A hypothesis test is conducted at the 0.05 level of significance to test whether or not the population correlation is zero. If the sample consists of 25 observations and the correlation coefficient is 0.60, what is the computed value of the test statistic? Round to two decimal places.

See the answers in explanation

Let's solve this problem as follows:

[tex]\bullet \ \frac{2x-8}{x-4} \\ \\ Common \ factor \ 2 \ from \ the \ numerator: \\ \\ \frac{2x-8}{x-4}=\frac{2(x-4)}{x-4} =2[/tex]

[tex]\bullet \ \frac{4x-8}{4x+20} \\ \\ Common \ factor \ 4 \ from \ the \ numerator \ and \ denominator: \\ \\ \frac{4x-8}{4x+20}=\frac{4(x-2)}{4(x+5)}=\frac{(x-2)}{(x+5)}[/tex]

[tex]\bullet \ \frac{x+7}{x^2+4x-21} \\ \\ Rearranging \ denominator: \\ \\ \frac{x+7}{x^2-3x+7x-21}=\frac{x+7}{x(x-3)+7(x-3)}=\frac{x+7}{x(x-3)+7(x-3)} \\ \\ Common \ factor \ x-3 \ from \ denominator: \\ \\ \frac{x+7}{(x-3)(x+7)}=\frac{1}{x-3}[/tex]

[tex]\bullet \ \frac{x^2-3x-10}{x+2} \\ \\ Rearranging \ numerator: \\ \\ \frac{x^2-3x-10}{x+2}=\frac{x^2-5x+2-10}{x+2}=\frac{x(x-5)+2(x-5)}{x+2} \\ \\ Common \ factor \ x-5 \\ \\ \frac{(x-5)(x+2)}{x+2}=x-5[/tex]

[tex]\bullet \ \frac{x^2-4}{2-x} \\ \\ Difference \ of \ squares \ from \ numerator: \\ \\ \frac{(x-2)(x+2)}{2-x} \\ \\ Common \ factor \ -1 \ from \ denominator: \\ \\ \frac{(x-2)(x+2)}{-(x-2)}=-(x+2)[/tex]

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Answer:

  B.  2010.62 ft³

Step-by-step explanation:

The formula for the volume of a cylinder is ...

  V = πr²h . . . . . where r is the radius and h is the height

Filling in the numbers and doing the arithmetic, we get ...

  V = π(8 ft)²(10 ft) = 640π ft³ ≈ 2010.6193 ft³ ≈ 2010.62 ft³

The volume of the cylinder is about 2010.62 ft³.

Answer:

a) The debris was 256 feet  into the air after 2 seconds of the explosion.

b)

[tex]h(t) = -16t(t-10)[/tex]

Step-by-step explanation:

We are given the following in the question:

Initial Velocity =  160 feet per second

[tex]h(t) = 160t-16t^2[/tex]

The above function gives the height in feet and t is seconds after the explosion.

a) Height of the debris 2 second(s) after the explosion.

We put t = 2 in the above function

[tex]h(2) = 160(2)-16(2)^2 = 256[/tex]

Thus, the debris was 256 feet  into the air after 2 seconds of the explosion.

b) Factor the polynomial

[tex]h(t) = 160t-16t^2\\= 16t(10-t)\\=-16t(t-10)[/tex]

Final answer:

The height of the debris 2 seconds after the explosion is 256 feet. The given polynomial can be factored completely as -16t(t - 10)

Explanation:

To find the height of the debris 2 seconds after the explosion, we can substitute t = 2 into the equation h(t) = 160t - 16t^2.

So, h(2) = 160(2) - 16(2)^2 = 320 - 16(4) = 320 - 64 = 256 feet.

Therefore, the height of the debris 2 seconds after the explosion is 256 feet.

To factor the given polynomial completely, we can rewrite it as:

h(t) = -16t^2 + 160t.

Now, we can factor out a common factor of -16t:

h(t) = -16t(t - 10).

This gives us the completely factored form of the polynomial.

Answer:

z= -0.968

We can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that  they would watch one of the television shows not differs from 0.5 or 50% .  

