Problem 2 (Vector and Matrix Refresh) Seven data points are arranged as columns of a data matrix X given as follows: 2 1 0 0 -1 0 -2 X = 2 0 1 0 0 -1 -2 a) Draw all data points on a 2D plane by hand. Properly label the two axes. Clearly provide important tick values to facilitate a precise graphical description of the data. b) Consider each point as a vector. Calculate the angle in between (2 27 and the five other points (excluding [0 07), respectively, using the inner/dot product formula that involves the angle. Note that the angle between two vectors can be negative. c) Calculate the matrix outer product for X, namely, R = XXT. Show the intermediate steps of calculating each element of the 2-by-2 matrix R. d) The matrix outer product can also be calculated via R =

[tex](\stackrel{x_1}{-12}~,~\stackrel{y_1}{15})\qquad (\stackrel{x_2}{18}~,~\stackrel{y_2}{-13}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-13}-\stackrel{y1}{15}}}{\underset{\textit{\large run}} {\underset{x_2}{18}-\underset{x_1}{(-12)}}} \implies \cfrac{-28}{18 +12} \implies \cfrac{ -28 }{ 30 } \implies - \cfrac{14 }{ 15 }[/tex]

The three positive numbers 72, 30, and 12 make up the total of 114.An object that can be counted, measured, or given a name is a number. The numbers, for instance, are 1, 2, 56, etc.

Suppose the three positive numbers are x, y and z.

If the summation of numbers is 114.

⇒ x + y + z = 114 --- (1)

⇒ z = 114 - x - y

Suppose the square sum of the number be S.

∴ S = x² + y² + z²

⇒ S = x²+ y² + (114 - x - y) ² --- (2)

To obtain the minimal value of S, we will optimize the function by partly differentiating it with respect to x and y and setting ds/dx = 0.

2x + 2(114- x - y) (-1) = 0

2x - 228 + 2x + 2y = 0

4x + 2y - 228 = 0

y = 114 - 2x --- (3)

Differentiate partially with respect to y as

2y + 2(114 - x - y) (-1) = 0

2y - 228 + 2x + 2y = 0

4y + 2x - 228 = 0

x = 228 - 2y --- (4)

Substitute the value of y = 114 - 2x in the equation (4),

x = 12 - 2(114 - 2x)

x = 12 - 228 + 4x

4x - x = 228 - 12

3x = 216

⇒ x = 72

By substituting x = 72 in equation (3)

y =2(72) - 114

y = 144 - 114

⇒ y = 30

Substitute the values of x and y in equation (1),

72 + (30) + z = 114

z = -72+114-30

z= 12

Thus, the three positive numbers whose sum is 114 will be 72, 30, and 12.

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The range of tails that can appear when a coin is tossed ten times is represented by the random variable X, and its expected value is 5.

The probability of getting a tail on a single toss of a fair coin is1/2. since each toss is independent of the others, we're capable to model the number of tails in 10 tosses as a binomial random variable with parameters n = 10 and p = 1/2.

The anticipated value of a binomial random variable is presented through the expression

E( X) = n * p

Exchanging n = 10 and p = 1/2, we get

E( X) = 10 *1/2

E( X) = 5

Thus, the expected value of the random variable X, which represents the range of tails that come up while a coin is tossed 10 times, is 5.

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Answer:

g(y) = √(3/2 y)

Step-by-step explanation:

To find the inverse of a function, we need to solve for x in terms of y and interchange x and y. That is, we need to write the given function f(x) = 2/3x^2 in the form y = 2/3x^2 and then solve for x in terms of y.y = 2/3x^2

Multiplying both sides by 3/2, we get:

3/2 y = x^2

Taking the square root of both sides, we get:x = ± √(3/2 y)

Note that we have two possible values of x for each value of y, because the square root can be either positive or negative. However, for a function to have an inverse, it must pass the horizontal line test, which means that each value of y can only correspond to one value of x.Therefore, we need to restrict the domain of the original function to ensure that it is one-to-one. The simplest way to do this is to take the range of the function and use it as the domain of the inverse function.The range of f(x) = 2/3x^2 is all non-negative real numbers, or [0, ∞). Therefore, we can define the inverse function g(y) as:

g(y) = ± √(3/2 y)

where we choose the positive square root to ensure that the function is one-to-one.Thus, the inverse of the function f(x) = 2/3x^2 is:

g(y) = √(3/2 y)

with domain [0, ∞).

