Mathematics High School

## Answers

**Answer 1**

its the solution I hope it's right good luck

## Related Questions

Which element of a test of a hypothesis is used to decide whether to reject the null hypothesis in favor of the alternative hypothesis? A. Test statistic B. Conclusion C. Rejection region D. Level of significance

### Answers

The element of a test of a hypothesis that is used to decide whether to reject the null **hypothesis** in favor of the alternative hypothesis is the test statistic. The test **statistic** is a **numerical value** that is calculated from the sample data and is used to compare against a critical value or rejection region to determine if the null hypothesis should be rejected. The level of significance is also important in determining the critical value or rejection region, but it is not the actual element used to make the decision to reject or fail to reject the null hypothesis.

About Hypothesis

The **hypothesis** or basic assumption is a temporary answer to a problem that is still presumptive because it still has to be proven true. The alleged answer is a temporary truth, which will be verified by data collected through research. Statistics is a science that studies how to plan, collect, analyze, then interpret, and finally present data. In short, statistics is the science concerned with data. The term statistics is different from statistics. A numeric value contains only numbers, a sign (leading or trailing), and a single decimal point.

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Leta,b, andcbe distinct points. If Pr({a,b}) = Pr({a,c}) = Pr({b,c}) and Pr({a,b,c}) = 1 what is Pr({a})

1/6

1/3

1/2

2/3

It cannot be determined from the information given.

### Answers

If **Probability** of ({a, b}) = Pr({a, c}) = Pr({b, c}) and Pr({a, b, c}) = 1 .

Pr({a}) = 1/3

**What is probability?**

The **probability** of an event is described as a number that indicates how likely the event is to occur which is expressed as a number in the range from 0 and 1, or, using** percentage notation**, in the range from 0% to 100%.

a, b, and c are different and we have that

Pr({a, b}) = Pr({a, c}) = Pr({b, c}) = 1/3,

Pr({a, b, c}) = **1. **

We now Substitute these values and get the following:

Pr({a}) = 1 - 2Pr({a, b}) - 2Pr({a, c}) + 2Pr({a, b, c})

Pr({a} = 1 - 2/3 - 2/3 + 2 = **1/3**

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Ellis and colleagues (2012) tested a new psychotherapy on depression. To study this, a sample of N = 20 inpatients at a psychiatric hospital completed a battery of measurements before and after treatment. Specifically, the sample rated their sense of hopelessness on the Beck Hopelessness Scale (BHS), where the lower the score, the less helpless the patient feels. Feelings of hopelessness are one major symptom of depression. Once psychotherapy was completed, the difference between before and after treatment was calculated, and the sample had M = -5. 34 on hopelessness. After conducting a two-tailed t test using 0. 05 significance level, the researchers calculated t = -2. 62 for the sample mean and d = 0. 83

### Answers

In a study conducted by Ellis and colleagues (2012), a new **psychotherapy** for depression was tested on a sample of 20 inpatients at a **psychiatric** hospital.

The participants rated their sense of hopelessness before and after treatment using the** Beck Hopelessness Scale **(BHS). The researchers found that after completing the psychotherapy, the sample had an average decrease in hopelessness score of -5.34. They conducted a two-tailed t-test with a significance level of 0.05 and calculated a t-value of -2.62 and an effect size (Cohen's d) of 0.83.

The researchers used the t-test to examine whether the difference in hopelessness scores before and after treatment was **statistically** significant. The calculated t-value of -2.62 represents the difference between the sample mean (-5.34) and the population mean (assumed to be 0) divided by the standard error of the mean. The negative t-value indicates that the sample **mean** is significantly lower than the assumed population mean.

The effect size, measured by Cohen's d, is a standardized measure of the difference between the means. A d-value of 0.83 indicates a moderate effect size, suggesting that the **psychotherapy** had a noticeable impact on reducing feelings of hopelessness.

Overall, the findings suggest that the new psychotherapy had a significant and meaningful effect on reducing hopelessness in the sample of **inpatients** with depression, as indicated by the significant t-value and moderate effect size.

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Douglas is saving up money for a down payment on a condominium. He currently has $2880 , but knows he can get a loan at a lower interest rate if he can put down $3774. If he invests the $2880 in an account that earns 5. 7% annually, compounded quarterly, how long will it take Douglas to accumulate the $3774 ? Round your answer to two decimal places, if necessary

### Answers

**Douglas **will need approximately **13.12 quarters**, or approximately 3 years and 4 months to accumulate $3774, with two decimal places.

To solve this problem

We can apply the **compound interest** formula:

A = P(1 + r/n)^(nt)

Where

A is the sum P is the principalr is the yearly interest raten is the frequency of compounding (quarterly means n = 4) t is the length of time in years

Douglas presently has $2880, thus in order to reach his goal of $3774, he must earn the following amount in interest:

$3774 - $2880 = $894

We can set up the equation as follows:

$2880(1 + 0.057/4)^(4t) = $3774

Simplifying the left side, we get:

$2880(1.01425)^(4t) = $3774

Dividing both sides by $2880, we get:

(1.01425)^(4t) = 1.31042

Taking the natural logarithm of both sides, we get:

4t * ln(1.01425) = ln(1.31042)

Dividing both sides by 4 ln(1.01425), we get:

t = ln(1.31042) / (4 ln(1.01425)) = 13.12 quarters

Therefore, Given that there are 4 quarters in a year, Douglas will need **approximately 13.12 quarters,** or approximately 3 years and 4 months, to accumulate $3774, with two decimal places.

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It will take Douglas approximately 3.02 years to accumulate $3,774 by investing his initial $2,880 in **an account **that earns 5.7% annually, compounded quarterly.

