we find t=2.73 with 5 degrees of freedom. what is the appropriate p-value.

If Leg 1 is 2 leg 2 is 2 then the hypotenuse of the triangle is 2.828.

Using Pythagoras theorem to find the hypotenuse c of the right angled triangle with base b and height a,

c² = a² + b²

In this case, leg 1 and leg 2 have lengths of 2, so we can substitute,

c² = 2² + 2²

c² = 4 + 4

c² = 8

c = √(8)

We can simplify this by factoring out a 2 from the square root,

c = √(4 x 2)

c = √(4) x √(2)

c = 2 x √(2)

c = 2.828

Hence the hypotenuse is found to be 2.828.

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The term var2 × var2 specifies that both the linear and quadratic terms for var2 should be included in the model.

Now, Let's an example code for fitting an MLR model with a linear and quadratic term for var2 using proc glm in SAS as;

proc glm data = your_dataset;

model var1 = var2 var2 × var2;

run;

Hence, In this code, your _ dataset refers to the name of the dataset that you are using.

The model statement specifies the variables in the model, where var1 is the dependent variable and var2 is the independent variable.

Thus, The term var2 × var2 specifies that both the linear and quadratic terms for var2 should be included in the model.

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For an independent-measures t statistic, the statement "the estimated standard error measures how much difference is reasonable to expect between the two sample means if the null hypothesis is true" is True.

Your answer: True. The estimated standard error in an independent-measures t statistic indeed measures the reasonable difference between the two sample means, assuming the null hypothesis is true. This value helps to determine if the observed difference in means is significantly different from what is expected by chance alone.

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102 Degrees

Angle A and Angle B are supplementary to each other.

Angle A measure = 78 degrees

Find Angle B

We know that in a supplementary relationship between two angles, the sum of both of the angles are equal to 180 degrees.

Here we know that Angle A  = 78 degrees

Angle B = 180 - Angle A = 180 - 78 = 102 degrees.

A circle is the locus of points in a plane that are equidistant from one point, known as the center.

In geometry, a circle is a two-dimensional figure that consists of all points in a plane that are at an equal distance, called the radius, from a fixed point called the center. The distance from the center to any point on the circumference of the circle is always the same.

This property makes circles useful in various fields, such as mathematics, physics, and engineering. Circles have several important characteristics, including a diameter (the longest chord that passes through the center), a circumference (the boundary of the circle), and an area (the measure of the space enclosed by the circle).

The equation of a circle in the Cartesian coordinate system is (x - a)^2 + (y - b)^2 = r^2, where (a, b) represents the center coordinates and r represents the radius of the circle.

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The function that represents the number of minutes Alejandro drove to reach mile marker m is f(m) = 4(m - 136).

The function that represents the number of minutes Alejandro drove to reach mile marker m on his route is:
f(m) = 4(m - 136)
This is because he drove at a constant speed, so the time it took to reach each mile marker was the same. From the table, we can see that he drove 5 miles in 4 minutes, so his speed was 5/4 miles per minute. Using this speed, we can write the equation:
distance = rate x time
where distance is (m - 136) miles (the distance from his starting point to the mile marker m), rate is 5/4 miles per minute, and time is the number of minutes it took to drive that distance.
Solving for time, we get:
time = distance / rate = (m - 136) / (5/4) = 4(m - 136)

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The Wason task is a classic problem in cognitive psychology that involves conditional reasoning. In the version of the task that you mentioned, participants are given the following information:

"If a person is drinking alcohol, then they must be at least 21 years old." They are then presented with four cards that show a person's age on one side and whether or not they are drinking alcohol on the other side. The task is to determine which cards need to be flipped over to test the conditional rule.

Research has shown that people perform better on the Wason task when it is presented in a meaningful context, such as the one you described involving age and drinking alcohol, rather than using abstract symbols or letters and numbers. This is because the meaningful context helps people to better understand and remember the conditional rule being tested. When the task is presented in terms of letters and numbers, it can be more difficult for people to make sense of the information and apply the conditional rule correctly.

