After the 8-week program, those who participated in the aquarobic program had their ending cholesterol measured, and the change in cholesterol was recorded for each participant. Estimate the mean cholesterol change using 95% confidence.

A researcher looking to estimate the percentage of teenagers in the US who support the legalization of recreational marijuana should employ descriptive statistics, specifically using a proportion or percentage to summarize the data collected from a representative sample of teenagers.

The researcher should employ inferential statistics to obtain the answer to her question. She could use survey sampling methods to collect data from a representative sample of teenagers in the US and use statistical analysis techniques such as hypothesis testing and confidence intervals to estimate the percentage of teenagers who support the legalization of recreational marijuana in the entire population of US teenagers.

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The probability that more than 3 of the entry forms will include an order is approximately 0.9896 or 98.96%.

This is a binomial distribution problem, where:

n = 18 (number of trials)

p = 0.3 (probability of success, i.e., an order being placed)

q = 1 - p = 0.7 (probability of failure, i.e., no order being placed)

We want to find the probability of more than 3 successes, which can be written as:

P(X > 3) = 1 - P(X ≤ 3)

To calculate this, we need to use the binomial cumulative distribution function or a binomial probability table. Alternatively, we can use the complement rule:

P(X > 3) = 1 - P(X ≤ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

Using the binomial probability formula, we can calculate each of these probabilities:

P(X = k) = (n choose k) * [tex]p^k * q^{(n-k)[/tex]

where (n choose k) = n! / (k! * (n-k)!) is the binomial coefficient.

P(X = 0) = (18 choose 0) * [tex]0.3^0 * 0.7^1^8[/tex] = 0.000005

P(X = 1) = (18 choose 1) * [tex]0.3^1 * 0.7^1^7[/tex] = 0.00016

P(X = 2) = (18 choose 2) *[tex]0.3^2 * 0.7^1^6[/tex]= 0.0017

P(X = 3) = (18 choose 3) *[tex]0.3^3 * 0.7^1^5[/tex] = 0.0086

Therefore

P(X > 3) = 1 - [0.000005 + 0.00016 + 0.0017 + 0.0086] ≈ 0.9896

So the probability that more than 3 of the entry forms will include an order is approximately 0.9896 or 98.96%.

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The  distance between the ships is increasing at a rate of approximately 18.71 km/h.

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the two sides are the distance traveled by ship A and the distance traveled by ship B.

Let's start by calculating the distance traveled by ship A from noon to 4:00 p.m., which is 4 hours:

distance = rate × time = 40 km/h × 4 h = 160 km

Now let's calculate the distance between the two ships at noon:

distance = √(170² + 0²) = √28900 ≈ 170.13 km

At 4:00 p.m., ship A has traveled 160 km east, and ship B has traveled 25 km/h × 4 h = 100 km north. We can use the Pythagorean theorem to calculate the new distance between the two ships:

distance = √(170² + 160² + 100²) ≈ 244.95 km

Therefore, the distance between the ships at 4:00 p.m. is approximately 244.95 km. To find the rate of change of this distance, we can subtract the initial distance from the final distance and divide by the time interval:

rate of change = (244.95 km - 170.13 km) / 4 h ≈ 18.71 km/h

Therefore, the distance between the ships is increasing at a rate of approximately 18.71 km/h.

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The efficiency of a probability sampling technique may be assessed by comparing it to that of non-probability sampling techniques.

Probability sampling is a method of selecting participants randomly from a population, which ensures that every individual has an equal chance of being included in the sample.

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The probability that the second light is green, given that the first light is green, is 0.695 or approximately 69.5%.

We can use Bayes' Theorem to find the probability that the second light is green, given that the first light is green. Let G1 and G2 denote the events that the first and second lights are green, respectively. Then we have:

P(G2 | G1) = P(G1 and G2) / P(G1)

We are given that P(G1 and G2) = 0.41, and P(G1) = 0.59. Substituting these values, we get:

P(G2 | G1) = 0.41 / 0.59 = 0.695

Therefore, the probability that the second light is green, given that the first light is green, is 0.695 or approximately 69.5%.