Step-by-step explanation:

1) Data given and notation n  

n=106 represent the random sample taken

X=48 represent the people who says that  they would watch one of the television shows.

[tex]\hat p=\frac{48}{106}=0.453[/tex] estimated proportion of people who says that  they would watch one of the television shows.

[tex]p_o=0.5[/tex] is the value that we want to test

[tex]\alpha[/tex] represent the significance level  

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that 50% of people who says that  they would watch one of the television shows.:  

Null hypothesis:[tex]p=0.5[/tex]  

Alternative hypothesis:[tex]p \neq 0.5[/tex]  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.453 -0.5}{\sqrt{\frac{0.5(1-0.5)}{106}}}=-0.968[/tex]  

4) Statistical decision  

P value method or p value approach . "This method consists on determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level is not provided, but we can assume [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z<-0.968)=0.333[/tex]  

So based on the p value obtained and using the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we fail to reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that  they would watch one of the television shows not differs from 0.5 or 50% .  

Answer: The weight of tallest person would be 607.16 pounds.

Step-by-step explanation:

Since we have given that

height of the tallest person = 8 feet 11 inches = 107 inches

If all men had identical body​ types, their weight would vary directly as the cube of their height.

If a man who is 5 feet 10 inches tall ​(70 ​inches) with the same body type as the tallest person weighs 170 ​pounds,

So, it becomes,

[tex]W=xh^3[/tex]

[tex]170=x(70)^3\\\\\dfrac{170}{70^3}=x\\\\x=\dfrac{170}{343000}[/tex]

So, weight of tallest person would be

[tex]W=\dfrac{170}{343000}\times 107^3\\\\W=607.16\ pounds[/tex]

Hence, the weight of tallest person would be 607.16 pounds.

First, let's consider the information we're given and set up an equation. We know that, given identical body types, a person's weight would vary directly as the cube of their height. What this means is that the ratio between the weight of two people and the cube of their respective heights will be the same regardless of their individual heights or weights. We can represent this as:

W_tall / W_normal = (Height_tall ^ 3) / (Height_normal ^ 3)

Where:
W_tall is the weight of the tallest person.
W_normal is the weight of the normal person (which we know is 170 pounds).
Height_tall is the height of the tallest person (which we know is 107 inches).
Height_normal is the height of the standard person (which we know is 70 inches).

We are looking to find the weight of the tallest person, so, to isolate W_tall in the equation above, we will multiply both sides of the equation by W_normal:

W_tall = W_normal * (Height_tall ^ 3) / (Height_normal ^ 3)

Substituting the known values into the equation, we get:

W_tall = 170 * (107 ^ 3) / (70 ^ 3)

Computing the values on the right-hand side of the equation, we find that the tallest person weighed approximately 607.16 pounds just before his death.

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the correct answer is C. The interval estimate is above 70%, so infer that it will be supported.The 95% confidence interval (0.75, 0.85) indicates that we can expect around 75% to 85% of students to support the fee increase, which is more than the presumed 70%. So, we can infer that the fee increase will be supported.

In the scenario given, the 95% confidence interval for the proportion of students supporting the fee increase is [0.75, 0.85].

This means that we are 95% confident that the true proportion of students who support the fee increase falls within this interval. Since the entire range is above 70%, we can be pretty confident that more than 70% of the student population supports the fee increase.

The confidence interval is a statistical method that gives us a range in which the true parameter likely falls based on our sample data. This shows us that statistical inference can give us insights about a population based on a sample.

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Answer:

the answer is 3

7x + 9= 30

7x = 21

x = 3

Answer:

x = 3

Step-by-step explanation:

Collect like terms;

7x + 9 = 30

Subtract 9 from both sides;

7x = 21

Divide both sides by 7;

x = 3

Nonresponse can cause bias in survey results due to differences in characteristics, the representation of strata, and the smaller sample size.

Nonresponse can cause the results of a survey to be biased in several ways:

Overall, nonresponse can introduce bias into a survey by excluding certain individuals or groups from the sample, leading to potentially inaccurate and biased results.

Height of the pole (DC) is 57.2709m

Step-by-step explanation:

Here, Wire DA and Wire DB supports a pole.

Given that Angle, A=54 , B= 72.

