Kate surveyed 16 politicians from her state about their views on certain issues. is this sample of citizens of the state likely to be representative?

The correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

On a unit circle, the terminal side of Angle 0 intersects the circle at points (x,y). We can use trigonometric ratios to relate the coordinates (x,y) to the angle θ.

A) sin θ = the ratio of y to 1, or simply y.

B) tan θ = the ratio of y to x.

C) cos θ = the ratio of x to 1, or simply x.

Therefore, the correct expressions are:

A) sin θ = y

B) tan θ = y/x

C) cos θ = x

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Therefore, the solutions of tan x = -1/2 in the interval (0, 2) are:

x ≈ 2.034, 5.176

We can simplify the given equation as follows:

4 tan 2x - 4 cot x = 0

4(tan 2x - cot x) = 0

4[(2tan x)/(1 - tan^2 x) - (1)/(tan x)] = 0

Multiplying both sides by (1 - tan^2 x) * (tan x), we get:

8tan^3 x - 4tan^2 x - 8tan x + 4 = 0

Dividing both sides by 4 and rearranging, we get:

2tan^3 x - tan^2 x - 2tan x + 1 = 0

Factorizing, we get:

(tan x - 1)(2tan^2 x - tan x - 1) = 0

Using the quadratic formula to solve for the roots of 2tan^2 x - tan x - 1 = 0, we get:

tan x = [1 ± sqrt(1 + 8)] / 4 = [1 ± sqrt(9)] / 4 = 1, -1/2

Therefore, the solutions of the given equation in the interval (0, 2) are the values of x such that tan x = 1 or tan x = -1/2.

We know that tan (π/4) = 1 and tan (-π/4) = -1, so the solutions of tan x = 1 in the interval (0, 2) are:

x = π/4, 5π/4

We can find the solutions of tan x = -1/2 in the interval (0, 2) by finding the reference angle and using the signs of sine and cosine in the corresponding quadrants. We have:

tan x = -1/2

Let θ be the reference angle such that tan θ = 1/2. We know that θ is in the second or fourth quadrant.

In the second quadrant, sine is positive and cosine is negative, so we have:

sin θ = sqrt(1/(1 + tan^2 θ)) = sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = π - θ = π + arctan(1/2) ≈ 2.034

In the fourth quadrant, both sine and cosine are negative, so we have:

sin θ = -sqrt(1/(1 + tan^2 θ)) = -sqrt(1/5)

cos θ = -tan θ = -1/2

Therefore, we get:

x = 2π - θ = 2π + arctan(1/2) ≈ 5.176

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Answer:

To find (f + g)(x), we need to add the two functions f(x) and g(x), and then evaluate the sum at x.

So, we have:

(f + g)(x) = f(x) + g(x)

Substituting the given functions, we get:

(f + g)(x) = (2x + 3) + (3x + 2)

Simplifying the expression, we get:

(f + g)(x) = 5x + 5

Therefore, (f + g)(x) is equal to 5x + 5.

The area of ΔOPQ is approximately 18.9 square inches, rounding off to nearest 10th.

To find the area of ΔOPQ, we can use the formula:

Area = (1/2) * base * height

We know that p = 9.5 inches, q = 7.6 inches, and ∠O = 31°.

Now, using  trigonometry the height h of the triangle can be found using the sin function.

              sin(θ) = perpendicular/hypotenuse

perpendicular = hypotenuse* sin(θ)

                        = 7.6 * sin(31°)

                        ≈ 3.98 inches

Now, we can use the formula for the area:

Area = (1/2) * base * height

Putting in the values, we get:

Area = (1/2) * 9.5 * 3.98

Area ≈ 18.93 square inches

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For water [OH⁻] equals 8.70 x 10⁻²³ M when [H⁺] = 1.15 x 10⁸ M .