We use the **formula** for **compound interest** to estimate how long it will take Douglas to accumulate the needed amount.

What is the formula for compound interest?

The **compound interest **formula we shall to solve the problem is:

A = P(1 + r/n)[tex]^(nt)[/tex]

where:

A = amount of money after t years

P = **principal** amount (or initial investment)

r = annual interest rate (as a decimal)

n = number of compound interest per year

t = number of years

Filling in the values:

P = $2880

r = 0.057 (5.7% as a decimal)

n = 4 (compounded quarterly)

A = $3774

$3774 = $2880 (1 + 0.057/4)[tex]^(4t)[/tex]

Simplifying the **equation**, we get:

1.308125 = (1.01425)[tex]^(4t)[/tex]

We take the natural log from both sides:

ln(1.308125) = ln((1.01425)[tex]^(4t)[/tex]

Using the **logarithm**, we can simplify the right-hand side:

ln(1.308125) = 4t * ln(1.01425)

Now we can solve for t by dividing both sides by 4ln(1.01425):

t = ln(1.308125) / (4 * ln(1.01425))

t ≈ 3.02

Therefore, it will take approximately 3.02 years, for Douglas to accumulate $3,774.

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1. compute coeﬃcient of variation (c.v.) for orders arrived. the potential answers are: a: 0.64. b: 0. c: 0.66. d: 0.65. e: 0.75. 2. compute the average inventory. the potential answers are:

### Answers

The coefficient of **variation **(c.v.) for orders arrived is not provided.The average inventory cannot be calculated without further information.

How to compute the average inventory?

To compute the **coefficient **of variation (C.V.) for orders arrived, we need the standard deviation (SD) and the mean (average) of the orders.

Unfortunately, the given options do not provide the necessary information to **calculate **the C.V. Therefore, none of the provided answers (a, b, c, d, or e) can be considered as the correct coefficient of variation.

Without any specific information regarding the inventory levels or their fluctuations, it is not possible to accurately calculate the **average** inventory. Therefore, no potential answer can be provided for the average inventory as the question lacks essential details such as the inventory turnover rate, stock levels, or any other relevant information necessary for calculating the average inventory.

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Use the properties of the definite integral

Question

If ∫51f(x)dx=3615, what is the value of ∫15f(x)dx?

### Answers

The **value** of given definite integral [tex]\int\limits^5_1 {f(x)} \, dx[/tex] is 3615.

In calculus, the **definite integral** is a mathematical concept used to calculate the area under a curve between two points on the x-axis. The properties of definite integrals allow us to make certain calculations and transformations to integrals to **simplify** their evaluation.

In this problem, we are given the definite integral of f(x) between 5 and 1 and asked to find the definite integral of f(x) between 1 and 5.

We are given that [tex]\int\limits^1_5 {f(x)} \, dx[/tex] = 3615, which represents the area under the curve of f(x) between the limits of 5 and 1 on the x-axis. We are asked to find the area under the same curve between the limits of 1 and 5 on the x-axis, which is represented by the definite integral [tex]\int\limits^5_1 {f(x)} \, dx[/tex].

One of the **properties** of definite integrals is that if we **reverse** the limits of integration, the sign of the integral changes. Therefore, we can write:

[tex]\int\limits^5_1 {f(x)} \, dx[/tex] = [tex]-\int\limits^1_5 {f(x)} \, dx[/tex]

We already know that [tex]\int\limits^1_5 {f(x)} \, dx[/tex] = 3615, so we can **substitute** this value into the above equation:

[tex]\int\limits^5_1 {f(x)} \, dx[/tex] = -3615

However, this is not the final answer because the question asks for the value of [tex]\int\limits^5_1 {f(x)} \, dx[/tex], not [tex]-\int\limits^1_5 {f(x)} \, dx[/tex]. To obtain the actual **value**, we need to multiply the above result by -1:

[tex]\int\limits^5_1 {f(x)} \, dx[/tex] = 3615

Therefore, the value of [tex]\int\limits^5_1 {f(x)} \, dx[/tex] is 3615.

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**Complete Question**

Use the properties of the definite integral

Question :

If [tex]\int\limits^1_5 {f(x)} \, dx[/tex] = 3615, what is the value of [tex]\int\limits^5_1 {f(x)} \, dx[/tex] ?

Solve the given initial-value problem. X' = 13 11 16 0 4 0 X, 1 1 3 X(O) = 5 X(t) = X(t) =

### Answers

To solve the given initial-value problem, we need to use matrix **calculus. **We have the following system of **differential **equations: Therefore, the solution to the initial-value problem is: X(t) = (9/5) e^(16t) [2; 0; 1] + (7/2) e^(2t) [1; 0; -1] + (3/2) e^(2t) [0; 1; 1]

X' = [13 11 16; 0 4 0; 1 1 3] X

Where X is a 3x1 matrix and X' is its **derivative**. We are also given the initial condition X(0) = [5; 1; 2].

To solve this system, we need to find the eigenvalues and eigenvectors of the coefficient matrix [13 11 16; 0 4 0; 1 1 3]. The eigenvalues are λ1 = 16, λ2 = 2, and λ3 = 2, with corresponding eigenvectors v1 = [2; 0; 1], v2 = [1; 0; -1], and v3 = [0; 1; 1].

