prove that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other

Answer:

I'm sorry, your question is not clear. Please provide more information or context so I can better understand what you are asking.

Step-by-step explanation:

Use the line segment that connects (3, 42.5) and (5, 44) to estimate the vine length after 4 days.

Given the table of values

To do this, we make use of linear interpolation

Such that the points closest to day 4 are (3, 42.5) and (5, 44)

The slope of the line passing through the two points can be found using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) = (3, 42.5) and (x2, y2) = (5, 44).

Thus, we have:

m = (44 - 42.5) / (5 - 3) = 0.75

So, we have

y = 0.75x + b

Using the point (3, 42.5), we get:

42.5 = 0.75(3) + b

Solving for b, we get:

b = 42.5 - 0.75(3)

b = 40.25

So, the equation is

y = 0.75x + 40.25

To estimate the vine length after 4 days, we substitute x = 4 into the equation and solve for y:

y = 0.75 * 4 + 40.25

y = 43.25

Therefore, the vine length after 4 days is estimated to be about 43.25 units.

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The measure of ∠J in ΔJKL is approximately 57.5 degrees.

The measure of ∠J in ΔJKL can be found using the trigonometric function tangent, which is defined as the ratio of the opposite side to the adjacent side.

The straight line that "just touches" the plane curve at a given point is called the tangent line in geometry. It was defined by Leibniz as the line that passes through two infinitely close points on the curve.

tan(∠J) = JK/KL

tan(∠J) = 7.3/4.7

∠J = arctan(7.3/4.7)

∠J = 57.5 degrees (rounded to the nearest tenth of a degree)

Therefore, the measure of ∠J in ΔJKL is approximately 57.5 degrees.

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According to Moore's Law, the number of transistors that can be placed inexpensively on a silicon chip doubles every two years, a typical CPU contained about 1,000,000 transistors in 1990.

Let’s first calculate the number of doublings from 1990 to 2000. Number of years from 1990 to 2000 = 2000 - 1990 = 10 yearsDoublings from 1990 to 2000 = [tex]$\dfrac{10 \text{ years}}{2 \text{ years per doubling}} = 5$[/tex] doublingsNow, we can calculate the number of transistors in a typical CPU in the year 2000:

[tex]$$\begin{aligned} \text{Number of transistors in 2000} &= \text{Number of transistors in 1990} \times 2^{\text{number of doublings}} \\ &= 1,\!000,\!000 \times 2^5 \\ &= 32,\!000,\!000 \end{aligned}$$[/tex]

Therefore, a typical CPU contained about 32,000,000 transistors in the year 2000.

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Step-by-step explanation:

Area of a square = s x s   and this equals  10 x s

 this is :  

     s^2 = 10 s    divide both sides of the equation by 's'

  s = 10   units    ( or zero....but that makes no sense)

Using the t-distribution, the 99% confidence interval for the true average number of alcoholic drinks all UF female students (over 21) have in a one week period is (3.25, 4.85).

First, we find the number of degrees of freedom, which is the sample size subtracted by 1, thus:

df = 170-1 = 169

Using the following formulas the lower and upper limits of the Interval are calculated,

n = 170

x = 4.05

s = 4.66 = 99% = 0.99

Because the population standard deviation is unknown, the Student T-distribution should be used. Yet, because the sample is huge, some books will utilise the normal distribution. I'll provide solutions for both techniques.

Error = z x s/√n

= 1.99 x 4.66/√170

Error margin ≈ 0.8

Lower limit = 4.05 - Error

= 4.05 - 0.8

= 3.25

Upper limit = 4.05 + Error

= 4.05 + 0.8

= 4.85

Therefore, the upper limit and lower limit is 3.25 and 4.85.

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Answer:

operaciones vectoriales, Extensión de las leyes del álgebra elemental a los vectores. Incluyen suma, resta y tres tipos de multiplicación. La suma de dos vectores es un tercer vector, representado como la diagonal del paralelogramo construido con los dos vectores originales como lados.

Answer:

operaciones vectoriales, Extensión de las leyes del álgebra elemental a los vectores. Incluyen suma, resta y tres tipos de multiplicación. La suma de dos vectores es un tercer vector, representado como la diagonal del paralelogramo construido con los dos vectores originales como lados.

