a racing car consumes a mean of 103 gallons of gas per race with a variance of 36 . if 38 racing cars are randomly selected, what is the probability that the sample mean would differ from the population mean by less than 2 gallons? round your answer to four decimal places.

Answer: Prove that the triangles are similar, and therefore the lines have the same slope and are parallel.

Answer:

[tex]h(x)=2x^2-12x+9[/tex],  [tex]p(x)=-5x^2-20x-5[/tex]

Step-by-step explanation:

To get to the standard form of a quadratic equation, we need to expand and simplify. Recall that standard form is written like so:

[tex]ax^2+bx+c[/tex]

Where a, b, and c are constants.

Let's expand and simplify h(x).

[tex]2(x-3)^2-9=\\2(x^2+9-6x)-9=\\2x^2+18-12x-9=\\2x^2+9-12x=\\2x^2-12x+9[/tex]

Thus, [tex]h(x)=2x^2-12x+9[/tex]

Let's do the same for p(x).

[tex]-5(x+2)^2+15=\\-5(x^2+4+4x)+15=\\-5x^2-20-20x+15=\\-5x^2-5-20x=\\-5x^2-20x-5[/tex]

Thus, [tex]p(x)=-5x^2-20x-5[/tex]

The area of the triangle is: 15 square feet

The area of a rectangle is: 96 square feet

The formula for the area of a triangle is:

Area = ¹/₂ * base * height

This triangle has the dimensions:

base = 12 ft

height = 2.5 ft

Thus:

Area = ¹/₂ * 12 * 2.5

Area = 15 square feet

The area of a rectangle is:

A = length * width

A = 12 * 8

A = 96 square feet

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Answer:

y = x - 5

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Points (-1, -6) and (6, 1)

We see the y increase by 7, and the x increase by 7, so the slope is

m = 7/7 = 1

Y-intercept is located at (0, -5)

So, the equation is y = x - 5

[tex](\stackrel{x_1}{-1}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{(-6)}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{(-1)}}} \implies \cfrac{1 +6}{6 +1} \implies \cfrac{ 7 }{ 7 } \implies 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-6)}=\stackrel{m}{ 1}(x-\stackrel{x_1}{(-1)}) \implies y +6 = 1 ( x +1) \\\\\\ y+6=x+1\implies {\Large \begin{array}{llll} y=x-5 \end{array}}[/tex]

The value of x in the intersecting chords that extend outside circle is 5

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

intersecting chords that extend outside circle

Using the theorem of intersecting chords, we have

4 * (x + 6 + 4) = 6 * (x - 1 + 6)

Evaluate the like terms

So, we have

4 * (x + 10) = 6 * (x + 5)

Using a graphing tool, we have

x = 5

Hence. the value of x is 5

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The chance that the number of heads will be in the range 4850 to 5150 is approximately 0.9973, or about 99.73%.

The number of heads in 10,000 coin tosses follows a binomial distribution with parameters n = 10,000 (the number of trials) and p = 0.5 (the probability of heads on a single toss).

We can approximate this binomial distribution using the normal distribution, with mean μ = np = 5000 and variance σ² = np(1-p) = 2500.

To find the probability that the number of heads is in the range 4850 to 5150, we can use the normal distribution and standardize the range using the z-score formula:

z = (x - μ) / σ

where x is the number of heads in the range we're interested in.

For the lower bound of 4850, we have:

[tex]z_lower = (4850 - 5000) / \sqrt{(2500)}[/tex]

= -3

For the upper bound of 5150, we have:

[tex]z_upper = (5150 - 5000) / \sqrt{(2500)} = 3[/tex]

Using a standard normal distribution table or calculator, we can find the probability of being within 3 standard deviations of the mean:

P([tex]z_lower[/tex]  < Z < [tex]z_upper[/tex] ) ≈ P(-3 < Z < 3)

= 0.9973.

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The equation x = 10 tan (32°) could be used to find x in ∆ABC.

A triangle is classified as a right triangle when it presents one of your angles equal to 90º.  The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.

The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.