Additionally, using a meaningful context can make the task more relevant and engaging to participants, which can improve their motivation and performance. Overall, presenting the Wason task with examples rather than letters and numbers can help to make the task more accessible and easier to understand for participants.

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77 is the mode of the data set represented by the stem and leaf plot.

Looking at the stem and leaf plot given, we can see that the most frequently occurring value, or mode, is 77. This can be determined by examining the plot and identifying the largest group of leaves, which in this case is the group of sevens under the stem of 5.

To further explain this mathematically, we can define mode as the value that occurs most frequently in a data set. In the stem and leaf plot, the leaves represent the individual values of the data set.

By counting the number of times each value appears in the plot, we can determine the frequency of each value.

The mode is then the value with the highest frequency. In this case, the value with the highest frequency is 77, which occurs five times.

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Answer:A

Step-by-step explanation:

Answer:

3/7

Step-by-step explanation:

Total spins: 9 + 7 + 5 = 21

Number of times landing on orange: 9

p(orange) = 9/21 = 3/7

Answer: 3/7

So, Nayan plays games for a total of 90 minutes or 1 hour and 30 minutes.

The time interval is the span of time between two specified times. To put it another way, it is the amount of time that has elapsed between the event's start and finish. A different name for it is elapsed time.  A larger span of time can be broken up into several shorter, equal-length segments. These are referred to as time periods.

Since there is no "true zero" value for time, it is regarded as an interval variable. However, differences between all time points are equal.

To calculate how long Nayan plays games, we need to add up the time intervals:

Morning: 7:00 - 6:15 = 45 minutes

Evening: 8:15 - 7:30 = 45 minutes

So, Nayan plays games for a total of 90 minutes or 1 hour and 30 minutes.

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A 95% confidence interval for the mean length of the slide pins  is (3.03, 3.27).

We are given:

sample size, n = 16

sample mean, x = 3.15

sample standard deviation, s = 0.2

confidence level, C = 95%

Since the sample size is less than 30, we use a t-distribution with n-1 degrees of freedom.

The formula for the confidence interval for the population mean is:

x ± tα/2 * s/√n

where tα/2 is the t-score with (n-1) degrees of freedom for the given confidence level and √n is the square root of the sample size.

Substituting the given values, we get:

Lower limit = x - tα/2 * s/√n

Upper limit = x + tα/2 * s/√n

From the t-distribution table with 15 degrees of freedom and a 95% confidence level, we find that the t-score is approximately 2.131.

Substituting the values, we get:

Lower limit = 3.15 - 2.131 * 0.2/√16 = 3.03

Upper limit = 3.15 + 2.131 * 0.2/√16 = 3.27

Therefore, the 95% confidence interval for the mean length of the slide pins is (3.03, 3.27).

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Terry should take the shortest path across the river and then travel downstream to Sophie's house. To find the optimal path, we can use the Pythagorean theorem and the speed of Terry's travel on land and water.

Let x be the distance Terry travels downstream along the riverbank before crossing the river. Then, the remaining distance to travel downstream on the water is (10-x) km. The distance across the river (the hypotenuse) is 2 km.

By the Pythagorean theorem, the distance Terry travels on land is sqrt(x^2 + 4), and the distance on water is sqrt((10-x)^2 + 4). Now, we can find the time Terry spends traveling on land and water:

Time on land = (sqrt(x^2 + 4))/4 km/h
Time on water = (sqrt((10-x)^2 + 4))/2 km/h

Terry's goal is to minimize the total time spent, so we need to find the value of x that minimizes the sum of these times:

Total time = (sqrt(x^2 + 4))/4 + (sqrt((10-x)^2 + 4))/2

By applying calculus and finding the derivative of this function with respect to x, we can determine the optimal value for x. In this case, the optimal path would be to travel approximately 2.07 km downstream along the riverbank and then cross the river, taking approximately 1.35 hours in total. This way, Terry can reach Sophie's house in the shortest possible time.