Step-by-step explanation:

Correct Option is Shifted 2 units left and 4 units up

A triangle is a geometric shape that is formed by three straight line segments that connect three non-collinear points. The three points where the segments intersect are called the vertices of the triangle, while the segments themselves are called the sides. The area enclosed by the sides of the triangle is called its interior, while the space outside the triangle is called its exterior.

The vertices of ABC are (4, 6), (7, 6), and (5,9)

The transformed image A'B'C' has vertices as

(2,10) (5,10) (3,13)

We see a pattern when we compare the matching vertices.

The y coordinate is raised by 4, while the x coordinate is shrunk by 2.

This implies the transformation is

Shifted 2 units left and 4 units up

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Answer:

Shifted 2 units left and 4 units up

Step-by-step explanation:

hope this helps

Answer:

The sum of the first n terms in an arithmetic sequence is (n/2)⋅(a₁+aₙ). It is called the arithmetic series formula.

Step-by-step explanation:

An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. Following is a simple formula for finding the sum: Formula 1: If S nrepresents the sum of an arithmetic sequence with terms , then. This formula requires the values of the first and last terms and the number of terms.

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Answer:

[tex]\bf a^2* (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

Step-by-step explanation:

Factorize:

 First take out the common term a².

[tex]a^5 - 0.064a^2= a^2*(a^3 - 0.064)\\\\[/tex]

Now, factorize using the identity a³ - b³

a³ - b³ = (a - b) (a²  + ab + b²)

[tex]a^2 * (a^2 - 0.064) = a^2 * (a^3 - 0.4^3)[/tex]

                          [tex]= a^2 * (a - 0.4) * (a^2 + a*0.4 + 0.4^2)\\\\=a^2 * (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

Answer: 180,000 g

Step-by-step explanation:

1st add all masses together:

80kg + 40kg + 60 kg = 180 kg

then we need to convert kilograms to grams:

1 kg = 1000g

180kg * (1000g / 1kg) = 180,000 g

To calculate the total mass (in grams) of the three boxes, we first need to convert their masses from kilograms to grams and then add them together.

The mass of the first box is 80 kg, which is equivalent to 80,000 grams (because 1 kg = 1000 grams).

The mass of the second box is 40 kilograms, or 40,000 grams.

The mass of the third box is 60 kilograms, or 60,000 grams.

To find the total mass of the three boxes, we add these values:

80,000 grams + 40,000 grams + 60,000 grams = 180,000 grams

Therefore, The total mass of the three boxes in gram is 180,000

The correct answer is D) 180,000

The slope of a linear equation is an important skill to have, as it can be used to identify the rate of change of the equation, and can be used to predict future values of the equation.

An equation is an expression that states the equality of two things. It typically consists of an equal sign (=) and two expressions on either side of the equal sign that represent the same thing. Equations are used to describe relationships between different variables and can be used to solve mathematical problems. They can also be used to show the relationships between different quantities in physics and chemistry.

a)The slope of this linear equation can be determined by using the slope formula, which is rise over run. In this equation, the rise is 6, and the run is 2, so the slope is 3.

b) The slope of this linear equation can be determined by using the slope formula, which is rise over run. In this equation, the rise is 3, and the run is -7, so the slope is -3/7.

c) The slope of this linear equation can be determined by using the slope formula, which is rise over run. In this equation, the rise is 4, and the run is 6, so the slope is 2/3.

Finding slope in tables and graphs is a common mathematical skill that is used to identify the rate of change of a linear equation. This is determined by finding the change in the dependent variable (the y-axis) divided by the change in the independent variable (the x-axis). This is what is referred to as the slope of the equation. To find the slope in tables and graphs, you must look at the differences between the points on the x-axis and y-axis, and divide the change in the y-axis by the change in the x-axis. This will give you the slope of the equation. Finding the slope of a linear equation is an important skill to have, as it can be used to identify the rate of change of the equation, and can be used to predict future values of the equation.

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The domain and range of the function f(x) =a^x, then option (c)  Domain = positive real numbers, range = positive real numbers.

The function f(x) = a^x is an exponential function with a base of a, where a is a positive real number. The domain of the function is all real numbers, because we can raise a positive number to any real power.

However, since a is positive, a^x will always be positive, which means that the range of the function is also positive real numbers. Therefore, the correct option is c. Domain = positive real numbers, range = positive real numbers.

Therefore, the correct option is (c) Domain = positive real numbers, range = positive real numbers.

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The distance from the point (15,-21) to the line is 5.39 units.

