Example #2
You want to compare the average number of months looking for jobs after graduation in your sample of GMU students to a sample of students from University of Alaska.
Information on samples:
xgmu = 3.6 xua = 2.7 sgmu = 2.1 sua = 2.3 ngmu = 100 nua = 100
1. State Hypotheses (1 point each)
H0:
Ha:
2. Choose alpha = .05
3. Find Critical t.
2 sample, 2 tailed t test (1 point each blank)
df = ngmu + nua - 2 = ________
t* = _______
4. Calculate tobt: (3 points)
Step 5. Compare Obtained t to Critical t (2 points)
___________________ the null hypothesis and conclude that ___________________________________
________________________________________________________________________________.
Review:
Z test: know population standard deviation and are comparing a sample mean to a known value.
T test (1 sample): do NOT have population standard dev. and are comparing a sample mean to a known value.
T test (2 sample): comparing two sample means.

(a) The box can be represented as a population of 200,000 registered voters, with 120,000 of them supporting the ballot proposition and 80,000 opposing it. Each voter can be represented by a ticket, and the box would contain 120,000 tickets labeled "Support" and 80,000 tickets labeled "Oppose."

(b) If we drew 200 times with replacement from this box, there is a 95% chance that the sum of draws (number of voters contacted who support the ballot proposition) would be between 106 and 134. Expressed as a percentage of the 200 people, this is between 53% and 67%.

(a) The box can be represented as a collection of 200,000 balls, where each ball corresponds to a registered voter. Among these, 120,000 balls are labeled as "support" to indicate that the voter supports the ballot proposition, and the remaining 80,000 balls are labeled as "oppose" to indicate that the voter does not support the proposition. When we randomly select 200 voters without replacement and count the number of supporters among them, we are essentially drawing 200 balls from this box without replacement and counting the number of balls labeled as "support".

(b) Since we are drawing with replacement, each draw is independent and has a Bernoulli distribution with a probability of success p = 0.6 (since 60% of the voters support the proposition). The sum of these draws has a binomial distribution with parameters n = 200 (the number of trials) and p = 0.6 (the probability of success). The mean of this distribution is μ = np = 200 x 0.6 = 120, and the standard deviation is σ = sqrt(np(1-p)) = sqrt(200 x 0.6 x 0.4) ≈ 7.75.

To find the range of values within which the sum of draws is likely to fall with 95% confidence, we can use the normal approximation to the binomial distribution, which is appropriate when np > 10 and n(1-p) > 10. In this case, we have np = 120 and n(1-p) = 80, so the normal approximation is valid.

We want to find the values of k such that P(μ - kσ < X < μ + kσ) = 0.95, where X is the sum of draws. Using the standard normal distribution, we have:

P(-k < Z < k) = 0.95,

where Z = (X - μ)/σ is a standard normal variable. From standard normal tables, we find that k ≈ 1.96.

Therefore, the 95% confidence interval for the sum of draws is:

120 - 1.96 x 7.75 ≈ 104.1 to 120 + 1.96 x 7.75 ≈ 135.9.

As a percentage of the 200 people polled, this is between:

104.1/200 x 100 ≈ 52.05% and 135.9/200 x 100 ≈ 67.95%.

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If two samples from the same population have the same mean and are both n = 100, they will probably still have different sample variances.

We have,

If two samples from the same population have the same mean and are both n=100, they will probably still have different sample variances.

This is because sample variance refers to the dispersion or spread of data points within each individual sample, and this can vary even if the samples have the same mean and sample size (n=100).

It's important to note that the other options (t-statistics, standard errors, and alpha values) are related to the sampling distribution or hypothesis testing, which may not be different simply due to having the same mean and sample size.

Thus,

If two samples from the same population have the same mean and are both n = 100, they will probably still have different sample variances.

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The required, monthly payment on the loan would be $84.00.

