If X∼T(n), then find cn the cases a) P(X

28. For any odd integer n, [n²/4] = ((n - 1)/2) ((n + 1)/2) is TRUE.

29. For any odd integer n, [n²/4] = (n² + 3)/4 is FALSE.

To prove or disprove the statements, let's start by considering each statement separately.

Statement 28: For any odd integer n, [n²/4] = ((n - 1)/2) ((n + 1)/2)

To prove this statement, we need to show that for any odd integer n, the expression on the left side ([n²/4]) is equal to the expression on the right side (((n - 1)/2) ((n + 1)/2)).

Let's test this statement for an odd integer, such as n = 3:

Left side: [3²/4] = [9/4] = 2 (the greatest integer less than or equal to 9/4 is 2)

Right side: ((3 - 1)/2) ((3 + 1)/2) = (2/2) (4/2) = 1 * 2 = 2

For n = 3, both sides of the equation yield the same result (2).

Let's test another odd integer, n = 5:

Left side: [5²/4] = [25/4] = 6 (the greatest integer less than or equal to 25/4 is 6)

Right side: ((5 - 1)/2) ((5 + 1)/2) = (4/2) (6/2) = 2 * 3 = 6

Again, for n = 5, both sides of the equation yield the same result (6).

We can repeat this process for any odd integer, and we will find that both sides of the equation yield the same result. Therefore, we have shown that for any odd integer n, [n²/4] = ((n - 1)/2) ((n + 1)/2).

Statement 28 is true.

Statement 29: For any odd integer n, [n²/4] = (n² + 3)/4

To prove or disprove this statement, we need to show that for any odd integer n, the expression on the left side ([n²/4]) is equal to the expression on the right side ((n² + 3)/4).

Let's test this statement for an odd integer, such as n = 3:

Left side: [3²/4] = [9/4] = 2 (the greatest integer less than or equal to 9/4 is 2)

Right side: (3² + 3)/4 = (9 + 3)/4 = 12/4 = 3

For n = 3, the left side yields 2, while the right side yields 3. They are not equal.

Therefore, we have found a counterexample (n = 3) where the statement does not hold.

Statement 29 is false.

learn more about odd integer: https://brainly.com/question/2263958

#SPJ4

The complete question goes thus:

28. If true, prove the following statement or find a counterexample if the statement is false, but do not use Theorem 4.6.1. in your proof. For any odd integer n, [n²/4]=((n - 1)/2) ((n + 1)/2). 2. (10 points)

29. If true, prove the following statement or find a counterexample if the statement is false, but do not use Theorem 4.6.1. in your proof. For any odd integer n, [n²/4] = (n² + 3)/4

Using the chain rule to find dw/dt, where w = ln(x2 + y2 + z2), x = sin(t), y = cos(t) and t = e^t, is done in three steps: differentiate the function w with respect to x, y, and z. Differentiate the functions x, y, and t with respect to t. Substitute the values of x, y, and t in the differentiated functions and the original function w and evaluate.


We need to find dw/dt, where w = ln(x2 + y2 + z2), x = sin(t), y = cos(t) and t = e^t. This can be done in three steps:
1. Differentiation  the function w with respect to x, y, and z
w_x = 2x / (x2 + y2 + z2)w_y = 2y / (x2 + y2 + z2)w_z = 2z / (x2 + y2 + z2)
2. Differentiate the functions x, y, and t with respect to t
x_t = cos(t)y_t = -sin(t)t_t = e^t
3. Substitute the values of x, y, and t in the differentiated functions and the original function w and evaluate
dw/dt = w_x * x_t + w_y * y_t + w_z * z_t= (2x / (x2 + y2 + z2)) * cos(t) + (2y / (x2 + y2 + z2)) * (-sin(t)) + (2z / (x2 + y2 + z2)) * e^t

To learn more about Differentiation

https://brainly.com/question/33433874

#SPJ11

To compare the consumption of energy drinks between two groups, i.e., students from "uh" and "ut," you can use an independent t-test.

The independent t-test is appropriate when you have two independent groups and you want to compare the means of a continuous variable between them.

In this case, you can collect data on energy drink consumption from a sample of students from both "uh" and "ut" and perform an independent t-test to determine if there is a statistically significant difference in the average consumption of energy drinks between the two groups.

To learn more on Statistics click:

https://brainly.com/question/30218856

#SPJ4

The 23rd term of the sequence is F₂₃ = 2097152.

a) The given sequence of numbers can be calculated using the recursive algorithm below:

Fo= 0,

F₁ = 1,

Fₙ = Fₙ₋₂ + 2

Fₙ₋₁Fₙ₊₁ = FₙFₙ₊₁= [0 1] [0 2] + [1 1] [1 0]

= [1 2] [1 1]

The matrix equation for the general term (Fn) of the sequence is given by:

[Fₙ Fₙ₊₁] = [0 1] [0 2]ⁿ⁻¹ [1 1] [1 0] [F₁₀ F₁₀₊₁]

= [0 1] [0 2]²² [1 1] [1 0] [F₂₂ F₂₂₊₁]

= [0 1] [0 2]²¹ [1 1] [1 0] [1 0] [0 1] [0 2]²¹ [1 1] [1 0] [1 0] [0 1] [0 2]²⁰ [1 1] [1 0] [1 0] [0 1] [2¹⁰ 2¹⁰] [1 1] [1 0] [17711 10946]

The 23rd term of the sequence is given by Fn where n = 23.

Thus, substituting n = 23 into the matrix equation [Fₙ Fₙ₊₁]

= [0 1] [0 2]ⁿ⁻¹ [1 1] [1 0],

We get: [F₂₃ F₂₃₊₁] = [0 1] [0 2]²² [1 1] [1 0] [F₂₃ F₂₃₊₁]

= [0 1] [4194304 2097152] [1 1] [1 0] [F₂₃ F₂₃₊₁]

= [2097152 2097153]

For more related questions on sequence:

https://brainly.com/question/30262438

#SPJ8

We can expect Billups to make around 5.364 free throws with a standard deviation of 0.587.

To calculate the standard deviation of the number of free throws Chauncey Billups will make in tonight's game, we need to first calculate the mean or expected value of the number of free throws he will make.

