Write the following another way: 11/15

we can expect there to be approximately 696 rats (rounded to the nearest whole) in the bay area of the city after 18 months of growth at a rate of 6% per month.

To solve this problem, we can use the formula for exponential growth:
[tex]P(t) = P(0)e^{rt}[/tex]

where P(0) is the initial population, r is the growth rate (in decimal form), t is the time period, and e is the mathematical constant approximately equal to 2.718.

In this case, P(0) = 310, r = 0.06 (since the population grows at a rate of 6% per month), and t = 18 months. So we have:

[tex]P(18) = 310 e^{(0.06 * 18)}[/tex]
P(18) =696

Therefore, we can expect there to be approximately 696 rats (rounded to the nearest whole) in the bay area of the city after 18 months of growth at a rate of 6% per month.

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This equation tells us that the distance Alanna ran (ddd) is equal to her average speed (vvv) multiplied by the time she spent running (ttt - 0.25).

To find the equation that relates ddd, vvv, and ttt, we need to use the formula for average speed, which is:

Average speed = distance ÷ time

In this case, Alanna ran ddd kilometers at an average speed of vvv kilometers per hour, so we can write:

vvv = ddd ÷ t1

where t1 is the time it took Alanna to run ddd kilometers.

After running, Alanna walked to cool down for 0.25 hours, so the total time for the trip was ttt = t1 + 0.25. We can substitute this into our equation to get:

vvv = ddd ÷ (ttt - 0.25)

Finally, we can rearrange this equation to solve for ddd:

ddd = vvv × (ttt - 0.25)

So the equation that relates ddd, vvv, and ttt is:

ddd = vvv × (ttt - 0.25)

This equation tells us that the distance Alanna ran (ddd) is equal to her average speed (vvv) multiplied by the time she spent running (ttt - 0.25).

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In British Columbia, Canada, the number y of purple martin nesting pairs x years since 2000 can be modeled by the equation y = 0.63x² +51.8x + 144 there were about 1000 nesting pairs around the time of year 2011.5.

To find when there were about 1000 nesting pairs, we need to solve the given equation for x.

y = 0.63x² +51.8x + 144

We substitute y = 1000 and solve for x:

1000 = 0.63x² +51.8x + 144

0.63x² + 51.8x - 856 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± sqrt(b² - 4ac)) / 2a

where a = 0.63, b = 51.8, and c = -856.

x = (-51.8 ± sqrt(51.8² - 4(0.63)(-856))) / 2(0.63)

x ≈ -86.3 or x ≈ 11.5

Since we are interested in the time after 2000, we discard the negative solution and get:

x ≈ 11.5

This means that there were about 1000 nesting pairs approximately 11.5 years after 2000. To find the actual year, we add 11.5 to 2000:

2000 + 11.5 = 2011.5

Therefore, there were about 1000 nesting pairs around the time of  year 2011.5.

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If a pilot elects to proceed to the selected alternate, the landing minimums used at that airport should be equal to or higher than the minimums required for the original destination airport.

When a pilot chooses to proceed to the selected alternate airport, they must consider the weather conditions at that airport, particularly the landing minimums. The landing minimums are the lowest weather conditions in which an aircraft can safely land at an airport. They are determined by various factors, such as the visibility range and the height of the cloud ceiling.

Therefore, if a pilot decides to go to the alternate airport, they should use the landing minimums for that airport to ensure a safe landing. The landing minimums for the alternate airport should be equal to or higher than the minimums required for the original destination airport.

It's important to note that the pilot must have knowledge of the landing minimums for the alternate airport before deciding to proceed there. If they don't have the necessary information, they should consider other options, such as returning to the departure airport or diverting to a different airport with suitable weather conditions.

In summary, when a pilot chooses to proceed to an alternate airport, they should use the landing minimums for that airport to ensure a safe landing. The pilot must have knowledge of the minimums before proceeding to the alternate airport.

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Normal probability distribution is applied to a continuous random variable. The correct option is D.

The normal probability distribution, also known as the Gaussian distribution, is a probability distribution that is commonly used in statistics and probability theory. It is a continuous probability distribution that is often used to model the behavior of a wide range of variables, such as physical measurements like height, weight, and temperature.

The normal distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ). It is a bell-shaped curve that is symmetrical around the mean, with the highest point of the curve being located at the mean. The standard deviation determines the width of the curve, and 68% of the data falls within one standard deviation of the mean, while 95% falls within two standard deviations.

The normal distribution is widely used in statistical inference and hypothesis testing, as many test statistics are approximately normally distributed under certain conditions. It is also used in modeling various phenomena, including financial markets, population growth, and natural phenomena like earthquakes and weather patterns.