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The 95%-confidence interval (45.76%, 50.23%) is a confidence interval for the percentage of listeners in the sample who would accept the invitation to enter the competition.

This means that if we were to repeat the same process of randomly selecting 500 listeners and offering them the chance to enter the competition, we would expect the true percentage of listeners who would accept the offer to fall within this interval 95% of the time. It is important to note that this interval only applies to the sample of 500 listeners who were selected for the promotion, and we cannot generalize these results to all listeners of the radio station. However, this information can still be useful for the radio station in terms of understanding the response rate to promotions among a specific group of its listeners.

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The value of C from the column value table is C = 40

Given data ,

Let the table be represented as T

where ,

6 B D

10 25 35

A C 56

Now , the ratio of the table values is r = 35 / 56

r = 5 / 8

So , from the proportion , the value of C is

25 = ( 5/8 ) C

Multiply by 8 on both sides , we get

5C = 200

Divide by 5 on both sides ,we get

C = 40

Hence , the proportion is solved and C = 40

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a. The median age in the study is  6 years.

b. The mean age in the study is 10.9 years.

(a) To find the median age, we need to find the age at which 50% of the children are younger and 50% are older. Adding up the percentages provided in increasing order of age until the total just exceeds 50%, we have:

7% (age 3) + 19% (age 4) + 17% (age 5) + 39% (age 6) = 82%

This means that 82% of the children are three, four, five, or six years old. To find the median age, we need to find the age at which 41 out of the 175 children (50% of 175) are younger and 134 are older. Since 82% of the children are younger than age 7, and 7 is the oldest age group listed, we know that the median age is age 6.

Therefore, the median age in the study is 6 years.

(b) To find the mean age, we can use the formula:

mean = (sum of values) / (number of values)

We can calculate the sum of values by multiplying each age by the number of children with that age, and adding up the results:

(12 x 3) + (33 x 4) + (29 x 5) + (69 x 6) + (32 x 7) = 1902

So the sum of values is 1902.

The number of values is the total number of children in the sample, which is 175.

Therefore, the mean age is:

mean = 1902 / 175 ≈ 10.9

Rounding to one decimal place, the mean age in the study is 10.9 years.

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The situation you are describing, in which the value of the solution may be made infinitely large in a maximization linear programming problem or infinitely small in a minimization problem without violating any constraints, is known as (b) unbounded.

In linear programming, unboundedness occurs when there is no upper limit on the value of the objective function in a maximization problem or no lower limit in a minimization problem. This happens because the feasible region (i.e., the set of points that satisfy all the constraints) extends indefinitely in the direction that improves the objective function value.

To better understand this concept, let's break it down step-by-step:

1. Linear programming problems involve an objective function (which needs to be maximized or minimized) and a set of constraints.
2. The feasible region is formed by the intersection of all constraint boundaries and represents the solution space where all constraints are satisfied.
3. If the feasible region is unbounded, it means that there is no limit to the value of the objective function in the direction of optimization.
4. For a maximization problem, unboundedness means the solution value can be increased infinitely, while for a minimization problem, it can be decreased infinitely without violating any constraints.

It's important to note that unboundedness is not the same as infeasibility, semi-optimality, or infiniteness. Infeasibility occurs when there are no solutions that satisfy all constraints, semi-optimality refers to a situation where the optimal solution lies at the boundary of the feasible region, and infiniteness is not a standard term used in linear programming.

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We can answer the questions based on the given triangles in this way:

a) sin(BAC) depends only on AB, BC, and BCA, and are the same in both drawings, we have sin(BAC) = 3/5.

b) ∠BAC in the drawing where it is acute is ≈ 36.9°.

c) The ∠BAC in the drawing where it is obtuse is ≈ 143.1°.

The angles of a triangle when added together is always 180°.