The equation of a line in point-slope form is:

y - y1 = m(x - x1) (x - x1)

where (x1, y1) is a point on the line and m is the slope of the line. When we are unsure of the y-intercept but are aware of the line's slope and a point on the line, we can utilise this form of the equation.

To calculate the equation of a line using the point-slope method, we must first determine the slope of the line using the following formula:

m = (y2 - y1)/(x2 - x1) (x2 - x1)

where the two points on the line are (x1, y1) and (x2, y2). We may enter the slope, along with one of the line's points, into the point-slope form to obtain the equation of the line.

Given that, the line has the following values f(4)=-8 and f(8)=-18.

The coordinates of the line are (4, -8) and (8, -18)

Thus, the slope of the line is:

m = y2 - y1/ x2 - x1

m = -18 + 8 / 8 - 4

m = -10/4 = -5/2

Now the slope intercept form is given as:

y - y1 = m (x - x1)

Substitute the values:

y + 8 = -5/2(x - 4)

2y + 16 = -5x + 20

2y = -5x + 20 - 16

2y = -5x + 4

The distance from the line to point is given as:

Distance = |ax + by + c| / √(a² + b²)

Substituting the values:

Distance = |-5(15) + -2(-21) + 4| √(-5² + -2²)

Distance = |-29|/ 5.38

Distance = 5.39

Hence, the distance from the point (15,-21) to the line is 5.39 units.

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The work required to pump all the water out of the rectangular swimming pool over the top is approximately 2,323,200 ft-lb.

We have,

To find the work required to pump all the water out of the rectangular swimming pool, we can use the concept of work as the force multiplied by the distance.

First, let's calculate the weight of the water in the pool.

The weight of an object is given by the formula:

Weight = mass x gravitational acceleration

Since the density of water is given as 1 lb/ft³, we need to find the volume of water in the pool.

The volume of the pool is given by the formula:

Volume = length x width x depth

Volume = 50 ft x 30 ft x 6 ft = 9000 ft³

Now, let's calculate the weight of the water:

Weight = density x volume x gravitational acceleration

Weight = 1 lb/ft³ x 9000 ft³ x 32.2 ft/s² ≈ 290,400 lb

To pump all the water out over the top, we need to raise it to the height of the pool, which is 8 ft.

The work required to pump the water out is given by the formula:

Work = weight x height

Work = 290,400 lb x 8 ft = 2,323,200 ft-lb

Therefore,

The work required to pump all the water out of the rectangular swimming pool over the top is approximately 2,323,200 ft-lb.

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intersection of A and B events, P(A ∩ B) is 0. So, P(A U B) = P(A) + P(B) = 0.3 + 0.6 = 0.9Hence, P(AUB) = 0.9.

Probability of P(ANC)We know that events A, B and C are mutually exclusive.

Therefore, if A, B, and C are mutually exclusive events, then P(A U B U C) = P(A) + P(B) + P(C). Given, P(A) = 0.3,P(B) = 0.6,P(C) = 0.1

Therefore, P(A U B U C) = P(A) + P(B) + P(C) = 0.3 + 0.6 + 0.1 = 1Now, P(ANC) = 1 - P(A U B U C) = 1 - 1 = 0

Probability the intersection of A and B events, P(A ∩ B) is 0. So, P(A U B) = P(A) + P(B) = 0.3 + 0.6 = 0.9

Hence, P(AUB) = 0.9.ility of P(AUB)We know that events A, B and C are mutually exclusive.

Therefore, if A, B, and C are mutually exclusive events, then P(A U B U C) = P(A) + P(B) + P(C)

Now, we need to find P(AUB). If two events A and B are not mutually exclusive events, then the probability of their union P(A U B) can be found as follows; [tex]P(A U B) = P(A) + P(B) - P(A ∩ B)[/tex]We know that events A, B and C are mutually exclusive.

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Answer:

y=(3/4)x+6

Step-by-step explanation:

Point 1 (-4,3)

Point 2 (0,6)

To obtain the slope, we apply the formula:

m= (6-3) / (0- (-4))

m= 3 / 4

We replace in the equation y=mx+b.

We take any point.

=> (0,6)

y=mx+b

6=(3/4)(0)+b

b=6

Joining all the terms:

y=(3/4)x+6

Consequently, the area of triangle $AMN$ is equal to $A = \frac{1}{2}bh = \frac{1}{2}(4)(2) = 4$ cm2.