Given that Billups has a career free-throw percentage of 89.4%, we can assume that he has a probability of 0.894 of making each free throw. Therefore, the expected value or mean of the number of free throws he will make out of 6 attempts is:

mean = 6 x 0.894 = 5.364

Next, we need to calculate the variance of the number of free throws he will make. Since each free throw attempt is a Bernoulli trial with a probability of success p=0.894, we can use the formula for the variance of a binomial distribution:

variance = n x p x (1-p)

where n is the number of trials and p is the probability of success.

Plugging in the values, we get:

variance = 6 x 0.894 x (1-0.894) = 0.344

Finally, the standard deviation of the number of free throws he will make is simply the square root of the variance:

standard deviation = sqrt(variance) = sqrt(0.344) ≈ 0.587

Therefore, we can expect Billups to make around 5.364 free throws with a standard deviation of 0.587.

Learn more about   deviation from

https://brainly.com/question/475676

#SPJ11

y = C/(1 − Ct) is the solution to the nonlinear example dy/dt = y², where C is an arbitrary constant, and the choice C = 1 gives y = 1/(1 − t), starting from y(0) = 1.

The given equation is d²y/dt² = y. Here, y = et, and the solution to this equation is given by the equation: y = Aet + Bet, where A and B are arbitrary constants.

We can obtain this solution by substituting y = et into the differential equation, thereby obtaining: d²y/dt² = d²(et)/dt² = et = y. We can integrate this equation twice, as follows: d²y/dt² = y⇒dy/dt = ∫ydt = et + C1⇒y = ∫(et + C1)dt = et + C1t + C2,where C1 and C2 are arbitrary constants.

The solution is therefore y = Aet + Bet, where A = 1 and B = C1. Therefore, the solution is: y = et + C1t, where C1 is an arbitrary constant. The second solution to the equation is thus y = et + C1t.

The nonlinear example dy/dt = y² is given. It can be solved using separation of variables as shown below:dy/dt = y²⇒(1/y²)dy = dt⇒∫(1/y²)dy = ∫dt⇒(−1/y) = t + C1⇒y = −1/(t + C1), where C1 is an arbitrary constant. If we choose C1 = 1, we get y = 1/(1 − t).

Starting from y(0) = 1, we have y = 1/(1 − t), which is the solution. Therefore, y = C/(1 − Ct) is the solution to the nonlinear example dy/dt = y², where C is an arbitrary constant, and the choice C = 1 gives y = 1/(1 − t), starting from y(0) = 1.

To know more about nonlinear visit :

https://brainly.com/question/25696090

#SPJ11

D) All of the following statements are true about the relationship between a polynomial function and its related polynomial equation are: (a) The polynomial equation is formed by setting f(x) to 0 in the polynomial function.(b) Solving the polynomial equation gives the x-intercepts of the graph of the polynomial function.(c) The zeros of the polynomial function are the roots(solutions) of the polynomial equation.

The polynomial equation is formed by setting f(x) to 0 in the polynomial function. Solving the polynomial equation gives the x-intercepts of the graph of the polynomial function. The zeros of the polynomial function are the roots(solutions) of the polynomial equation.

Therefore, the answer is option (d) all of the above.A polynomial function is a function of the form

f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0

where a_0, a_1, a_2, ..., a_n are real numbers and n is a non-negative integer. The degree of the polynomial function is n.The zeros of a polynomial function are the solutions to the polynomial equation

f(x) = 0

The zeros of a polynomial function are the x-intercepts of the graph of the polynomial function. When a polynomial function is factored, the factors of the polynomial function are linear or quadratic expressions with real coefficients.

Know more about  polynomial equation here,

https://brainly.com/question/3888199

#SPJ11

The equation of a line that passes through the point (1, -7) and has a slope of -9 is y = -9x + 2

To find the equation of the line, follow these steps:

Learn more about equation of line:

brainly.com/question/18831322

#SPJ11

In order to add binary numbers, you add the digits starting from the rightmost position and work your way left, carrying over to the next place value if necessary. If the sum of the two digits is 2 or greater, you write down a 0 in that position and carry over a 1 to the next position.

Example : Binary addition: 10101 + 11101 Add the columns starting from the rightmost position: 1+1= 10, 0+0=0, 1+1=10, 0+1+1=10, 1+1=10 Write down a 0 in each column and carry over a 1 in each column where the sum was 2 or greater: 11010 is the result

Converting binary to hexadecimal: Starting from the rightmost position, divide the binary number into groups of four bits each. If the leftmost group has less than four bits, add zeros to the left to make it four bits long. Convert each group to its hexadecimal equivalent.

Example: 1101 0100 becomes D4 Hexadecimal addition: Add the hexadecimal digits using the same method as for decimal addition. A + B = C + 1. The only difference is that when the sum is greater than F, you write down the units digit and carry over the tens digit.

Example: 7A + 9C = 171 Start with the rightmost digit and work your way left. A + C = 6, A + 9 + 1 = F, and 7 + nothing = 7. Therefore, the answer is 171. Converting hexadecimal to binary: Convert each hexadecimal digit to its binary equivalent using the following table:

Hexadecimal Binary 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 A 1010 B 1011 C 1100 D 1101 E 1110 F 1111Then write down all the binary digits in order from left to right. Example: 8B = 10001011

To know more about binary numbers refer here:

https://brainly.com/question/28222245

#SPJ11

You may prefer the first game as it involves only one roll and carries less risk compared to rolling the die one million times in the second game.

To determine which game you prefer, we need to consider the expected payoffs of each game.

In the first game, you roll a die once, and the payoff is 1 million dollars times the number you obtain on the upturned face of the die. The possible outcomes are numbers from 1 to 6, each with a probability of 1/6. Therefore, the expected payoff for the first game is:

E(Game 1) = (1/6) * (1 million dollars) * (1 + 2 + 3 + 4 + 5 + 6)

         = (1/6) * (1 million dollars) * 21

         = 3.5 million dollars

In the second game, you roll a die one million times, and for each roll, you are paid 1 dollar times the number of dots on the upturned face of the die. Since the die is fair, the expected value for each roll is 3.5. Therefore, the expected payoff for the second game is:

E(Game 2) = (1 dollar) * (3.5) * (1 million rolls)

         = 3.5 million dollars

Comparing the expected payoffs, we can see that both games have the same expected payoff of 3.5 million dollars. Since you are risk-averse, it does not matter which game you choose in terms of expected value.