To calculate the angles of a triangle, we use the formulas like the Law of Cosines, the Law of Sines, or trigonometric functions like sine, cosine, and tangent.

a) To find sin(BAC), we shall use the Law of Cosines to first find the length of AC:

(AC)² = (AB)² + (BC)² - 2(AB*BC)cos(BCA)

AC² = 15² + 18² - 2(15*18)cos(30°)

AC² = 729

AC = 27

Next, we use the Law of Sines to find sin(BAC):

sin(BAC) / AB = sin(BCA) / AC

sin(BAC) / 15 = 1/2 / 27

sin(BAC) = 3/5

Since sin(BAC) only depends on AB, BC, and BCA, which are the same in both drawings, we have sin(BAC) = 3/5 in both drawings.

b) In the acute triangle, we have:

sin(BAC) / AB = sin(BCA) / AC

sin(BAC) / 15 = 1/2 / 27

sin(BAC) = 3/5

BAC = arc sin(3/5)

BAC ≈ 36.9°

c) In the obtuse triangle, we have:

sin(BAC) / AB = sin(BCA) / AC

sin(BAC) / 15 = 1/2 / 27

sin(BAC) = 3/5

Since sin(BAC) is positive and ≤ 1, we know that BAC is an acute angle or a reflex angle.

But we are told that BAC is obtuse angle, meaning:

BAC = 180° - arc sin(3/5)

BAC ≈ 143.1°

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For y-hat for x = 10 and d = 1 is 2.51(rounded to 2 decimal places) and for  y-hat for x = 10 and d = 0 is 0.16(rounded to 2 decimal places).

A. To compute y-hat for x = 10 and d = 1: - First, calculate the numerator: exp(-1.9885 + 0.2099(10) + 0.4498(1)) = 4.0412 - Then, calculate the denominator: 1 + exp(-1.9885 + 0.2099(10)) = 1.6117 -

Finally, divide the numerator by the denominator: y-hat = 4.0412/1.6117 = 2.5087 Therefore, y-hat for x = 10 and d = 1 is 2.51 (rounded to 2 decimal places).

B. To compute y-hat for x = 10 and d = 0: - First, calculate the numerator: exp(-1.9885 + 0.2099(10)) = 0.1835 - Then, calculate the denominator: 1 + exp(-1.9885 + 0.2099(10)) = 1.1835 -

Finally, divide the numerator by the denominator: y-hat = 0.1835/1.1835 = 0.155 Therefore, y-hat for x = 10 and d = 0 is 0.16 (rounded to 2 decimal places).

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If one or both samples have a sample size less than 30, you can still conduct a two sample test if the population standard deviation is known. In such cases, you can use a Z-test to compare the means of two samples.

If the population standard deviation is unknown, you would need to use a T-test instead. The T-test is preferred over the Z-test when the sample size is small because it is more appropriate for small sample sizes and it accounts for the uncertainty of the sample standard deviation. When conducting a two sample test, it is important to ensure that the samples are independent and representative of the population. The sample size should also be large enough to ensure that the results are statistically significant and accurate. If the sample size is too small, the results may not be reliable or representative of the population. In summary, if one or both samples have a sample size less than 30, you can still conduct a two sample test if the population standard deviation is known. However, it is important to ensure that the samples are independent, representative of the population, and large enough to yield accurate results. Additionally, the appropriate statistical test should be used based on whether the population standard deviation is known or unknown.

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The complex partial fraction for the rational functions are: 16/(z⁴+4) = (-4i/√2)/(z² + 2i) + (4i/√2)/(z² - 2i)

To find the complex partial fractions, we first factor the denominator as follows:

z⁴ + 4 = (z² + 2i)(z² - 2i)

Then we can write the rational function as:

16/(z⁴ + 4) = A/(z² + 2i) + B/(z² - 2i)

where A and B are constants to be determined.

We now need to find the values of A and B. To do this, we multiply both sides of the equation by the common denominator (z⁴ + 4), which gives:

16 = A(z² - 2i) + B(z² + 2i)

We can now substitute z = i√2 into this equation, which gives:

16 = A(-2) + B(2i√2)

Solving for A, we get:

A = -4i/√2

Similarly, substituting z = -i√2 gives:

A = 4i/√2

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The function f(theta) is:

f(theta) = sin(theta) + cos(theta) + 3

To find the function f, we will integrate the given second derivative with respect to theta twice, and use the initial conditions to determine the constants of integration.