The Pythagorean Theorem says: (hypotenuse)²= (leg1)²+(leg2)² . And the main trigonometric ratios are: sin (x) , cos  (x) and tan  (x) , where:

[tex]sin(x)=\frac{opposite\ side}{hypotenuse} \\ \\ cos(x)=\frac{adjacent\ side}{hypotenuse}\\ \\ tan(x)=\frac{sin(x)}{cos(x)} =\frac{opposite\ side}{adjacent\ side}[/tex]

The question gives the value of the two sides and the value of an angle. From the trigonometric ratios presented before, you can write:

[tex]tan(32)=\frac{opposite\ side}{adjacent\ side}=\frac{x}{10} \\ \\ x=10\ tan (32\°)[/tex]

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Answer:

x = -5, x = 1

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

Step-by-step explanation:

The x-intercepts are the x-values of the points at which the curve crosses the x-axis, so when y = 0.

From inspection of the given graph, we can see that the parabola crosses the x-axis at x = -5 and x = 1.

Therefore, the x-intercepts of the parabola are:

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

Sarah then purchased 9 coach seats as by increasing the first equation by 190 and deducting it from the second equation.

An equation is a logical statement that utilises the equal sign to demonstrate the equality of two expressions. Factors, constants, and mathematical like addition, reduction, multiply, division, and exponentiation can all be found in it. Equations are utilised to find solutions for problems in both mathematics and the real world.

given

Let's use the letters "c" for the quantity of coach tickets and "f" for the quantity of first-class tickets. We are aware that there were 12 travellers in all, so

c + f + 1 = 12

We also know that the entire cost of the airfare was $4650, with coach tickets costing $190 and first-class tickets costing $980. With this knowledge, we can construct the equation shown below:

[tex]190c + 980f = 4650 - 980[/tex]

When we simplify this equation, we obtain:

[tex]190c + 980f = 3670[/tex]

Elimination can now be used to find either "c" or "f." By increasing the first equation by 190 and deducting it from the second equation, let's get rid of "c":

[tex]190c + 190f + 190 = 2280[/tex]

-190c - 980f = -3670

-790f = -1390

f = 1.76

We can round "f" up to 2 because we cannot have a fractional number of persons.

c + 2 + 1 = 12

c = 9

Sarah then purchased 9 coach seats as by increasing the first equation by 190 and deducting it from the second equation.

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Without knowing which specific function you're referring to, the answer to this question may depend on the type of function and the nature of the iterative process applied to it. In some cases, the function value may converge towards a limiting value as the number of iterations increases, while in other cases it may oscillate or diverge.

For example, in the case of the fixed-point iteration method used to find the root of a function, the value of the function typically converges towards the root as the number of iterations increases. More specifically, if we have a function f(x) and a starting guess x0 for its root, we can use the iterative formula x(+1)=g(x()), where g(x) is some function that we set based on f(x), to generate a sequence of increasingly accurate approximations to the root. As the number of iterations increases, this sequence of approximations typically converges towards the root of the function, unless some conditions are not met (e.g., the method is not well-suited for some functions, or the iteration formula is not properly set.)

In the case of other types of iterative methods or other functions, however, the behavior of the function value as the number of iterations increases may differ. For instance, in some cases, the function value may oscillate between two or more values or diverge to infinity as the number of iterations increases.

Therefore, the specific behavior of the function value as the number of iterations increases may depend on the specific function being evaluated and the iterative method used.

The least number of arc measures needed to determine the measures of each angle in the inscribed quadrilateral is 2.

To determine the measures of each angle in the quadrilateral, we need to find the central angles of the arcs that intersect the quadrilateral's vertices. Since the quadrilateral is not a rectangle, it is not a cyclic quadrilateral, which means that its opposite angles do not add up to 180 degrees.

Therefore, we need to use the fact that the sum of the measures of the opposite angles in an inscribed quadrilateral is 360 degrees. Let the angles of the quadrilateral be A, B, C, and D, with opposite angles A and C, and B and D. We can find the measure of arc AC by drawing a chord connecting the endpoints of AC and finding the central angle that intercepts it. Similarly, we can find the measure of arc BD.

Now, we can use the fact that the sum of the central angles that intercept arcs AC and BD is equal to 360 degrees. Let these angles be x and y, respectively. Then, we have:

x + y = 360

We can solve for one of the variables, say y, in terms of the other:

y = 360 - x

Substituting this into the equation for arc BD, we have:

2x + 2(360 - x) = arc BD

Simplifying this equation, we get:

arc BD = 720 - 2x

Now, we can use the fact that the sum of the measures of angles A and C is equal to the measure of arc AC, and the sum of the measures of angles B and D is equal to the measure of arc BD. Therefore, we have:

A + C = arc AC
B + D = arc BD = 720 - 2x

We need to find the least number of arc measures needed to determine the measures of A, B, C, and D. Since we have two equations and two variables (x and A), we can solve for both variables. Then, we can use the equations for B and D to find their measures.