To calculate the simple interest on a loan, we use the formula:

Simple Interest = Principal * Interest Rate * Time

In this case, the Principal is $1800, the Interest Rate is 6% (or 0.06 as a decimal), and the Time is 2 years. So, the simple interest on the loan would be:

Simple Interest = $1800 * 0.06 * 2 = $216

To calculate the monthly payments, we need to add the interest to the principal and divide by the number of months in the loan term. Since the loan term is 2 years or 24 months, the monthly payment would be:

Monthly Payment = (Principal + Simple Interest) / Number of Months

= ($1800 + $216) / 24

= $84.00

Therefore, the monthly payment on the loan would be $84.00.

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One property of the inverse of a Hilbert Matrix is that its entries grow very large as the size of the matrix increases. This can lead to numerical instability when computing the inverse, as the values become too large for the computer to handle.

The condition number of a matrix is a measure of its sensitivity to numerical errors during computation. It is defined as the ratio of the largest and smallest singular values of the matrix. A high condition number indicates that the matrix is ill-conditioned, meaning small perturbations in the input can result in large changes in the output. It is important to compute the condition number of a matrix when solving numerical problems, such as linear systems of equations or matrix inversions, to ensure the accuracy and stability of the solution.
Let's calculate the condition number of two 3x3 matrices:
Matrix A = [1 2 3; 4 5 6; 7 8 9]
Matrix B = [1 0 0; 0 1 0; 0 0 1]
Using MATLAB, we can compute the condition number of each matrix:
cond(A) = 2.96e+16
cond(B) = 1
We can conclude that Matrix A is ill-conditioned, while Matrix B is well-conditioned.
To create a Hilbert Matrix in MATLAB, we can use the hilb function. Let's create a 4x4 Hilbert Matrix and find its condition number:
H = hilb(4)
cond(H) = 15513.7387
We can see that the Hilbert Matrix is highly ill-conditioned.
One property of the inverse of a Hilbert Matrix is that its entries grow very large as the size of the matrix increases. This can lead to numerical instability when computing the inverse, as the values become too large for the computer to handle. In fact, for large n, the entries of the inverse approach infinity, making it effectively impossible to compute accurately.

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The As Fed weight of the bale is 733.33 lbs and in  1 ton (2000 lbs) of that hay there would be  1740 lbs of DM.

You have a bale of hay that is 87% DM, and there is 638 lbs of DM in that bale. To find the As Fed weight of the bale, you can use the following formula:

As Fed weight = (DM weight) / (% DM)

Step 1: Convert the percentage to a decimal:
87% DM = 0.87

Step 2: Plug the values into the formula:
As Fed weight = (638 lbs) / (0.87)

Step 3: Calculate the As Fed weight:
As Fed weight ≈ 733.33 lbs

So, the As Fed weight of the bale is approximately 733.33 lbs.

Now, to find how many pounds of DM would be in 1 ton (2000 lbs) of that hay, you can use the following formula:

DM in 1 ton = (2000 lbs) * (% DM)

Step 1: We already have the percentage as a decimal (0.87).

Step 2: Plug the values into the formula:
DM in 1 ton = (2000 lbs) * (0.87)

Step 3: Calculate the DM in 1 ton:
DM in 1 ton = 1740 lbs

Therefore, there would be 1740 lbs of DM in 1 ton (2000 lbs) of that hay.

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The probability of getting a 6 on a loaded die is not provided, so I cannot give an exact answer. However, if we assume that the probability of getting a 6 on each roll is p, then the probability of getting exactly seven 6's in ten rolls can be calculated using the binomial distribution.

The formula for the binomial distribution is:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where P(X = k) is the probability of getting exactly k successes in n trials, p is the probability of success on each trial, and (n choose k) is the number of ways to choose k successes from n trials.

In this case, we want to find the probability of getting exactly seven 6's in ten rolls, so n = 10 and k = 7. We don't know p, so we can't calculate the exact probability, but we can use a range of values for p to see how it affects the probability.