First, integrating f ''(theta) = sin(theta) + cos(theta) with respect to theta gives:

f '(theta) = -cos(theta) + sin(theta) + C1

where C1 is a constant of integration.

Next, integrating f '(theta) = -cos(theta) + sin(theta) + C1 with respect to theta gives:

f(theta) = sin(theta) + cos(theta) + C1*theta + C2

where C2 is another constant of integration.

To determine the values of C1 and C2, we use the initial conditions:

f(0) = 4 gives us:

4 = sin(0) + cos(0) + C1*0 + C2

4 = 1 + C2

so C2 = 3.

f '(0) = 1 gives us:

1 = -cos(0) + sin(0) + C1

1 = 1 + C1

so C1 = 0.

Therefore, the function f(theta) is:

f(theta) = sin(theta) + cos(theta) + 3

Note that there are other ways to express this function, such as using trigonometric identities to simplify the expression, but this is the most general form.

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The null hypothesis (H0) is that the proportion of taxpayers who do not pay income taxes in the region is equal to 47%. The alternative hypothesis (Ha) is that the proportion of taxpayers who do not pay income taxes in the region is not equal to 47%.

In other words, the null hypothesis assumes that the politician's claim is true, while the alternative hypothesis assumes that the claim is not true. To test these hypotheses, Makayla can use a one-sample proportion test to determine whether the proportion of taxpayers who do not pay income taxes in her sample is significantly different from 47%.
It's important to note that the results of the test will not definitively prove or disprove the politician's claim, but rather provide evidence for or against it. Additionally, the test may have limitations and assumptions that need to be taken into account when interpreting the results.

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According to the formula for the area of a rectangle, which is A = l x w, where A is the area, l is the length, and w is the width. The length of the rectangle is 12 feet and the width is 8 feet.

To solve this problem, we can use the formula for the area of a rectangle, which is A = l x w, where A is the area, l is the length, and w is the width.

We know that the area of the rectangle is 96 ft^2, so we can plug that in for A:

96 = l x w

We also know that the width is 4 feet less than the length, so we can write:

w = l - 4

Now we can substitute this expression for w into our equation for the area:

96 = l x (l - 4)

Expanding the right side, we get:

96 = l^2 - 4l

Rearranging this equation, we get a quadratic equation in standard form:

l^2 - 4l - 96 = 0

We can solve this equation by factoring or using the quadratic formula, but in this case, it's easier to factor:

(l - 12)(l + 8) = 0

This gives us two possible values for l: l = 12 or l = -8. Since the length of a rectangle can't be negative, we discard the second solution and conclude that the length of the rectangle is 12 feet.

To find the width, we can use the equation we had earlier:

w = l - 4

Substituting l = 12, we get:

w = 12 - 4 = 8

Therefore, the length of the rectangle is 12 feet and the width is 8 feet.

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Answer:

45.5

Step-by-step explanation:

i passed

The interval for the FEUT to apply is: t ∈ (-∞, 0) U (0, ∞)

a) To transform the given DE into a system of two first order DE in matrix form, we first define:

y1 = y
y2 = y'

Then, we can rewrite the given DE as:

[t^2 - 5t + 4] y2' + [t] y2 + [2] y1 = cot(t)

Now, we can express this system in matrix form as:

[0   1] [y1']   [0]
[-2/t 5/t-4] [y2'] = [cot(t)/(t^2-5t+4)]

Therefore, the system of two first order DE in matrix form is:

y' = A(t) y + b(t)

where A(t) = [0   1; -2/t 5/t-4] and b(t) = [0; cot(t)/(t^2-5t+4)]

b) To determine the interval of t for the FEUT (Finite Element Unfitted Taylor) method to apply, we need to consider the singularities of the system matrix A(t). In this case, the singularity occurs at t = 0, which is also the initial point. Therefore, the interval of t for the FEUT method to apply is [2, ∞), which includes the initial point t = 2.