To answer your question, we will use the conditional probability formula:

P(A and B) = P(A) * P(B|A)

In this case, A represents the first light being green, B represents the second light being green, and P(A and B) is the probability of both lights being green. We are given the following:

P(A and B) = 0.41
P(A) = 0.59
We need to find P(B|A), which is the probability that the second light is green given that the first light is green.

Using the formula, we have:

0.41 = 0.59 * P(B|A)

To solve for P(B|A), divide both sides by 0.59:

P(B|A) = 0.41 / 0.59 ≈ 0.6949

Therefore, the probability that the second light is green, given that the first light is green, is approximately 0.6949.

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Therefore, The probability that the customer likes neither hot nor iced coffee is 0. This can be calculated by subtracting the probability of the customer liking hot coffee or iced coffee from 1.

The probability that the customer likes neither hot nor iced coffee can be calculated by subtracting the probability of the customer liking hot coffee or iced coffee from 1. Let A be the event that the customer likes neither hot nor iced coffee. Then, P(A) = 1 - P(H) - P(I). If P(H) = 0.6 and P(I) = 0.4, then P(A) = 1 - 0.6 - 0.4 = 0. Therefore, the probability that the customer likes neither hot nor iced coffee is 0.
To find the probability of an event, we need to divide the number of favorable outcomes by the total number of possible outcomes. Here, the customer can either like hot coffee, iced coffee, or neither. Since the customer can only like one of the two options, we can use the complement rule to find the probability of the customer not liking either. We subtract the sum of probabilities of the customer liking hot and iced coffee from 1.
Therefore, The probability that the customer likes neither hot nor iced coffee is 0. This can be calculated by subtracting the probability of the customer liking hot coffee or iced coffee from 1.

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A 99% confidence interval for the true proportion of teens aged 13-17 who have their own desktop or laptop computer can be calculated using the information provided in the nationwide random survey of 1,500 teens. In the survey, approximately 65% (or 0.65) of the respondents indicated that they have their own computer.

To calculate the confidence interval, we'll use the formula:

CI = p-hat ± Z * √(p-hat * (1 - p-hat) / n)

where:
- CI represents the confidence interval
- p-hat is the sample proportion (0.65)
- Z is the Z-score corresponding to the desired confidence level (2.576 for a 99% confidence interval)
- n is the sample size (1,500)

Now, let's plug in the values:

CI = 0.65 ± 2.576 * √(0.65 * 0.35 / 1,500)

CI = 0.65 ± 2.576 * 0.0128

CI = 0.65 ± 0.0330

So, the 99% confidence interval is (0.617, 0.683).

This means that, based on the survey results, we can be 99% confident that the true proportion of teens aged 13-17 who have their own desktop or laptop computer is between 61.7% and 68.3%.

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The equations have been solved below.

We can se that we have two equations that we should solve at the same time in both cases.

We know that;

x = -4 --- (1)

3x + 2y = 20 ----- (2)

Substitute (1) into (2)

3(-4) + 2y = 20

-12 + 2y = 20

y = 20 + 12/2

y = 16

The solution set is (-4, 16)

Now in the second case;

x - y = 1 ----- (1)

x + y = 3 ----- (2)

Then;

x = 1 + y ---(3)

Substitute (3) into (2)

1 + y + y = 3

2y = 2

y = 1

Then substitute y = 1 into (1)

x - 1 = 1

x = 2

The solution set is (2, 1)

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The total cost for Cara is $15,756.86

Given that, Cara leased a convertible by making a $3,000 deposit and paying $349 per month for 36 months, and an $80 title fee and a $112.86 license fee.

We need to find the total lease cost.

(36x349) +3000+80+112.86

= $15,756.86

Hence, the total leased cost is $15,756.86.