For example, if we assume that p = 0.5 (i.e. the loaded die has an equal chance of rolling 6 and any other number), then the probability of getting exactly seven 6's in ten rolls is:

P(X = 7) = (10 choose 7) * 0.5^7 * 0.5^3
          = 0.117

So there is about an 11.7% chance of getting exactly seven 6's in ten rolls if the die has an equal chance of rolling 6 and any other number. If the probability of rolling a 6 is higher or lower than 0.5, the probability will be different.

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The location of S" after translating and rotating the rectangle is (-3, -4).

To find the location of S" after translating and rotating the rectangle, we need to follow the two steps in order:

Translation: Apply the rule (x, y) → (x − 2, y − 4) to each vertex of the rectangle to get its new location. The new vertices are:

S'(-4, -3), T'(-4, -1), U'(-2, -1), V'(-2, -3)

Rotation: Rotate the translated rectangle 90° counterclockwise. This means that each vertex will swap its x and y coordinates and the new x-coordinate will be negated. The new vertices after rotation are:

S"(-3, -4), T"(-1, -4), U"( -1, -2), V"(-3, -2)

Therefore, the location of S" is (-3, -4).

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Answer: the point is (5,4)

Part(a),

The average rate of change in annual sales between 2001 and 2002 was $14$ million per year.

Part(b),

The average rate of change in annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

a) To find the rate of change between 2001 and 2002, we need to calculate the difference in sales between those two years and divide it by the number of years:

Rate of change = (Sales in 2002 - Sales in 2001) / (2002 - 2001)

Rate of change = (255 - 241) / 1 = 14 million/year

Therefore, the average rate of change of annual sales between 2001 and 2002 was $14$ million per year.

b) To find the rate of change between 2001 and 2004, we need to calculate the difference in sales between those two years and divide it by the number of years:

Rate of change = (Sales in 2004 - Sales in 2001) / (2004 - 2001)

Rate of change = (231 - 241) / 3 = -3.33 million/year

Therefore, the average rate of change of annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

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The graph of the function is given above.

The graph of the function f(x) = x + 3x + 2 is a parabola that opens upwards.

We have,

The graph of the function f(x) = x + 3x + 2 is a parabola.

The coefficient of x² is positive, so the parabola opens upwards.

To sketch the graph of the function, we can use the vertex formula.

The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients of x^2 and x, respectively.

In this case, a = 1 and b = 3, so the x-coordinate of the vertex is -3/2.

To find the y-coordinate of the vertex, we can substitute this value of x into the function to get:

f(-3/2) = (-3/2)^2 + 3(-3/2) + 2 = 1/4 - 9/2 + 2 = -15/4

So the vertex is at (-3/2, -15/4).

We can also find the y-intercept by setting x = 0:

f(0) = 0² + 3(0) + 2 = 2

So the y-intercept is at (0, 2).

Thus,

The graph of the function is given below.

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15 milliliters of antibiotic must be drawn out of the vial for a 3-gram dose.

To determine how many milliliters of the antibiotic must be drawn out of the vial for a 3-gram dose, follow these steps:

1. Identify the total amount of antibiotic in the vial (4 grams) and the volume after dilution (20 milliliters of sterile water).
2. Calculate the concentration of the antibiotic solution after dilution: 4 grams / 20 milliliters = 0.2 grams/mL.
3. Determine the required dose of the antibiotic (3 grams) and divide it by the concentration to find the volume needed: 3 grams / 0.2 grams/mL = 15 milliliters.

So, you will need to draw out 15 milliliters of the diluted antibiotic solution from the vial to administer the 3-gram dose intravenously to the patient over 1 hour.

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The area of the region between the curves y = x^2 and y = x^3 is 1/12 sq units. The correct answer is option a.

To find the area of the region between the curves y = x^2 and y = x^3, we need to integrate the difference between the two curves with respect to x from the point where they intersect.

