Consider the joint PDF of two random variables X, Y given by fx,y (x, y) = C, where 0 < x < 9 and 0

The upper and lower quartiles of the discounts are £45 and £20 respectively.

The interquartile range of the discounts is £25.

The upper and lower quartiles of the discounts can be calculated by using the following mathematical expressions;

Upper quartile, P₇₅ = 80 × 75/100

Upper quartile, P₇₅ = 60, which corresponds to £45.

Lower quartile, P₂₅ = 80 × 25/100

Lower quartile, P₂₅ = 20, which corresponds to £20.

Mathematically, the interquartile range (IQR) of a data set is typically calculated as the difference between the first quartile (Q₁) and third quartile (Q₃):

Interquartile range (IQR) = Q₃ - Q₁ = P₇₅ - P₂₅

Interquartile range (IQR) = 45 - 20

Interquartile range (IQR) = £25.

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A function f: A--B is injective if each element in A maps to a unique element in B. It is surjective if every element in B has a corresponding element in A.

A bijective function is both injective and surjective, meaning that every element in A maps to a unique element in B, and every element in B has a corresponding element in A.  If IAI = IBI, then there are the same number of elements in A and B. Therefore, if f is injective, every element in A must map to a unique element in B, leaving no elements in B without a corresponding element in A. This means that f is also surjective. Similarly, if f is surjective, then every element in B has a corresponding element in A, which means that f is also injective. Thus, f is bijective if and only if it is injective and surjective.

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Answer:

kite

4x + 1 = 17 6y - 3 = 21

4x = 16 6y = 24

x = 4 y = 4

The simple interest rate of the loan is 8.8%. We can use the formula for simple interest to find the interest rate:

I = P * r * t

where I is the interest charged, P is the principal amount (the amount loaned), r is the interest rate, and t is the time period.

In this case, we know that P = $5,000, t = 2 years, and I = $880. Plugging these values into the formula, we get:

$880 = $5,000 * r * 2

Simplifying this expression, we get:

r = $880 / ($5,000 * 2)

r = 0.088 or 8.8%

Therefore, the simple interest rate of the loan is 8.8%.

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a) A possible differential equation with this characteristic equation is:

y'''' - 6y''' + 13y'' - 12y' + 4y = 0

b) The general solution of the differential equation is:

[tex]y = (c1 + c2x)e^x + (c3 + c4x)e^2x[/tex]

a) To find such a differential equation, we can use the fact that the roots of the characteristic equation correspond to the solutions of the homogeneous linear differential equation.

The characteristic equation is given by:

[tex](r - 1)^2 (r - 2)^2 = 0[/tex]

Expanding the terms, we get:

[tex]r^4 - 6r^3 + 13r^2 - 12r + 4 = 0[/tex]

Therefore, a possible differential equation with this characteristic equation is:

y'''' - 6y''' + 13y'' - 12y' + 4y = 0

b) To find the general solution of this differential equation, we can use the method of undetermined coefficients or the method of variation of parameters.

However, since the roots of the characteristic equation have a multiplicity of 2, we know that the general solution will involve terms of the form:

[tex]y = (c1 + c2x)e^x + (c3 + c4x)e^2x[/tex]

where c1, c2, c3, and c4 are constants to be determined based on initial or boundary conditions.

Therefore, the general solution of the differential equation is:

[tex]y = (c1 + c2x)e^x + (c3 + c4x)e^2x[/tex]

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The values of t, when the ball's height is 15 feet, are 0.42 and 1.33.

What is a quadratic equation?

Any equation in algebra that can be written in the standard form where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.

Here, we have

Given: A ball is thrown from an initial height of 6 feet with an initial upward velocity of 28 ft/s.

The ball's height, h (in feet), after t seconds is given by the following:

h = 6 + 28t - 16t²

When h = 15, we have

15 = 6 + 28t - 16t²

6 + 28t - 16t² - 15 = 0

-16t² + 28t + 6 - 15 = 0

-16t² + 28t - 9 = 0

16t² - 28t + 9 = 0

Using a graphing tool, we have:

t = 0.42 and 1.33

Hence, the values of t when the ball's height is 15 feet are 0.42 and 1.33.