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There are 51,121,423 different committees that can be formed from 12 teachers and 32 students if the committee consists of both teachers and students.

To find the number of different committees that can be formed from 12 teachers and 32 students if the committee consists of both teachers and students, we need to use the combination formula. We can choose k members from a group of n members by using the formula:

n choose k = n! / (k! * (n-k)!)

In this case, we want to choose a committee that consists of both teachers and students, so we need to choose at least one teacher and at least one student. We can do this by choosing 1, 2, 3, ..., 11, or 12 teachers, and then choosing the remaining members of the committee from the students.

Let's start with choosing 1 teacher. There are 12 choices for the teacher, and we need to choose the remaining members of the committee from the 32 students. We can choose k students from a group of 32 students using the combination formula:

32 choose k = 32! / (k! * (32-k)!)

So the total number of committees that can be formed with 1 teacher and k students is:

12 * 32 choose k

To find the total number of committees that can be formed with at least one teacher and at least one student, we need to sum up the number of committees for each possible number of teachers:

total = 12 * (32 choose 1) + 12 * (32 choose 2) + ... + 12 * (32 choose 31) + 12 * (32 choose 32)

This is a bit cumbersome to calculate, but fortunately there is a shortcut: we can use the complement rule to find the number of committees that do not include any teachers, and then subtract this from the total number of committees. The number of committees that do not include any teachers is simply the number of committees that can be formed from the 32 students:

32 choose k = 32! / (k! * (32-k)!)

So the total number of committees that can be formed with at least one teacher and at least one student is:

total = 12 * (32 choose 1) + 12 * (32 choose 2) + ... + 12 * (32 choose 31) + 12 * (32 choose 32)
     = 12 * (2^32 - 1) - 32 choose 0
     = 12 * (2^32 - 1) - 1
     = 51,121,423

Therefore, there are 51,121,423 different committees that can be formed from 12 teachers and 32 students if the committee consists of both teachers and students.

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Reject the null hypothesis at the 0.020 level of significance and conclude that the population mean is less than 7.

To find the critical value for the test statistic, we first need to determine the level of significance or alpha (α). In this case, the significance level is given as 0.020.

Since this is a one-tailed test (alternative hypothesis is less than 7), we will use a z-test and look up the critical value from the z-table.

Using a standard normal distribution table, we can find the z-score that corresponds to a probability of 0.020 in the left-tail. The critical value is the negative z-score that corresponds to the probability of 0.020.

Looking up the z-score in the table or using a calculator, we find that the critical value for a significance level of 0.020 is -2.054.

This means that if our calculated test statistic falls below -2.054, we can reject the null hypothesis at the 0.020 level of significance and conclude that the population mean is less than 7.

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The length of the sides are;

AB = 78 = BC

AC = 100

We need to know that the two sides of an isosceles triangle are equal.

Then, we have that from the information;

Line AB = line BC

Now, substitute the values

20x- 2 = 12x + 30

collect the like terms, we get;

20x - 12x = 30 + 2

Add the values

8x = 32

Make 'x' the subject

x = 4

Then,

Line AB = 20(4) - 2 = 80 - 2 = 78

Line BC = 12(4) + 30 = 48 + 30 = 78

Line AC = 25(4) = 100

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Answer:

According to the Triangle Inequality Theorem, the range of possible lengths for the third side of the triangle, x, is 1 < x < 11.

Step-by-step explanation:

To determine the range of possible lengths for the third side of the triangle, we need to use the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

If a, b, and c are the lengths of the sides of a triangle, then:

We have been told that two sides of the triangle are 5 cm and 6 cm.

Let "x" be the length of the third of the triangle.

Using the Triangle Inequality Theorem, we can write the following inequalities:

Simplify the inequalities:

The first inequality tells us that x should be less than 11 cm.

The second inequality tells us that x should be greater than zero (since length cannot be negative).

The third inequality tells us that the x should be greater than 1 cm.