First, we need to find where the curves intersect by setting them equal to each other:
x^2 = x^3
x^3 - x^2 = 0
x^2(x-1) = 0
x = 0 or x = 1

So the curves intersect at x = 0 and x = 1.

Now, we can set up the integral to find the area between the curves:
A = ∫[0,1] (x^3 - x^2) dx

Evaluating the integral, we get:
A = [x^4/4 - x^3/3] from 0 to 1
A = (1/4 - 1/3) - (0 - 0)
A = 1/12 sq units

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The measures of the following angles are; Angle 1 =44

Given that HC is the diameter of the circle, then:

HA = 83°, BC= 50°, HD = 135°, GF = 32°, HG = 45°, and FE = 55°

From the given figure, HC can be expressed as:

HC = HA + BC + AB

Substituting with HC = 180°, HA = 83°, and BC = 50°, and solving for AB:

180 = 83 + AB + 50

AB = 47

The relation between the angle outside the circle, ∠1, and the intersected arcs AB and HD are:

Angle = 1/2(HD - AB)

Substituting with HD = 135°, and AB = 47°:

Angle 1 = 1/2(135 - 47)

Angle 1 =44

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The greatest common divisor of the sequence 16 +10n-1, n = 1,2,... is 5.

To find the greatest common divisor of the sequence 16 +10n-1, n = 1,2,..., we can start by finding the values of the sequence for the first few terms:

When n = 1, the sequence is 16 + 10(1) - 1 = 25
When n = 2, the sequence is 16 + 10(2) - 1 = 35
When n = 3, the sequence is 16 + 10(3) - 1 = 45

We can see that all the terms in the sequence are odd numbers. This means that the greatest common divisor of the sequence must be an odd number.

To find the greatest common divisor, we can use the Euclidean algorithm. Let's start by finding the greatest common divisor of the first two terms:

gcd(25, 35) = gcd(25, 35 - 25) = gcd(25, 10) = gcd(5 x 5, 2 x 5) = 5

Now, let's find the greatest common divisor of the third term and the greatest common divisor of the first two terms:

gcd(45, 5) = gcd(5 x 9, 5) = 5

Therefore, the greatest common divisor of the sequence 16 +10n-1, n = 1,2,... is 5.

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The diameter of the headlight's opening is 4 inches.

To find the diameter of the headlight's opening, we need to determine the distance between the two points on the paraboloid where the depth is 2 inches.

Using the formula for the paraboloid of revolution, we know that the equation for the shape of the headlight is:
[tex]y^2 = 4px[/tex]

where p is the distance from the vertex to the focus. In this case, p = 1 inch.

To find the distance between two points on the paraboloid where the depth is 2 inches, we can set y = ±2 and solve for x:

[tex]4p^2 = 4x(\pm2)\\x = p^2/\pm1[/tex]

Since we're interested in the distance between two points, we can subtract the two x-values:

[tex]x^2 - x^1 = p^{2/1} - p^{2/-1}\\x^2 - x^1 = 2p^2[/tex]
Substituting in the value of p, we get:

x2 - x1 = 2(1^2) = 2

So the distance between the two points where the depth is 2 inches is 2 inches.

To find the diameter of the headlight's opening, we need to double this distance and then multiply by the focal length:

d = 2(2)(1) = 4 inches

Therefore, the diameter of the headlight's opening is 4 inches.

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To assign the sum of extra credit received to the total extra credit, you need to follow these steps:

1. Make a list of all the test grades, and mark the ones that have extra credit with a "+" sign. For example, if the test scores are: 95, 110+, 80, 120+, 100, you would mark the 110+ and the 120+.

2. Add up all the extra credit marks. In this example, that would be 110 + 120 = 230.

3. Count the number of extra credit marks. In this example, there are two.

4. Multiply the number of extra credit marks by the maximum extra credit value. If full credit is 100, then the maximum extra credit value is the amount that a student can exceed the maximum score by. In this case, that would be 20 (since 120 - 100 = 20). So, 2 x 20 = 40.