Overall, the normal probability distribution is a powerful tool for modeling and analyzing a wide range of continuous random variables in a variety of fields.

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Answer:

Yes, it is TRUE

Hope my answer helps you ✌️

There is about a 26.5% chance of randomly selecting one fork, one spoon, and one knife from the drawer of 18 pieces of silverware.

To find the probability of selecting one fork, one spoon, and one knife out of the 18 pieces of silverware in the drawer, we can use the formula for probability:

Probability = (number of favorable outcomes) / (total number of possible outcomes)

First, let's find the total number of possible outcomes. Since we are selecting three pieces of silverware without replacement, the total number of possible outcomes is the number of ways to choose three pieces from 18:

Total possible outcomes = 18C3 = (18 x 17 x 16) / (3 x 2 x 1) = 816

Next, let's find the number of favorable outcomes, i.e., the number of ways to choose one fork, one spoon, and one knife. We can break this down into three steps:

Step 1: Choose one fork from the 6 available forks
Step 2: Choose one spoon from the 6 available spoons
Step 3: Choose one knife from the 6 available knives

The number of ways to perform each step is simply the number of available items, so:

Number of favorable outcomes = 6 x 6 x 6 = 216

Therefore, the probability of selecting one fork, one spoon, and one knife is:

Probability = 216 / 816 = 0.265

This means that there is about a 26.5% chance of randomly selecting one fork, one spoon, and one knife from the drawer of 18 pieces of silverware.

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we need to first calculate the deviation of a GPA less than 1.85 from the mean GPA of 2.3, Deviation = 1.85 - 2.3 = -0.45 Next, we need to calculate the z-score for this deviation using the formula: z = (x - μ) / σ .

where x is the value we want to convert to a z-score, μ is the population mean, and σ is the population standard deviation. z = (-0.45 - 2.3) / 0.7 = -3.07, We can use a z-score table or calculator to find the percentage of the population that falls below this z-score.

According to the table, the percentage of the population with a z-score less than -3.07 is approximately 0.001. Therefore, the percentage of students who will be dropped from college is approximately 0.1% (or 0.001 x 100).
The Z-score formula is: Z = (X - μ) / σ.

Where X is the GPA value (1.85), μ is the mean (2.3), and σ is the standard deviation (0.7). Z = (1.85 - 2.3) / 0.7 = -0.64
Now, we'll use the standard normal distribution table to find the percentage of students corresponding to a Z-score of -0.64. The table gives us a value of 0.2611.Thus, approximately 26.11% of the students with a GPA less than 1.85 will be dropped from college.

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The answer is: ∫c xy^2 ds = 0To evaluate the line integral of c xy2 ds, we need to first parameterize the given curve c, which is the right half of the circle x2 y2 = 4 oriented counterclockwise.

Let's use the parameterization x = 2cos(t) and y = 2sin(t) for t in [0, pi], which traces out the right half of the circle as t varies from 0 to pi.

Next, we need to find ds, which represents an infinitesimal length element along the curve c. We can use the formula ds = sqrt(dx/dt)^2 + (dy/dt)^2 dt, which simplifies to ds = 2dt.

Substituting our parameterization and ds into the line integral, we get:

∫c xy^2 ds = ∫0^π (2cos(t) * 2sin(t)^2)(2dt)

Simplifying, we get:

∫c xy^2 ds = 8 ∫0^π sin^2(t)cos(t) dt

Using the identity sin^2(t) = (1/2)(1-cos(2t)), we can rewrite the integral as:

∫c xy^2 ds = 8 ∫0^π (1/2)(1-cos(2t))cos(t) dt

Expanding and simplifying, we get:

∫c xy^2 ds = 4 ∫0^π cos(t) dt - 4 ∫0^π cos(2t)cos(t) dt

Evaluating the first integral gives us:

4sin(π) - 4sin(0) = 0

For the second integral, we can use the identity cos(2t)cos(t) = (1/2)cos(3t) + (1/2)cos(t) and simplify:

∫0^π cos(2t)cos(t) dt = (1/2) ∫0^π cos(3t) dt + (1/2) ∫0^π cos(t) dt

The first integral evaluates to 0, and the second integral evaluates to 2sin(π) - 2sin(0) = 0.

Therefore, the final answer is:

∫c xy^2 ds = 0

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Q1 = median(2,4,4,5,7,8) = (4+5)/2 = 4.5

The third quartile (Q3) is the median of the 7th through 12th values:

Q3 = median(8,9,12,15,17,28) = (12+15)/2 = 13.5

Thus, the first quartile is 4.5 and the third quartile is 13.5.