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The solution is, simplification of the expression 6(5+5i) / (-2i) (3i⁵) is:

-5 - 5i.

Here, we have,

given that,

the expression is:

6(5+5i) / (-2i) (3i⁵)

now, we have to simplify this expression.

we know that,

i² = 1

so, (i²)³ = 1²

or, i⁶ = 1

we have,

6(5+5i) / (-2i) (3i⁵)

=30 + 30i / (-6i⁶)

=6(5+5i) / -6 * 1

=(5+5i) / -1

= - (5+5i)

= -5 - 5i

Hence, The solution is, simplification of the expression 6(5+5i) / (-2i) (3i⁵) is: -5 - 5i.

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Right endpoint estimate with n = 6 gives an estimated value of 8648.89 for the integral.

Using the right endpoint estimate with n = 6, we have:

Δx = (b - a)/n = (4 - 0)/6 = 2/3

The right endpoints are x1 = 2/3, x2 = 4/3, x3 = 2, x4 = 8/3, x5 = 10/3, x6 = 4.

So, the integral can be estimated by:

∫604x2dx ≈ Δx [f(x1) + f(x2) + f(x3) + f(x4) + f(x5) + f(x6)]

where f(x) = 604x2.

Plugging in the values, we get:

∫604x2dx ≈ (2/3) [604(2/3)2 + 604(4/3)2 + 604(2)2 + 604(8/3)2 + 604(10/3)2 + 604(4)2]

Simplifying, we get:

∫604x2dx ≈ 8648.89

Therefore, the right endpoint estimate with n = 6 gives an estimated value of 8648.89 for the integral.

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The differential of the function m = p³q⁹ is dm = 3p²q⁹ dp + 9p³q⁸ dq.

To find the differential of the function m = p3q9, we use the formula:
dm = (∂m/∂p) dp + (∂m/∂q) dq
Where ∂m/∂p is the partial derivative of m with respect to p, and ∂m/∂q is the partial derivative of m with respect to q.
Taking the partial derivative of m with respect to p, we get:
∂m/∂p = 3p2q9
Taking the partial derivative of m with respect to q, we get:
∂m/∂q = p3(9q8) = 9p3q8
Substituting these partial derivatives into the formula for the differential, we get:
dm = (3p2q9) dp + (9p3q8) dq
Therefore, the differential of the function m = p3q9 is:
dm = 3p2q9 dp + 9p3q8 dq
find the differential of the function m = p³q⁹. To find the differential, we'll use the product rule, which states that if m = uv, then dm = u(dv) + v(du). Here, u = p3 and v = q9. Now, let's find the differentials:
For u = p3, we have du = 3p² dp.
For v = q9, we have dv = 9q⁸ dq.
Now, let's apply the product rule:
dm = p3(dq⁹) + q9(dp³) = p3(9q⁸ dq) + q9(3p² dp) = 3p²q⁹ dp + 9p³q⁸ dq.
So, the differential of the function m = p3q9 is dm = 3p²q⁹ dp + 9p³q⁸ dq.

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Answer:

The expression 3(x-10) can be simplified by distributing the 3 to both x and -10. This gives us:

3(x-10) = 3x - 30

So, the expression equivalent to 3(x-10) is 3x - 30.

Step-by-step explanation:

To compute the inverse Laplace transform of L^-1 {3s + 2/s^2 - s - 12 e^-4s}, we first need to break it up into simpler terms using partial fraction decomposition. We have:

L^-1 {3s + 2/s^2 - s - 12 e^-4s}

= L^-1 {3s} + L^-1 {2/s^2 - s} + L^-1 {12 e^-4s}

= 3 δ(t) + (2 u(t) - 1) - (1 - u(t - 4)) \* 3/2 e^(4(t-4))

where δ(t) is the Dirac delta function and u(t) is the Heaviside step function.

The first term, 3 δ(t), comes from the L^-1 {3s} term, which corresponds to a constant function.