5. Add the extra credit value to the total score. In this example, the total score is 505 (95 + 110 + 80 + 120 + 100). So, you would add 40 to that to get 545.

Therefore, the answer is: To assign sum extra with the total extra credit received given list test grades, you need to add the extra credit value to the total score by multiplying the number of extra credit marks by the maximum extra credit value, and then adding that value to the total score.

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Answer:

3(a-0.3')^2 -5.3'

Step-by-step explanation:

you can factorized as i explaind you in the pic

Therefore, any equation of the form y = kx, where k is a constant, has the same constant of proportionality as the equation y = (1/3)x.

Proportionality, equivalence between two ratios in mathematics. A and B are in the same ratio as C and D in the formula a/b = c/d. When one of a proportion's four quantities is unknown, a proportion is often built up to resolve the word problem.  

According to the principle of proportionality, any incidental human casualties and property damage must not outweigh the tangible benefits to the military that may be expected from the destruction of a military objective.

The equation y = (1/3)x represents a proportional relationship between y and x with a constant of proportionality of 1/3.

Any equation of the form y = kx, where k is a constant, represents a proportional relationship between y and x with a constant of proportionality of k.

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The probability that there is one or less theft in a minute is 0.09161 or 9.161%.

To find the probability that there is one or less theft in a minute, given that recent crime reports indicate that 4.0 motor vehicle thefts occur each minute in the USA, we can use the Poisson probability distribution formula.

The Poisson probability formula is:
P(x) = (e^(-λ) * λ^x) / x!

where λ (lambda) represents the average rate of occurrences (4.0 thefts per minute in this case), x is the number of occurrences we're interested in (0 or 1 theft), and e is the base of the natural logarithm (approximately 2.71828).

We want to find the probability of having 0 or 1 theft, so we'll calculate the probabilities for x=0 and x=1, and then add them together.

For x = 0:
P(0) = (e^(-4) * 4^0) / 0! = (0.01832 * 1) / 1 = 0.01832

For x = 1:
P(1) = (e^(-4) * 4^1) / 1! = (0.01832 * 4) / 1 = 0.07329

Now, we add the probabilities together:
P(0 or 1 theft) = P(0) + P(1) = 0.01832 + 0.07329 = 0.09161

So, the probability that there is one or less theft in a minute, given the Poisson distribution, is approximately 0.09161 or 9.161%.

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Answer:

24

Step-by-step explanation:

starting with the "top floor" :

3 single small triangles.

then the left 2 combined and the right 2 combined.

and then all 3 combined.

that is 3 + 2 + 1 = 6 triangles.

now we extend the triangles from the top floor to the next floor below.

we have the same number of triangles, they are just longer.

6 triangles there.

and we extend the triangles to the next floor below.

we have again the same number of triangles, they are just even longer.

6 triangles there.

and we extend the triangles to the next (and last) floor below.

we have again the same number of triangles, they are just very long.

6 triangles there.

that makes for the 4 floors

6×4 = 24 triangles.

Eastwood Enterprises offers horseback riding lessons and during the month of June, the company provided lessons on account totaling $5,100. By the end of the month, the company received on account $4,500 of this amount, meaning there is still $600 outstanding.

Eastwood Enterprises offered horseback riding lessons during the month of June, and the total value of lessons provided on account was $5,100. Here's a step-by-step explanation of the transactions:

1. Eastwood Enterprises provides horseback riding lessons worth $5,100 on account in June.
2. By the end of June, the company receives $4,500 on account from the customers who took lessons during that month.
3. In addition to the June payments, Eastwood also receives $500 on account from customers who took lessons in May.

To summarize, Eastwood Enterprises provided $5,100 worth of lessons on account in June, received $4,500 from those June lessons, and an additional $500 from customers who had taken lessons in May.