Solving for A in terms of x, we have:

A = arc AC - C
A = 360 - x - C

Substituting this into the equation for B + D, we have:

(360 - x - C) + B + D = 720 - 2x

Simplifying this equation, we get:

B + D = 360 + x - C

Now, we have three equations and three variables (x, A, and C). We can solve for each variable in terms of x, and then use the equation for B + D to find their measures.

Therefore, the least number of arc measures needed to determine the measures of each angle in the quadrilateral is two: arc AC and arc BD.

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Answer: Let the experienced one help you out! Therefore, the ball hits the ground after 4 seconds. Read the explanation down below:

Brainliest?

Step-by-step explanation:

To find when the ball hits the ground, we need to find the value of t when h=0, since at that point the height of the ball is zero, indicating that it has reached the ground.

We have the equation:

h = -16t^2 + 48t + 64

Setting h to zero, we get:

0 = -16t^2 + 48t + 64

Dividing both sides by -16, we get:

0 = t^2 - 3t - 4

Now we can use the quadratic formula to solve for t:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -3, and c = -4.

Plugging in these values, we get:

t = (-(-3) ± sqrt((-3)^2 - 4(1)(-4))) / 2(1)

t = (3 ± sqrt(9 + 16)) / 2

t = (3 ± 5) / 2

So we have two solutions:

t = (3 + 5) / 2 = 4

t = (3 - 5) / 2 = -1

The negative solution doesn't make sense in this context, so we discard it. Therefore, the ball hits the ground after 4 seconds.

Tim was driving for 5 hours before he caught up to Dora.

The answer is (a) 5 hours.

To solve this problem, we can use the formula:
distance = rate × time
Let's denote the time Tim drove as t hours.

Since Dora started driving one hour earlier, her driving time would be (t + 1) hours.
Dora's distance: 75 kph × (t + 1)
Tim's distance: 90 kph × t
Since Tim catches up to Dora, their distances will be equal:
75(t + 1) = 90t
Now we can solve for t:
75t + 75 = 90t
75 = 15t
t = 5.

The answer is (a) 5 hours.

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The expression for the area of the rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b is ad + bd + ac + bc.

The rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b can be visualized as a rectangle with width d+c and height a+b.

Area is a measurement of the size of a two-dimensional surface, such as a plane or a flat shape.

Therefore, the area of the rectangular region is given by

Area = Width x Height

Substitute the values in the equation and find the expression of area

= (d + c) x (a+b)

= ad + bd + ac + bc

So the expression for the area of the rectangular region is

ad + bd + ac + bc.

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The given question is incomplete, the complete question is:

Let a, b, c, and d be positive numbers. Find the expression for the area of the rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b?

the simplified  form  of expression is: -(x² + 2x - 2)/((x+2)*(x+4))

what is expression  ?

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that represents a mathematical quantity or relationship. Expressions can be simple or complex

In the given question,

To evaluate the expression 1/(x+2) - (x+1)/(x+4), we need to find a common denominator for the two terms. The least common multiple of (x+2) and (x+4) is (x+2)(x+4).

So, we can rewrite the expression as:

(1*(x+4) - (x+1)(x+2))/((x+2)(x+4))

Expanding the brackets, we get:

(x+4 - x² - 3x - 2)/((x+2)*(x+4))

Simplifying the numerator, we get:

(-x² - 2x + 2)/((x+2)*(x+4))

Therefore, the simplified expression is:

-(x² + 2x - 2)/((x+2)*(x+4))

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The terms arranged in order from smallest to biggest are: (-2)³, -√25, √11, 10, and 4² after comparing the values of the final numbers.

We shall first simplify the numbers to get their final values and then compare to which is smaller as follows:

4² = 4 × 4 = 16

-√25 = -5

10 = 10

√11 = 3.3166

(-2)³ = -2 × -2 × -2 = -8

In conclusion, we have by comparing the final values of the numbers the terms arranged from smallest to the biggest as: (-2)³, -√25, √11, 10, and 4².

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To form a suitable quadratic equation using the values from the table given in the question also  considering the event of forming a equation that represents f(x) is Option A.
In order to find the equation that is  represented by f(x), we have to implement the standard form of a quadratic function
f(x) = ax² + bx + c
here a, b and c = constants.
We can utilize the given table of values to evaluate these constants.