We can then write the general solution as:

X(t) = c1 e^(16t) [2; 0; 1] + c2 e^(2t) [1; 0; -1] + c3 e^(2t) [0; 1; 1]

Using the initial condition X(0) = [5; 1; 2], we can solve for the constants c1, c2, and c3. We get:

c1 = 1/5 [2; 0; 1] . [5; 1; 2] = 9/5

c2 = 1/2 [1; 0; -1] . [5; 1; 2] = 7/2

c3 = 1/2 [0; 1; 1] . [5; 1; 2] = 3/2

Therefore, the solution to the initial-value problem is:

X(t) = (9/5) e^(16t) [2; 0; 1] + (7/2) e^(2t) [1; 0; -1] + (3/2) e^(2t) [0; 1; 1]

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the base of a solid s is an elliptical region with boundary curve 49x2 4y2 = 196. cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

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The base of a solid s is given by the equation 49x² + 4y² = 196, which represents an **elliptical region **in the xy-plane. Cross-sections of the solid perpendicular to the x-axis are isosceles right triangles, meaning that they have two sides of equal length and a right angle.

To visualize this, imagine slicing the solid s with a plane** perpendicular** to the x-axis. This plane intersects the elliptical base and forms a triangle that is right-angled at the point where the plane meets the base. Since the cross-section is isosceles, the other two sides of the triangle must be of equal length. Therefore, the hypotenuse of the triangle must lie on the boundary curve 49x² + 4y² = 196.

As we move the **slicing plane** along the x-axis, the hypotenuse of each cross-section remains on the elliptical boundary curve, and the legs of the triangle get shorter or longer depending on the distance of the plane from the origin. Thus, the solid s has a varying height and a changing shape along the x-axis.

In summary, the solid s is formed by stacking** isosceles right triangles** with a common hypotenuse lying on the boundary curve of the elliptical base. The resulting shape of the solid changes along the x-axis and can be visualized by slicing it perpendicular to the x-axis.

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given a normal random variable x with mean 36 and variance 16, and a random sample of size n taken from the distribution, what sample size n is necessary in order that p(35.9≤x≤36.1)=0.95?

### Answers

Thus, a sample size of 615 is necessary in order to have a 95% **confidence interval **for the population mean that is within +/- 0.1 of the sample mean, given a normal random variable x with mean 36 and variance 16.

Use the formula for the **standard error **of the mean:

SE = σ / sqrt(n)

where σ is the standard deviation of the population, which is the square root of the variance (in this case, σ = sqrt(16) = 4), and n is the sample size.

We want to find the sample size n that will give us a 95% confidence interval for the population mean that is within +/- 0.1 of the sample mean. This means we need to find the z-score for a 95% confidence interval, which is 1.96 (from a standard **normal distribution** table).

So we have:

0.1 = 1.96 * SE

0.1 = 1.96 * (4 / sqrt(n))

0.1 = 7.84 / sqrt(n)

sqrt(n) = 78.4

n = 614.2

Rounding up to the nearest integer, we get a sample size of n = 615.

Therefore, a sample size of 615 is necessary in order to have a 95% confidence interval for the population mean that is within +/- 0.1 of the sample mean, given a normal random variable x with mean 36 and variance 16.

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The center field fence in a ballpark is 10 feet high and 400 feet from home plate. 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour. (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when $\theta=15^{\circ} .$ Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when $\theta=23^{\circ} .$ Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

### Answers

he parametric **equations **are: [tex]x(t)[/tex]= 100tcos(theta)

y(t) = [tex]-16t^2[/tex] + 100tsin(theta) + 3

How to determine the parametric equations for the path of the ball, graph the ball's path for different angles, and find the minimum angle required for a home run hit in the given scenario?

(a) To write the **parametric** equations for the path of the ball, we can use the following variables:

x(t): horizontal position of the ball at time ty(t): vertical position of the ball at time t

Considering the initial conditions, the equations can be defined as:

x(t) = 400t

y(t) = -16t^2 + 100t + 3

(b) To graph the path of the ball when θ = 15°, we substitute the value of θ into the parametric equations and plot the **resulting **curve. However, to determine if it's a home run, we need to check if the ball clears the 10-foot high fence. If the y-coordinate of the ball's path exceeds 10 at any point, it is a home run.

(c) Similarly, we graph the path of the ball when θ = 23° and check if it clears the 10-foot fence to determine if it's a home run.

(d) To find the **minimum **angle for a home run, we need to find the angle at which the ball's path reaches a maximum y-coordinate greater than 10 feet. We can solve for θ by setting the derivative of y(t) equal to zero and finding the corresponding angle.

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Drag and drop the answer into the box to match each multiplication problem. 0.38 × 10³ 0.38 × 100,000 0.38 × 10 The option "380" (3 of 5) has been grabbed. Press tab to choose where to drop it. Drop it by pressing the spacebar key. Cancel the operation by pressing escape.

### Answers

Option A corresponds to the **product** 3800, Option B **corresponds** to the product 38,000, and Option C corresponds to the product 3.8. Start with the correct answer:

Option A: 3800

Option B: 38,000

Option C: 3.8

The given **multiplication **problems are:

[tex]0.38 × 10³[/tex]

[tex]0.38 × 100,000[/tex]

[tex]0.38 × 10[/tex]

The answer to the given multiplication problems are:

0.38 × 10³ = 3800[tex]0.38 × 10³ = 3800[/tex]

[tex]0.38 × 100,000 = 38,000[/tex]

[tex]0.38 × 10 = 3.8[/tex]

Therefore, **the answer **is:

Option A: 3800

Option B: 38,000

Option C: 3.8

In** conclusion**, the correct answers to the given multiplication problems are as follows: The product of 0.38 multiplied by 10³ is 380. When 0.38 is multiplied by 100,000, the result is 38,000. Lastly, when 0.38 is multiplied by 10, the answer is 3.8.

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when a function is invoked with a list argument, the references of the list is passed to the functiontrue/false

### Answers

The answer is true. When a function is invoked with a list argument in Python, the reference to the list is **passed** to the function.

Is it true that when a list is passed as an argument to a function its reference is passed to the function?

This means that any changes made to the list within the** function** will affect the original list outside of the function as well.