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The term that must be added to the equation to make it a perfect square is 9, which makes the option D correct.

We have to apply completing the square method to know the term to be added as follows:

For the equation;

x² + 6x = 1

we first divide the coefficient of x by 2;

6/2 = 3

then we square the result;

3² = 9

and then add 9 to both sides of the equation to make it a perfect square

x² + 6x + 9 = 1 + 9

(x + 3)² = 10.

Therefore, the term that must be added to the equation to make it a perfect square is 9.

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The annual interest savings is the amount Toni would save each year in interest charges due to the lower APR. It would be around  0.375%

Let x be the original APR. Then the first purchase of a point reduced the APR to x - 0.125%, the second point reduced it further to x - 0.25%, and the third point reduced it to x - 0.375%. Since Toni's new APR is 3.2%, we have:

x - 0.375% = 3.2%

Solving for x, we get:

x = 3.2% + 0.375% = 3.575%

Therefore, Toni's original APR was 3.575%.

To check our answer, we can use the fact that Toni purchased 3 points at a cost of 1% each. Since her loan value is the total cost of the points ($8,100) divided by the cost per percent (1%), we have:

loan value = $8,100 / 1% = $810,000

The reduction in APR due to the 3 points is 0.375%, which is equivalent to a reduction in the annual interest rate of:

0.375% / 100% = 0.00375

The annual interest savings due to the reduction in APR is then:

$810,000 x 0.00375 = $3,037.50

The annual interest savings is the amount Toni would save each year in interest charges due to the lower APR. Dividing this by the loan value gives us the actual reduction in APR:

$3,037.50 / $810,000 = 0.00375 = 0.375%

This confirms that our answer for the original APR is correct.

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A Class A defect is considered very serious and will most likely cause operating failure.

Designed experiments are important tools for optimizing processes, identifying interactions among variables, and reducing variation in processes.

We have,

A Class A defect is typically the most severe type of defect in a manufacturing or quality control context.

It usually means that a product or component has a flaw that is likely to cause it to fail in normal use or operation, posing a significant risk to safety or functionality.

Designed experiments, also known as DOE (Design of Experiments), are a commonly used statistical tool in process optimization and improvement.

They involve careful planning and executing a series of controlled tests or trials to systematically investigate the effects of different process variables or factors on a given output or response variable.

Thus,
A Class A defect is considered very serious and will most likely cause operating failure. Designed experiments are important tools for optimizing processes, identifying interactions among variables, and reducing variation in processes.

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Answer: The probability of picking a red candy is 14/63, the probability of picking a blue candy is 14/63, the probability of picking a green candy is 9/63, the probability of picking a brown candy is 15/63, the probability of picking an orange candy is 3/63, and the probability of picking a yellow candy is 8/63. These probabilities can be calculated by dividing the number of candies of each color by the total number of candies in the bag.

Step-by-step explanation:

Answer: vvvvv

Step-by-step explanation:

The odds of getting a green M&M can be calculated by dividing the number of green M&M candies by the total number of candies in the bag.

Number of green M&Ms = 9

Total number of candies = 14 + 14 + 9 + 15 + 3 + 8 = 63

Odds of getting a green M&M = Number of green M&Ms / Total number of candies

Odds of getting a green M&M = 9/63 = 1/7

The probability of getting a green M&M can be calculated by dividing the number of green M&M candies by the total number of candies in the bag.

Probability of getting a green M&M = Number of green M&Ms / Total number of candies

Probability of getting a green M&M = 9/63 = 0.14285714285714285 or approximately 14.29% (rounded to two decimal places).

Also thanks a lot, now i want m&m's

We can draw the conclusion that there is enough data to demonstrate that, at a significance level of = 0.05, the mean forced expiratory volume in one second for the population of 13-year-old boys in the high-pollution community differs from the established mean of 2.6 liters.