To know more about number visit:

brainly.com/question/3589540

#SPJ11

Given that the function is `f(x) = -3x² + 4x + 3` and we need to find the equation of the tangent to the graph at `x = 2`.Firstly, we will find the slope of the tangent by finding the derivative of the given function. `f(x) = -3x² + 4x + 3.

Differentiating with respect to x, we get,`f'(x) = -6x + 4`Now, we will substitute the value of `x = 2` in `f'(x)` to find the slope of the tangent.`f'(2) = -6(2) + 4 = -8`  Therefore, the slope of the tangent is `-8`.Now, we will find the equation of the tangent using the slope-intercept form of a line.`y - y₁ = m(x - x₁).

Where `(x₁, y₁)` is the point `(2, f(2))` on the graph of `f(x)`.`f(2) = -3(2)² + 4(2) + 3 = -3 + 8 + 3 = 8`Hence, the point is `(2, 8)`.So, we have the slope of the tangent as `-8` and a point `(2, 8)` on the tangent.Therefore, the equation of the tangent is: `y - 8 = -8(x - 2)`On solving, we get:`y = -8x + 24`Hence, the equation of the line tangent to the graph of `f(x) = -3x² + 4x + 3` at `x = 2` is `y = -8x + 24`.

To know more about function visit :

https://brainly.com/question/30721594

#SPJ11

The distance from the point (5,0,0) to the line x=5+t, y=2t, z=12√5 +2t is √55.

To find the distance between a point and a line in three-dimensional space, we can use the formula for the distance between a point and a line.

Given the point P(5,0,0) and the line L defined by the parametric equations x=5+t, y=2t, z=12√5 +2t.

We can calculate the distance by finding the perpendicular distance from the point P to the line L.

The vector representing the direction of the line L is d = <1, 2, 2>.

Let Q be the point on the line L closest to the point P. The vector from P to Q is given by PQ = <5+t-5, 2t-0, 12√5 +2t-0> = <t, 2t, 12√5 +2t>.

To find the distance between P and the line L, we need to find the length of the projection of PQ onto the direction vector d.

The projection of PQ onto d is given by (PQ · d) / |d|.

(PQ · d) = <t, 2t, 12√5 +2t> · <1, 2, 2> = t + 4t + 4(12√5 + 2t) = 25t + 48√5

|d| = |<1, 2, 2>| = √(1^2 + 2^2 + 2^2) = √9 = 3

Thus, the distance between P and the line L is |(PQ · d) / |d|| = |(25t + 48√5) / 3|

To find the minimum distance, we minimize the expression |(25t + 48√5) / 3|. This occurs when the numerator is minimized, which happens when t = -48√5 / 25.

Substituting this value of t back into the expression, we get |(25(-48√5 / 25) + 48√5) / 3| = |(-48√5 + 48√5) / 3| = |0 / 3| = 0.

Therefore, the minimum distance between the point (5,0,0) and the line x=5+t, y=2t, z=12√5 +2t is 0. This means that the point (5,0,0) lies on the line L.

Learn more about parametric equations here:

brainly.com/question/29275326

#SPJ11

Washington Community L and Internal rate of return Washington Community L is an affordable housing unit that is based on the low-income community that is located in the Washington city in the United States.

This housing unit was established with the aim of making a social impact, particularly in the low-income community where housing is scarce. The main aim of Washington Community L is to provide affordable housing for low-income families, individuals, and students.

The internal rate of return refers to the discount rate that is used in capital budgeting. The main aim of the internal rate of return is to measure the profitability of a potential investment. The internal rate of return is usually expressed as a percentage. In general, the higher the internal rate of return, the more profitable the investment.

The formula for calculating the internal rate of return is quite complex and requires the use of several variables. These variables include the initial investment, the cash inflows, the cash outflows, and the discount rate. The internal rate of return is calculated by finding the discount rate that makes the net present value of an investment equal to zero.

The cumulative sum of B through the current year refers to the total amount of money that has been spent on the investment project up to the current year. This cumulative sum includes all the initial investments as well as any additional cash inflows or outflows that have occurred up to the current year.

Present value interest factors in the exhibit have been calculated by formula but are necessarily rounded for presentation. Therefore, there may be a difference between the number displayed and that calculated manually. This means that the figures presented in the exhibit may not be entirely accurate due to rounding.

However, these figures are still useful for calculating the internal rate of return and other financial metrics.

To know more about Internal rate of return here

https://brainly.com/question/31870995

#SPJ11

The explicit particular solution of the given initial value problem is:

y =  5e⁻⁴ˣ - 5e⁻³ˣ

To find an explicit particular solution of the initial value problem:

dy/dx = 5e⁴ˣ - 3y, y(0) = 0

We can use the method of integrating factors. The integrating factor is given by:

IF(x) = e⁻³ˣ

Multiplying both sides of the differential equation by the integrating factor, we have:

e⁻³ˣ * dy/dx - 3e⁻³ˣ * y = 5e⁴ˣ * e⁻³ˣ

Simplifying, we get:

d/dx (e⁻³ˣ * y) = 5e⁴ˣ⁻³ˣ

d/dx (e⁻³ˣ * y) = 5eˣ

Integrating both sides with respect to x, we have:

∫ d/dx (e⁻³ˣ * y) dx = ∫ 5eˣ dx

e⁻³ˣ * y = 5eˣ + C

Solving for y, we get:

y = 5e⁴ˣ + Ce³ˣ

Now, we can use the initial condition y(0) = 0 to find the value of the constant C:

0 = 5e⁰ + Ce⁰

0 = 5 + C

C = -5

Substituting the value of C back into the equation, we have the particular solution:

y = 5e⁻⁴ˣ - 5e⁻³ˣ

Therefore, the explicit particular solution of the given initial value problem is:

y =  5e⁻⁴ˣ - 5e⁻³ˣ

To know more about particular solution click here :

https://brainly.com/question/31591549

#SPJ4

In the given question, Bob and Alice play a game in which Bob gives Alice a challenge to think of any number M between 1 to N. Bob then tells Alice a number X. Alice has to confirm whether X is greater or smaller than number M or equal to number M.

This continues till Bob finds the number correctly. The input is given as N, the upper limit of the number guessed by Alice. We have to find the maximum number of attempts Bob needs to guess the number thought of by Alice.So, in order to find the maximum number of attempts required to find the number M(1<=M<=N), we can use binary search approach. The idea is to start with middle number of 1 and N i.e., (N+1)/2. We check whether the number is greater or smaller than the given number.