The second term, (2 u(t) - 1), comes from the L^-1 {2/s^2 - s} term, which we can decompose as:

2/s^2 - s

= 2/s^2 - 2s/s^2 + s/s^2

= 2 (1/s - 1/s^2) - s/s^2

Taking the inverse Laplace transform of each term separately gives:

L^-1 {2 (1/s - 1/s^2)} = 2 (u(t) - 1) L^-1 {-s/s^2} = -(t u(t))

Putting these together, we get: L^-1 {2/s^2 - s} = 2 (u(t) - 1) - (t u(t))

The third term, (1 - u(t - 4)) \* 3/2 e^(4(t-4)), comes from the L^-1 {12 e^-4s} term, which corresponds to an exponentially decaying function.

We use the time-shifting property of the Laplace transform to shift the function by 4 units to the right, giving: L^-1 {12 e^-4s} = 3/2 e^(4(t-4)) u(t-4)

But we want the function to be 0 for t < 4, so we subtract off the Heaviside step function u(t - 4), giving: L^-1 {12 e^-4s} = (1 - u(t - 4)) \* 3/2 e^(4(t-4))

Putting everything together, we get: L^-1 {3s + 2/s^2 - s - 12 e^-4s} = 3 δ(t) + 2 (u(t) - 1) - (t u(t)) - (1 - u(t - 4)) \* 3/2 e^(4(t-4))

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Answer:

=25

Step-by-step explanation:

Perimeter of a square = 4L

20 = 4L

divide both sides by 4

L = 5[ length is 5cm]

Area of a square = L*L

Area = 5cm times 5cm

Area = 25cm^2

Answer:

Step-by-step explanation:

Just like a lot of our other curve transformations

The number with x => (x+2) is the shift in x direction and you take opposite sign so it's been shifted -2 or left 2

bring the 4 to the back of teh equation and it will be +4, that controls the y direction which is +4 or up 4

That would leave a - in the front which means the graph  wil be reflected.

(a) The velocity vector is v(t) = 3i + j + 1/2tk, the speed is

∥v(t)∥ = √(10t² + 1), and the acceleration vector is a(t) = 1/2k.

(b) Plugging in t = 2, the velocity vector is v(2) = 3i + j + k, and the acceleration vector is a(2) = 1/2k.

(a) To find the velocity vector, we take the derivative of the position vector with respect to time:

To find the speed of the object, we take the magnitude of the velocity vector:

|v(t)| = √(9 + 1 + 1/4t²)

Now, to find the acceleration vector, we take the derivative of the velocity vector with respect to time:

a(t) = v'(t) = 1/2k

(b) To find the velocity vector and acceleration vector at t=2, we substitute t=2 into the expressions we found in part (a):

Therefore, at t=2, the object has a position vector of 6i + 2j + k, a velocity vector of 3i + j + 2k, a speed of√(10t² + 1), and an acceleration vector of 1/2k.

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If a plane is 111 mi north and 189 mi east of an airport, the pilot should turn 31.4 degrees to fly directly to the airport.

To find the angle that the pilot should turn in order to fly directly to the airport, we can use the trigonometric functions sine, cosine, and tangent. Specifically, we will use the tangent function, which relates the opposite side (in this case, the distance north) to the adjacent side (in this case, the distance east) of a right triangle:

tan(x) = opposite/adjacent

We can rearrange this formula to solve for x:

x = arctan(opposite/adjacent)

where arctan is the inverse tangent function.

In this case, the opposite side is 111 miles (the distance north) and the adjacent side is 189 miles (the distance east). Plugging these values into the formula, we get:

x = arctan(111/189)

x = 31.4 degrees

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In simple harmonic motion, the number of cycles per second of a point is its frequency. Here's a step-by-step explanation:

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Answer:

10 km/hr.

Step-by-step explanation:

Speed of the boat with the current = 20 + c   where c is the speed of the current.

Against the current the speed is 20 - c.