The area of triangle $AMN$ in square $ABCD$ can be calculated using the formula for area of a triangle, $A = \frac{1}{2}bh$, where $b$ is the length of the base and $h$ is the height of the triangle.

Since side $BC$ has a length of 4 cm, we can determine that $N$ is located 2 cm away from point $B$ and 2 cm away from point $C$.

Similarly, we can conclude that $M$ is located 2 cm away from point $C$ and 2 cm away from point $D$.

Therefore, the base of triangle $AMN$ is equal to 4 cm, and the height of triangle $AMN$ is equal to 2 cm.

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By using this value of x in the formula we previously discovered, we can get the patio's area Patio's size is equal to x2 + 4x + 4 = ((1 + 7)/3)2 + 4((1 + 7)/3) + 4 = 4.72 square meters.

A square is a 2D shape with equal-sized sides on each side. The area would be length times width, which is equal to side  side because all the sides are equal. As a result, a square's area is side square.

Let's first find the area of the entire rectangle:

A = lw = (3x + 6)(2x + 4) = 6x² + 30x + 24

Area of patio = (x + 2)² = x² + 4x + 4

Area of herb garden = (2x + 2)(x + 4) = 2x² + 10x + 8

Area of flower garden = (3x + 4)(x + 4) = 3x² + 16x + 16

Sum of areas = x² + 4x + 4 + 2x² + 10x + 8 + 3x² + 16x + 16

= 6x² + 30x + 28

0.25(6x² + 30x + 24) = 6x² + 30x + 28

Simplifying and solving for x, we get:

1.5x² - x - 1 = 0

Using the quadratic formula, we find that:

x = (1 ± √7)/3

x = (1 + √7)/3

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The area in square meters of the patio is 850 square meters.

What is a rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees) with opposite sides being parallel and equal in length.

Let's start by calculating the total area of the rectangle:

Area of rectangle = length x width = 100m x 40m = 4000 square meters

Now, let's denote the width of the herb garden as x meters. Then, the length of the herb garden would be 10 meters.

The area of the herb garden would be:

Area of herb garden = length x width = 10m x x = 10x square meters

The area of the patio can be calculated as:

Area of patio = (100 - x) x (40 - 2x) square meters

(100 - x) is the length of the patio, and (40 - 2x) is the width of the patio, since the herb garden takes up x meters of the width.

The area of the flower garden can be calculated by subtracting the area of the rectangle, the herb garden, and the patio from each other:

Area of flower garden = 4000 - 10x - (100 - x) x (40 - 2x) square meters

Now, we need to find the value of x so that the sum of the areas of the patio, herb garden, and flower garden is 25% of the area of the entire rectangle. In other words:

Area of herb garden + Area of patio + Area of flower garden = 0.25 x Area of rectangle

10x + (100 - x) x (40 - 2x) + 4000 - 10x = 0.25 x 4000

Simplifying this equation, we get:

-2x^2 + 30x + 1000 = 1000

-2x^2 + 30x = 0

-2x(x - 15) = 0

Therefore, x = 0 or x = 15. Since x cannot be 0 (since the herb garden would have no width), the value of x must be 15 meters.

Now we can calculate the area of the patio:

Area of patio = (100 - x) x (40 - 2x) = (100 - 15) x (40 - 2(15)) = 850 square meters

Therefore, the area in square meters of the patio is 850 square meters.

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Using the given percentages we can see that 150 people likes grapes.

To find this, we need to take the product between the percentage of people that likes grapes (in decimal form) and the total number of people surveyed.

To get the decimal form of the percentage we just need to divide it by 100%, we will get:

20%/100% = 0.2

And there were 750 people surveyed, then the total number of people that likes grapes is:

N = 750*0.2 = 150.

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Answer:

10

Step-by-step explanation:

Based on the given conditions, formulate: 40x16 divided by 64

Cross out the common factor: 40/4

Cross out common factor: 10

Get the result

Answer: 10

The Rico ride his bike at a speed of 6 miles per hour.

Speed is a measure of how quickly an object moves, calculated as the distance traveled per unit of time.

Let's take Rico's speed on his bicycle is 'r'.

So, Luis's speed can be expressed as 4r.

