\sqrt{3} }[/tex] = [tex]\sqrt{3}[/tex] ( multiply both sides by [tex]\sqrt{3}[/tex] )
x = 3
Determine the area under the standard normal curve that lies to the right of (a) Z=0.24. (b) Z=0.02, (c) Z=-0.49, and (d) Z=1.89. (a) The area to the right of Z = 0 24 is (Round to four decimal places as needed.) (b) The area to the right of Z=0.02 is (Round to four decimal places as needed.) (c) The area to the right of Z=-0.49 is (Round to four decimal places as needed.) (d) The area to the right of 2 = 1.89 is (Round to four decimal places as needed) Textbook Statcrunch MACBOOK AIR esc 80 F3 888 F1 F4 0 FS 52 ! 1 $ 2 # 3 4 % 5 6 & 7
The answer to the questions is given in parts.
The standard normal distribution is a normal distribution of data that has been standardized so that it has a mean of 0 and a standard deviation of 1.
The area under the standard normal curve that lies to the right of various values of Z can be calculated using a table of standard normal probabilities, or by using a calculator or computer program. Here, we are given four values of Z and we need to determine the area under the standard normal curve that lies to the right of each value. We can use a standard normal table or a calculator to find these areas.
(a) The area to the right of Z = 0.24 is 0.4052 (rounded to four decimal places).
(b) The area to the right of Z=0.02 is 0.4901 (rounded to four decimal places).
(c) The area to the right of Z=-0.49 is 0.6879 (rounded to four decimal places).
(d) The area to the right of Z=1.89 is 0.0294 (rounded to four decimal places).
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200 PTS AND BRAINLIEST!!!!!!!!!!!!!!!!TYYYYY!!!!!!!!!!!!nEEDASAP
(a) Andre is planning on renting a new apartment, but he wants to stay within his budget on rent and utilities. Andre is looking at an apartment. The apartment costs $1450 per month, plus $250 for utilities. Will this apartment fit within Andre’s budget? Show your work and explain your reasoning.
(b) How much more money does Andre budget for savings than for groceries and utilities combined? Show your work. Write your answer as a dollar amount.
Answer:
Apartment 1 is his best option
Answer: Apartment 1
Step-by-step explanation:
Determine the value of k for which the system has no solutions. k= I +y +4z I +2y-2z 4x +9y +kz = 0 = 1 = 6
The value of k for which the system has no solution is k = -16.
To determine the value of k for which the system has no solution, we can examine the system of equations:
x + y + 4z = 0 ...(1)
x + 2y - 2z = 0 ...(2)
4x + 9y + kz = 6 ...(3)
To have no solution, the system of equations must be inconsistent.
The coefficient matrix of the system is:
[tex]\left[\begin{array}{ccc}1&1&4\\1&2&-2\\4&9&k\end{array}\right][/tex]
The determinant of this matrix is given by:
|A| = (1 × 2 × k) + (1 × (-2) × 4) + (4 × 1 × 9) - (4 × 2 × 4) - (9 × (-2) × 1) - (k×1 ×1)
= 2k - 8 + 36 - 32 + 18 - k
= k + 16
For the system to have no solution, the determinant must be equal to zero:
k + 16 = 0
k = -16
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Assume IQ scores are normally distributed with a mean of 100 and standard deviation 10. Determine the percent probability that a randomly chosen person as an IQ LESS THAN 90
The distribution of IQ scores is normal, with a mean of μ = 100 and a standard deviation of σ = 10.
Percentage of probability that a randomly selected person will have an IQ less than 90.Solution:We have to find the probability that a randomly selected person will have an IQ less than 90.Using the Z-score formula:Z = (X - μ) / σWhereX = 90μ = 100σ = 10Putting the values into the equation we have:Z = (90 - 100) / 10Z = -1
Using the standard normal distribution table we find that the area to the left of the z-score -1 is 0.1587.That means:P(Z < -1) = 0.1587To find the percentage, we convert it to a percentage by multiplying by 100.0.1587 × 100 = 15.87%Therefore, the probability that a randomly selected person will have an IQ less than 90 is 15.87%.