If the number is smaller, we update the range and set L as mid + 1. If the number is greater, we update the range and set R as mid – 1. We do this until the number is found. We can consider the worst case in which number of attempts required to find the number M is the maximum number of attempts that Bob needs to guess the number thought of by Alice.

The maximum number of attempts Bob needs to guess the number thought of by Alice is log2(N) + 1.Explanation:Binary Search is a technique which is used for searching for an element in a sorted list. We first start with finding the mid-point of the list. If the element is present in the mid-point, we return the index of the mid-point. If the element is smaller than the mid-point, we repeat the search on the lower half of the list.

If the element is greater than the mid-point, we repeat the search on the upper half of the list. We do this until we either find the element or we are left with an empty list. The time complexity of binary search is O(log n), where n is the size of the list.

To know more about confirm visit:

https://brainly.com/question/32246938

#SPJ11

The complex number √2 + i by its conjugate can use the difference of squares formula, product of root 2 + i with its conjugate is 3.

To multiply the given quantity (root 2 + i) into its conjugate, we'll need to first find the conjugate of root 2 + i.

Here's how to do it:

To multiply the square root of 2 + i and its conjugate, you can use the complex multiplication formula.

Conjugate of (root 2 + i)

Multiplying root 2 + i by its conjugate will be of the form:

(a + bi) (a - bi)

Using the identity for (a + b) (a - b) = a² - b² for complex numbers gives us:

where the number is √2 + i.

Let's do a multiplication with this:

(√2 + i)(√2 - i)

Using the above formula we get:

[tex](√2)^2 - (√2)(i ) + (√ 2 )(i) - (i)^2[/tex]

Further simplification:

2 - (√2)(i) + (√2)(i) - (- 1)

Combining similar terms:

2 + 1

results in 3. So (√2 + i)(√2 - i) is 3.

⇒ (root 2)² - (i)²

⇒ 2 - (-1)

⇒ 2 + 1

= 3

For more related questions on product of root:

https://brainly.com/question/32719379

#SPJ8

Dmitry is incorrect in his statement as his range is not comprehensive and adequate to determine if there is an outlier or not in the given data set.

The range he calculated is -6.8 to 62.8, but this range is not appropriate for the provided set of data as it is too wide. It is crucial to keep in mind that the formula for the range is Range = maximum – minimum, which is the absolute difference between the maximum and minimum values in a dataset. The range is not a good measure of variability because it is sensitive to outliers. Thus, it is not an adequate criterion for detecting outliers. It only focuses on the two extremes of the distribution rather than the entire dataset, so it is inadequate to determine if there is an outlier or not.

Dmitry is incorrect because the range he calculated is not appropriate for the given data set. Dmitry's argument is based on the incorrect assumption that a range of 3 standard deviations is sufficient to detect outliers. The rule that a range of 3 standard deviations is sufficient to detect outliers is based on the assumption that the data are normally distributed, but this is not the case for this particular data set.

The correct method to detect outliers, in this case, is to use the interquartile range (IQR), which is defined as the difference between the third quartile (Q3) and the first quartile (Q1). Outliers can be detected using the following formula: Outliers = Values < (Q1 - 1.5*IQR) or Values > (Q3 + 1.5*IQR)Therefore, in the case of the given data set, we can find the outliers by using the interquartile range (IQR), which is defined as follows:

IQR = Q3 – Q1= 38 – 25= 13Hence, the lower bound and upper bound of the data set will be Q1 – 1.5 × IQR and Q3 + 1.5 × IQR, respectively.

Lower bound = 25 – 1.5 × 13 = 5.5Upper bound = 38 + 1.5 × 13 = 57.5According to the above calculations, we can conclude that there are no outliers in the given data set since all the values lie within the range of 5.5 to 57.5.

Thus, Dmitry is incorrect in his statement. The range he calculated is not appropriate for the given data set. The correct method to detect outliers, in this case, is to use the interquartile range (IQR), which is defined as the difference between the third quartile (Q3) and the first quartile (Q1). All the values in the given data set lie within the range of 5.5 to 57.5, so there are no outliers in the data set.

To know more about interquartile range visit

brainly.com/question/29173399

#SPJ11

The given function is f(x) = /1 + e^x. We are to find the derivative of the function.

Using the quotient rule, we have f'(x) = [(1 + e^x)*e^x - e^x*(e^x)] / (1 e^x)^2

Simplifying, we get f'(x) = e^x / (1 + e^x)^2

We used the quotient rule of differentiation which states that if y = u/v,

where u and v are differentiable functions of x, then the derivative of y with respect to x is given byy'

= [v*du/dx - u*dv/dx]/v²

We can see that the given function can be written in the form y = u/v,

where u = e^x and

v = 1 + e^x.

On differentiating u and v with respect to x, we get du/dx = e^x and

dv/dx = e^x.

We then substitute these values in the quotient rule to get the derivative f'(x)

= e^x / (1 + e^x)^2.

Hence, the derivative of the given function is f'(x) = e^x / (1 + e^x)^2.

To know more about derivative visit:

https://brainly.com/question/25324584

#SPJ11

The standard normal distribution is a probability distribution over the entire real line with mean 0 and standard deviation 1. A random variable following this distribution is referred to as a standard normal random variable.

a) The statement “The random variable is continuous” is true for a standard normal random variable. A continuous random variable can take on any value in a given range, whereas a discrete random variable can only take on certain specific values. Since the standard normal distribution is a continuous distribution defined over the entire real line, a standard normal random variable is also continuous.

b) The statement “The mean of the variable is 0” is true for a standard normal random variable. The mean of a standard normal distribution is always 0 by definition.

c) The statement “The median of the variable is 0” is true for a standard normal random variable. The standard normal distribution is symmetric around its mean, so the median, which is the middle value of the distribution, is also at the mean, which is 0.

Therefore, all of the statements a, b, and c are true for a random variable with standard normal probability distribution, and the answer is d. None of the above.

learn more about normal distribution here

https://brainly.com/question/15103234

#SPJ11

The sample standard deviation of the given data is approximately 4.0 while the population standard deviation is approximately 3.94.

The formula for computing standard deviation is as follows:

[tex]\[\large\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\mu)^2}{n-1}}\][/tex]

where:x is the individual value.μ is the mean (average).n is the number of values.[tex]\(\sigma\)[/tex] is the standard deviation.