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In the graph, Student B and C are both 10 years old and student C has a shoe size of 5. The coordinates of D are (12,6)

In graph theory, a graph is a framework that consists of a collection of objects, some of which are paired together to form "related" objects. The objects are represented by mathematical abstractions known as vertices (also known as nodes or points), and each set of connected vertices is known as an edge (also called link or line). A graph is typically shown diagrammatically as a collection of dots or circles representing the centres and lines or curves representing the edges.

Both directed and undirected lines are possible. For instance, if the edges between two individuals are handshakes, then the graph is undirected because any individual A can only shake hands with an individual B if B also holds hands with A. The graph is directed, however, if an edge from person A to person B indicates that A owes money to B because borrowing money is not always returned.

In the given graph,

Student B and C are both 10 years old and student C has a shoe size of 5.

The coordinates of D are (12,6)

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Step-by-step explanation:

1. 2/10

2.3/15

3.4/20

4. 5/25

5.6/30

6.7/35

Answer:

2.

a) log base 10 of 100

b) The expression means that 10 to the 2nd power equals 100.

c) 2

3.

4. It would make sense that the value is between 1 and 2 because 10 to the 1st power is 10, and 10 to the 2nd power is 100. 50 is between 10 and 100 so the value would have to be between 1 and 2. This works because logs are the inverse function to exponentiation.

a) 2.6021

b) 3 and the value is exact because the base 10 in the log expression is technically the exponent when you convert it to exponent form. This works because logs are the inverse function to exponentiation. So 10 to the 3rd power would give you exactly 1000.

The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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Use the vector law of addition to get the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

What is the resulting vector and its magnitude?

Vector addition can be defined as the sum of two or more vectors of corresponding components.

It is given that,

[tex]\vec{r}= < 4, -2 > \\\vec{s}= < 3,4,-1 >[/tex]

Given vectors can also be written as,

[tex]\vec{r}=4i-2j\\\vec{s}=3i+4j-k[/tex]

Add above vectors as follows:

[tex]\vec{r}+\vec{s}=(4i-2j)+(3i+4j-k)\\\\=(7i+2j-k)[/tex]

Therefore,

[tex]\vec{r}+\vec{s}= < 7,2,-1 >[/tex]

Show the resulting vector as follows:

Now calculate the magnitude of the vector [tex](\vec{r}+\vec{s})[/tex]

[tex]|\vec{r}+\vec{s}|=\sqrt{(7)^2+(2)^2+(-1)^2}\\=\sqrt{49+4+1}\\=\sqrt{54}\\=3\sqrt{6}[/tex]

Hence the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

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The volume of the square pyramid is 9.12 m.

A square pyramid is a pyramid, in geοmetry, that has a square base and fοur lateral faces. A Pyramid is a pοlyhedrοn that has a base and 3 οr greater triangular faces that meet at a pοint abοve the base (the apex). A square pyramid is a three-dimensiοnal shape that has a tοtal οf five faces, hence called a pentahedrοn.  A mοst famοus example οf such a pyramid in real life is the Great Pyramid οf Giza.

[tex]$ \rm V = \frac{1}{3} a^2H[/tex]

Where v is the volume

a is the base side and

H is the height

Here,

a = 12 cm

H = 19 cm

Then,

[tex]$ \rm V = \frac{1}{3} 12 \times 12 \times 19[/tex]

[tex]$ \rm V = 4 \times 12 \times 19[/tex]

V = 912 cm or 9.12  m

Thus, The volume of the square pyramid is 9.12 m.

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Answer:

Angle AIC is vertical.

Step-by-step explanation:

Defn of vertical angles

Answer:

Step-by-step explanation:

|x-a|=b

x-a=±b

x=a±b

|x-2|=6

x-2=±6

either x-2=6

x=2+6=8

or

x-2=-6

x=2-6

x=-4

d

Answer:

  C. RP = 4.5

Step-by-step explanation:

You want to know what condition is sufficient to show ∆ABC ~ ∆QPR, given  three sides and 2 angles in ∆ABC, and 2 sides in ∆QPR.