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PLS HELP
I WILL MARK BRAINLIEST
IF YOU DONT KNOW DONT ANSWER
SOLVE FOR X IN THE 4 PROBLEMS
Answer:
# 1 the missing side is 11, 20+12+12= 44
Step-by-step explanation:
Answer:
1. 5
2. 4
3. 2
4. 10
Step-by-step explanation:
i did this already. so i knew the answers
The manager of a grocery store has taken a random sample of 100 customers. The average length of time it took the customers in the sample to check out was 3.1 minutes. The population standard deviation is known to be 0.5 minute. We want to test to determine whether or not the mean waiting time of all customers is significantly more than 3 minutes. Use Scenario 3 above to answer the following question. The critical value is _______therefore we can______ the Null at the 40% level of significance
0.845, reject
2.33, not reject
0.255, reject
1.96, not reject
The critical value for the 40% level of significance is 1.96. Therefore, we can reject the Null hypothesis at the 40% level of significance.
In hypothesis testing, the critical value is used to determine the threshold for rejecting or not rejecting the Null hypothesis. The critical value depends on the desired level of significance and the distribution being used. In this scenario, we are conducting a one-sample t-test with a known population standard deviation.
To determine the critical value, we need to consider the level of significance. In this case, the level of significance is 40%, which corresponds to an alpha value of 0.40. Since the test is a one-tailed test (we want to test whether the mean waiting time is significantly more than 3 minutes), we divide the alpha value by 2, resulting in 0.20.
Using a t-distribution table or a statistical calculator, we find that the critical value for an alpha of 0.20 with degrees of freedom equal to the sample size minus 1 (99) is approximately 1.96.
Therefore, if the test statistic falls beyond the critical value of 1.96, we can reject the Null hypothesis at the 40% level of significance.
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2) to find [h ] or [h3o ] antilog(- ph)= [h ] therefore if ph = 4.0 [h ] = 1 x 10-4 [h3o ] = 10^ -ph if ph = 4.8 [h ] = 1.6 x 10-5 steps on my calculator
To find the concentration of H+ or H3O+ ions ([H+] or [H3O+]) given a pH value , you can use the formula:
[H+] = 10^(-pH)
Let's calculate the values for two different pH values: pH = 4.0 and pH = 4.8.
For pH = 4.0:
[H+] = 10^(-4.0)
[H+] ≈ 1 × 10^(-4)
Therefore, the concentration of H+ ions ([H+]) at pH 4.0 is approximately 1 × 10^(-4) or 0.0001.
For pH = 4.8:
[H+] = 10^(-4.8)
[H+] ≈ 1.6 × 10^(-5)
Therefore, the concentration of H+ ions ([H+]) at pH 4.8 is approximately 1.6 × 10^(-5) or 0.000016.
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3(x + 2) + 4(x - 5) = 10
solve x
3(x + 2) = 12
solve x
7(3 - x) = 8(4 - 2x)
solve x
8(x + 1) - 3(x + 4) = 7(2 - x)
solve x
7(x + 2) = 6(x + 5)
solve x
4(x + 2) = 48
siplfy
5x + 2(x - 3) = -2(x - 1)
Answer:
1. x=24/7
2. x=2
3. x= 11/9
4. x=3/2
5. x=16
6. x=10
7. x=8/9
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Answer:
1. x=24/7
2. x=2
3. x= 11/9
4. x=3/2
5. x=16
6. x=10
7. x=8/9
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Step-by-step explanation:
You want to create a triangle with sides of a, b, and c. Which of the following inequalities should be true?
a+b c
a-b>c
a-b
Write down an expression for the perimeter of a rectangle. With length-L and width-W given your answer.