The mean of the given data is:

mean = (2+4+4+5+7+8+8+9+12+15+17+28)/12 = 10.5

Thus, the mean is 10.5.

To find the quartiles and median, we first need to order the data:

2, 4, 4, 5, 7, 8, 8, 9, 12, 15, 17, 28

The median is the middle value of the ordered data. Since we have 12 data points, the median is the average of the 6th and 7th values:

median = (7+8)/2 = 7.5

To find the quartiles, we need to divide the ordered data into four equal parts. Since we have 12 data points, the first quartile (Q1) is the median of the 1st through 6th values:

Q1 = median(2,4,4,5,7,8) = (4+5)/2 = 4.5

The third quartile (Q3) is the median of the 7th through 12th values:

Q3 = median(8,9,12,15,17,28) = (12+15)/2 = 13.5

Thus, the first quartile is 4.5 and the third quartile is 13.5.

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The critical points of the function f(x) = 4sin(x)cos(x) in the interval (0, 2π) are:
x = π/4, 3π/4, 5π/4, 7π/4

To find the critical points of the function f(x) = 4sin(x)cos(x) in the interval (0, 2π), we need to find the values of x where the first derivative, f'(x), is equal to 0 or undefined.

First, let's find the derivative using the product rule:

f'(x) = (4sin(x))'(cos(x)) + (4sin(x))(cos(x))'
f'(x) = (4cos(x))(cos(x)) - (4sin(x))(sin(x))
f'(x) = 4(cos^2(x) - sin^2(x))

Now, we set f'(x) = 0 and solve for x:

4(cos^2(x) - sin^2(x)) = 0
cos^2(x) = sin^2(x)

Recall that sin^2(x) + cos^2(x) = 1. So, cos^2(x) = 1 - sin^2(x). Substitute this into the\:

1 - sin^2(x) = sin^2(x)
2sin^2(x) = 1
sin^2(x) = 1/2
sin(x) = ±√(1/2)

So, x = arcsin(±√(1/2)).

We need to find the solutions for x in the interval (0, 2π):

x = π/4, 3π/4, 5π/4, 7π/4

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The probability that the second inspector will find a flaw in an item that has been passed by the first inspector is simply the True Positive Rate, which is "b".

To find the probability that the second inspector will find a flaw in an item that has been passed by the first inspector, we need to consider the following terms:

1. Probability of the first inspector passing a flawed item (False Negative Rate)
2. Probability of the second inspector finding a flaw when examining a flawed item (True Positive Rate)

Let's assume the following probabilities:
- Probability of the first inspector passing a flawed item: P(False Negative) = a
- Probability of the second inspector finding a flaw in a flawed item: P(True Positive) = b

Now, let's calculate the probability that the second inspector will find a flaw in an item that has been passed by the first inspector:

P(Second Inspector finds flaw | First Inspector passes item) = P(True Positive | False Negative) = b

In this case, we don't need to account for the probability of the first inspector passing the flawed item, as we're given that the second inspector only examines items passed by the first inspector.

Therefore, the probability that the second inspector will find a flaw in an item that has been passed by the first inspector is simply the True Positive Rate, which is "b".

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The correct answer to the question "The sampling distribution of a statistic probability distribution." is option c) is a probability distribution.

To understand the concept of a sampling distribution, we need to understand what a statistic is. A statistic is a numerical value that describes some aspect of a population. For example, the mean, median, mode, variance, and standard deviation are all examples of statistics.

In statistics, we are often interested in making inferences about a population based on a sample of that population. A sampling distribution is a probability distribution that shows all possible values that a statistic can take on, and the probability of getting each value under the assumption that it had resulted due to chance alone.

The sampling distribution is an important concept in statistics because it allows us to make inferences about a population based on a sample. By knowing the sampling distribution of a statistic, we can calculate the probability of getting a particular value of the statistic under the assumption of random sampling. This information can be used to make decisions about the population, such as whether a hypothesis is supported or rejected.

In conclusion, the sampling distribution of a statistic is a option c) It is a probability distribution that gives the probability of getting each value of a statistic under the assumption it had resulted due to chance alone. It is a critical tool for making inferences about populations based on samples.

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The perimeter of the triangle is 23 ft.

We have,

We can use the triangle inequality theorem to check whether this triangle can exist.

According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check whether this condition holds for the sides given:

5 + 7 = 12, which is greater than 11, so the condition holds.

5 + 11 = 16, which is greater than 7, so the condition holds.

7 + 11 = 18, which is greater than 5, so the condition holds.

Therefore, this triangle can exist.