Assuming that the ion product constant for water is being questioned, Kw:

Kw = [H⁺] [OH⁻]

where Kw = ion product constant for water

kW at 25°C = 1.0 x 10⁻¹⁴

Since Kw is constant at a given temperature, if we know the concentration of [H⁺] or [OH⁻], we can find it with Kw the concentration of the new ion

given that [H⁺] = 1.15 x 10⁸ M, [OH⁻] can be determined as follows.

Kw = [H⁺] [ OH⁻]

plugging in the values

1.0 x 10⁻¹⁴ M² = (1.15 x 10⁸ M) * ([OH⁻])

[OH⁻] = (1.0 x 10⁻¹⁴ M² ) / (1.15 x 10⁸ M )

[OH⁻] = 8.70 x 10⁻²³ M

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If a teacher has an annual salary of 98,500, the teacher makes  $947.12 per biweekly period.

To calculate the teacher's biweekly salary, we need to divide their annual salary by the number of weeks in a year, and then divide that result by 2 (since there are 2 weeks in a biweekly period).

There are a few different ways to approach this calculation, but one common method is to use the following formula:

Biweekly Salary = (Annual Salary / Number of Weeks in a Year) / 2

Using this formula, we can calculate the teacher's biweekly salary as follows:

Biweekly Salary = (98,500 / 52) / 2

Biweekly Salary = (1,894.23) / 2

Biweekly Salary = 947.12

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The ball will rebound to maximum height of 25 centimetres or 0.25 meters after 7 bounces.

Firstly perform the unit conversion. As known, 1 meter is 100 cm. So, 25 centimetres is 0.25 meters.

Now, the formula to be used to find the number of bounces is -

New height × [tex] {2}^{n} [/tex] = old height, where n refers to number of bounces.

Keeping the values in formula

0.25 × [tex] {2}^{n} [/tex] = 32

Rearranging the equation

[tex] {2}^{n} [/tex] = 32/0.25

Divide the values

[tex] {2}^{n} [/tex] = 128

Converting the result into exponent form

[tex] {2}^{n} [/tex] = 2⁷

Thus, n will be 7 bounces.

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The demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

The demand equation is given as:

q = 55.1 - 0.023p

where q is the quantity demanded and p is the price of the airfare.

This equation shows an inverse relationship between price and quantity demanded. As the price of the airfare increases, the quantity demanded decreases, and vice versa.

For example, if the airfare price is $100, we can calculate the quantity demanded as:

q = 55.1 - 0.023(100) = 52.8

This means that at a price of $100, the quantity demanded is approximately 52.8 units.

Similarly, if the airfare price is $200, we can calculate the quantity demanded as:

q = 55.1 - 0.023(200) = 50.4

This means that at a price of $200, the quantity demanded is approximately 50.4 units.

So, demand equation can be used to estimate the demand for air travel at different price levels, and can help airlines make pricing decisions based on the expected demand.

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The volume of the larger box is 200 cubic inches, and the volume of the smaller box is 150 cubic inches.

To find the volume of each box, we use the formula for the volume of a rectangular prism, which is V = lwh, where l is the length, w is the width, and h is the height.

For the larger box, we have l = 8 inches, w = 2.5 inches, and h = 10 inches, so

Volume of large box = = 8 x 2.5 x 10 = 200 cubic inches.

For the smaller box, we have l = 0.75 x 8 = 6 inches, w = 2.5 inches, and h = 10 inches, so

Volume of small box= 6 x 2.5 x 10 = 150 cubic inches.

The difference in the volumes of the two boxes is

Volume of large box - Volume of small box = 200 - 150 = 50 cubic inches.

The units for the volumes are cubic inches, since we are dealing with three-dimensional space.

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The inequality which can be used to determine amount the club needs to raise during remaining months is 400 + 4n ≥ 750.

The goal of the school chess club is to raise at least $750 in total so that they can attend a state competition. They have already raised $400, but they still need to raise more money. Let's call the amount they need to raise each month "n".

Since the club has 4 months remaining until the competition, they will need to raise a total of "4n" dollars during that time period.

To determine the minimum amount they need to raise each month, the inequality can be written as : 400 + 4n ≥ 750,

4n ≥ 350 ; n ≥ 87.5.