Time = Distance / Speed

For Luis, the time it takes to travel 24 miles is:

Time = 24miles / 4r

For Rico, the time it takes to travel 24miles is:

Time = 24miles / r

Since Rico takes 3 hours longer than Luis to travel the same distance (24miles), we can set up an equation:

24miles / r = (24miles / 4r) + 3hour

Simplifying this equation, we get:

⇒ 24 = 6 + 3r

⇒ r = 6

Therefore, Rico can ride his bike at a speed of 6 miles per hour.

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Answer:

Step-by-step explanation:

The answer is true

The interval does not include the null hypothesis value, then the results are significant.

When calculating a confidence interval for the difference between two means or proportions, the significance level must be considered to determine whether the results suggest a significant difference.What is a confidence interval?A confidence interval (CI) is an interval estimate that quantifies the uncertainty associated with the unknown population parameter. Confidence intervals are used to express how confident we are about the accuracy of an estimated population parameter.The significance level is the level at which the results of a statistical test are considered statistically significant. The significance level is frequently represented as alpha, and its value is usually set to 0.05 (5%) in most statistical analyses. This implies that there is a 5% chance that the statistical test findings will indicate a significant difference when, in fact, there is no such difference. The significance level is the probability of rejecting a true null hypothesis.Therefore, in order to determine whether or not the results of a confidence interval for the difference between two means or proportions indicate a significant difference, we must compare the interval with the significance level (alpha) that was established before the test. If the interval contains the null hypothesis value (usually 0), then the results are not significant. If the interval does not include the null hypothesis value, then the results are significant.

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(a) The recursive formula for An is: An = (1.015)An-1 - 100.

(b) The explicit formula for An in summation notation is: An = 4000(1.015)^n - 100[1 + (1.015) + (1.015)^2 + ... + (1.015)^(n-1)].

(a) This formula says that to find the balance for month n, we take the balance for month n-1, multiply it by 1.015 (to account for the interest rate), and subtract 100 (to account for the withdrawal). This formula is consistent with the results for n=1,2,3 on the previous page.

(b) The explicit formula for An in summation notation is: An = 4000(1.015)^n - 100[1 + (1.015) + (1.015)^2 + ... + (1.015)^(n-1)].

This formula says that to find the balance for month n, we take the starting balance of 4000 multiplied by the interest rate raised to the power of n, and then subtract the sum of 100 multiplied by the geometric series (1 + r + r^2 + ... + r^(n-1)), where r = 1.015. This formula is consistent with the results for n=1,2,3 on the previous page.

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Answer: 8

Step-by-step explanation:

6 + 2 is a simple arithmetic operation, which results in 8. When adding 6 and 2, you are combining two numbers to find their total. In other words, you are asking "if I have 6 items and someone gives me 2 more, how many items do I have in total?" The answer is 8, because you would have a total of 8 items.

Answer:

8

Step-by-step explanation:

[tex]6+2[/tex]

We can find the value of this by counting up 2 from 6.

7, 8

So the answer is 8

The total value of both bonds is $704,367,500.

Coupon payment = [tex]\frac{Coupon rate * Par value}{2}[/tex]

Coupon payment = [tex]\frac{11 * $1,000}{2}[/tex]

Coupon payment = $55

PV = [tex]55 * [1 - (1 + 0.04)^{^-40} ] / 0.04 + $1,000 / (1 + 0.04)^40[/tex]

[tex]PV = $1,026.45[/tex]

[tex]Total value = PV * Number of bonds * Par value\\Total value = $1,026.45 * 150,000 * $1,000\\Total value = $153,967,500[/tex]

[tex]PV = \frac{Price}{(1 + r)^n}[/tex]

[tex]PV = \frac{16}{(1 +0.03)^60}\\PV = $1.72[/tex]

[tex]Total value = PV * Number of bonds * Par value\\Total value = $1.72 * 320,000 * $1,000\\Total value = $550,400,000[/tex]

Therefore, the total value of both bonds is:

[tex]Total value = Value of coupon bonds + Value of zero coupon bonds\\Total value = $153,967,500 + $550,400,000\\Total value = $704,367,500[/tex]

A coupon rate is the annual interest rate paid by a bond or other fixed-income security to its bondholders or investors. It is typically expressed as a percentage of the bond's face value, also known as its par value. For example, if a bond has a face value of $1,000 and a coupon rate of 5%, the bond will pay $50 in interest each year to its bondholders. The coupon payments are usually made semi-annually or annually, depending on the terms of the bond.