Therefore, the range of possible lengths for the third side is 1 < x < 11.

Thus, the radius of the right circular cylinder as 2 inches for the given values of lateral surface area (LSA) and volume.

To find the radius of the right circular cylinder, we will use the given lateral surface area (LSA) and volume. The formulas for these are:

LSA = 2 * pi * r * h

Volume = pi * r^2 * h

Where r is the radius, and h is the height of the cylinder.

We are given LSA = 3.5 square inches and Volume = 3.5 cubic inches. Let's plug these values into the formulas:

3.5 = 2 * pi * r * h (1)

3.5 = pi * r^2 * h (2)

Now, we want to isolate the radius. To do this, we can solve equation (1) for h:

h = 3.5 / (2 * pi * r)

Now, substitute this expression for h into equation (2):

3.5 = pi * r^2 * (3.5 / (2 * pi * r))

Simplify the equation by cancelling out pi and 3.5:

1 = r * (1 / (2 * r))

Multiply both sides by 2 * r:

2 * r = r^2

Now, solve for r:

r^2 - 2 * r = 0

r(r - 2) = 0

This gives us two possible values for r: r = 0 and r = 2. Since the radius cannot be 0, we have the radius of the right circular cylinder as 2 inches.

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The probability that at least 1 patient will arrive during this time is approximately 0.9967 or 99.67%. This information will help you staff the ER with the optimal number of doctors and nurses to handle the patient load.

As the human resource manager for the Cookeville Regional Medical Center's Emergency room, we need to calculate the probability that at least 1 patient will arrive between 11pm and midnight on Friday night.

Since the number of patients follows a Poisson distribution with a mean of 5.7, we can use the Poisson distribution formula:

P(X ≥ 1) = 1 - P(X = 0)

where X represents the number of patients arriving during this time.

To calculate P(X = 0), we can use the Poisson distribution formula:

P(X = 0) = (e^-λ * λ^0) / 0!

where λ is the mean number of patients, which is 5.7 in this case.

Plugging in the values, we get:

P(X = 0) = (e^-5.7 * 5.7^0) / 0! = 0.0030

Therefore,

P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.0030 = 0.9970

So the probability that at least 1 patient will arrive between 11pm and midnight on Friday night is 0.9970 or approximately 99.7%.

This information can be used by the Human Resources Director to staff the ER with the optimal number of doctors and nurses to handle the expected patient volume during this time.

As the human resource manager for the Cookeville Regional Medical Center's Emergency room, you need to calculate the probability that at least 1 patient will arrive between 11pm and midnight on a Friday night. The number of patients follows a Poisson distribution with a mean of 5.7 patients.

To find the probability that at least 1 patient will arrive, we will first calculate the probability that no patients arrive (P(X=0)) and then subtract it from 1. The formula for the Poisson distribution is:

P(X = k) = (e^(-λ) * λ^k) / k!

where λ is the mean (5.7 patients in this case), k is the number of patients, and e is the base of the natural logarithm (approximately 2.71828).

To calculate P(X=0):

P(X = 0) = (e^(-5.7) * 5.7^0) / 0!
         = (e^(-5.7) * 1) / 1
         ≈ 0.0033

Now, to find the probability of at least 1 patient arriving, we will subtract the probability of no patients arriving from 1:

P(X ≥ 1) = 1 - P(X = 0)
        = 1 - 0.0033
        ≈ 0.9967

So, the probability that at least 1 patient will arrive during this time is approximately 0.9967 or 99.67%. This information will help you staff the ER with the optimal number of doctors and nurses to handle the patient load.

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The injury probability p that makes the decision maker indifferent between entering now and waiting until next year is approximately 0.26.