Here's an example to illustrate this behavior:

def add_element(lst, element):

lst.append(element)

my_list = [1, 2, 3]

add_element(my_list, 4)

print(my_list) # Output: [1, 2, 3, 4]

In this example, the add_element function takes a list (lst) and an element (element) as arguments and** appends** the element to the end of the list.

When the function is called with my_list as the first argument, the reference to my_list is passed to the function.

Therefore, when the function modifies lst by appending element to it, the original my_list list is also modified. The output of the program confirms that the original list has been changed.

It's important to keep this behavior in mind when working with functions that take list arguments, as** unexpected modifications** to the original list can lead to bugs and errors in your code.

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A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5. Using a sample of 75 students, find: a. The probability that the mean stress score for the 75 students is less than 2. b. The probability that the total of the 75 stress scores is less than 200. c. The 90th percentile for the total stress score for the 75 students. d. The probability that a student with stress scores is less than

### Answers

a. The **probability **that the mean stress score for the 75 students is less than 2 is 0.

b. The probability that the total of the 75 stress scores is less than 200 is approximately 0.

c. The 90th percentile for the total stress score for the 75 students is approximately 232.4.

d. The probability that a student with stress score less than 5 is is 1.

a. The **mean **stress score for a sample of 75 students can be modeled by a normal distribution with a mean of 3 and a standard deviation of σ/√n, where σ is the standard deviation of the uniform distribution and n is the sample size. Since the lowest stress score is 1 and the highest is 5, the standard deviation is (5-1)/√12 = 2/√3. Thus, the mean stress score for a sample of 75 students is normally distributed with a mean of 3 and a standard deviation of 2/√225 = 2/15.

Using the z-score formula, we have:

z = (2 - 3)/(2/15) = -15/2

P(mean stress score < 2) = P(z < -15/2) = 0 (since the probability of a z-score less than -4 or greater than 4 is very close to 0)

Therefore, the probability that the mean stress **score **for the 75 students is less than 2 is 0.

b. The total stress score for the 75 students can be modeled by a normal distribution with a mean of 3 * 75 = 225 and a standard deviation of √(75/12) * (5-1) = √25 = 5.

Using the z-score formula, we have:

z = (200 - 225)/5 = -5

P(total stress score < 200) = P(z < -5) ≈ 0

Therefore, the probability that the total of the 75 stress scores is less than 200 is approximately 0.

c. The 90th percentile for the total stress score for the 75 students corresponds to the value for which 90% of the total stress scores are less than or equal to that value.

Using a standard normal distribution table, we find that the z-score corresponding to the 90th percentile is approximately 1.28.

Thus, the total stress score corresponding to the 90th percentile is:

X = 225 + 1.28 * 5 ≈ 232.4

Therefore, the 90th percentile for the total stress score for the 75 students is approximately 232.4.

d. Since the stress scores follow a uniform distribution, the probability that a student with stress scores is less than a certain value x is given by (x-1)/(5-1) = (x-1)/4.

Therefore, the probability that a student with stress scores is less than x is:

P(stress score < x) = (x-1)/4

For example, the probability that a student with stress score less than 3 is:

P(stress score < 3) = (3-1)/4 = 0.5

Similarly, the probability that a student with stress score less than 4 is:

P(stress score < 4) = (4-1)/4 = 0.75

And the probability that a student with stress score less than 5 is:

P(stress score < 5) = (5-1)/4 = 1

Note that these probabilities are only for individual students, and do not necessarily apply to the mean or total stress scores for a sample of students.

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find the values of p for which the series is convergent. [infinity] 9 n(ln(n)) p n = 2 p -?-

### Answers

The **series** converges for p > 1/2.

To determine the **convergence** of the series, we can use the integral test. The integral test states that if the function f(n) is positive, continuous, and decreasing for n ≥ N, and if the series Σ f(n) converges, then the series Σ a(n) also converges, where a(n) = f(n) for all n.

In this case, we have a(n) = 9n(ln(n))^p. To check the convergence, we will consider the **function** f(n) = 9n(ln(n))^p and evaluate the integral of f(n) from N to infinity, where N is a positive integer.

∫[N,∞] 9n(ln(n))^p dn = 9∫[N,∞] n^(1+p)ln(n)^p dn

Using integration by parts with u = ln(n)^p and dv = n^(1+p) dn, we get du = p(ln(n))^(p-1)/n dn and v = n^(2+p)/(2+p).

Applying the integration by parts formula, the integral becomes:

9[(ln(n))^p * n^(2+p)/(2+p) - p/(2+p) ∫[N,∞] (ln(n))^(p-1) n^(2+p)/(n) dn]

Simplifying further, we have:

9[(ln(n))^p * n^(2+p)/(2+p) - p/(2+p) ∫[N,∞] (ln(n))^(p-1) n^(1+p) dn]

Since ln(n) is positive for n > 1, we can drop the absolute value signs.

The **term** p/(2+p) ∫[N,∞] (ln(n))^(p-1) n^(1+p) dn will be finite for p > 1/2. This is because (ln(n))^(p-1) approaches 0 as n approaches infinity, and n^(1+p) is a convergent power series for p > -1.

Therefore, the integral ∫[N,∞] 9n(ln(n))^p dn converges if p > 1/2. Consequently, the series Σ 9n(ln(n))^p converges for p > 1/2.

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using generating functions to prove vandermonde's identityC (m +n, r) = ∑r k=0 C(m,r- k) C(n,k) whenever m, n and r are nonnegative integers with r not exceeding either m or n

### Answers

Using generating functions, **Vandermonde's identity** can be proven as C(m+n,r) = ∑r k=0 C(m,r-k) C(n,k), where C(n,k) denotes the binomial coefficient. This identity is useful in combinatorics and probability theory, as it provides a way to calculate the number of combinations of r objects that can be chosen from two sets of m and n objects.