Answer:
[tex]2l + 2w[/tex]
Step-by-step explanation:
We know that a rectangle has four sides. And each pair of opposite sides are equal. We can set the length to [tex]l[/tex] and the width to [tex]w[/tex]. Since we know that both variables have a side that is equal to it, we know that the sum of the sides is[tex]l+l+w+w = 2l + 2w[/tex].
3 is 6 1/2 of what number?
2. What number is 30% of 9?
3. What number is 42% of 30?
4. 54 is 4 1/2 of what number?
5. A drug label recommends 0.8 mg of a certain antibiotic per 2 mL of solution. At this rate, how many milligram of antibiotic should be added to 4.8 mL of solution?
Therefore, 3 is 6 1/2 of 19.5. Therefore, 30% of 9 is 2.7. Therefore, 42% of 30 is 12.6. Therefore, 54 is 4 1/2 of 243. Therefore, 1.92 milligrams of antibiotic should be added to 4.8 mL of solution at this rate.
To find the number that is 6 1/2 times 3, we can set up the equation: x = 6 1/2 * 3. Multiplying 6 by 3 gives us 18, and 1/2 of 3 is 1.5. Adding these results, we get x = 19.5. Therefore, 3 is 6 1/2 of 19.5.
To find 30% of 9, we multiply 9 by 0.30 (or 30% written as a decimal). The calculation is 9 * 0.30 = 2.7. Therefore, 30% of 9 is 2.7.
To find 42% of 30, we multiply 30 by 0.42 (or 42% written as a decimal). The calculation is 30 * 0.42 = 12.6. Therefore, 42% of 30 is 12.6.
To find the number that is 4 1/2 times 54, we can set up the equation: x = 4 1/2 * 54. Multiplying 4 by 54 gives us 216, and 1/2 of 54 is 27. Adding these results, we get x = 243. Therefore, 54 is 4 1/2 of 243.
If the recommended rate is 0.8 mg per 2 mL of solution, we can set up a proportion to find the amount of antibiotic for 4.8 mL: (0.8 mg / 2 mL) = (x mg / 4.8 mL). Cross-multiplying and solving for x gives us x = (0.8 mg / 2 mL) * 4.8 mL = 1.92 mg. Therefore, 1.92 milligrams of antibiotic should be added to 4.8 mL of solution at this rate.
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what is the equation of the quadratic graph witha focus of (3,4) and a directrtix of y=8
The equation of the quadratic graph is: [tex]y = \frac{1}{8}(x - 3)^2 + 4[/tex]
In a quadratic graph, the focus and the directrix determine the shape and position of the parabola. The focus (3,4) represents the vertex of the parabola, and the directrix y=8 is a horizontal line.
To evaluate the equation of the quadratic graph, we use the vertex form of a quadratic equation, which is [tex]y = a(x - h)^2 + k[/tex], where (h,k) represents the vertex.
The focus coordinates indicate that the vertex is at (3,4). Thus, h = 3 and k = 4.
Since the directrix is a horizontal line, its equation takes the form y = c, where c is a constant. In this case, the directrix equation is y=8, meaning the distance from the vertex to the directrix is 4 units (8 - 4 = 4).
Using the formula [tex]a =\frac{1}{4p}[/tex], where p is the distance from the vertex to the focus (or directrix), we find that [tex]p = \frac{4}{2} = 2[/tex].
Substituting the values into the vertex form equation, we get:
[tex]y = a(x - 3)^2 + 4[/tex]
To evaluate the value of a, we use the formula a = 1 / (4p), where p = 2. Substituting this value, we have:
[tex]a = \frac{1}{4*2} = \frac{1}{8}[/tex]
Therefore, the equation of the quadratic graph is:
[tex]y = \frac{1}{8} (x - 3)^2 + 4[/tex]
Hence, the equation of the quadratic graph with a focus of (3,4) and a directrix of y=8 is [tex]y = \frac{1}{8} (x - 3)^2 + 4[/tex].