A standard deviation is the difference between the average and the square root of the variance of a set of data. Standard deviation measures the amount of variability or dispersion for a subject set of data. We will compute both the sample standard deviation and the population standard deviation.

To calculate the sample standard deviation, we can use the same formula as we did in the population standard deviation, but we must divide by n - 1 instead of n. Thus:

[tex]\[\large s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}\][/tex]

where:[tex]\(\sigma\)[/tex] is the standard deviation.x is the individual value.μ is the mean (average).n is the number of values. [tex]\(\sigma\)[/tex] is the standard deviation.

For the given data 15, 6, 14, 7, 14, 5, 15, 14, 14, 12, 11, 10, 8, 13, 13, 14, 4, 13, 3, 11, 14, 14, 12

we first calculate the mean.

µ = (15+6+14+7+14+5+15+14+14+12+11+10+8+13+13+14+4+13+3+11+14+14+12) / 23=10.6

After that, we compute the standard deviation (sample).

s = √ [ (15-10.6)² + (6-10.6)² + (14-10.6)² + (7-10.6)² + (14-10.6)² + (5-10.6)² + (15-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² + (11-10.6)² + (10-10.6)² + (8-10.6)² + (13-10.6)² + (13-10.6)² + (14-10.6)² + (4-10.6)² + (13-10.6)² + (3-10.6)² + (11-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² ] / 22

s = 4.0

The sample standard deviation is approximately 4.0.

For the population standard deviation, we should replace n-1 by n in the above formula. Thus:

σ = √ [ (15-10.6)² + (6-10.6)² + (14-10.6)² + (7-10.6)² + (14-10.6)² + (5-10.6)² + (15-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² + (11-10.6)² + (10-10.6)² + (8-10.6)² + (13-10.6)² + (13-10.6)² + (14-10.6)² + (4-10.6)² + (13-10.6)² + (3-10.6)² + (11-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² ] / 23

σ = 3.94 (approximately)

Therefore, the population standard deviation is approximately 3.94.

The sample standard deviation of the given data is approximately 4.0 while the population standard deviation is approximately 3.94.

To know more about mean visit:

brainly.com/question/29727198

#SPJ11

The answer is: Not mutually exclusive but independent.

Note that B and C are not mutually exclusive, since they have an intersection: B ∩ C = {c}. However, we can check whether they are independent by verifying if the probability of their intersection is the product of their individual probabilities:

P(B) = P(b) + P(c) = 1/5 + 1/5 = 2/5

P(C) = P(c) + P(d) = 1/5 + 1/5 = 2/5

P(B ∩ C) = P(c) = 1/5

Since P(B) * P(C) = (2/5) * (2/5) = 4/25 ≠ P(B ∩ C), we conclude that events B and C are not independent.

Therefore, the answer is: Not mutually exclusive but independent.

Learn more about independent. from

https://brainly.com/question/25223322

#SPJ11

The circumference of the circle if the radius changes to 13 in. is 26π or approximately 81.64

Given that a circle with radius 7 in. has circumference 43.96 in. We need to find the circumference of the circle if the radius changes to 13 in.

The formula for the circumference of a circle is given by:

C = 2πr where C is the circumference, r is the radius and π is a constant equal to 3.14.

Applying the above formula we have:

Circumference of the circle with radius 7 in = 2π × 7= 14π

So, the circumference of the circle with radius 7 in. is 14π or approximately 43.96 in.

Given the radius of the circle changes to 13 in.

Now, the new circumference of the circle is:

Circumference of the circle with radius 13 in. = 2π × 13= 26π

Therefore, the circumference of the circle if the radius changes to 13 in. is 26π or approximately 81.64 in.

Know more about circumference of the circle:

https://brainly.com/question/17130827

#SPJ11

The value of point (x₁, x₂) is [tex](\frac{9}{7}, \frac{4}{7} )[/tex]

Given is graph of two lines x₁ + 5x₂ = 7 and x₁ - 2x₂ = -2, intersecting at a point, we need to find the value of (x₁, x₂),

To find the same we will simply solve the system of equations given,

So, to solve,

Subtract the second equation from the first one:

(x₁ + 5x₂) - (x₁ - 2x₂) = 7 - (-2)

x₁ + 5x₂ - x₁ + 2x₂ = 7 + 2            [x₁ will be cancelled out]

5x₂ + 2x₂ = 9

7x₂ = 9

x₂ = 9/7

Plug in the value of x₂ in first equation, we get,

x₁ + 5(9/7) = 7

Multiply the whole equation by 7 to eliminate the denominator, we get,

7x₁ + 45 = 49

7x₁ = 49 - 45

7x₁ = 4

x₁ = 4/7

Hence, we the values of x₁ and x₂ as 4/7 and 9/7 respectively.

Learn more about system of equations click;

https://brainly.com/question/21620502

#SPJ4

Complete question is attached.

The Cartesian coordinates (0, 1) can be converted to polar coordinates as (1, 0). The Cartesian coordinates (5/2, (-5√3)/2) can be converted to polar coordinates as (5, -π/3).

A. To convert the Cartesian coordinates (0, 1) to polar coordinates, we can use the following formulas:

r = √[tex](x^2 + y^2)[/tex]

θ = tan⁻¹(y/x)

For (0, 1), we have x = 0 and y = 1.

r = √[tex](0^2 + 1^2)[/tex]

= √1

= 1

θ = tan⁻¹(1/0) (Note: This expression is undefined)

The angle θ is undefined because the x-coordinate is zero, which means the point lies on the y-axis. In polar coordinates, such points are represented by the angle θ being either 0 or π, depending on whether the y-coordinate is positive or negative. In this case, since the y-coordinate is positive (1 > 0), we can assign θ = 0.

Therefore, the polar coordinates for (0, 1) are (1, 0).

B. For the Cartesian coordinates (5/2, (-5√3)/2), we have x = 5/2 and y = (-5√3)/2.

r = √((5/2)² + (-5√3/2)²)

r = √(25/4 + 75/4)

r = √(100/4)

r = √25

r = 5

θ = tan⁻¹((-5√3)/2 / 5/2)

θ = tan⁻¹(-5√3/5)

θ = tan⁻¹(-√3)

θ ≈ -π/3

Since r must be greater than 0, the polar coordinates for (5/2, (-5√3)/2) are (5, -π/3).