Also, AB = 23m

Now, Taking triangle BCD and Using basic trigonometry

Height of pole H = DC

[tex]TanB = \frac{DC}{BC}[/tex]

[tex]BC= \frac{DC}{TanB}[/tex]

Now, Taking triangle ACD and Using basic trigonometry

[tex]TanA = \frac{DC}{AC}[/tex]

[tex]AC= \frac{DC}{TanA}[/tex]

From figure, we know that

AC = AB + BC

AC - BC = AB = 23

Replacing values of AC and BC

[tex]\frac{DC}{TanA} - \frac{DC}{TanB}=23\\

DC(\frac{1}{TanA}-\frac{1}{TanB})=23[/tex]

Now, TanB= Tan72 =3.0776 and TanA = Tan54=1.3763

[tex]DC (\frac{1}{1.3763} - \frac{1}{3.0776})_= 23[/tex]

[tex]DC ( 0.7265-0.3249)= 23[/tex]

[tex]DC ( 0.4016 )= 23[/tex]

[tex]DC = 57.2709 [/tex]

Thus, Height of thepole is 57.2709m

Answer:

B) 2 hours

Step-by-step explanation:

If machine  A complete a job in 3 1/2 hours or 7/2 of an hour

means that in one hour finished 1÷ 7/2     or  2/7

If machine  B complete a job in 4 2/3 hours or  14/3 of an hour

means that in one hour finished 1÷ 14/3    or  3/14 of an hour

Then the two machines working together in one hour will make

2/7 + 3/14    =  (4 + 3)/ 14

or   7/14   = 1/2

half of the job. Therefore these two machines working together will take two hours

Answer:

4(3x-2)

Step-by-step explanation

6x+12/2x-8

6x+6x-8

12x-8

4(3x-2)

Answer:

Step-by-step explanation:

2x-8 )  6x+12 ( 3

           6x-24

         -     +

         -----------

                 36

quotient=3

remainder=36

Answer:

The required probability is [tex]\frac{3}{4}[/tex].

Step-by-step explanation:

Consider the provided information.

There are 8 leaders.

Thus, the total number of ways to arrange 8 leaders are 8!.

Assume that Obama and Berlusconi is one person.

Therefore the total number of leaders are 7 (As Obama and Berlusconi is one person).

The number of ways in which 7 leader can be arranged: 7!

Although Obama and Berlusconi is one unit but they can interchange their place in 2 ways. Like Obama and Berlusconi or Berlusconi and Obama

That means the total number of ways : 7!×2

The number of ways in which they are not next to each other = Total number of ways - The number of ways in which they are next to each other

Number of ways they are not next to each other = 8!-7!×2

The probability that they are not next to each other = [tex]\frac{8!-7!\times2}{8!}=\frac{3}{4}[/tex]

Hence, the required probability is [tex]\frac{3}{4}[/tex].

You have 31 nickels and 44 dimes.

Step-by-step explanation:

Total coins = 75

Worth of coins = $5.95 = 5.95*100 = 595 cents

1 nickel = 5 cents

1 dime = 10 cents

Let,

Number of nickels = x

Number of dimes = y

According to given statement;

x+y=75   Eqn 1

5x+10y=595    Eqn 2

Multiplying Eqn 1 by 5

[tex]5(x+y=75)\\5x+5y=375\ \ \ Eqn\ 3\\[/tex]

Subtracting Eqn 3 from Eqn 2

[tex](5x+10y)-(5x+5y)=595-375\\5x+10y-5x-5y=220\\5y=220[/tex]

Dividing both sides by 5

[tex]\frac{5y}{5}=\frac{220}{5}\\y=44[/tex]

Putting y=44 in Eqn 1

[tex]x+44=75\\x=75-44\\x=31[/tex]

You have 31 nickels and 44 dimes.

Keywords: linear equations, subtraction

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Answer:

Step-by-step explanation:

Answer:

  0.91

Step-by-step explanation:

The total of all numbers in the diagram is 35 +5 +10 +5 = 55.

The total of the numbers inside one or both circles is 35 +5 +10 = 50.

The probability of choosing a random student from inside one or both circles (plays some instrument) is 50 out of 55, or ...