Distance = speed * time :

With the current:

180 = (20 + c)t..........A

Against the current:

20 = (20 - c)(8- t)......B

Solving this system of equations:

From A:

t = 180/(20 + c)

So

20 = (20 - c)(8 - 180/(20+c))

160 - 3600/(20 + c) - 8c  + 180c/(20 + c) = 20

Mltiplying throagh by (20 + c):

160(20 + c) - 3600 - 8c(20 + c) + 180c = 20(20 + c)

3200 + 160c - 3600 - 160c - 8c^2 + 180c - 400 - 20c = 0

-8c^2 + 160c  - 800 = 0

8c^2 - 160c + 800 = 0

c^2 - 20c + 100 = 0

(c - 10)*2 = 0

c = 10

So the speed of the crrent = 10 km/hr

The calculated test statistic z = -0.56 is not in the rejection region, we fail to reject the null hypothesis H0: p = 0.30.

(a) The test statistic for testing the null hypothesis H0: p = 0.30 against the alternative hypothesis Ha: p ≠ 0.30 using the sample proportion p = 0.285 is given by:

z = (p - μ) / (σ / sqrt(n))

where μ = 0.30, σ = sqrt(μ(1-μ)/n) = sqrt(0.30(0.70)/400) = 0.027,

and n = 400.

Substituting the values, we get:

z = (0.285 - 0.30) / (0.027)

z = -0.56 (rounded to two decimal places)

(b) The p-value is the probability of getting a test statistic as extreme as the observed, assuming the null hypothesis is true. Since this is a two-tailed test, we need to find the probability of getting a z-score less than -0.56 or greater than 0.56, given a standard normal distribution.

Using a standard normal table or calculator, we find that the probability of getting a z-score less than -0.56 is 0.2881, and the probability of getting a z-score greater than 0.56 is also 0.2881. Therefore, the p-value is the sum of these probabilities:

p-value = P(z < -0.56 or z > 0.56) = 0.2881 + 0.2881 = 0.5762 (rounded to four decimal places)

(c) The rejection rule using the critical value depends on the level of significance α and the type of test (one-tailed or two-tailed). Assuming a two-tailed test with α = 0.05, the critical values for the test statistic are ±1.96, which are obtained from a standard normal table or calculator.

Therefore, the rejection rule is:

if z ≤ -1.96 or z ≥ 1.96, reject H0.

Otherwise, fail to reject H0.

Since the calculated test statistic z = -0.56 is not in the rejection region, we fail to reject the null hypothesis H0: p = 0.30.

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The sample size needed for the sampling distribution of sample proportions to have a standard deviation of 0.02 is approximately 628 children.



The formula for the standard deviation of the sampling distribution of sample proportions is:
[tex]σp = sqrt[p(1-p)/n][/tex]

Where p: population proportion (0.49 in this case) and n: sample size.

We are given that σp = 0.02. So, we can set up the equation:
[tex]0.02 = sqrt[0.49(1-0.49)/n][/tex]

Simplifying:
0.0004 = 0.24/n
n = 0.24/0.0004
n = 627.5 = 628

However, this is only an estimate because the sample size must be a whole number. Since we cannot have a fractional sample size, we round up to the nearest whole number:

n = 628

To calculate the sample size (n) needed to achieve a standard deviation of 0.02 for the sampling distribution of sample proportions, we'll use the formula:

[tex]Standard Deviation = sqrt[(P * (1 - P)) / n][/tex]

where P: proportion of children that catch a cold while at school (0.49), and n: sample size we want to find. We're given the desired standard deviation as 0.02. Now, let's solve for n:

[tex]0.02 = sqrt[(0.49 * (1 - 0.49)) / n][/tex]

Square both sides to get rid of the square root:
0.0004 = (0.49 * 0.51) / n

Now, solve for n:

n = (0.49 * 0.51) / 0.0004
n = 627.75

Since we can't have a fraction of a child, we'll round up to ensure the standard deviation is no greater than 0.02:

n = 628 children

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The car travels 220.5 feet before stopping.

To find the distance the car travels before stopping, we first need to determine the constant negative acceleration. We can use the formula vf = vi + at, where vf is the final velocity (0 ft/sec), vi is the initial velocity (63 ft/sec), a is the acceleration, and t is the time (7 seconds).