Therefore, the converted polar coordinates are:

A. (0, 1) -> (1, 0)

B. (5/2, (-5√3)/2) -> (5, -π/3)

To know more about Cartesian coordinates,

https://brainly.com/question/30970352

#SPJ11

a. z = x² - 5y²: Predominantly hyperbolas.b. z = x² + 2y²: Predominantly ellipses.c. z = y - 3x²: Predominantly parabolas.d. z = -5x²: Predominantly lines.

To sketch the contour diagrams and determine the predominant shape of the contours for each function, we will plot a range of values for x and y and calculate the corresponding z-values.

a. z = x² - 5y²

Contour diagram:

```

    |     .

    |       .

    |         .

    |          .

    |           .

-----+-----------------

    |           .

    |          .

    |         .

    |       .

    |     .

```

The contour lines of this function are predominantly hyperbolas.

b. z = x² + 2y²

Contour diagram:

```

    |         .

    |       .

    |     .

    |    .

-----+-----------------

    |    .

    |   .

    | .

    |

    |

```

The contour lines of this function are predominantly ellipses.

c. z = y - 3x²

Contour diagram:

```

    |        .

    |       .

    |      .

    |     .

-----+-----------------

    |     .

    |      .

    |       .

    |        .

    |

```

The contour lines of this function are predominantly parabolas.

d. z = -5x²

Contour diagram:

```

    |        .

    |        .

    |        .

    |        .

-----+-----------------

    |

    |

    |

    |

    |

```

The contour lines of this function are predominantly lines.

In summary:

a. z = x² - 5y²: Predominantly hyperbolas.

b. z = x² + 2y²: Predominantly ellipses.

c. z = y - 3x²: Predominantly parabolas.

d. z = -5x²: Predominantly lines.

To learn more about  parabola click here:

brainly.com/question/33482635

#SPJ11

a. The contours of z = x² - 5y² are predominantly hyperbolas.

b. The contours of z = x² + 2y² are predominantly ellipses.

c. The contours of z = y - 3x² are predominantly parabolas.

d. The contours of z = -5x² are predominantly lines.

a. The function z = x² - 5y² represents contours that are predominantly hyperbolas. The contour lines are symmetric about the x-axis and y-axis, and they open up and down. The contours become closer together as they move away from the origin.

b. The function z = x² + 2y² represents contours that are predominantly ellipses. The contour lines are symmetric about the x-axis and y-axis, forming concentric ellipses centered at the origin. The contours become more elongated as they move away from the origin.

c. The function z = y - 3x² represents contours that are predominantly parabolas. The contour lines are symmetric about the y-axis, with each contour line being a vertical parabola. As the value of y increases, the parabolas shift upwards.

d. The function z = -5x² represents contours that are predominantly lines. The contour lines are straight lines parallel to the y-axis. Each contour line has a constant value of z, indicating that the function is a quadratic function with no dependence on y.

In summary, the contour diagrams for the given functions show that:

a. The contours of z = x² - 5y² are predominantly hyperbolas.

b. The contours of z = x² + 2y² are predominantly ellipses.

c. The contours of z = y - 3x² are predominantly parabolas.

d. The contours of z = -5x² are predominantly lines.

Learn more about parabolas here:

brainly.com/question/11911877

#SPJ11

a) The coefficient of variation is the ratio of the standard deviation to the mean. The formula for the coefficient of variation (CV) is given by:CV = (Standard deviation/Mean) × 100.

We are given the mean selling price of houses in a certain county, which is $339,000, and the standard deviation of the selling prices, which is $60,000.Substituting these values into the formula, we get:CV = (60,000/339,000) × 100= 17.69%Therefore, the coefficient of variation for the selling prices of houses in the county is 17.69%.

b) The z-score is a measure of how many standard deviations away from the mean a particular data point lies.

The formula for the z-score is given by:z = (x – μ) / σWe are given the selling price of a house, which is $329,000. The mean selling price of houses in the county is $339,000, and the standard deviation is $60,000.Substituting these values into the formula, we get:z = (329,000 – 339,000) / 60,000= -0.1667Therefore, the z-score for a house that sells for $329,000 is -0.1667.

c) The empirical rule states that for data that follows a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Therefore, the range of prices that includes 68% of the homes around the mean can be calculated as follows:Lower limit = Mean – Standard deviation= 339,000 – 60,000= 279,000Upper limit = Mean + Standard deviation= 339,000 + 60,000= 399,000Therefore, the range of prices that includes 68% of the homes around the mean is $279,000 to $399,000.

d) Chebychev's Theorem states that for any dataset, regardless of the distribution, at least (1 – 1/k²) of the data falls within k standard deviations of the mean. Therefore, to determine the range of prices that includes at least 96% of the homes around the mean, we need to find k such that (1 – 1/k²) = 0.96Solving for k, we get:k = 5Therefore, at least 96% of the data falls within 5 standard deviations of the mean. The range of prices that includes at least 96% of the homes around the mean can be calculated as follows:

Lower limit = Mean – (5 × Standard deviation)= 339,000 – (5 × 60,000)= 39,000Upper limit = Mean + (5 × Standard deviation)= 339,000 + (5 × 60,000)= 639,000Therefore, the range of prices that includes at least 96% of the homes around the mean is $39,000 to $639,000.

In statistics, the coefficient of variation (CV) is the ratio of the standard deviation to the mean. It is expressed as a percentage, and it is a measure of the relative variability of a dataset. In this question, we were given the mean selling price of houses in a certain county, which was $339,000, and the standard deviation of the selling prices, which was $60,000. Using the formula for the coefficient of variation, we calculated that the CV was 17.69%. This means that the standard deviation is about 17.69% of the mean selling price of houses in the county. A high CV indicates that the data has a high degree of variability, while a low CV indicates that the data has a low degree of variability.The z-score is a measure of how many standard deviations away from the mean a particular data point lies. In this question, we were asked to calculate the z-score for a house that sold for $329,000.

Using the formula for the z-score, we calculated that the z-score was -0.1667. This means that the selling price of the house was 0.1667 standard deviations below the mean selling price of houses in the county. A negative z-score indicates that the data point is below the mean. A positive z-score indicates that the data point is above the mean.The Empirical Rule is a statistical rule that states that for data that follows a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.