Similarity can be shown if all three sides are proportional, or if two angles are congruent.

The offered answer choices only list one angle, so none of those will work. The answer choice that makes the third side of ∆QPR be in the same proportion as the corresponding side of ∆ABC is the condition of interest.

  C. RP = 4.5

__

Additional comment

The side ratios in the two triangles are ...

  AB : BC : CA = 10 : 9 : 6

  QP : PR : RQ = 5 : PR : 3

For these ratios to be the same, PR must be half of BC, just as the other segments in ∆QPR are half their counterparts in ∆ABC.

Answer:

g h hh h

Step-by-step explanation:

Answer: We can first find the dot product of vec r with hat j and hat k:

vec r * hat j = (3 hat i - 2 hat j + 6 hat k) * (- hat j) = -2

vec r * hat k = (3 hat i - 2 hat j + 6 hat k) * hat k = 6

Substituting these values into the expression given, we get:

(vec r * hat j).(vec r * hat k) - 12 = (-2) * (6) - 12 = -24

Therefore, the value of (vec r * hat j).(vec r * hat k) - 12 is -24.

Step-by-step explanation:

The values of x representing side length for which triangle is acute is given by 4, 5, and 6 .

For a triangle to be acute,

Sum of the squares of the two shorter sides must be greater than the square of the longest side.

Mathematically,

a^2 + b^2 > c^2

where a, b, and c are the side lengths of the triangle,

With c being the longest side.

The sides are given as 3, 6, and x.

Without loss of generality,

Assume that 3 and 6 are the shorter sides,

3^2 + 6^2 > x^2

Simplifying this inequality, we get,

45 > x^2

Taking the square root of both sides, we get,

6.71 > x

Since x must be an integer, the possible values of x are 4, 5, and 6.

Triangle inequality is satisfied,

Sum of any two sides of a triangle must be greater than the third side.

3 + 6 > x

3 + x > 6

6 + x > 3

Inequalities are all satisfied for x = 4, 5, and 6.

Therefore, there are 3 possible values of x is 4, 5, and 6 for which the triangle is acute.

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Answer: 1704 square inches

Step-by-step explanation:

The surface area of a square pyramid is given by the formula:

Surface Area = base area + 4 × (1/2 × slant height × base length)

The base area of the pyramid is:

base area = length × width = 24 in × 24 in = 576 in²

The slant height of the pyramid can be found using the Pythagorean theorem:

slant height = sqrt(20² + 12²) = 236/5 in

Therefore, the surface area of the whole pyramid is:

Surface Area = 576 in² + 4 × (1/2 × 236/5 in × 24 in) = 1152 in² + 2256 in² = 3408 in²

Half of this area is black, so the area that is yellow is:

Yellow Area = 1/2 × 3408 in² = 1704 in²

Therefore, the surface area of the toy that is yellow is 1704 square inches.

Dale will receive $1200, Evelyn will receive $1800, and Frank will receive $1500 of the $4500 profit.

What is profit ?

Profit is a financial term that refers to the difference between the revenue earned by a business or individual and the expenses incurred in producing or delivering goods or services. In other words, profit is the amount of money that is left over after all the costs of doing business have been paid.

Profit is an important indicator of the financial health of a business or individual, as it measures how much money is being earned from a particular activity or investment. A positive profit indicates that revenues are higher than expenses, while a negative profit indicates that expenses are higher than revenues.

According to the question:
The total investment is $4000 + $6000 + $5000 = $15000.

The proportion of the profit that each person will receive is equal to their investment divided by the total investment:

Dale's proportion = $4000 / $15000 = 4/15

Evelyn's proportion = $6000 / $15000 = 2/5

Frank's proportion = $5000 / $15000 = 1/3

To find each person's share of the profit, we multiply the total profit by their proportion:

Dale's share = (4/15) x $4500 = $1200

Evelyn's share = (2/5) x $4500 = $1800

Frank's share = (1/3) x $4500 = $1500

Therefore, Dale will receive $1200, Evelyn will receive $1800, and Frank will receive $1500 of the $4500 profit.