0 = 63 + 7a
-63 = 7a
a = -9 ft/sec²

Now, we can use the formula d = vi*t + 0.5*a*t² to find the distance (d).

d = (63 ft/sec)(7 sec) + 0.5*(-9 ft/sec²)(7 sec)²
d = 441 + (-220.5)
d = 220.5 ft

To graph the velocity from t=0 to t=7, plot a straight line with an initial velocity of 63 ft/sec and a constant negative slope of -9 ft/sec². The line will reach 0 ft/sec at t=7 seconds.

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The probability cannot be negative, there must be an error in the given information. So, the correct answer is:
f) None of the above.

We are given the following probabilities: P(A) = 17/50, P(B) = 17/25, and P(A ∪ Bc) = 2/5. We are asked to find P(A ∩ Bc).

Using the formula for the union of two events: P(A ∪ Bc) = P(A) + P(Bc) - P(A ∩ Bc)

Since Bc is the complement of B, we have P(Bc) = 1 - P(B) = 1 - (17/25) = 8/25.

Now we can plug in the given probabilities into the formula:
2/5 = (17/50) + (8/25) - P(A ∩ Bc)

To solve for P(A ∩ Bc), we first find a common denominator for the fractions, which is 50. So, we have:
20/50 = (17/50) + (16/50) - P(A ∩ Bc)

Combine the fractions:
20/50 = 33/50 - P(A ∩ Bc)

Subtract 33/50 from both sides to isolate P(A ∩ Bc):
P(A ∩ Bc) = -13/50

Since the probability cannot be negative, there must be an error in the given information. So, the correct answer is:
f) None of the above.

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Abdul gave 7 correct answers in the maths paper.

To find the number of correct answers given by Abdul in a maths paper with 10 questions, let's use the given information about the scoring system and set up an equation.
Let x be the number of correct answers and y be the number of wrong answers. We know that:
x + y = 10 (total questions)
5x - 2y = 29 (marks scored)
We can solve this system of linear equations step-by-step:
Solve the first equation for x:
x = 10 - y
Substitute this expression for x in the second equation:
5(10 - y) - 2y = 29
Simplify and solve for y:
50 - 5y - 2y = 29
50 - 7y = 29
7y = 21
y = 3
Substitute the value of y back into the expression for x:
x = 10 - 3
x = 7
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The two variables that have the highest correlation coefficient are Lot and House, with a correlation coefficient of 0.83. Therefore, the correlation between Lot and House is the most concerning when it comes to multicollinearity.

Multicollinearity is a common problem in regression analysis, which occurs when the independent variables in a regression model are highly correlated with each other. This means that the explanatory power of each independent variable is shared with other independent variables in the model, which can lead to biased and unstable estimates of the regression coefficients. In other words, multicollinearity makes it difficult to determine the individual effect of each independent variable on the dependent variable.

In this case, the correlation matrix shows that there are high correlations between several independent variables. However, the correlation coefficient between Lot and House is the highest, which suggests that these two variables are highly correlated with each other. Therefore, if both Lot and House are included in the regression model, it may be difficult to determine the individual effect of each variable on the Selling Price of a Home (Price). This can result in biased and unreliable estimates of the regression coefficients. Hence, it is important to check for multicollinearity before running the regression model and consider removing one of the highly correlated variables from the model.

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The OLS assumption most likely violated by omitted variables bias is: E(μ₁|X₁) = 0.

Omitted variables bias occurs when a relevant variable is left out of the regression model, leading to biased and inconsistent estimates. This violates the OLS assumption that the expected value of the error term, given the independent variable (E(μ₁|X₁)), is equal to zero.

When omitted variables bias is present, the error term captures the effect of the omitted variable, resulting in a non-zero expected value of the error term conditional on the independent variable. This can cause misleading inferences and incorrect conclusions about the relationships between the variables in the model.

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The p-value for the given analysis of variance (ANOVA) test is approximately 0.0012.

To find the p-value for this ANOVA test, we need to consider the F-statistic (5.71) and degrees of freedom for groups (4) and error (35). Using an F-distribution table or an online calculator, input the F-statistic value and the degrees of freedom to obtain the p-value.

The p-value represents the probability of observing an F-statistic as extreme as the one calculated from your data, assuming the null hypothesis is true (i.e., all group means are equal).