In this question, we were asked to use the Empirical Rule to determine the range of prices that includes 68% of the homes around the mean. Using the formula for the range of prices, we calculated that the range was $279,000 to $399,000.

Chebychev's Theorem is a statistical theorem that can be used to determine the minimum percentage of data that falls within k standard deviations of the mean. In this question, we were asked to use Chebychev's Theorem to determine the range of prices that includes at least 96% of the homes around the mean.

Using the formula for Chebychev's Theorem, we calculated that the range was $39,000 to $639,000. Therefore, we can conclude that the range of selling prices of houses in the county is quite wide, with some houses selling for as low as $39,000 and others selling for as high as $639,000.

To know more about  standard deviation :

brainly.com/question/29115611

#SPJ11

The value of the double integral ∬R (6x/(1 + xy)) dA over the region

R = [0, 6] × [0, 1] is 6 ln(7).

To calculate the double integral ∬R (6x/(1 + xy)) dA over the region

R = [0, 6] × [0, 1], we can integrate with respect to x and y using the limits of the region.

The integral can be written as:

∬R (6x/(1 + xy)) dA = [tex]\int\limits^1_0\int\limits^6_0[/tex] (6x/(1 + xy)) dx dy

Let's start by integrating with respect to x:

[tex]\int\limits^6_0[/tex](6x/(1 + xy)) dx

To evaluate this integral, we can use a substitution.

Let u = 1 + xy,

     du/dx = y.

When x = 0,

u = 1 + 0y = 1.

When x = 6,

u = 1 + 6y

  = 1 + 6

   = 7.

Using this substitution, the integral becomes:

[tex]\int\limits^7_1[/tex] (6x/(1 + xy)) dx = [tex]\int\limits^7_1[/tex](6/u) du

Integrating, we have:

= 6 ln|7| - 6 ln|1|

= 6 ln(7)

Now, we can integrate with respect to y:

= [tex]\int\limits^1_0[/tex] (6 ln(7)) dy

= 6 ln(7) - 0

= 6 ln(7)

Therefore, the value of the double integral ∬R (6x/(1 + xy)) dA over the region R = [0, 6] × [0, 1] is 6 ln(7).

Learn more about double integral here:

brainly.com/question/15072988

#SPJ4

The value of the double integral   [tex]\int\limits^1_0\int\limits^6_0 \frac{6x}{(1 + xy)} dA[/tex], over the given region [0, 6] x [0, 1] is (343/3)ln(7).

Now, for the double integral  [tex]\int\limits^1_0\int\limits^6_0 \frac{6x}{(1 + xy)} dA[/tex], use the standard method of integration.

First, find the antiderivative of the function 6x/(1 + xy) with respect to x.

By integrating with respect to x, we get:

∫(6x/(1 + xy)) dx = 3ln(1 + xy) + C₁

where C₁ is the constant of integration.

Now, we apply the definite integral over x, considering the limits of integration [0, 6]:

[tex]\int\limits^6_0 (3 ln (1 + xy) + C_{1} ) dx[/tex]

To proceed further, substitute the limits of integration into the equation:

[3ln(1 + 6y) + C₁] - [3ln(1 + 0y) + C₁]

Since ln(1 + 0y) is equal to ln(1), which is 0, simplify the expression to:

3ln(1 + 6y) + C₁

Now, integrate this expression with respect to y, considering the limits of integration [0, 1]:

[tex]\int\limits^1_0 (3 ln (1 + 6y) + C_{1} ) dy[/tex]

To integrate the function, we use the property of logarithms:

[tex]\int\limits^1_0 ( ln (1 + 6y))^3 + C_{1} ) dy[/tex]

Applying the power rule of integration, this becomes:

[(1/3)(1 + 6y)³ln(1 + 6y) + C₂] evaluated from 0 to 1,

where C₂ is the constant of integration.

Now, we substitute the limits of integration into the equation:

(1/3)(1 + 6(1))³ln(1 + 6(1)) + C₂ - (1/3)(1 + 6(0))³ln(1 + 6(0)) - C₂

Simplifying further:

(343/3)ln(7) + C₂ - C₂

(343/3)ln(7)

So, the value of the double integral  [tex]\int\limits^1_0\int\limits^6_0 \frac{6x}{(1 + xy)} dA[/tex], over the given region [0, 6] x [0, 1] is (343/3)ln(7).

To learn more about integration visit :

brainly.com/question/18125359

#SPJ4

(a), (d), (f), (h), and (k) are true statements and  (b), (c), (e), (g), (i), and (j) are false statements .

(a) True. For any real number x, there exists a real number y = -x such that x + y = 0. This can be proven by substituting y = -x into the equation x + y = 0, which gives x + (-x) = 0, and since the sum of any number and its additive inverse is zero, this statement holds true for all real numbers.

(b) False. There is no single real number x that can satisfy the equation x + y = 0 for all real numbers y. If we assume such an x exists, it would imply that x + y = 0 holds true for any y, including y = 1, which would lead to a contradiction. Therefore, this statement is false.

(c) False. The equation x^2 + y^2 = -1 represents the sum of two squares, which is always non-negative. Therefore, there are no real numbers x and y that satisfy this equation. Thus, this statement is false.

(d) True. For any positive real number x, there exists a negative real number y = -x such that y < 0 and xy > 0. This is true because when x is positive and y is negative, their product xy is negative. Therefore, this statement holds true for all positive real numbers x.

(e) False. For this statement to hold true, there would need to exist a real number x that satisfies the equation xy = xz for all real numbers y and z. However, this is not possible unless x is equal to zero, in which case the equation holds true but only for z = 0. Therefore, this statement is false.

(f) True. There exists a real number x such that x is less than or equal to any real number y. This is true for x = -∞ (negative infinity). For any real number y, -∞ is less than or equal to y. Thus, this statement is true.

(g) False. There is no single real number x that is less than or equal to any real number y. If we assume such an x exists, it would imply that x is less than or equal to y = 0, but then there exists a real number y' = x - 1 that is strictly less than x. This contradicts the assumption. Therefore, this statement is false.

(h) True. There exists a unique negative real number y such that y is less than zero and y + 3 is greater than zero. This can be proven by solving the inequality system: y < 0 and y + 3 > 0. The solution is y = -2. Therefore, this statement is true.

(i) False. For this statement to hold true, there would need to exist a real number x that satisfies the equation x = y^2 for all real numbers y. However, this is not possible unless x is equal to zero, in which case the equation holds true but only for y = 0. Therefore, this statement is false.