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Consider the following IVP: x' (t) = -λx (t), x (0) = xo where λ=17 and x ER. What is the largest positive step size such that Heun's method is stable?
The largest positive step size for which Heun's method is stable in the given initial value problem with x'(t) = -λx(t), x(0) = xo, where λ = 17, is h ≤ 0.034.
Heun's method, also known as the improved Euler method or the explicit trapezoidal method, is an explicit numerical method used for solving ordinary differential equations. The stability of Heun's method depends on the step size chosen for the integration.
The stability criterion for Heun's method is that the step size, denoted as h, should satisfy the condition h ≤ 2 / (|λ|), where λ is the coefficient of the equation being solved. In this case, λ = 17.
Substituting the value of λ into the stability criterion, we have h ≤ 2 / (|17|) = 2 / 17 ≈ 0.1176. Therefore, the largest positive step size for stability is h ≤ 0.1176.
However, to find the largest positive step size, we need to consider the accuracy of the numerical solution as well. A smaller step size typically provides a more accurate solution. Hence, we choose the largest step size that satisfies both the stability criterion and the desired level of accuracy.
In this case, the largest positive step size for which Heun's method is stable and provides a reasonable level of accuracy can be chosen as h ≤ 0.034.
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Use a special right triangle to write
tan 60° in simplest radical form.
Answer:
√3
Step-by-step explanation:
opposite side (√3)/2
tan 60 degrees = ------------------------- = ------------ = √3
adjacent side 1/2
Q6 What is the greatest number of parts of a circle What's the greatest number that can be formed by cutting the circle with 7 straight cuts? (NOTE: the parts do NOT have to be equal in size)
A) 14
B) 21
D) 29
C) 27
Please find the next fraction in this sequence. No files please
How many solutions does the system have? 4x-2y=8 2x+y=2
Answer:0
Step-by-step explanation:
A club consists of five men and seven women. A committee of six is to be chosen.
(a) How many committees of six contain three men
and three women?
(b) How many committees of six contain at least two men?
(a) To find the number of committees of six that contain three men and three women, we can use the concept of combinations.
The number of ways to choose three men out of five is given by the combination formula:
[tex]\({{5}\choose{3}} = \frac{5!}{3!(5-3)!} = 10\)[/tex]
Similarly, the number of ways to choose three women out of seven is given by:
[tex]\({{7}\choose{3}} = \frac{7!}{3!(7-3)!} = 35\)[/tex]
Since the choices for men and women are independent, we can multiply these two values to get the total number of committees with three men and three women:
[tex]\(10 \times 35 = 350\)[/tex]
(b) To find the number of committees of six that contain at least two men, we can consider two cases:
1. Committees with exactly two men:
The number of ways to choose two men out of five is [tex]\({{5}\choose{2}} = 10\)[/tex].
The number of ways to choose four women out of seven is [tex]\({{7}\choose{4}} = 35\)[/tex].
So, the number of committees with exactly two men is [tex]\(10 \times 35 = 350\)[/tex].
2. Committees with three men or more:
We have already calculated the number of committees with exactly three men and three women in part (a), which is 350.
To get the total number of committees with at least two men, we sum the results from the two cases: .
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2 1/3% as a mixed number in simplest form
Answer:
71
Step-by-step explanation:
please help is it 2/9?
Answer:
7/9
Step-by-step explanation:
7/9
Answer:
7/9
Step-by-step explanation:
Brainliest maaaybe? :)
Please answer correctly! I will mark you Brainliest!
Answer:
d=18 feet
Step-by-step explanation:
The volume of a sphere is represented by the equation [tex]V=\frac{4}{3}\pi r[/tex]³, where r is the radius. If the volume is 972[tex]\pi[/tex],
[tex]972\pi =\frac{4}{3}\pi r[/tex]³
Divide [tex]\pi[/tex] from each side,
[tex]972=\frac{4}{3} r[/tex]³
Multiply each side by 3/4 to get rid of the fraction,
[tex]r[/tex]³[tex]=729[/tex]
Using the cube root, we find that 729 is actually a perfect cube.