  50/55 ≈ 0.9090909... ≈ 0.91

P(A∪B) ≈ 0.91

Answer:

We conclude that  the mean transaction time exceeds 60 seconds.

Step-by-step explanation:

We are given the following in the question:  

Population mean, μ = 60 seconds

Sample mean, [tex]\bar{x}[/tex] = 67 seconds

Sample size, n = 16

Alpha, α = 0.05

Sample standard deviation, s = 12 seconds

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 60\text{ seconds}\\H_A: \mu > 60\text{ seconds}[/tex]

We use One-tailed(right) t test to perform this hypothesis.

Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex] Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{67 - 60}{\frac{12}{\sqrt{16}} } = 2.34[/tex]

Now, [tex]t_{critical} \text{ at 0.05 level of significance, 15 degree of freedom } = 1.753[/tex]

Since,                  

[tex]t_{stat} > t_{critical}[/tex]

We fail to accept the null hypothesis and reject it. Thus, we conclude that  the mean transaction time exceeds 60 seconds.

Final answer:

The hypothesis test for the mean ATM transaction time given a sample mean of 67 seconds and a standard deviation of 12 seconds involves a one-tailed Z-test, where the null hypothesis H0: μ = 60 is rejected in favor of the alternative hypothesis Ha: μ > 60 since the test statistic of 2.33 exceeds the critical value of 1.645 at an α = .05 significance level.

Explanation:

Conducting a Hypothesis Test for Mean Transaction Time

For the sample of 16 ATM transactions with a mean of 67 seconds and a standard deviation of 12 seconds, the goal is to test the hypothesis that the mean transaction time exceeds 60 seconds.

a. Determine your hypotheses

The null hypothesis (H0) is that the mean transaction time is 60 seconds (H0: μ = 60). The alternative hypothesis (Ha) is that the mean transaction time is greater than 60 seconds (Ha: μ > 60).

b. Compute the test statistic

To compute the test statistic, we use the following formula for a sample mean with a known standard deviation: Z = (Xbar - μ0) / (s / sqrt(n)) where Xbar is the sample mean, μ0 is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Plugging in the values we get: Z = (67 - 60) / (12 / sqrt(16)) = 7 / (12 / 4) = 2.33.

c. Rejection rule

The rejection rule is if the computed test statistic is greater than the critical value at α = .05 significance level, we reject H0.

d. Critical Value at α = .05

The critical value for a one-tailed Z-test at α = .05 is approximately 1.645.

e. Conclusion

Because our test statistic of 2.33 exceeds the critical value of 1.645, we reject the null hypothesis, concluding that there is sufficient evidence at the 0.05 significance level to suggest that the mean transaction time exceeds 60 seconds.

Answer:

  c. uses the average of the most recent data values in the time series as the forecast for the next period.

Step-by-step explanation:

We assume you want to complete the description of moving averages method.

The moving in moving averages refers to the fact that the data points used to compute the average are some number of most recent data points. As data is accumulated, the data used to compute the average "moves" to include the newest data and exclude the oldest data.

Answer:

Step-by-step explanation:

Set this up according to the Triangle Proportionality Theorem:

[tex]\frac{3x}{4x}=\frac{3x+7}{5x-8}[/tex]

Cross multiply to get

[tex]3x(5x-8)=4x(3x+7)[/tex]

and simplify to get

[tex]15x^2-24x=12x^2+28x[/tex]

Get everything on one side of the equals sign and solve for x:

[tex]3x^2-52x=0[/tex] and

[tex]x(3x-52)=0[/tex]

By the Zero Product Property,

x = 0 or 3x - 52 = 0 so x = 17 1/3

Answer:

atleast 52

Step-by-step explanation:

Given that an employment agency requires applicants average at least 70% on a battery of four job skills tests.

An applicant scored 70%, 77%, and 81% on the first three exams,

Since weightages are not given we can assume all exams have equal weights

Let x be the score on the 4th test

Then total of all 4 exams = [tex]70+77+81+x\\= 228+x[/tex]

Average should exceed 70%

i.e.[tex]\bar X \geq 70\\Total\geq 70(4) =280[/tex]

Comparing the two totals we have

[tex]228+x\geq 280\\x\geq 280-228 = 52[/tex]

Hemust  score on the fourth test a score atleast 52 to maintain a 70% or better average.