To calculate this probability, we need to set the expected values of entering now and waiting until next year equal to each other and solve for p. Let EMV₁ be the expected value of entering now and EMV₂ be the expected value of waiting until next year. Then we have:

EMV₁ = -1000p + 5000(1-p)

EMV₂ = 0.9(-1000p) + 0.1(5000)

Setting EMV₁ equal to EMV₂, we get:

-1000p + 5000(1-p) = 0.9(-1000p) + 0.1(5000)

Solving for p, we get:

p ≈ 0.26

Therefore, if the probability of injury is greater than 0.26, the decision maker should wait until next year to enter the market, and if the probability of injury is less than 0.26, the decision maker should enter the market now.

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Answer:

x + 2x + 3x = 20

6x = 20

x = 3 1/3 cm, so 2x = 6 2/3 cm and

3x = 10 cm

The lengths are 3 1/3 cm, 6 2/3 cm, and 10 cm.

Answer: 4x+6

Step-by-step explanation:

3x-y+8+x+y-2

3x+x +y-y  +8-2

4x+6

The equation which matches the graph of the greatest integer function given below is A) y = [x] - 4.

Given a graph of the greatest integer function.

Greatest integer functions are functions which are also called step functions.

It rounds off the number to the nearest integer.

When x = 2, y = -2

y = x - 4 = 2 - 4 = -2

This is the case for every values.

So the equation is,

y = [x] - 4

Hence the given function is A) y = [x] - 4.

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In a control chart, when a data point falls outside the control limits (upper and lower), it must be concluded that the process is out of control, indicating the presence of special-cause variation.

A control chart is a statistical tool used to monitor and analyze process performance over time. It has three main components: the centerline, which represents the process average; and the upper and lower control limits, which define the acceptable range of variability.

When all data points fall within the control limits, the process is considered to be in control, suggesting that the variation is due to common causes (normal variation inherent in the process). However, if a data point falls outside the control limits, it indicates that the process is out of control, and the variation is likely due to special causes (unusual events or circumstances affecting the process).

When an out-of-control point is identified, it is essential to investigate the cause of the variation to determine if it is due to an assignable cause or just random chance. If an assignable cause is found, corrective action should be taken to eliminate the source of the special-cause variation and bring the process back into control.

Once the process is stable and in control, continuous improvement efforts can be made to reduce common-cause variation and enhance process performance.

In summary, when a data point falls outside the control limits in a control chart, it must be concluded that the process is out of control, and special-cause variation is likely present. The next step is to investigate and address the cause of this variation to bring the process back into control.

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For multivariate statistical techniques where there is more than one dependent variable, multivariate analysis of variance and covariance and canonical correlation and multiple discriminant analysis can be used.

In data analysis, we look at different variables (or factors) and how they can affect certain situations or outcomes. For example, in marketing, you can look at how the "money spent on advertising" variable affects the "number of sales" variable. A multivariate statistical technique known as factor analysis or multivariate analysis is used to look for patterns among related variables. Multivariate analysis is based on the observation and analysis of more than one statistical outcome variable simultaneously. Many different multivariate statistical techniques such as discriminant analysis, cluster analysis, principal component analysis (PCA) and factor analysis (FA). Therefore, multivariate analysis of variance (MANOVA) is used to measure the effect of multiple independent variables on two or more dependent variables.

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Answer:

the first one because Albert sat on apple tree and discovered u don't have a father

Answer: A

Step-by-step explanation:

Once you complete the 50 trials, you can calculate the total number of heads and tails by adding the counts from the corresponding columns.

Since I'm a text-based AI and cannot physically flip pennies, I'll provide you with a general understanding of the possible outcomes.
When you simultaneously flip two pennies 50 times, there are four possible outcomes for each flip:
1. Both pennies show heads (HH)
2. The first penny shows heads, and the second penny shows tails (HT)
3. The first penny shows tails, and the second penny shows heads (TH)
4. Both pennies show tails (TT)
Each outcome has a probability of 1/4. After flipping the two pennies 50 times, you'll have a total of 100 coin flips. To record the number of heads and tails you get, create a table with four columns: 'HH', 'HT', 'TH', and 'TT'. Then, mark each outcome as you perform the 50 trials.