To test whether the mean forced expiratory volume in one second (FEV1) for the population of 13-year-old boys in the high-pollution community differs from the known mean of 2.6 liters, we can use a one-sample t-test.

Given that the sample size is not provided, we assume it to be large enough for the sample mean to follow a normal distribution by the central limit theorem.

The null hypothesis is: H0: μ = 2.6 (the population mean is equal to 2.6 liters)

The alternative hypothesis is: Ha: μ ≠ 2.6 (the population mean is not equal to 2.6 liters)

We will use a significance level of α = 0.05.

To find the critical value, we need to determine the degrees of freedom. Since the sample size is not given, we can assume it to be large enough (say, n > 30) and use a t-distribution with degrees of freedom approximately equal to n - 1.

Using a t-table with 30 degrees of freedom (which is conservative), the critical values for a two-tailed test with α = 0.05 are -2.042 and 2.042.

The test statistic can be calculated as:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

t = (sample mean - 2.6) / (population standard deviation / sqrt(sample size))

Since the sample standard deviation is not given, we can use the population standard deviation as an estimate (assuming that the sample is representative of the population). Thus,

t = (3.0 - 2.6) / (0.5 / sqrt(n))

t = 0.4 / (0.5 / sqrt(n))

We do not know the sample size, but we can solve for n using the given sample mean and standard deviation:

standard error = population standard deviation / sqrt(n)

0.5 / sqrt(n) = (3.0 - 2.6) / t

n = (0.5 / ((3.0 - 2.6) / t))^2

n = (0.5 / (0.4 / 2.042))^2

n = 107

Thus, the sample size is 107. Now we can calculate the test statistic:

t = (3.0 - 2.6) / (0.5 / sqrt(107))

t = 4.89

The calculated t-value of 4.89 is greater than the critical value of 2.042, so we reject the null hypothesis.

We can conclude that there is sufficient evidence to suggest that the mean forced expiratory volume in one second for the population of 13-year-old boys in the high-pollution community differs from the known mean of 2.6 liters at a significance level of α = 0.05.

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The Mean Value Theorem applies over the interval (-5, 5) and (5, ∞).

To determine the interval(s) where the Mean Value Theorem (MVT) applies for the function y=√(x^2-25), we need to ensure that the function is continuous and differentiable on the given interval.

1. The function is continuous when the expression under the square root is non-negative, which means x^2-25≥0. Solving for x, we get x≥5 or x≤-5. In interval notation, the domain for continuity is (-∞,-5] U [5,∞).

2. To check for differentiability, we need to find the derivative of the function. The derivative of y=√(x^2-25) is:

y' = (1/2)(x^2-25)^(-1/2) * 2x
y' = x/√(x^2-25)

Now, we need to ensure that the derivative is defined on the given interval. Since x=5 or x=-5 makes the denominator zero, we should exclude these points. Hence, the interval for differentiability is (-∞,-5) U (5,∞).

Since the MVT requires both continuity and differentiability, the applicable interval(s) for the Mean Value Theorem are (-∞,-5) U (5,∞).

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The legitimate probability distribution for the discrete random variable X is:

Value of x P(X= )

1 0.10

2 0.22

4 0.14

5 0.18

6 0.36

To create a legitimate probability distribution, the sum of all the probabilities should be equal to 1. So, we can use the fact that the sum of all probabilities must equal 1 to find the missing probability for X = 5.

Value of x P(X= )

1 0.10

2 0.22

4 0.14

5 ?

6 0.36

To find P(X = 5), we can subtract the sum of the probabilities for X = 1, 2, 4, and 6 from 1:

P(X = 5) = 1 - (0.10 + 0.22 + 0.14 + 0.36) = 0.18

Therefore, the legitimate probability distribution for the discrete random variable X is:

Value of x P(X= )

1 0.10

2 0.22

4 0.14

5 0.18

6 0.36

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