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Answer:

D) Kite

Step-by-step explanation:

A kite is a 4 sides figure that a pair of adjacent sides have the same length.

Helping in the name of Jesus.

Answer:

D) kite.

Step-by-step explanation:

Refer to the picture, when you combine the quadrilateral, it's more precise to make a kite.

Answer: To solve this problem, we can use the formula:

Total = Stocks + Bonds - Both

where "Total" represents the total number of people who invested in either stocks or bonds.

Plugging in the given values, we get:

Total = 700 + 400 - 300

Total = 800

Therefore, 800 people invested in stocks or bonds.

Step-by-step explanation:

So, in this case, the length of each of the equal sides is: 17 / [tex]\sqrt{2}[/tex] ≈ 12.02

An isosceles triangle is a triangle in which two sides have the same length. This means that two of the three angles in the triangle are also equal in measure. The side that is not congruent to the other two sides is called the base of the triangle, and the angles opposite the congruent sides are called the base angles. In an isosceles triangle, the base angles are always equal in measure.

Isosceles triangles are a common shape in geometry and can be found in many real-world applications. They have special properties that make them useful in various fields of mathematics, such as trigonometry and geometry. For example, the base angles of an isosceles triangle are always equal, so if you know the measure of one of the base angles, you can determine the measure of the other base angle using the fact that the sum of the angles in a triangle is 180 degrees.

given by the question.

In an isosceles triangle with two equal angles of 45 degrees and a third angle of 90 degrees, the two equal sides are opposite the 45 degree angles, and the third side is opposite the 90 degree angle. Let's call the length of the missing side "x".

Using the Pythagorean theorem, we know that in a right triangle with legs of equal length (which is the case for the two 45-degree angles), the length of the hypotenuse is. [tex]\sqrt{2}[/tex] times the length of the legs.

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Answer:

d

Step-by-step explanation:

the theoretical probability that the coin lands on the same side every time is 0.0625.

What is Probability?

The area of mathematics known as probability is concerned with how random events turn out. The definition of probability is chance or potential for a result. It clarifies the likelihood of a specific occurrence. We regularly use words like - 'It will probably rain today, 'he will probably pass the test', 'there is very less possibility of receiving a storm tonight', and 'most certainly the price of onion will go high again. In essence, probability is the forecasting of an outcome that is either based on the analysis of past data or the variety and quantity of alternative outcomes.

The theoretical probability of getting the same side every time in five coin tosses is:

Since the coin has two sides, there are 2^5 = 32 possible outcomes in total. Out of these outcomes, there are only two ways to get the same side every time (either all heads or all tails). Therefore, the probability of getting the same side every time is:

P(E) = favorable outcomes / total outcomes

     =  2/32

     = 1/16 = 0.0625 or 6.25%

So, the theoretical probability of getting the same side every time in five coin tosses is 0.0625 or 6.25%.

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Answer:

efrtrreeeeed

ggggffffff

Step-by-step explanation:

ggffffddeeeereeeer hhhhfdvjuhhhhhhhhjjjhhhhhhgfxd you and your family are doing well and your family are doing well and your dad said he had a good night with you don't have stich clock I WINNNNNN have stich clock out of it and your mom are the best ones that have stich you and Darshanie is a good night with my mom and my mom are the only ones who have a good weekend too and your dad said you and I WINNNNNN you and I will get it you and your family have a good day I love your family and your mom are doing well too and my mom I WINNNNNN and your dad said for you guys to come back to work tomorrow and your family are doing well and your family are doing well and your family are doing well and your family are

The area of the outdoor carpet is 63 square foot.

What does a math area mean?

The amount of space occupied by a flat (2-D) surface or an object's shape is known as its area. An object's area is the space that its boundary encloses. The quantity of unit squares that completely encircle the surface of a closed figure is its area.

The dimensions of the outdoor carpet which is rectangle in shape are:

L=6 ft and w=12 ft.

The dimensions of the indoor carpet that is triangle in shape are:

b=3 ft and h=9 ft.