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

F_Y(y) = P(Y <= y) = 1 - P(Y > y) = 1 - P(X1 > y, X2 > y, ..., Xn > y)

Since X1, X2, ..., Xn are independent and uniformly distributed on [θ1, θ2], we have:

P(Xi > y) = (θ2 - y) / (θ2 - θ1) for θ1 <= y <= θ2

P(Xi > y) = 0 for y < θ1

P(Xi > y) = 1 for y > θ2

Therefore,

F_Y(y) = 1 - P(X1 > y, X2 > y, ..., Xn > y)

= 1 - P(X1 > y) * P(X2 > y) * ... * P(Xn > y)

= 1 - [(θ2 - y) / (θ2 - θ1)]^n for θ1 <= y <= θ2

The density function of Y is the derivative of the CDF:

f_Y(y) = d/dy [1 - [(θ2 - y) / (θ2 - θ1)]^n]

= n(θ2 - y)^(n-1) / (θ2 - θ1)^n for θ1 <= y <= θ2

(b) The expectation of Y is:

E(Y) = ∫θ1^θ2 y * f_Y(y) dy

= ∫θ1^θ2 y * n(θ2 - y)^(n-1) / (θ2 - θ1)^n dy

= n/(n+1) * (θ2 - θ1) / 2

(c) Since Y is an unbiased estimator of θ2, we have:

E(Y) = θ2

n/(n+1) * (θ2 - θ1) / 2 = θ2

θ2 = 2/n * (n/(n+1) * (θ2 - θ1) + θ1)

Therefore, an unbiased estimator for θ2 is:

θ2_hat = 2/n * (n/(n+1) * (X_bar - θ1) + θ1)

where X_bar is the sample mean of X1, X2, ..., Xn.

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Answer: $4$

Step-by-step explanation:

The correct option is b) 3. The average processing time is 4 minutes. A denial of service probability of no more than 0.01 is desired. The average interarrival time is 10 minutes. The system need 3 servers.

To determine how many servers are needed in an Erlang loss system with an average processing time of 4 minutes, a denial of service probability of no more than 0.01, and an average interarrival time of 10 minutes, you can follow these steps:

1. Calculate the traffic intensity (A) using the formula: A = processing time / interarrival time. In this case, A = 4 minutes / 10 minutes = 0.4 Erlangs.

2. Use Erlang's B formula to find the number of servers (s) required to achieve a desired probability of denial of service (P): P = ([tex]A^s[/tex] / s!) / Σ([tex]A^k[/tex] / k!) from k = 0 to s.

3. Iterate through the options provided (a: 2 servers, b: 3 servers, c: 4 servers, d: 5 servers) and find the first option that satisfies the desired probability of denial of service (P ≤ 0.01).

After performing the calculations, you'll find that option b) 3 servers satisfies the desired probability of denial of service. Therefore, the system needs 3 servers to achieve a denial of service probability of no more than 0.01.

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With 90% confidence, we can say that the population mean falls between 20.96 and 23.04.

To construct the confidence interval for a population mean, we can use the formula:

Confidence interval = sample mean ± (critical value) x (standard error)

where the standard error is the standard deviation of the sample mean, which is calculated as:

standard error = sample standard deviation / √sample size

The critical value depends on the desired level of confidence and the degrees of freedom (df), which is the sample size minus 1. For a 90% confidence interval and 62 degrees of freedom, the critical value from a t-distribution is 1.667 (found using a t-table or calculator).

Plugging in the values, we get:

standard error = 5 / √64 = 5 / 8 = 0.625

Confidence interval = 22 ± 1.667 x 0.625 = (20.96, 23.04)

Therefore, with 90% confidence, we can say that the population mean falls between 20.96 and 23.04.

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The length of the third side is BC ≈ 1315.5.

Let the two sides of the triangle with given altitudes be AB=1010 and AC=1515. Let h be the length of the altitude from A to BC.