To use **generating functions** to prove Vandermonde's identity, we can start by defining two generating functions:

f(x) = (1+x)^m

g(x) = (1+x)^n

Using the **binomial theorem**, we can expand these generating functions as:

f(x) = C(m,0) + C(m,1)x + C(m,2)x^2 + ... + C(m,m)x^m

g(x) = C(n,0) + C(n,1)x + C(n,2)x^2 + ... + C(n,n)x^n

Now, let's multiply these two generating functions together and look at the coefficient of x^r:

f(x)g(x) = (1+x)^m (1+x)^n = (1+x)^(m+n)

Expanding this using the binomial theorem gives:

f(x)g(x) = C(m+n,0) + C(m+n,1)x + C(m+n,2)x^2 + ... + C(m+n,m+n)x^(m+n)

So, the coefficient of x^r in f(x)g(x) is equal to C(m+n,r).

Now, let's rearrange the terms in f(x)g(x) to isolate the term involving C(m,r-k) and C(n,k):

f(x)g(x) = (C(m,0)C(n,r) + C(m,1)C(n,r-1) + ... + C(m,r)C(n,0))x^r

+ (C(m,0)C(n,r+1) + C(m,1)C(n,r) + ... + C(m,r+1)C(n,0))x^(r+1)

+ ...

So, the coefficient of x^r in f(x)g(x) is also equal to the sum:

∑r k=0 C(m,r- k) C(n,k)

Therefore, we have shown that C(m+n,r) = ∑r k=0 C(m,r- k) C(n,k), which is Vandermonde's identity.

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Consider the following competing hypotheses:

H0: rhoxy = 0 HA: rhoxy ≠ 0

The sample consists of 18 observations and the sample correlation coefficient is 0.15. [You may find it useful to reference the t table.]

a-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

a-2. Find the p-value.

0.05 p-value < 0.10

0.02 p-value < 0.05

0.01 p-value < 0.02

p-value < 0.01

p-value 0.10

b. At the 10% significance level, what is the conclusion to the test?

Reject H0; we can state the variables are correlated.

Reject H0; we cannot state the variables are correlated.

Do not reject H0; we can state the variables are correlated.

Do not reject H0; we cannot state the variables are correlated.

### Answers

a) The correct answer is:** p-value** 0.10.

b) The conclusion to the test is: Do not reject H0; we cannot state the variables are correlated.

a-1. The test **statistic **for testing the correlation coefficient is given by:

t = r * sqrt(n-2) / sqrt(1-r^2)

where r is the sample correlation** coefficient **and n is the sample size.

Substituting the given values, we get:

t = 0.15 * sqrt(18-2) / sqrt(1-0.15^2) ≈ 1.562

Rounding to 3** decimal **places, the test statistic is 1.562.

a-2. The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming that the null hypothesis is true. Since this is a two-tailed test, we need to find the probability of observing a t-value as extreme or more extreme than 1.562 or -1.562. Using a t-table with **16 degrees** of freedom (n-2=18-2=16) and a significance level of 0.05, we find the critical values to be ±2.120.

The p-value is the area under the t-distribution curve to the right of 1.562 (or to the left of -1.562), multiplied by 2 to account for the two tails. From the t-table, we find that the area to the right of 1.562 (or to the left of -1.562) is between 0.10 and 0.20. Multiplying by 2, we get the p-value to be between 0.20 and 0.40.

Therefore, the correct **answer** is: p-value 0.10.

b. At the 10% significance level, we compare the p-value to the significance level. Since the p-value is greater than the significance level of 0.10, we fail to reject the null **hypothesis**. Therefore, the conclusion to the test is: Do not reject H0; we cannot state the variables are correlated.

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27. A particle moves along a coordinate line so that x, its distance from the origin at time t, 0 is given by: x(t) = cos' t. The first time interval in which the point is moving to the right is (A) 0

### Answers

The **answer **is (C) (π/2, 3π/2).

Where is the particle moving?

The **particle **is moving to the right when its velocity is positive.

The **velocity** of the particle is given by:

x'(t) = -sin(t)

The particle is moving to the right on the time **intervals **where x'(t) > 0.

x'(t) > 0 when -sin(t) > 0, which means sin(t) < 0.

The sine function is negative in the second and third quadrants.

So the first time interval in which the particle is moving to the right is (π/2, 3π/2).

Therefore, the answer is (C) (π/2, 3π/2).

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show that if f is integrable on [a, b], then f is integrable on every interval [c, d] ⊆ [a, b].

### Answers

To show that if f is **integrable **on [a, b], then f is integrable on every interval [c, d] ⊆ [a, b], we need to use the definition of **integrability**.

Recall that a function f is **integrable** on an interval [a, b] if and only if for any given ε > 0, there exists a partition P of [a, b] such that the difference between the upper and lower **Riemann sums **of f over P is less than ε. That is,

|U(f, P) - L(f, P)| < ε,

where U(f, P) is the upper Riemann sum of f over P and L(f, P) is the lower Riemann sum of f over P.

Now, suppose f is integrable on [a, b]. We want to show that f is also integrable on every interval [c, d] ⊆ [a, b]. Let ε > 0 be given. Since f is integrable on [a, b], there exists a **partition **P of [a, b] such that

|U(f, P) - L(f, P)| < ε/2.

Now, since [c, d] ⊆ [a, b], we can refine the partition P to obtain a partition Q of [c, d] by only adding or removing **points **from P. More formally, we can define Q as follows:

Q = {x0 = c, x1, x2, ..., xn-1, xn = d},

where x1, x2, ..., xn-1 are points in P that are also in [c, d].