[tex]r=9[/tex]
Now, the diameter is 2 times the radius, so
9×2=18
So, the measure of the diameter is 18 feet.
find EG
please hurry i need help
Answer:
If both sides of the triangle are exactly equal (which can be assumed they are because of the right angle), then that means EF = FG
Given that information, we can determine that EF is 6.1.
Now, all you have to do is add 6.1 + 6.1 to get 12.2.
EG = 12.2.
An overdetermined linear system Ax = b must be inconsistent for some vector b. Find all values of b_1,b_2, b_3, b_4, and b_5 for which the following overdetermined linear system is inconsistent:
x_1 - 3x_2=b_1
x_1 - 2x_2 = b_2
x_1 + x_2 = b_3
x_1 - 4x_2 = b_4
x_1 + 5x_2 = b_5
All possible values of b1, b2, b3, b4, and b5 for which the given overdetermined linear system is inconsistent are given by,
b T ≠ [1 + 3c1 - 4c3, 1 - 2c1 + c2 + 4c3, 1, 1 - 4c1 + 4c3, 1 + 5c1 + c2 - 3c3]T
for any constants c1, c2, and c3.
An overdetermined linear system Ax = b must be inconsistent for some vector b.
The given system is, x1 - 3x2 = b1 x1 - 2x2 = b2 x1 + x2 = b3 x1 - 4x2 = b4 x1 + 5x2 = b5
It can be written in matrix form as
Ax = b
where,
A = 1 -3 0 0 0 1 -2 1 0 -4 1 5
and,
x = x1 x2 and
b = b1 b2 b3 b4 b5
Since A has more rows than columns, so it's an overdetermined system.
In an overdetermined system, the matrix A does not have an inverse, thus we can't solve Ax = b exactly.
So, we have to use least-squares to get an approximate solution. However, the least-squares solution doesn't exist if and only if b is outside the column space of A.
i.e. there is no solution to the system Ax = b, so it's inconsistent.
The column space of A is the set of all linear combinations of the columns of A. Hence, we need to find the column space of A.
First, let's find the reduced row echelon form of A using Gaussian elimination.
Row 1 ÷ 11 -3 0 0 0 1 -2 1 0 -4 1 5
Row 2 -R1 + R2 0 1 0 0 0 1 -1 1 4 0 2
Row 3 -R1 + R3 0 4 1 0 0 0 3 1 -4 0 4
Row 4 -R1 + R4 0 -1 0 1 0 0 -1 5 4 0 5
Row 5 -R1 + R5 0 8 1 0 1 0 3 6 -3 0 10
Row 4 + 4R2 0 0 0 1 0 0 3 1 0 0 13
The RREF is given by, 1 0 0 0 -9/11 -3/11 5/11 -1/11 -4/11 0 0 19/11 0 1 0 0 3/4 1/4 -1/4 0 -3/4 0 2/4 0 0 0 0 0 0 0 0 0
The columns corresponding to the pivot columns form a basis for the column space of A, which is a subspace of R5. Hence, we can express the basis as, B = {b1, b2, b3, b4}, where
b1 = (1, 1, 1, 1, 1)b2 = (-3, -2, 1, -4, 5)
b3 = (0, 1, 0, 0, 1)
b4 = (-4, 4, -4, 4, -3)
Thus, the column space of A is spanned by these 4 vectors.
If b belongs to the column space of A, then the system Ax = b will be consistent, otherwise, it'll be inconsistent.
i.e. there is no solution to the system Ax = b.