Let D and E be the feet of the perpendiculars from A to BC and from B to AC, respectively. Then we have:

BD = AB - AD = 1010 - h

CE = AC - AE = 1515 - h

Since the length of the altitude from A to BC is the average of the lengths of the altitudes to the two given sides, we have:

h = (BD + CE) / 2

h = [(1010 - h) + (1515 - h)] / 2

2h = 2525 - h

3h = 2525

h = 2525 / 3

Now we use the Pythagorean theorem on the right triangle ABD:

[tex]AD^2 + BD^2 = AB^2[/tex]

[tex]h^2 + (1010 - h)^2 = 1010^2[/tex]

Expanding and simplifying, we get:

[tex]2h^2 - 2020h + 515 = 0[/tex]

Solving for h using the quadratic formula, we get:

h = [tex](2020 ± sqrt(2020^2 - 4(2)(515))) / (2(2))[/tex]

h ≈ 808.6 or h ≈ 1511.4

We take the smaller value of h since the altitude is shorter than the length of the side opposite to it. Therefore, h ≈ 808.6.

Now we can use the Pythagorean theorem on the right triangle ABC:

[tex]BC^2 = AC^2 - h^2[/tex]

[tex]BC^2 = 1515^2 - (808.6)^2[/tex]

BC ≈ 1315.5

Therefore, the length of the third side is BC ≈ 1315.5.

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After the fifth day, Mary will have $32 left.

1. Start with the initial amount of money: $243.
2. For each day, calculate the amount spent and the remaining amount.
3. Repeat steps 1 and 2 for the five days.

Here's the step-by-step explanation:

Day 1:
- Money spent: $243 x [tex]\frac{1}{3}[/tex] = 81$
- Remaining money: $243 - 81 = 162$

Day 2:
- Money spent: $162 x [tex]\frac{1}{3}[/tex] = 54$
- Remaining money: $162 - 54 = 108$

Day 3:
- Money spent: $108 \times \frac{1}{3} = 36$
- Remaining money: $108 - 36 = 72$

Day 4:
- Money spent: $72 x [tex]\frac{1}{3}[/tex]= 24$
- Remaining money: $72 - 24 = 48$

Day 5:
- Money spent: $48  x [tex]\frac{1}{3}[/tex]  = 16$
- Remaining money: $48 - 16 = 32$

After the fifth day, Mary will have $32 left.

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To find the equation of the tangent plane to the surface x^2 + y^2 - 4z^2 = 41 at the point (-3, -6, 1), we need to first find the partial derivatives of the surface equation with respect to x, y, and z:

f_x = 2x

f_y = 2y

f_z = -8z

Then, we can evaluate these partial derivatives at the given point (-3, -6, 1):

f_x(-3, -6, 1) = -6

f_y(-3, -6, 1) = -12

f_z(-3, -6, 1) = -8

So the normal vector to the tangent plane at the given point is:

n = <f_x(-3, -6, 1), f_y(-3, -6, 1), f_z(-3, -6, 1)> = <-6, -12, -8>

To find the equation of the tangent plane, we can use the point-normal form of the equation of a plane:

n · (r - P) = 0

where n is the normal vector, P is the given point, and r is a general point on the plane. Substituting in the values we have, we get:

<-6, -12, -8> · (r - <-3, -6, 1>) = 0

Simplifying and expanding the dot product, we get:

-6(r - (-3)) - 12(r - (-6)) - 8(r - 1) = 0

Simplifying further, we get:

-6r + 18 - 12r + 72 - 8r + 8 = 0

Combining like terms, we get:

-26r + 98 = 0

Dividing both sides by -26, we get:

r = -3

So the equation of the tangent plane to the surface x^2 + y^2 - 4z^2 = 41 at the point (-3, -6, 1) is:

-6(x + 3) - 12(y + 6) - 8(z - 1) = 0

Simplifying, we get:

6x + 12y + 8z = -66

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