Now, we have to  place each x value into f(x) to get the concerning y value. Then we can utilize these points to create three equations with three undetermined (a, b and c).
Evaluating these equations will give us the values of a, b and c.

Now, the table of values given in the question is

f(-2) = 48 = 4a - 4b + c
f(-1) = 50 = a - b + c
f(0) = 48 = c
f(1) = 42 = a + b + c
f(2) = 32 = 4a + 4b + c
f(3) = 18 = 9a + 3b + c
f(4) = 0 = 16a + 4b + c


Calculating these equations

a = -2
b = -4
c = 48

Hence, the equation that represents f(x) is f(x) = -2x² - 4x + 48.

The correct option for the given question after considering the given conditions is Option A.

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The complete question is
The table of values forms a quadratic function f(x). X f(x)−2 48
f(−1) = 50
f(0) = 48
f(1) = 42
f(2) = 32
f(3) = 18
f(4) = 0

What is the equation that represents f(x)?
a) f(x) = –2x² – 4x + 48
b) f(x) = 2x² + 4x – 48
c) f(x) = x² + 2x – 24
d) f(x) = –x² – 2x + 24

The value of angle GFH is 18°

A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.

A theorem in circle geometry starts that angle in the same segment are equal. In triangle EFG, angle F and G are on the same segment, this means that angle F and G are equal.

Represent angle F as x

therefore 144+2x = 180° ( sum of angle in a triangle)

2x = 180-144

2x = 36

x = 36/2 = 18°

Therefore the measure of angle GFH is 18°

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The points on the surface z² = xy + 16 closest to the origin are: (-4,4,0) and (4, -4, 0)

We know that the distance between an arbitrary point on the surface and the origin is d(x, y, z) = √(x² + y² + z²)

Using Lagrange multipliers,

L(x, y, z, λ) = x² + y² + z² + λ(z² - xy - 16)

We have partial derivatives.

[tex]L_x[/tex] = 2x - λy

[tex]L_y[/tex] = 2y - λx

[tex]L_z[/tex] = 2z + 2zλ

[tex]L_\lambda[/tex] = z² - xy - 16

Now we set each partial derivative to zero to find critical points.

[tex]L_x[/tex] = 0

2x - λy = 0

[tex]L_y[/tex] = 0

2y - λx = 0

After solving above equations simultaneously we get (x + y)(x - y) = 0

i.e., x = -y   OR   x = y

[tex]L_z[/tex] = 0

2z + 2zλ = 0

z = 0  OR  λ = 0

Consider [tex]L_\lambda[/tex] = 0

z² - xy - 16 = 0

-xy = 16                  ............(as z = 0)

when x = y then -y² = 16 which is not true.

So, consider x = -y

-(-y)y = 16

y² = 16

y = ±4

when y = 4 then we get x = -4

and when y = -4 then we get x = 4

Therefore, the closest points are:(-4,4,0) and (4, -4, 0)

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Just use the pythagorean theorem to solve the hypotenuse!

(3^2)+(2^2)=x^2

9+4=13^2

[tex]\sqrt{13}[/tex] = [tex]\sqrt{x}[/tex]

[tex]13^{2}[/tex] km

Hope this helps <3

The sides and the angle of the right triangle are a = 10√2, b = 10√2 and B = π / 4.

In this problem we need to determine the values of two sides and an angle of the right triangle. This can be done by means of the following properties:

A + B + C = π

sin A = a / c

cos A = b / c

tan A = a / b

Where:

If we know that A = π / 4, C = π / 2 and c = 20, then the missing angle and missing sides are, respectively:

B = π - π / 4 - π / 2

B = π / 4

cos (π / 4) = b / 20

b = 20 · cos (π / 4)

b = 10√2

sin (π / 4) = a / 20

a = 20 · sin (π / 4)

a = 10√2

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The probability of getting 'p' success when a die rolled 12 times with 'x' success that has binomial distribution is equal to ¹²Cₓ pˣ ( 1 - p )¹²⁻ˣ.

Number of times die to be rolled = 12

In this scenario, a trial refers to a single roll of the 20-sided die.

Here , 'x' represents the the number times the die lands on a green square.

A success would be defined as landing on a green square,

And a failure would be landing on any other color.

Since the die will be rolled 12 times, there are 12 trials in total.

This implies, the number of times the die lands on a green square, x, has a binomial distribution.

With parameters n = 12 the number of trials and p the probability of success  which is landing on a green square.