Answer:

f(g(x)) = g(f(x))  = x and f and g are the inverses of each other.

Step-by-step explanation:

Here, the given functions are:

[tex]f(x) = x^2 - 3, g(x) = \sqrt{({3+x)} }[/tex]

To Show:  f (g(x))  = g (f (x))

(1)  f (g(x))

Here, by the composite function:

[tex]f (g(x)) = f (\sqrt{3+x} )  = \sqrt{(3+x)} ^2 - 3  =  (3 + x) - 3  =  x[/tex]

⇒ f (g(x))  = x

(2) g (f(x))

Here, by the composite function:

[tex]g(f(x)) = g(x^2 -3)   = \sqrt{3 +(x^2 -3) }  = \sqrt{x^2}   = x[/tex]

⇒ g (f(x))  = x

Hence, f(g(x)) = g(f(x))  = x

⇒ f and g are the inverses of each other.

Answer:

$22,508

Step-by-step explanation:

Edmentum

The total cost that she pays for car will be $22,508.

Simple interest is the concept that is used in many companies such as banking, finance, automobile, and so on.

A = P + (PRT)/100

Where P is the principal, R is the rate of interest, and T is the time.

Mia as of late purchased a vehicle worth $20,000 borrowed with a financing cost of 6.6%. She made an initial investment of $1,000 and needs to reimburse the credit in the span of two years (two years).

Then the total cost that she pays will be calculated as,

A = $20,000 + ($19,000 x 6.6 x 2) / 100

A = $20,000 + $2,508

A = $22,508

The total cost that she pays will be $22,508.

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Answer:

a) 0.394

b) 0.430

c) 0.981

Step-by-step explanation:

Use binomial probability:

P = nCr pʳ (1−p)ⁿ⁻ʳ

where n is the number of trials,

r is the number of successes,

and p is the probability of success.

Here, n = 3 and p = 4/15.

r is the number of defective tablets.

a) If r = 0:

P = ₃C₀ (4/15)⁰ (1−4/15)³⁻⁰

P = 1 (1) (11/15)³

P = 0.394

b) If r = 1:

P = ₃C₁ (4/15)¹ (1−4/15)³⁻¹

P = 3 (4/15) (11/15)²

P = 0.430

c) If r ≠ 3:

P = 1 − ₃C₃ (4/15)³ (1−4/15)³⁻³

P = 1 − 1 (4/15)³ (1)

P = 0.981

To find the probability of selecting no defective tablets, multiply the probabilities of selecting non-defective tablets.

a) To find the probability of selecting no defective tablets, we need to find the probability of selecting 3 non-defective tablets. There are 11 non-defective tablets out of the total of 15 tablets. So, the probability is:

Multiplying these probabilities together:

(11/15) * (10/14) * (9/13) = 990/2730 ≈ 0.362 = 36.2%

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Answer:

Step-by-step explanation:

The quadrilateral has 4 sides and only two of them are equal.

A) to find PR, we will consider the triangle, PRQ.

Using cosine rule

a^2 = b^2 + c^2 - 2abcos A

We are looking for PR

PR^2 = 8^2 + 7^2 - 2 ×8 × 7Cos70

PR^2 = 64 + 49 - 112 × 0.3420

PR^2 = 113 - 38.304 = 74.696

PR = √74.696 = 8.64

B) to find the perimeter of PQRS, we will consider the triangle, RSP. It is an isosceles triangle. Therefore, two sides and two base angles are equal. To determine the length of SP,

We will use the sine rule because only one side,PR is known

For sine rule,

a/sinA = b/sinB

SP/ sin 35 = 8.64/sin110

Cross multiplying

SPsin110 = 8.64sin35

SP = 8.64sin35/sin110

SP = (8.64 × 0.5736)/0.9397

SP = 5.27

SR = SP = 5.27

The perimeter of the quadrilateral PQRS is the sum of the sides. The perimeter = 8 + 7 + 5.27 + 5.27 = 25.54 cm

deltax = (56-0)/4 = 14

number of intervals 4

The intervals are:

(0,14),(14,28),(28,42),(42,56)

The Midpoint Rule =

f(7)+f(21)+f(35)+f(49)

[0.47577]+[-0.99159]+[-0.35892]+[0.65699]

deltax =14

sum = -0.21774

Multiplying by deltax = -3.0484

To approximate the integral of sin(x^2) dx from 0 to 5 using the Midpoint Rule with n=5, first calculate Δx = 1. Next, perform calculations with x values 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, use the Midpoint Rule formula to calculate the approximated integral.