Then, we have

L(f, Q) ≤ L(f, P),

since L(f, Q) is computed using a smaller set of partitions than L(f, P).

Similarly,

U(f, Q) ≥ U(f, P),

since U(f, Q) is computed using a larger set of partitions than U(f, P).

Now, we can use the triangle inequality to get

|U(f, Q) - L(f, Q)| ≤ |U(f, Q) - U(f, P)| + |U(f, P) - L(f, P)| + |L(f, P) - L(f, Q)|.

By the definition of Q, we know that

|U(f, Q) - U(f, P)| ≤ M(d-c)ε/2,

where M is the maximum value of f on [a, b]. Similarly,

|L(f, Q) - L(f, P)| ≤ M(d-c)ε/2.

Therefore, we have

|U(f, Q) - L(f, Q)| ≤ M(d-c)ε/2 + ε/2 + M(d-c)ε/2 = ε.

Thus, f is integrable on [c, d].

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You and your pen pal record the weather in your respective countries on weekend days over the summer. Complete parts a through b

### Answers

We recorded the temperature in degrees Celsius and Fahrenheit, the **precipitation **(if any), and the overall weather conditions (sunny, cloudy, rainy, etc.).b) By comparing the weather in our respective countries over the summer, we were able to note any similarities or differences in our **climates **and weather patterns.

As per the given scenario, you and your pen pal record the **weather **in your respective countries on weekend days over the summer. There are a couple of details you need to record in order to get accurate **information **regarding the weather. These are as follows:Temperature: It is one of the most essential factors of weather and measured in degrees Celsius or Fahrenheit.Precipitation: It refers to the amount of water that falls from the sky in the form of rain, hail, sleet, or snow. The amount of precipitation varies on a daily basis.Overall Weather Conditions: It refers to the condition of the weather. For example, it can be sunny, cloudy, rainy, or any other conditions.You must record these factors in both Celsius and Fahrenheit since both countries have different **measuring **systems. To analyze the weather patterns of both countries, you need to compare the data and note any similarities or differences.

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Iready quiz on linear models. When you answer can you provide an explanation please. Thank you much!

### Answers

**Linear models** are mathematical representations used to describe the relationship between two variables. They can be expressed in the form of a linear equation, y = mx + b, where y represents the dependent variable, x represents the **independent variable**, m represents the slope, and b represents the y-intercept.

In mathematics, a linear model is a way to represent the relationship between two **variables **using a straight line. The equation of a linear model is typically written as y = mx + b, where y is the **dependent variable**, x is the independent variable, m is the slope of the line, and b is the y-intercept (the point where the line crosses the y-axis).

The **slope**, m, determines the steepness of the line. It represents how much the dependent variable (y) changes for each unit increase in the independent variable (x). A positive slope indicates a positive relationship, where y increases as x increases. A negative slope indicates a negative relationship, where y decreases as x increases. A slope of zero represents a horizontal line, indicating no relationship between the variables.

The **y-intercept**, b, is the value of y when x is zero. It represents the starting point of the line on the y-axis. It gives an initial value for the dependent variable before considering the effect of the independent variable.

Overall, linear models are useful for analyzing and predicting the relationship between two variables in a simple and straightforward manner. They provide insights into how changes in the independent variable affect the dependent variable and help make predictions based on the observed data.

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harvesting at the mathematical maximum sustainable yield (msy) can be risky for the long term sustainability of a fishery.

T/F

### Answers

True, **Harvesting **at the mathematical maximum sustainable yield (MSY) can be risky for the long-term sustainability of a fishery.

The concept of MSY is based on the idea of maximizing the catch of fish without depleting the population. However, it assumes that fish populations can be managed as single-species entities and that they can be harvested at a constant rate.

In reality, ecosystems are complex and interconnected, and fish populations interact with other species and the environment in various ways. Harvesting at the MSY level may not consider the broader ecological impacts and can lead to unintended consequences.

While maximizing the catch in the short term may seem beneficial, it can result in overfishing and the depletion of fish **stocks **over time.

This can disrupt the balance of the ecosystem, impact other species that rely on the fish population, and threaten the long-term sustainability of the fishery.

It is important to consider factors such as the reproductive capacity of fish, their life **history **traits, and the overall health of the ecosystem when setting sustainable fishing limits.

Sustainable fisheries management practices often involve adopting precautionary approaches that prioritize the **conservation **and responsible use of fishery resources to ensure their long-term viability.

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evaluate the integral by interpreting it in terms of areas. part 1 of 3 we are concerned with the segment of the line y = 3 2 x − 6 that begins at (0, −6) and that ends at 5, 3/2 3/2

### Answers

Therefore, The **integral** would be ∫[0,5] (3/2)x - 6 dx. Integrating this **equation** would give us the area of the region under the curve.

Explanation: To **evaluate** the integral by interpreting it in terms of areas, we need to find the area of the region under the curve. For part 1 of 3, we are given a **segment** of the line y = (3/2)x - 6 that begins at (0, -6) and ends at (5, 3/2).

To find the **area** of this region, we need to integrate the equation from x = 0 to x = 5. The integral would be:

∫[0,5] (3/2)x - 6 dx

Integrating this equation would give us the area of the region under the **curve**.

To evaluate the integral by interpreting it in terms of areas, we need to find the area of the region under the curve. For part 1 of 3, we are given a segment of the line y = (3/2)x - 6 that begins at (0, -6) and ends at (5, 3/2). To find the area of this region, we need to integrate the **equation** from x = 0 to x = 5.

Therefore, The **integral** would be ∫[0,5] (3/2)x - 6 dx. Integrating this **equation** would give us the area of the region under the curve.