Probability = ⁿCₓ pˣ ( 1 - p )ⁿ⁻ˣ

Therefore, the probability of rolling a die 12 times with 'x' success which has binomial distribution and 'p' probability of success is equal to ¹²Cₓ pˣ ( 1 - p )¹²⁻ˣ.

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Therefore, the roots of the polynomial equation x² + x - 72 = 0 are -9 and 8.

A quadratic equation is a type of polynomial equation of the second degree, which means it has one or more terms in which the variable is raised to the power of two, but no higher powers.Quadratic equations can have zero, one, or two real solutions, depending on the values of a, b, and c. These solutions are also called the roots or zeros of the equation.

Here,

To find the roots of the polynomial equation x² + x - 72 = 0, we can set y = 0 and solve for x. This is equivalent to finding the x-intercepts of the graph of the function f(x) = x² + x - 72.

We can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

In this case, a = 1, b = 1, and c = -72, so we have:

x = (-1 ± √(1² - 4(1)(-72))) / (2(1))

x = (-1 ± √(1 + 288)) / 2

x = (-1 ± √(289)) / 2

x = (-1 ± 17) / 2

Therefore, the roots of the polynomial equation x² + x - 72 = 0 are:

x = -9 or x = 8

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A p-value is a probability.

A p-value is the probability of obtaining a test statistic as extreme or more extreme.

The observed value, assuming the null hypothesis is true.

It ranges in value from 0 to 1 and represents the strength of evidence against the null hypothesis.

A p-value cannot be negative, as it is a probability and probabilities are always between 0 and 1.

A p-value also cannot be larger than 1, as it represents a probability.

A probability cannot exceed 1.

Finally, a p-value cannot be smaller than 0, as it represents a probability.

A probability cannot be negative.

the correct option is b. is a probability.

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The Cube root of the resulting number is 8.

The smallest number that 40000 can be divided by so that the result is a perfect cube, we need to factorize 40000 into its prime factors:

[tex]40000 = 2^6 \times 5^4[/tex]

To make this a perfect cube, we need to ensure that the powers of each prime factor are multiples of 3.

The smallest number we can divide 40000 by so that the result is a perfect cube is:

[tex]40000 = 2^6 \times 5^4[/tex]

Now we can find the cube root of the resulting number:

[tex]3\sqrt (40000 \div 100) = 3\sqrt400 = 8.[/tex]

Factories 40000 into its prime components in order to determine.

The least number that the result may be divided by while still producing a perfect cube.

The powers of each prime factor must be multiples of three in order for this to be a perfect cube.

The least number that 40000 may be divided by to produce a perfect cube is:

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The 'footprint' of CO2 emissions for a person in 1830 would be 818,199 tons of CO2 emissions per person.

To find the 'footprint' of CO2 emissions for a person in 1830, we need to substitute the value of x = 1830 - 1800 = 30 into the given function C(x) = 0.0365 (1.758)^x.

Plugging in x = 30 into the function, we get:

C(30) = 0.0365 * (1.758)^30

Substituting this value back into the function, we get:

C(30) = 0.0365 * 22416413.1381

C(30) = 818199.079541

C(30) ≈ 818,199.08

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The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is approximately [-6.62, 22.02].

The critical value that should be used in constructing the confidence interval.

Since we are looking for a 90% confidence interval, we need to find the critical value associated with a 5% level of significance in a two-tailed test.

Using a t-distribution with (n1-1) + (n2-1) degrees of freedom and a significance level of 0.05, we find the critical value to be:

t-critical = 1.717 (using a t-distribution table or a calculator)

Step 2 of 3:

Next, we need to find the standard error of the sampling distribution to be used in constructing the confidence interval.

Since the population variances are not equal, we need to use the Welch-Satterthwaite equation to calculate the standard error:

SE = sqrt[([tex]s1^2[/tex]/n1) + ([tex]s2^2[/tex]/n2)]

where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Substituting the given values, we get:

SE = sqrt[([tex]19.25^2[/tex]/17) + ([tex]26.25^2[/tex]/12)]

SE ≈ 8.35

Step 3 of 3:

To construct the 90% confidence interval, we can use the formula:

(mean1 - mean2) ± t-critical * SE

where mean1 and mean2 are the sample means, and t-critical and SE are the values calculated in steps 1 and 2.

Substituting the given values, we get:

= (84.80 - 77.10) ± 1.717 x 8.35

= 7.70 ± 14.32

Therefore,

The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is (approx) [-6.62, 22.02].

We can be 90% confident that the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases falls within this interval.

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