This means that the chess-club needs to raise at least $87.50 each month in order to reach their goal of $750 in 4 months.

Therefore, the required inequality is 400 + 4n ≥ 750.

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The given question is incomplete, the complete question is

A school chess club needs to raise at least $750 to attend a state competition. The club has already raised $400 and there are 4 months remaining until the competition. Write an inequality which can be used to determine the dollar amount the club will need to raise during the remaining months?

x² + 10x + 25 is the expression of (x+5)^2 in trinomial in standard form

A trinomial is a polynomial that consists of three terms. It is a type of algebraic expression that contains three algebraic terms separated by either plus or minus signs.

For example, the expression 2x^2 + 5x - 3 is a trinomial because it has three terms: 2x^2, 5x, and -3. Similarly, the expression 4a^3 - 7a^2 + 2a is also a trinomial because it has three terms: 4a^3, -7a^2, and 2a.

We have to expand this

x² + 2(x)(5) + 5²

= x² + 10x + 25

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The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

To find the scale factor of figure KLMN to figure PQRS, given KN = 3 cm, MN = 7 cm, RS = 14 cm, and PS = 6 cm, follow these steps:

1. First, find the length of a side in figure KLMN. We can use KN since it's given: KN = 3 cm.
2. Next, find the corresponding side length in figure PQRS. Since KN corresponds to PS, we have: PS = 6 cm.
3. Now, find the scale factor by dividing the length of the side in figure PQRS by the length of the corresponding side in figure KLMN: scale factor = PS/KN = 6 cm / 3 cm.

The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.

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Answer:

--------------------

Let the original number be n.

We are given that 30% of n is 14.99.

Set up an equation to reflect this relation:

Solve it for n:

Answer:

Isolate the variable by dividing each side by factors that don't contain the variable.

Exact Form:

x = −1/2

Decimal Form:

x = −0.5

Step-by-step explanation:

Answer:

x = 24

Step-by-step explanation:

using the rule of logarithms

[tex]log_{b}[/tex] x = n ⇒ x = [tex]b^{n}[/tex]

note that log x represents [tex]log_{10}[/tex] x

given

log(4x + 4) = 2 , then

4x + 4 = 10² = 100 ( subtract 4 from both sides )

4x = 96 ( divide both sides by 4 )

x = 24

Option C is correct i.e. A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

Given integers are m, n, p and q

And q ≠ 0 and p ≠ 0

A = m / p

B = n/ q

Adding A and B

A + B = m / p + n / q

A + B = (mq + np) / pq

as p ≠ 0 and q ≠ 0 so, pq ≠ 0

So, A + B = (mq + np) / pq is a rational number

Therefore, option C is correct i.e. A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

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Given question is incomplete, the complete question is below:

Keisha's teacher gives her the following information:

• m, n, p, and q are all integers, and p≠ 0 and q≠0

•A= m/q and B= n/p

What conclusion can Keisha make?

A: A +B = (mp + nq)/pq, so the sum of a rational number and an irrational number is an irrational number.

B: A•B= (mp + nq)/pq, so the product of two irrational numbers is an irrational number.

C: A+B= (mp + nq)/pq, so the sum of two rational numbers is a rational number.

D: A•B= (mp + nq)/pq, so the product of two rational numbers is a rational number.​

Answer:

w

Step-by-step explanation:

w

Answer:

(5.5, -1.5)

Step-by-step explanation:

(x, y)midpoint = (x1 + x2)/2 , (y1 + y2)/2

= (6 + 5)/2, (3 - 6)/2

= (11/2, -3/2)

= (5.5, -1.5)

The size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

To determine the function f given that f'(x)=6x² – 5 sin x+eˣ and f(0) = 20, we need to integrate f'(x) with respect to x to obtain f(x), and then use the initial condition f(0) = 20 to find the value of the constant of integration.

Integrating f'(x) with respect to x, we have:

f(x) = 2x³ + 5 cos x + eˣ + C

where C is the constant of integration.