To solve this problem, you'll need to use the Midpoint Rule, which is a numerical method used to approximate definite integrals. In this case, we want to approximate ∫ from 0 to 5 of sin(x^2) dx with an n value of 5.

The Midpoint Rule can be represented as: Mn = Δx[f(x1) + f(x2) + ... + f(xn)], where Δx = (b-a)/n and each xi = a + (Δx/2) + (i-1)Δx.

Here, we'll first find our Δx = (5-0)/5 = 1, then calculate each xi and plug those into our function. Our xi values will be 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, we plug into our formula and solve to find our approximated integral value.

A detailed and step-by-step solution would involve calculating each f(xi) with the xi values given above, and then adding these up

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To find the difference between the number of tetras and minnows, set up a system of equations using the given ratios. Solve the system to find the number of tetras, guppies, and minnows. Finally, subtract the number of minnows from the number of tetras to find the difference.

To determine the difference between the number of tetras and minnows in the fish tank, we need to first find the number of each type of fish. We can do this by setting up a system of equations using the given ratios. Let T represent the number of tetras, G represent the number of guppies, and M represent the number of minnows.

From the first ratio, we have T/G = 4/2. Simplifying this equation, we get T = 2G.

From the second ratio, we have M/G = 1/3. Simplifying this equation, we get M = (1/3)G.

Since we know there are a total of 60 fish in the tank, we can create the equation T + G + M = 60. Substituting the previous equations into this equation, we get 2G + G + (1/3)G = 60. Solving for G, we find G = 9. Plugging this value into the equations for T and M, we get T = 2(9) = 18 and M = (1/3)(9) = 3.

Therefore, there are 18 tetras and 3 minnows in the fish tank. The difference between the number of tetras and minnows is 18 - 3 = 15.

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Answer:

c. 4.27

Step-by-step explanation:

We have been given the regression equation of for predicting number of speeding tickets (Y) from information about driver age (X) as [tex]Y=-0.065(X)+5.57[/tex].

To find the predicted number of tickets for a twenty-year-old, we will substitute [tex]X=20[/tex] in our given equation.

[tex]Y=-0.065(20)+5.57[/tex]

[tex]Y=-1.3+5.57[/tex]

[tex]Y=4.27[/tex]

Therefore, the predicted number of tickets for a twenty-year-old would be 4.27 tickets.

Priya practiced the violin for 114 minutes

Given that  

Han spent 75 minutes practicing the piano over the weekend

Priya practiced the violin for 152% as much as Han practiced the piano

Need to determine how long did priya practice  

Duration of Han practicing the piano over weekend = 75 minutes

As given that Priya practiced the violin for 152% as much as Han practiced the piano

=> Duration of Priya practicing the piano = 152% of Han practicing the piano

=> Duration of Priya practicing the piano = 152% of 75 minutes

We know that a % of b is written in fraction as [tex]\frac{a}{100} \times b[/tex]

[tex]\Rightarrow \text { Duration of Priya practicing the piano }=\frac{152}{100} \times 75=114 \text { minutes }[/tex]

Hence Priya practiced the violin for 114 minutes

Answer:

At x = 1 and maximum area  = 0.0499

Step-by-step explanation:

The hypotenuse of a right triangle has one end at the origin and other end on the curve, [tex]y=x^2e^{-3x}[/tex]  with x ≥ 0.

One leg of right triangle is x-axis and another leg parallel to y-axis.

Length of base of right triangle =  x

Height of right triangle = y

Area of right triangle, [tex]A=\dfrac{1}{2}xy[/tex]

[tex]A=\dfrac{1}{2}x^3e^{-3x}[/tex]

For maximum/minimum value of area.