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Determine which of the four levels of measurement (nominal, ordinal, interval ratio) is most appropriate Ages of survey respondents. A. Ordinal B. Interval C. Ratio D. Nominal

### Answers

The most appropriate level of **measurement** for the Ages of survey respondents would be the interval ratio.

This is because age is a **quantitative variable** that can be measured on a continuous scale with equal intervals between each value. The nominal level of measurement is used for categorical variables with no inherent order, the ordinal level of measurement is used for variables with a specific order, but the differences between values are not meaningful, and the ratio level of measurement is used for variables with a true zero point and meaningful ratios between values.

Since age can be measured on a continuous scale with a meaningful zero point (birth), the interval ratio is the most appropriate level of measurement. Hence, the answer is B) Interval

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In a class 50 students, three-fifths are girls. Each girl brings a ribbon of length 2 three-fourths metre and each boy brings 3 one-fourths metre. What is the total length of ribbon collected by all 50 students?

### Answers

**Answer:**

total ribbon collected is** 147.5 meters.**

**Step-by-step explanation:**

Total students = 50

3/5 are girls

girls = 3/5*50 = **30**

boys= 50-30 = **20**

length of ribbon brought by girls = 30*2.75 = 82.5

length of ribbon brought by boys = 20*3.25 = 65

total length of ribbon = 82.5+65 = 147.5 metres

David has a credit card with an APR of 13. 59% and a 30-day billing cycle. The table below details David’s transactions with that credit card in the month of November. Date Amount ($) Transaction 11/1 1,998. 11 Beginning balance 11/5 43. 86 Purchase 11/16 225. 00 Payment 11/23 61. 21 Purchase Between the previous balance method and the daily balance method, which method of calculating David’s November finance charge will result in a greater finance charge, and how much greater will it be? a. The daily balance method will have a finance charge $1. 59 greater than the previous balance method. B. The daily balance method will have a finance charge $0. 40 greater than the previous balance method. C. The previous balance method will have a finance charge $0. 96 greater than the daily balance method. D. The previous balance method will have a finance charge $2. 55 greater than the daily balance method.

### Answers

The previous **balance **method will have a **finance **charge of $2.55 greater than the daily balance method.

Here, we have

Given:

Between the previous balance method and the daily balance method, the previous balance method will have a finance charge of $2.55 greater than the daily balance method.

The difference between the two methods lies in the way in which **interest **is calculated. In the previous balance method, finance charges are based on the beginning balance of the month; on the other hand, in the daily balance method, interest is based on the **average** daily balance of the month.

The formula used to calculate the daily balance method is:

Average Daily Balance (ADB) = (Total of all balances during billing period ÷ Number of days in billing period)

So, the first step is to compute **David's **average daily balance using the formula mentioned above:

ADB = ((1,998.11 x 30) + (43.86 x 21) + (225 x 7) + (61.21 x 2)) ÷ 30 = $1,153.03

The finance charge using the daily balance method would be:($1,153.03 x 13.59% ÷ 365) x 30 = $5.41

The previous balance method charges interest based on the initial amount. As a result, the finance charge is equal to the balance at the end of the **billing **period multiplied by the APR divided by 12.

The finance charge using the previous balance method would be:($152.65 x 13.59% ÷ 12) = $1.71

Therefore, the previous balance method will have a finance **charge **of $2.55 greater than the daily balance method.

The previous balance method will have a finance charge of $2.55 greater than the daily balance method.

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The average driving distance (yards) and driving accuracy (percent of drives that land in the fairway) for 8 golfers are recorded in the table to the right. Complete parts a through e below.

Player Distance (yards) Accuracy (%)

1 316.4 46.2

2 303.8 56.9

3 310.7 51.8

4 312.2 53.2

5 295.5 61.8

6 290.8 66.1

7 295.1 60.4

8 295.9 61.6

a. Write the equation of a straight-line model relating driving accuracy (y) to driving distance (x). Choose the correct answer

below.

A. y = β1x2 + β0

B. y = β0 + β1x + ε

C. y = β1x + ε

D. y = β1x

b. Fit the model, part a, to the data using simple linear regression. Give the least squares prediction equation.

^y = (1)________ + (2) __________x

(1) a. 232.4 b. 258.2 c. 271.1 d. 296.9 (2) a.− 0.7639 b. − 0.6975 c. − 0.5979 d. − 0.6643

c. Interpret the estimated y-intercept of the line. Choose the correct answer below.

A. Since a drive with distance 0 yards is outside the range of the sample data, the y-intercept has no practical interpretation.

B. For each additional percentage in accuracy, the distance is estimated to change by the value of the y-intercept.

C. Since a drive with 0% accuracy is outside the range of the sample data, the y-intercept has no practical interpretation.

D. For each additional yard in distance, the accuracy is estimated to change by the value of the y-intercept.

d. Interpret the estimated slope of the line. Choose the correct answer below.

A. Since a drive with distance 0 yards is outside the range of the sample data, the slope has no practical interpretation.

B. For each additional yard in distance, the accuracy is estimated to change by the value of the slope.

C. For each additional percentage in accuracy, the distance is estimated to change by the value of the slope.

D. Since a drive with 0% accuracy is outside the range of the sample data, the slope has no practical interpretation.

e. A golfer is practicing a new swing to increase her average driving distance. If the golfer is concerned that her driving accuracy will be lower, which of the two estimates, y-intercept or slope, will help determine if the golfer's concern is valid?

The (3)_____________ will help determine if the golfer's concern is valid because the (4)________________ determines whether the accuracy increases or decreases with distance.