To find the perimeter of the triangle, we simply add up the lengths of the three sides:

Perimeter = 5 + 7 + 11 = 23 ft

Therefore,

The perimeter of the triangle is 23 ft.

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The triangle inequality theorem states that a triangle with sides of 5, 7, and 11 feet is feasible.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

Any two triangle sides' lengths added together must be bigger than the third side's length.

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Answer: (x+7)(x+7), 7 plus 7 is 14 and 7 times 7 is 49

The final cost if paid within 15 days is $51.91 ($53.52 - $1.61).

The electrician purchased 30 double-pole, double-branch cutouts listed at $6.40 per box of 5, with a 25% discount, and 8 surface panels listed at $4.75 each, with a 35% discount. If the bill is paid within 15 days, an additional 3% discount is applied.

First, we need to calculate the cost of cutouts and surface panels separately. For the cutouts, 30 cutouts are equivalent to 6 boxes (30/5). With the 25% discount, the price per box becomes $4.80 ($6.40 * 0.75). So, the total cost for cutouts is $28.80 (6 * $4.80).

For the surface panels, with the 35% discount, the price per panel becomes $3.09 ($4.75 * 0.65). The total cost for 8 panels is $24.72 (8 * $3.09).

Now, we add the costs together: $28.80 (cutouts) + $24.72 (panels) = $53.52. Lastly, we apply the 3% discount for paying the bill within 15 days, which is $1.61 ($53.52 * 0.03).

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Thus, there are 18 ways to choose one book, 153 ways to choose two books, and 360 ways to choose three books, one from each category.

To find the number of ways a student can choose from the given library, we need to use the formula for combinations.

The formula for combinations is:
nCr = n! / r!(n-r)!

where n is the total number of items, r is the number of items to be chosen, and ! denotes factorial.

Let's consider the different ways a student can choose:

1. Choosing one book from any category:
Total number of books = 5+3+6+4 = 18
n = 18, r = 1
Number of ways = 18C1 = 18! / 1!(18-1)! = 18

2. Choosing two books from any category:
n = 18, r = 2
Number of ways = 18C2 = 18! / 2!(18-2)! = 153

3. Choosing three books, one from each category:
For this, we need to choose one book from each category, and the order of selection does not matter.
n1 = 5, r1 = 1 (for history)
n2 = 3, r2 = 1 (for sociology)
n3 = 6, r3 = 1 (for anthropology)
n4 = 4, r4 = 1 (for psychology)
Number of ways = (5C1 x 3C1 x 6C1 x 4C1) = (5 x 3 x 6 x 4) = 360

Therefore, there are 18 ways to choose one book, 153 ways to choose two books, and 360 ways to choose three books, one from each category.

In summary, a student can choose:
- 18 ways to choose one book
- 153 ways to choose two books
- 360 ways to choose three books, one from each category

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The mean, of the sampling distribution of the mean, for samples of size 48 equals 65.

The central limit theorem states that the sampling distribution of the mean of a large number of random samples taken from a population will be approximately normally distributed, regardless of the shape of the population distribution. The mean of the sampling distribution of the mean is equal to the population mean, which in this case is 65.

The standard deviation of the sampling distribution of the mean is equal to the population standard deviation divided by the square root of the sample size. Therefore, the standard deviation of the sampling distribution of the mean for samples of size 48 is 10.2 / √48 ≈ 1.47.

Since the sampling distribution of the mean is approximately normally distributed with a mean of 65 and a standard deviation of 1.47, the mean of the sampling distribution of the mean for samples of size 48 is 65.

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The component of statistical methodology that encompasses the collection, organization, and summarization of data is referred to as descriptive statistics.

Descriptive statistics is a fundamental aspect of statistical methodology that focuses on the initial stages of data analysis. It involves the collection of data through various methods such as surveys, experiments, or observational studies. This data is then organized and structured in a systematic manner to facilitate a comprehensive understanding of its characteristics.

The first step in descriptive statistics is data collection. This can involve various techniques such as sampling, where a subset of the population is selected to represent the entire group. The collected data can be quantitative (numerical) or qualitative (categorical), depending on the nature of the study.

Once the data is collected, it needs to be organized in a meaningful way. This can involve sorting, categorizing, and arranging the data into different groups or categories. For quantitative data, measures such as frequency distributions, histograms, and scatter plots can be used to organize and display the data. For qualitative data, methods such as tables, charts, or graphs may be employed to summarize the information.

Finally, descriptive statistics involves summarizing the collected and organized data to extract key insights and characteristics. This can include measures such as central tendency (mean, median, mode), variability (range, variance, standard deviation), and correlation. These measures provide a concise summary of the data, enabling researchers to gain a better understanding of its overall patterns, trends, and distribution.