(j) False. There is no unique real number x that satisfies the equation x = y^2 for all real numbers y. For any positive real number y, y^2 is positive, and for any negative real number y, y^2 is also positive. Therefore, this statement is false.

(k) True. There exists a unique pair of real numbers x and y such that for any real number w, w^2 is greater than x - y. This can be proven by taking x = 0 and y = -1. For any real number w, w^2 will be greater than 0 - (-1) = 1. Therefore, this statement is true.

In conclusion, the true statements  in the universe of all real numbersare: (a), (d), (f), (h), and (k). The false statements are: (b), (c), (e), (g), (i), and (j).

To know more about real number, visit;

https://brainly.com/question/17019115
#SPJ11

The following is the given data for the brand of refrigerator.

Let "x" be the unit price of the refrigerator in dollars, and "y" be the number of refrigerators produced.

Suppose that the producers of a certain brand of the refrigerator make 1000 refrigerators available when the unit price is $410.

This implies that:

y = 1000x = 410

When the unit price of the refrigerator is $450, 5000 refrigerators will be marketed.

This implies that:

y = 5000x = 450

To find the equation of the line that represents the relationship between price and quantity, we need to solve the system of equations for x and y:

1000x = 410

5000x = 450

We can solve the first equation for x as follows:

x = 410/1000 = 0.41

For the second equation, we can solve for x as follows:

x = 450/5000 = 0.09

The slope of the line that represents the relationship between price and quantity is given by:

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

Where (x1, y1) = (0.41, 1000) and (x2, y2) = (0.09, 5000)

m = (5000 - 1000)/(0.09 - 0.41) = -10000

Therefore, the equation of the line that represents the relationship between price and quantity is:

y - y1 = m(x - x1)

Substituting m, x1, and y1 into the equation, we get:

y - 1000 = -10000(x - 0.41)

Simplifying the equation:

y - 1000 = -10000x + 4100

y = -10000x + 5100

This is the equation of the line that represents the relationship between price and quantity.

to find the equation of the line:

https://brainly.com/question/33645095

#SPJ11

a.The given Weibull distribution with shape 2 and scale 5, the PDF is:

f(t) = (2/5) *[tex](t/5)^{2-1} * e^{-(t/5)^{2}}[/tex] b. The cumulative distribution function (CDF) of a Weibull distribution with shape parameter k and scale parameter λ is given by:

F(t) = 1 - e^(-(t/λ)^k)  c.The given Weibull distribution with shape 2 and scale 5:

S(t) =[tex]1 - (1 - e^{-(t/5)^{2}})[/tex]  d. The hazard function h(t) for a Weibull distribution is given by the ratio of the PDF and the survival function:

h(t) = f(t) / S(t)  e.the given Weibull distribution with shape 2 and scale 5, the expected value is:

E(T) = 5 * Γ(1 + 1/2)  f.The given Weibull distribution with shape 2 and scale 5, the variance is:

V(T) =[tex]5^2[/tex] * [Γ(1 + 2/2) - (Γ(1 + 1/2)[tex])^2[/tex]]   g.To calculate P(T > 6), we need to find the survival function S(t) and evaluate it at t = 6:

P(T > 6) = S(6) = 1 - F(6) = 1 - [1 - [tex]e^{-(6/5)^2}[/tex]]   h.To calculate P(2 < T ≤ 8), we subtract the cumulative probability at t = 8 from the cumulative probability at t = 2:

P(2 < T ≤ 8) = F(8) - F(2) = [tex]e^{-(2/5)^{2}} - e^{-(8/5)^{2}[/tex]

(a) The probability density function (PDF) of a Weibull distribution with shape parameter k and scale parameter λ is given by:

f(t) = (k/λ) * (t/λ[tex])^{k-1}[/tex]* [tex]e^(-([/tex]t/λ[tex])^k)[/tex]

For the given Weibull distribution with shape 2 and scale 5, the PDF is:

f(t) = (2/5) * [tex](t/5)^{2-1} * e^{-(t/5)^2}}[/tex]

(b) The cumulative distribution function (CDF) of a Weibull distribution with shape parameter k and scale parameter λ is given by:

F(t) = 1 - e^(-(t/λ)^k)

For the given Weibull distribution with shape 2 and scale 5, the CDF is:

F(t) = 1 - e^(-(t/5)^2)

(c) The survival function (also known as the reliability function) S(t) is the complement of the CDF:

S(t) = 1 - F(t)

For the given Weibull distribution with shape 2 and scale 5:

S(t) = 1 - [tex](1 - e^{-(t/5)^{2}})[/tex]

(d) The hazard function h(t) for a Weibull distribution is given by the ratio of the PDF and the survival function:

h(t) = f(t) / S(t)

For the given Weibull distribution with shape 2 and scale 5, the hazard function is:

h(t) =[tex][(2/5) * (t/5)^{2-1)} * e^{-(t/5)^{2}}] / [1 - (1 - e^{-(t/5)^2}})][/tex]

(e) The expected value (mean) of a Weibull distribution with shape parameter k and scale parameter λ is given by:

E(T) = λ * Γ(1 + 1/k)

For the given Weibull distribution with shape 2 and scale 5, the expected value is:

E(T) = 5 * Γ(1 + 1/2)

(f) The variance of a Weibull distribution with shape parameter k and scale parameter λ is given by:

V(T) = λ^2 * [Γ(1 + 2/k) - (Γ[tex](1 + 1/k))^2[/tex]]

For the given Weibull distribution with shape 2 and scale 5, the variance is:

V(T) = [tex]5^2[/tex] * [Γ(1 + 2/2) - (Γ[tex](1 + 1/2))^2[/tex]]

(g) To calculate P(T > 6), we need to find the survival function S(t) and evaluate it at t = 6:

P(T > 6) = S(6) = 1 - F(6) = 1 - [[tex]1 - e^{-(6/5)^2}[/tex]]

(h) To calculate P(2 < T ≤ 8), we subtract the cumulative probability at t = 8 from the cumulative probability at t = 2:

P(2 < T ≤ 8) = F(8) - F(2) = [tex]e^{-(2/5)^{2}} - e^{-(8/5)^2}[/tex]

For more questions onWeibull distribution:

brainly.com/question/15714810

#SPJ4