(3) a.slope b. y-intercept (4) a. sign of the slope b. sign of the y-intercept c. magnitude of the slope d. magnitude of the y-intercept

### Answers

A. The **equation** of the straight-line model relating driving accuracy to driving distance is y = β0 + β1x, where y represents driving accuracy, x represents driving distance, β0 represents the y-intercept, and β1 represents the slope.

B. Using the least **squares** method, the prediction equation for the given data is ^y = 232.4 - 0.7639x, where ^y represents the predicted accuracy for a given distance x.

C. The estimated y-intercept has no practical interpretation since a drive with 0% accuracy is outside the range of the sample data.

D. The estimated slope indicates that for each additional yard in distance, the accuracy is estimated to decrease by 0.7639%.

E. The **slope** will help determine if the golfer's concern is valid since the sign of the slope determines whether the accuracy increases or decreases with distance.

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Given f(x) = -x² - 7x, find f(-10)

### Answers

**Answer:**

- 30

**Step-by-step explanation:**

Given

f (x) = - x² - 7x

To find : f (- 10)

- x²

= - (- 10)²

= - [ (- 10)×(- 10) ]

= - [ 100 ]

= - 100

- 7x

= - 7 × - 10

= 70

f (- 10) = - 100 + 70

f (- 10) = - 30

Kitchenaid will discontinue the bisque color for its dishwashers due to reports suggesting it is not popular west of the Mississippi unless more than 30% of its customers in states east of the Mississippi prefer it to make up for lost sales elsewhere). As part of the decision process, a random sample of 500 customers east of the Mississippi is selected and their preferences are recorded. of the 500 interviewed, 185 said they prefer the bisque color. a. (3 pts) Define the parameter of interest in words and notation.

### Answers

The** parameter** of interest in words and notation is the proportion of Kitchen aid dishwasher customers east of the Mississippi who prefer the bisque color (p).

The parameter of interest in word and notation is the proportion of Kitchen aid dishwasher customers east of the Mississippi who prefer the bisque color. It can be denoted as p. The **null hypothesis** is that the proportion of customers east of the Mississippi who prefer the bisque color is less than or equal to 0.3, i.e., p ≤ 0.3. The alternative hypothesis is that the **proportion** of customers east of the Mississippi who prefer the bisque color is greater than 0.3, i.e., p > 0.3. This is based on the condition that if less than 30% of customers east of the Mississippi prefer the bisque color, then the color will be discontinued unless more than 30% of its customers in states east of the Mississippi prefer it to make up for lost sales elsewhere.

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If f(x)=3x+2 and g(x)=x^2+1 which expression is equivalent to (f.g)(x)?

### Answers

If f(x) = 3x + 2 and g(x) = x² + 1, we need to find out which of the **expressions **is equal to (f.g)(x). Solution: To solve the given problem, we need to use the **formula **of composition of two functions:f.g(x) = f[g(x)] = 3[x² + 1] + 2f.g(x) = 3x² + 3 + 2f.g(x) = 3x² + 5

Therefore, the expression 3x² + 5 is **equivalent **to (f.g)(x).That is, (f.g)(x) = 3x² + 5In the above solution, we have used the formula of composition of two functions, which is given below:If f(x) and g(x) are two functions, then the composition of two functions f(x) and g(x) is defined as

f[g(x)].If f(x) = 3x + 2 and g(x) = x² + 1, then (f.g)(x) = f[g(x)] = 3[x² + 1] + 2 = 3x² + 3 + 2 = 3x² + 5, which means the expression 3x² + 5 is equivalent to (f.g)(x).The explanation of the **solution **is written in more than 100 words.

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use the power series method to determine the general solution to the equation. (1 − x 2 )y ′′ − xy′ 4y = 0.

### Answers

The values of the **coefficients **is y = 1 - x^2/3 + x^4/30 - x^6/630 + ... and this is the general solution to the **differential equation**.

To use the **power series **method to determine the **general solution **to the equation (1-x^2)y'' - xy' + 4y = 0, we assume that the solution y can be written as a **power series**:

y = a0 + a1x + a2x^2 + ...

Then, we **differentiate **y to obtain:

y' = a1 + 2a2x + 3a3x^2 + ...

And differentiate again to get:

y'' = 2a2 + 6a3x + 12a4x^2 + ...

Substituting these expressions into the original equation and collecting terms with the same powers of x, we get:

[(2)(-1)a0 + 4a2] + [(6)(-1)a1 + 12a3]x + [(12)(-1)a2 + 20a4]x^2 + ... - x[a1 + 4a0 + 16a2 + ...] = 0

Since this equation must hold for all x, we equate the **coefficients **of each power of x to zero:

(2)(-1)a0 + 4a2 = 0

(6)(-1)a1 + 12a3 - a1 - 4a0 = 0

(12)(-1)a2 + 20a4 + 4a2 - 16a0 = 0

...

Solving these equations recursively, we can obtain the coefficients a0, a1, a2, a3, a4, ... and hence obtain the power series solution y.

In this case, we can simplify the recursive equations by using the fact that a1 = (4a0)/(1!), a2 = (6a1 - 12a3)/(2!), a3 = (6a2 - 20a4)/(3!), and so on. Substituting these expressions into the equation for a0 and simplifying, we get:

a0 = 1

Using this as the starting point, we can compute the other coefficients recursively:

a1 = 0

a2 = -1/3

a3 = 0

a4 = 1/30

a5 = 0

a6 = -1/630

...

Thus, the power series solution to the equation (1-x^2)y'' - xy' + 4y = 0 is:

y = a0 + a1x + a2x^2 + a3x^3 + a4x^4 + a5x^5 + a6x^6 + ...

Substituting the values of the coefficients, we obtain:

y = 1 - x^2/3 + x^4/30 - x^6/630 + ...

This is the general solution to the differential equation.

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