[tex]\dfrac{dA}{dx}=\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}[/tex]

Now, find critical point, [tex]\dfrac{dA}{dx}=0[/tex]

[tex]\dfrac{3}{2}x^2e^{-3x}-\dfrac{3}{2}x^3e^{-3x}=0[/tex]

[tex]\dfrac{3}{2}x^2e^{-3x}(1-x)=0[/tex]

x =0,1

For x = 0, y = 0

For x = 1, [tex]y=e^{-3}[/tex]

using double derivative test:-

[tex]\dfrac{d^2A}{dx^2}=\dfrac{6}{2}xe^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^2e^{-3x}-\dfrac{9}{2}x^3e^{-3x}[/tex]

At x= 0 , [tex]\dfrac{d^2A}{dx^2}=0[/tex]

Neither maximum nor minimum

At x = 1, [tex]\dfrac{d^2A}{dx^2}=-0.14<0[/tex]

Maximum area at x = 1

The maximum area of right triangle at x = 1

Maximum area, [tex]A=\dfrac{1}{e^3}\approx 0.0499[/tex]

The point of maxima will be x=3 and the maximum area will be 0.002 square units.

According to the diagram attached

The area of the given triangle will be = 0.5*base*height

As one end of the hypotenuse is on the curve [tex]y = x^2e^(-3x)[/tex], Coordinates of one end of the hypotenuse will be [tex](x, x^2e^(-3x)[/tex].

Area A(x) of the given triangle = 0.5*base*  height

Base =  x

Height = [tex]x^2e^(-3x)[/tex]

So A(x) = [tex]0.5*x*x^{2} *e^(-3x)[/tex]

[tex]A(x) = 0.5*x*x^{2} *e^(-3x)\\\\A(x) = 0.5 x^3e^(-3x)[/tex]

For the maximum area,

[tex]A'(x) = 0\\\\x^2e^(-3x) (x-3) = 0\\x = 0 and x=3[/tex]will be the points of extremum.

Points of extremum are the values of x for which a function f(x) attains a maximum or minimum value.

A(0) = 0

A(3) = 0.002

Therefore, The point of maxima will be x=3, and the maximum area will be 0.002 square units.

To get more about maxima and minima visit:

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Jacob's survey is a study in statistics, specifically looking at the correlation between the amount of time spent on sports and student's mood ratings. A line is fit to the data to determine the relationship, with the direction of the line offering insights into how these two variables correlate, but this does not imply causation.

From your question, Jacob performed a survey asking about the number of hours students spent playing sports in the past day and asked them to rate their mood. It's a study of basic statistics, specifically focusing on correlation between two variables, here those are the number of hours spent on sports and mood ratings. To determine a relationship between these variables, a line is often fit to the data, using methods like linear regression.

For example, if the line on the graph is rising, it indicates a positive correlation between the amount of sports played and a student's mood, meaning that as sports playtime goes up, so does mood ratings. A falling line means there's a negative correlation. If there's no clear direction, it's likely that there's no significant correlation between the two variables. But remember that correlation doesn't mean causation: just because two things correlate doesn't mean that one causes the other.

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To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for z-score. By calculating the z-scores for both values, we can find the areas under the curve and subtract them to get the probability.

To find the probability that a randomly selected utility bill is between $110 and $130, we can use the formula for the z-score:

z = (x - μ) / σ

where x is the value we are looking for, μ is the mean, and σ is the standard deviation.

In this case, x = $110, μ = $121, and σ = $23.

Substituting these values into the formula, we get:

z = (110 - 121) / 23 = -0.4783

Using a z-table or a calculator, we can find that the area to the left of -0.4783 is approximately 0.3186.

Next, we repeat the process for $130:

z = (130 - 121) / 23 = 0.3913

Using a z-table or a calculator, we can find that the area to the left of 0.3913 is approximately 0.6480.

To find the probability that the utility bill is between $110 and $130, we subtract the area to the left of $110 from the area to the left of $130:

P(110 ≤ X ≤ 130) = 0.6480 - 0.3186 = 0.3294

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