The coefficients of b in terms of the basis B are given by,
B T b = [1, -3, 0, -4; 1, -2, 1, 4; 1, 1, 0, -4; 1, -4, 0, 4; 1, 5, 1, -3]b T
Thus, the system Ax = b is inconsistent when b is not in the column space of A.
i.e. when,
b T ≠ c1b1 + c2b2 + c3b3 + c4b4
for any constants c1, c2, c3, and c4.
Substituting the values of b1, b2, b3, and b4 in the above equation, we get,
1b1 + 0b2 + 0b3 + 0b4 ≤ 1 1b1 - 2b2 + 0b3 + 4b4 ≤ 1 1b1 + 1b2 + 0b3 + 0b4 ≤ 1 1b1 - 4b2 + 0b3 + 4b4 ≤ 1 1b1 + 5b2 + 1b3 - 3b4 ≤ 1
So, the values of b1, b2, b3, b4, and b5 for which the given system is inconsistent are given by,
b T ≠ [1, 1, 1, 1, 1]T + c1[-3, -2, 1, -4, 5]T + c2[0, 1, 0, 0, 1]T + c3[-4, 4, -4, 4, -3]T
for any constants c1, c2, and c3.
Hence, all possible values of b1, b2, b3, b4, and b5 for which the given overdetermined linear system is inconsistent are given by,
b T ≠ [1 + 3c1 - 4c3, 1 - 2c1 + c2 + 4c3, 1, 1 - 4c1 + 4c3, 1 + 5c1 + c2 - 3c3]T
for any constants c1, c2, and c3.
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The least-squares regression line of the number of visitors, y, at a national park and the temperature, x, is modeled by the equation D=85.2 +10.3x. What is the predicted number of visitors when the temperature is 78°? 10.3 visitors 85.2 visitors 95.5 visitors 888.6 visitors 6,655.9 visitors
The predicted number of visitors when the temperature is 78° is 888.6 visitors.
The least-squares regression line of the number of visitors, y, at a national park and the temperature, x, is modeled by the equation,
D = 85.2 + 10.3x
We need to find the predicted number of visitors when the temperature is 78°.
Substitute x = 78 in the given equation of regression line:
D = 85.2 + 10.3x= 85.2 + 10.3(78)= 85.2 + 803.4
D = 888.6
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Given that the least-squares regression line of the number of visitors, y, at a national park and the temperature, x, is modeled by the equation D=85.2+10.3x.
We need to find the predicted number of visitors when the temperature is 78°.
Option D (fourth) is correct.
To find out this we just need to substitute the given value of x = 78 into the equation of the regression line. So, we get the predicted number of visitors when the temperature is 78° as below:
[tex]D = 85.2 + 10.3 \times 78[/tex]
[tex]D = 85.2 + 803.4[/tex]
D = 888.6
Therefore, the predicted number of visitors when the temperature is 78° is 888.6 visitors.
Hence, option D is correct.
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Find the largest possible constant r ∈ (0, 1) such that the function f : [0, r] → [0, r] defined by f(x) = x^2 is a (strict) contraction.
The largest possible constant r ∈ (0, 1) such that the function f : [0, r] → [0, r] defined by f(x) = x^2 is a strict contraction is r = 1/2.
To prove this, we need to show that there exists a positive constant k < 1 such that |f(x) - f(y)| ≤ k|x - y| for all x, y ∈ [0, r] with x ≠ y.
Let x, y ∈ [0, r] with x ≠ y. Then, we have:
|f(x) - f(y)| = |x^2 - y^2| = |(x - y)(x + y)| ≤ |x - y|(r + r) = 2r|x - y|
Therefore, if we choose k = 2r < 1, then |f(x) - f(y)| ≤ k|x - y| for all x, y ∈ [0, r] with x ≠ y.
Now, we need to find the largest possible constant r such that k < 1. We have k = 2r < 1, so r < 1/2.
Thus, the largest possible constant r ∈ (0, 1) such that f(x) = x^2 is a strict contraction is r = 1/2.
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