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According to the attached graph

The roots of the parabola (-3, 0)and (7, 0)

the vertex is at (2, 25)

two other points are (0, 21) and (-2.5, 5)

The vertex of a parabola signifies the highest or lowest point depending on the direction it opens.

To find the vertex, we use the formula with regards to a parabola in the format of y = ax^2 + bx + c:

Vertex x-coordinate = -b / (2a)

Vertex y-coordinate is equal to f(x) = ax²+bx+c, keeping x as the found vertex x-coordinate

Vertex x-coordinate = -b / (2a) for y = -x^2 + 4x + 21

b = 4

a = -1

Vertex x-coordinate = -4 / (2 * -1) = 2

substituting x = 2 into  y = -x^2 + 4x + 21  gives y as 25

hence the vertex is (2, 25

)

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Therefore, there are 12,650 distinct tetrahedra that can be formed using the given 25 points.

To form a tetrahedron using the given 25 points, we need to select 4 points from the total of 25. Since the order in which the points are selected does not matter, we can use the combination formula to calculate the number of ways to choose 4 points out of 25.

The combination formula is:

C(n, r) = n! / (r! * (n - r)!)

where n is the total number of items and r is the number of items to be selected.

In this case, we have 25 points and we want to select 4 points to form a tetrahedron. So we can plug these values into the combination formula as follows:

C(25, 4) = 25! / (4! * (25 - 4)!)

= (25 * 24 * 23 * 22 * 21!) / (4 * 3 * 2 * 1 * 21!)

= (25 * 24 * 23 * 22) / (4 * 3 * 2 * 1)

= 12,650

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The derivative of f(x) is f'(x) = 3x^2 + 6x - 4. To find the inflection points, we need to find where the concavity changes.

To find the inflection points, we need to find where the concavity changes. This occurs where the second derivative, f''(x), equals zero or is undefined. Taking the derivative of f'(x), we get f''(x) = 6x + 6.
Setting f''(x) = 0, we get 6x + 6 = 0, which gives x = -1. This is the only inflection point since f''(x) is defined for all values of x.

To check whether this point is a point of inflection, we need to examine the concavity of the function around x = -1. Plugging in x = -2, we get f''(-2) = -6, which is negative, indicating that the function is concave down. Plugging in x = 0, we get f''(0) = 6, which is positive, indicating that the function is concave up. Therefore, the point x = -1 is a point of inflection.

Thus, the location of the inflection point is x = -1.

To find the location of all points of inflection of the function f(x)=(x^2-4)(x+3), we first need to find its second derivative.

1. Find the first derivative, f'(x):
f'(x) = (2x)(x+3) + (x^2-4)(1) = 2x^2 + 6x + x^2 - 4 = 3x^2 + 6x - 4

2. Find the second derivative, f''(x):
f''(x) = 6x + 6

Now, to find the points of inflection, we need to determine where f''(x) changes its sign. This occurs when f''(x) = 0.

3. Solve f''(x) = 0 for x:
6x + 6 = 0
6x = -6
x = -1

The location of the point of inflection is x = -1.

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If  a random variable T is Exponential with u = 102 then the probability that T is less than or equal to 153 is 0.632

If T is an exponential random variable with parameter u, then the probability density function of T is given by:

[tex]f(t) = (1/u) \times e^(^-^t^/^u^)[/tex] for t ≥ 0

The cumulative distribution function (CDF) of T is given by:

F(t) = P(T ≤ t)

= ∫[0, t] f(x) dx

[tex]= 1 - e^(^-^t^/^u^)[/tex] for t ≥ 0

In this case, we are given that T is Exponential with u = 102.

To find P(T ≤ 153), we can use the CDF formula with t = 153:

P(T ≤ 153) = F(153)

= [tex]1 - e^(^-^1^5^3^/^1^0^2^)[/tex]

P(T ≤ 153) = 0.632

Therefore, the probability that T is less than or equal to 153 is 0.632.

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