Using the initial condition f(0) = 20, we have:

f(0) = 2(0)³ + 5 cos 0 + e⁰ + C = 6 + C = 20

Therefore, the constant of integration is C = 14, and the function f(x) is:

f(x) = 2x³ + 5 cos x + eˣ + 14

To determine the size of the square that needs to be cut out of each corner of a standard piece of notebook paper to create a box of maximum volume, we can start by drawing a diagram of the box and labeling the sides as follows:

| |

| |

| | h

| |

|__________|

L

Let x be the length of each side of the square that is cut out of each corner. Then, the length and width of the base of the box will be L - 2x and 11 - 2x, respectively, and the height of the box will be x. Therefore, the volume V of the box can be expressed as:

V(x) = x(L - 2x)(11 - 2x)

Expanding and simplifying, we get:

V(x) = -4x³ + 46x² - 110x

To find the size of the square that maximizes the volume of the box, we need to find the value of x that maximizes V(x). This can be done by finding the critical points of V(x) and determining whether they correspond to a maximum or minimum.

Taking the derivative of V(x) with respect to x, we get:

V'(x) = -12x² + 92x - 110

Setting V'(x) = 0 and solving for x, we get:

x = 5/3 or x = 11/6

To determine whether these values correspond to a maximum or minimum, we can use the second derivative test. Taking the second derivative of V(x) with respect to x, we get:

V''(x) = -24x + 92

Evaluating V''(5/3) and V''(11/6), we find that:

V''(5/3) = -4 < 0, so x = 5/3 corresponds to a maximum.

V''(11/6) = 20 > 0, so x = 11/6 corresponds to a minimum.

Therefore, the size of the square that needs to be cut out of each corner to create a box of maximum volume is 5/3 inches.

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As per the growth function, the value of the quantity after 97 years would be $67,458.85.

In your problem, you have a quantity with an initial value of 8200 that grows continuously at a rate of 0.55% per decade. To find the value of the quantity after 97 years, we can use the following growth function:

A(t) = A₀[tex]e^{kt}[/tex]

In this formula, A(t) represents the value of the quantity after time t, A₀ represents the initial value of the quantity (in this case, 8200), e represents Euler's number (a mathematical constant equal to approximately 2.718), k represents the growth rate (in this case, 0.0055 per decade), and t represents the time elapsed (in this case, 97 years).

To solve for the value of the quantity after 97 years, we simply plug in the values we know and solve for A(t):

A(t) = 8200[tex]e^{(0.0055/10\times97)}[/tex]

= 8200[tex]e^{0.5285}[/tex]

≈ 67,458.85

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Answer 1: Together, Streetcar and Cable Car are the preferred transportation for 168 residents.

The circle graph shows the percentage of residents who prefer each transportation method, and the total sample size is 400.

For streetcar, (15/100) x 400 = 60 residents prefer it, and for cable car, (27/100) x 400 = 108 residents prefer it.

Together, Streetcar and Cable Car are the preferred transportation for 60 + 108 = 168 residents.

Answer 2: The median is the best measure of center, and it equals 19.

The box plot shows the distribution of the number of tickets sold for a school dance.

The median is the middle value of the data when arranged in order, and it is represented by the line in the box. In this case, the median is 19. The mean, on the other hand, can be influenced by extreme values, and we cannot determine it from the box plot alone.

Answer 3: Median, because Sunny Town is skewed.

When comparing the data, we need to consider the measure of center that is less affected by extreme values, and that is the median.

The median is the middle value of the data when arranged in order. The histogram for Sunny Town is skewed to the right, which means that there are some very high values that are affecting the mean.

Therefore, the median is the better measure of center to determine which location typically has the cooler temperature.

Answer 4: The IQR of 13 is the most accurate to use, since the data is skewed.

The histogram shows the frequency of donations received by a charity, and the data is skewed to the right.

The IQR (Interquartile Range) is the difference between the third quartile (Q3) and the first quartile (Q1), which represents the middle 50% of the data.