The coupon rate is set when the bond is issued and remains fixed throughout the life of the bond unless the bond issuer chooses to call the bond or the bond defaults. Coupon rates are determined by a variety of factors, including market conditions, the creditworthiness of the issuer, and the length of the bond's maturity.

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Complete Question: -

The IPO Investment Bank has the following financing outstanding,

Debt: 150,000 bonds with a coupon rate of 11 percent and a current price quote of 108; the bonds have 20 years to maturity. 320,000 zero coupon bonds with a price quote of 16 and 30 years until maturity. Both bonds have a par value of $1,000 and semiannual coupons.

Preferred stock: 240,000 shares of 9 percent preferred stock with a current price of $67, and a par value of $100.

Common stock: 3,500,000 shares of common stock; the current price is $53, and the beta of the stock is.9.

Market: The corporate tax rate is 24 percent, the market risk premium is 8 percent, and the risk-free rate is 5 percent.

What is the WACC for the company? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer:

Step-by-step explanation:

       [tex]x^2-5=-7x-1[/tex]

[tex]x^2+7x-5=-1[/tex]         (subtracted 7x from both sides of the equation)

[tex]x^2+7x-4=0[/tex]            (+1 both sides)

Use quadratic formula to solve for x:

   [tex]x=\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex]     where [tex]a=1,b=7,c=-4[/tex]

       [tex]=\frac{-7 \pm \sqrt{7^2 - 4\times1\times(-4)} }{2\times 1}[/tex]

       [tex]=\frac{-7 \pm \sqrt{49 +16} }{2}[/tex]

        [tex]=\frac{-7 \pm \sqrt{65} }{2}[/tex]

     [tex]x=\frac{-7 +\sqrt{65} }{2},\frac{-7 - \sqrt{65} }{2}[/tex]

     [tex]x=0.53,-7.53[/tex]

The probability that an 18-year-old man selected at random is between 70 and 72 inches tall is approximately 0.0793 and the probability that the mean height of a sample of eight 18-year-old men is between 70 and 72 inches is approximately 0.9057 and the probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

What do you mean by normally distributed data?

In statistics, a normal distribution is a probability distribution of a continuous random variable. It is also known as a Gaussian distribution, named after the mathematician Carl Friedrich Gauss. The normal distribution is a symmetric, bell-shaped curve that is defined by its mean and standard deviation.

Data that is normally distributed follows the pattern of the normal distribution curve. In a normal distribution, the majority of the data is clustered around the mean, with progressively fewer data points further away from the mean. The mean, median, and mode are all the same in a perfectly normal distribution.

Calculating the given probabilities :
(a) The probability that an 18-year-old man selected at random is between 70 and 72 inches tall can be found by standardizing the values and using the standard normal distribution table. First, we find the z-scores for 70 and 72 inches:

[tex]z-1 = (70 - 71) / 5 = -0.2[/tex]

[tex]z-2 = (72 - 71) / 5 = 0.2[/tex]

Then, we use the table to find the area between these two z-scores:

[tex]P(-0.2 < Z < 0.2) = 0.0793[/tex]

So the probability that an 18-year-old man selected at random is between 70 and 72 inches tall is approximately 0.0793.

(b) The mean height of a sample of eight 18-year-old men can be considered a random variable with a normal distribution. The mean of this distribution will still be 71 inches, but the standard deviation will be smaller, equal to the population standard deviation divided by the square root of the sample size:

[tex]\sigma_x = \sigma / \sqrt{n} = 5 / \sqrt{8} \approx 1.7678[/tex]

To find the probability that the sample mean height is between 70 and 72 inches, we standardize the values using the sample standard deviation:

[tex]z_1 = (70 - 71) / (5 / \sqrt{8}) \approx -1.7889[/tex]

[tex]z_2 = (72 - 71) / (5 / \sqrt{8}) \approx 1.7889[/tex]

Then, we use the standard normal distribution table to find the area between these two z-scores:

[tex]P(-1.7889 < Z < 1.7889) \approx 0.9057[/tex]

So the probability that the mean height of a sample of eight 18-year-old men is between 70 and 72 inches is approximately 0.9057.

(c) The probability in part (b) is much higher because the standard deviation is smaller for the x distribution. When we take a sample of eight individuals, the variability in their heights is reduced compared to the variability in the population as a whole. This reduction in variability results in a narrower distribution of sample means, with less probability in the tails and more probability around the mean. As a result, it becomes more likely that the sample mean falls within a given interval, such as between 70 and 72 inches.

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