In this case, the p-value of approximately 0.0012 indicates a very low probability of observing the given F-statistic if the null hypothesis were true, suggesting that there is a significant difference among the group means.

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The error message "x and y must have same first dimension but have shaped" typically occurs in programming languages such as Python or MATLAB when trying to operate on arrays or matrices with incompatible dimensions. In this case, the first dimension of the arrays or matrices must be the same, but they are not.

For example, if we have two arrays, x with shape (3, 4) and y with shape (2, 4), we cannot perform certain operations such as addition or multiplication between them because the first dimension, which represents the number of rows, is different.

To resolve this error, we can either reshape one of the arrays to have the same number of rows as the other, or we can transpose one of the arrays so that their dimensions match up. Another option is to adjust the code to ensure that the arrays being used have the same first dimension.

In summary, the "x and y must have the same first dimension but have shaped" error occurs when we attempt to operate on arrays or matrices with incompatible dimensions, and it can be resolved by reshaping, transposing, or adjusting the code.

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To calculate the value of the sample test statistic for the difference between two population means (μ1 - μ2), you need to use the following formula:

t = (M1 - M2 - 0) / √[(s1^2 / n1) + (s2^2 / n2)]

Here, M1 and M2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

To provide you with an accurate answer, please provide the required information (M1, M2, s1, s2, n1, and n2). Once you provide these values, I can help you calculate the test statistic rounded to three decimal places.

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The total cost of 4 units is b)240.

Cost refers to the amount of money, time, or resources that are required to produce or acquire something. It is a fundamental concept in business and economics and is used to evaluate the efficiency and profitability of a particular activity or venture. Cost can be divided into two main categories: direct costs and indirect costs.

From the given options, we can see that each unit costs the same amount. So, to find the total cost of 4 units, we simply need to multiply the cost of one unit by 4.

Using the given options, we can see that option b (240) is the result of multiplying the cost of one unit by 4. Therefore, the total cost of 4 units is b)240.

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La tableta de chocolate de Demetrio puede representarse como la unidad entera. Si Demetrio dejó 1/4 de la tableta para sí mismo, entonces se puede decir que se comió 3/4 de la tableta.

A continuación, si le dio 1/3 de la tableta a su hermano, entonces queda:

3/4 x 1/3 = 1/4

Por lo tanto, Demetrio le dio 1/4 de la tableta a su hermano.

Teniendo en cuenta que le quedaba 1/4 de la tableta, se puede deducir que eso es lo que le dio a su profesor.

En resumen, Demetrio se comió 3/4 de la tableta, le dio 1/4 de la tableta a su hermano y el otro 1/4 de la tableta se lo dio a su profesor.

To create a price prediction model for a used Toyota Corolla based on the given data, follow these steps:

1. Open Excel and import the dataset containing 1436 vehicles' selling price and odometer reading (km).

2. Click on the Data tab, then select Data Analysis, and choose Regression.

3. Select the appropriate input ranges for your dependent variable (selling price) and independent variable (odometer reading), and set alpha to 0.025.

4. Click on the Output Range to select the desired location for the regression statistics, then click OK.

Now, interpret the regression statistics:
- The R-squared value indicates how well the model fits the data.

- The p-value associated with the odometer reading will determine its significance in predicting the selling price. If p < 0.025, it's significant.

If the vehicle's KM driven is significant in predicting the selling price, the expected selling price, as well as the upper and lower limits (97.5% confidence interval), can be calculated using the regression coefficients.

To check for outliers in the dataset:
1. Calculate the residuals by subtracting the predicted selling prices from the actual selling prices.

2. Identify any outliers based on unusual residual values.

3. If outliers are present, remove them and perform the regression analysis again.

4. Compare the new R-squared value with the original model to determine if the model's explanatory power has improved.

In summary, use Excel's regression analysis tool with an alpha of 0.025 to create a simple regression model, interpret the regression statistics, determine if the vehicle's KM driven is significant, and calculate the expected selling price for a Toyota Corolla with 23500 km on the odometer.

Check for outliers and their impact on the model's explanatory power.

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