The IQR is less sensitive to extreme values and is a better measure of variability for skewed data. In this case, the IQR is 49 - 42 = 7, which is the most accurate measure of variability to use.

Answer 5: There were about 15 principals in attendance.

In the exhibit room, out of 80 people, 15 are principals.

We can assume that the proportion of principals in the exhibit room is the same as the proportion of principals in the conference.

Therefore, the estimated number of principals in the conference is (15/80) x 900 = 168.75, which is approximately 169.

Answer 6: Histogram

The teacher wants to represent the subject preferences of 100 students. A histogram would be the best graphical representation to use because it shows the frequency distribution of a continuous variable, which in this case could be the number of students who prefer each subject.

A stem-and-leaf plot is used for small datasets, and a box plot is used to display the distribution of a continuous variable across categories. A circle graph is more appropriate for displaying categorical data, such as the percentage of students who prefer each subject.

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The volume of the table tent 3 cmH8 cm12 cm is 288 cubic cm.

The volume of the table tent can be found by multiplying the length, width, and height of the tent.

Volume = length x width x height

In this case, the length is 12 cm, the width is 8 cm, and the height is 3 cm.

Volume = 12 cm x 8 cm x 3 cm

Volume = 288 cubic cm

Therefore, the volume of the table tent is 288 cubic cm.

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Answer:

0.91

30.4

850

Step-by-step explanation:

Laptop- 44, Tablet- 94, Desktop- 62, Percentage of tablets- 47%.

To calculate the percentage of devices that are tablets, we need to find the proportion of tablets among all devices. In this case, there are 94 tablets out of a total of 200 devices (44 laptops + 94 tablets + 62 desktops).

To find the proportion, we divide the number of tablets by the total number of devices: 94/200 = 0.47.

To convert this proportion to a percentage, we multiply by 100: 0.47 * 100 = 47%.

47% of the devices owned by the local company are tablets.

This means that nearly half of the devices fall into the tablet category, making it a significant portion of their device inventory.

It's important to note that this calculation assumes the given numbers accurately represent the actual device distribution and that there are no missing or unaccounted devices.

By determining the percentage of tablets, we gain insight into the composition of the device inventory, which can be useful for decision-making and resource allocation within the local company.

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The zero of the function is 14. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

To find the zero of the function, we need to determine the value of x when y equals 0.

0 = 10,080 - 720x

To solve for x, we will isolate the variable by following these steps:

1. Add 720x to both sides:
720x = 10,080

2. Divide both sides by 720:
x = 14

The zero of the function is 14, which means that after 14 hours, there will be no water remaining in the pool. In the context of the situation, this means it will take 14 hours for Libby to completely drain the pool for cleaning.

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Answer is 7/3 cups.

To determine the amount of raisins needed for the snack mix, subtract 3 5/12 cups from 5 3/4 cups of cereal.

First, convert the mixed numbers to improper fractions:
5 3/4 = (5 × 4 + 3)/4 = 23/4
3 5/12 = (3 × 12 + 5)/12 = 41/12

Next, subtract the two fractions:
23/4 - 41/12

To subtract, find a common denominator. The least common multiple of 4 and 12 is 12. Convert both fractions to equivalent fractions with a denominator of 12:
(23/4) × (3/3) = 69/12
(41/12) × (1/1) = 41/12

Now, subtract the fractions:
69/12 - 41/12 = 28/12

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (4):
28/12 = (28 ÷ 4)/(12 ÷ 4) = 7/3

So, you need 7/3 cups of raisins for the snack mix.

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The circle has horizontal tangent lines at the points (1, 2 + √(14)) and (1, 2 - √(14)) and the circle has vertical tangent lines at the points (1 + √(8), -2) and (1 - √(8), -2).

To find the points where the circle defined by x^2 + y^2 - 2x - 4y = 11 has horizontal and vertical tangent lines, we need to use implicit differentiation to find the derivatives of x and y with respect to each other.