Now, area of the outdoor= Area of the green region-Area of triangular region

 Area = l * b - 1/2 * b * h

         = 6* 12 - 1/2 * 3 * 6

         = 72 - 9

         = 63 square foot

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The complete question is -

How many square feet of outdoor carpet will we need for this hole? 72 ft 9 ft 3 ft 6 ft 6 ft 12 ft

To calculate the buoyant force that will be applied to a string of 2 inch OD 1.688 ID open end tubing, you will need to calculate the weight of the submerged portion of the tubing. This can be done using the following equation:

Weight (lbs) = (Height (ft) * Length (ft) * 0.052 lbs/ft3) * Fluid Density (ppg)

In your scenario, the Height is 10700 ft - 7000 ft = 3700 ft, the Length is 10700 ft, and the Fluid Density is 9.6 ppg.

Therefore, the Weight of the submerged portion of the tubing is:

Weight (lbs) = (3700 ft * 10700 ft * 0.052 lbs/ft3) * 9.6 ppg

Weight (lbs) = 10967 lbs

Now that you know the weight of the submerged portion of the tubing, you can calculate the Buoyant Force by subtracting the Weight of the submerged portion of the tubing from the Weight of the total Height of the tubing. This can be done using the following equation:

Buoyant Force (lbs) = (Height (ft) * Length (ft) * 0.052 lbs/ft3) * Fluid Density (ppg) - Weight (lbs)

Therefore, the Buoyant Force that will be applied to the string of tubing when it

Answer:

FIRST ONE "Deb sold vases for two years, neither sold nor bought the next year and then sold bases for two more years"

Step-by-step explanation:

Notice the number of bases in debs collection is DECREASING as the years passes for the first and third period. This is she is selling her vases but in the middle the number is the same (two point in the same horizontal line) this means she neither sold nor bought any vase in that period.

As per have proved that the metric space (x,d) has the property that every closed and bounded subset of x is compact.

First, let us define a sequence of positive real numbers εₙ as follows: εₙ = 1/2ⁿ. Since εₙ → 0 as n → ∞, we have that for each n ∈ ℕ, there exists an Nₙ ∈ ℕ such that if n, a > Nₙ, then d(xₙ, xₐ) < εₙ.

To see that A is bounded, note that for any two elements xₙ, x_{Nₙ} in A, we have d(xₙ, x_{Nₙ}) ≤ d(xₙ, x_{Nₙ-1}) + d(x_{Nₙ-1}, x_{Nₙ}) ≤ εₙ + ε_{Nₙ} ≤ 2εₙ. Therefore, the diameter of A is bounded by 2εₙ for any n ∈ ℕ. Since εₙ → 0 as n → ∞, we have that A is bounded.

To see that A is closed, let us consider a point x in the closure of A. If x is not in A, then there exists a positive real number δ such that the open ball B(x, δ) does not intersect A.

Let N be such that 1/2ⁿ < δ/2, and let y be an element in A. If y is equal to x_{Nₙ} for some n ∈ ℕ, then d(x, y) ≥ δ/2 > ε_{Nₙ}. Otherwise, y is equal to xₐ for some a ∈ ℕ.

If a > N, then d(x, y) > εₙ > δ/2. If a ≤ N, then d(y, xₙ) ≤ d(y, xₐ) + d(xₐ, xₙ) ≤ εₐ + εₙ < δ/2 + δ/2 = δ, which contradicts the assumption that B(x, δ) does not intersect A. Therefore, we must have x ∈ A, and so A is closed.

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Answer:

Genevieve should use a three digit code instead of a two digit code for her locker combination because a three digit code has more possible combinations than a two digit code, which makes it more secure.

With a two digit code, there are only 100 possible combinations (10 possible digits for the first number and 10 possible digits for the second number). On the other hand, with a three digit code, there are 1,000 possible combinations (10 possible digits for each of the three numbers). This means that it would take a potential thief 10 times longer to guess the correct combination for a three digit code compared to a two digit code, assuming they are using a brute force method of trying all possible combinations.

Therefore, using a three digit code increases the security of the locker and decreases the likelihood of someone gaining unauthorized access.

Answer:

more security.

Step-by-step explanation:

If you have a 2 digit code people will find it. but with 3, it makes it harder to find so people can't open her locker.

Answer:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

invNorm(0.484)= 1.96

Therefore, the Z-score that has 48.4% of the distribution's area to its left is approximately 1.96.