Taking the derivative of both sides of the equation with respect to x, we get:

2x + 2y(dy/dx) - 2 - 4(dy/dx) = 0

Simplifying, we get:

(dy/dx) = (x-1) / (y+2)

To find the points where the circle has horizontal tangent lines, we need to find where the derivative dy/dx is equal to zero. This occurs when x-1 = 0, or x = 1. Substituting this value of x back into the original equation, we get:

1 + y^2 - 4y = 11

Simplifying, we get:

y^2 - 4y - 10 = 0

Using the quadratic formula, we get:

y = 2 ± √(14)

To find the points where the circle has vertical tangent lines, we need to find where the derivative dy/dx is undefined, which occurs when y+2 = 0, or y = -2. Substituting this value of y back into the original equation, we get:

x^2 - 2x + 4 = 11

Simplifying, we get:

x^2 - 2x - 7 = 0

Using the quadratic formula, we get:

x = 1 ± √(8)

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For approximately 64% of the shows, the theater was half full or less than half full.

Since the seat capacity of the theater is 300, half full would be 150 seats or less. Looking at the histogram, we can see that there are 3 bars representing showings with 150 or less viewers.

The first bar represents showings with 0-50 viewers. From the histogram, it looks like there were about 5 showings with this number of viewers.

The second bar represents showings with 50-100 viewers. From the histogram, it looks like there were about 8 showings with this number of viewers.

The third bar represents showings with 100-150 viewers. From the histogram, it looks like there were about 3 showings with this number of viewers.

So the total number of showings with 150 or less viewers is 5+8+3 = 16.

Since the histogram shows a total of 25 showings, the fraction of shows that was half full or less than half is:

16/25 = 0.64 or 64%

Therefore, for approximately 64% of the shows, the theater was half full or less than half full.

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The total number of students in Mrs. Cunningham's class is 30.

From the question we know that the ratio of boys to girls in Mrs. Cunningham's class is 2 to 3 so we can write

no.of boys: no.of girls = 2:3

The total number of girls in the class is given as 18 so with this we can find out the number of boys in the class that is :

no.of boys= (2/3)*no.of girls in class

now after substituting the values in the equation, we get

no. of boys = (2/3) * 18

no.of boys = 12.

So, now we know the number of boys in the class that is 12 and the number of girls in the class is 18.

We can calculate the total number of students in the class which is equal to

= no.of boys + no.of girls.

= 12+18

=30

Therefore, the total number of students in Mrs. Cunningham's class is 30.

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The point estimate for μ is 15.3 years, based on the sample of 60 Grade 9 students.

Based on the statistical techniques given information, the point estimate for the population mean age of all Grade 9 students is 15.3 years. This means that if we assume that the sample is representative of the entire population of Grade 9 students, then we estimate that the average age of all Grade 9 students is 15.3 years.

To estimate the precision of this point estimate, we can calculate confidence intervals. For a 95% confidence interval, we can use the formula:

CI = X ± (Zα/2) * (σ/√n)

where X is the point estimate, Zα/2 is the critical value of the standard normal distribution for a 95% confidence level (1.96), σ is the population standard deviation (which we assume to be known as 4), and n is the sample size (which is 60).

Substituting the values, we get the 95% confidence interval as:

CI = 15.3 ± (1.96) * (4/√60) = (14.33, 16.27)

This means that we can be 95% confident that the true population mean age of Grade 9 students lies between 14.33 and 16.27 years.

For a 99% confidence interval, we can use the same formula with a different value of Zα/2 (2.58 for a 99% confidence level). Substituting the values, we get the 99% confidence interval as:

CI = 15.3 ± (2.58) * (4/√60) = (13.94, 16.66)

This means that we can be 99% confident that the true population mean age of Grade 9 students lies between 13.94 and 16.66 years.

Based on the confidence intervals, we can conclude that the sample provides evidence that the true mean age of all Grade 9 students is likely to be between 14.33 and 16.27 years with a 95% confidence level, and between 13.94 and 16.66 years with a 99% confidence level. However, we cannot be completely certain that the true population mean falls within these intervals as there is always some level of uncertainty associated with sample-based estimates.

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