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If a scatterplot shows a non-linear relationship between the response and explanatory variable, several options can be considered depending on the purpose of the analysis and the nature of the data.

Here are some possible actions:

Transform the data: One common approach is to transform the data to make the relationship linear. For example, if the relationship appears to be exponential, taking the logarithm of the response variable might make it linear. Similarly, taking the square root, cube root, or inverse of one or both variables might also help.

Use a non-linear model: Another option is to fit a non-linear model that can capture the curvature in the relationship. There are many types of non-linear models, such as quadratic, cubic, exponential, logistic, or spline models. The choice of model depends on the shape of the curve and the underlying theory or hypothesis.

Resample or subset the data: If the non-linear relationship is driven by outliers, influential points, or a subset of the data, it might be helpful to resample or subset the data to remove them. For example, trimming the extreme values, bootstrapping the data, or stratifying the data by a third variable might help.

Explore alternative variables or interactions: If the non-linear relationship is due to an unobserved or omitted variable, it might be useful to explore alternative variables or interactions that could explain the pattern. For example, if the response is sales and the explanatory variable is price, adding a competitor's price or a marketing variable might improve the fit.

Use caution in interpretation: Finally, if the non-linear relationship persists after exploring the above options, it might be necessary to acknowledge the non-linearity and use caution in interpreting the results. Non-linear relationships can be more difficult to interpret and extrapolate, and the statistical inference might be more uncertain or sensitive to assumptions.

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The correct answer is "the model of arrivals." A Poisson process is a statistical model used to describe the arrival times of events that occur randomly over time. These events are assumed to occur independently of each other and with a constant rate or intensity. The process is named after French mathematician Siméon Denis Poisson.

The model of departure times, on the other hand, describes the times at which objects or individuals leave a certain location or system. It is not necessarily a Poisson process and can depend on various factors such as the size of the system or the behavior of the individuals.

Dependent events are those that are influenced by previous events or conditions. They are not typically modeled using a Poisson process, as this assumes independence between events. However, there are other statistical models that can be used to describe dependent events.

Overall, it is important to understand the characteristics and assumptions of different statistical models in order to choose the appropriate one for a given situation.

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The probability is 28/45, which simplifies to 0.6222 or approximately 62.22%.

To calculate the probability of both light bulbs being good, we'll use the terms:

sample space, favorable outcomes, and probability.
Sample space:

This is the total number of possible outcomes when selecting two light bulbs.

Since there are 10 light bulbs in the drawer, the sample space is the number of ways to choose 2 light bulbs from 10, which can be calculated using combinations.

So, the sample space is C(10,2) = 10! / (2! * (10-2)!) = 45 possible outcomes.
Favorable outcomes:

These are the outcomes where both light bulbs are good.

There are 8 good light bulbs in the drawer,

so the number of favorable outcomes is the number of ways to choose 2 good light bulbs from the 8, which is C(8,2) = 8! / (2! * (8-2)!) = 28 favorable outcomes.
Probability: To find the probability of both light bulbs being good, divide the number of favorable outcomes by the sample space.

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Answer:  How do you know how many solutions an equation has?

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one soluti

Step-by-step explanation:

If x = 2, y = 3 and z = 4 then the value of ;

1. 3yz = 36

2. xy = 6

Substitution means replacing a particular variable with another term, either a constant or another variable. For example, if x = 2 and y = 1

3x²+y² will be calculated by replacing 2 for x and 1 for y in the expression.

Similarly;

if x = 2 , y = 3 and z = 4

then;

3yz = 3 × 3 × 4

= 9× 4

= 36

and xy = 2 × 3

= 6

therefore the value of 3yz and xy are 36 and 6; respectively.

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Answer:

C

Step-by-step explanation:

given the data in ascending order

1 , 3 , 5 , 6 , 7 , 9 , 100

              ↑ middle value

then the second quartile Q₂ ( the median ) is the middle value of the set

thus Q₂ = 6

the first quartile Q₁ is the middle value of the data to the left of the median

1 , 3 , 5

    ↑

Q₁ = 3

the third quartile Q₃ is the middle value of the data to the right of the median

7 , 9 , 100

     ↑

Q₃ = 9

the first , second and third quartiles are 3 , 6 and 9

The first, second, and third quartiles are 3, 6, and 9. The correct answer is option C) 3, 6, and 9.

The first quartile (Q1) is the value that divides the data set into quarters, with 25% of the data falling below this value. To find Q1, we need to locate the median of the first half of the data set. The first half of the data set consists of (1, 3, 5). The median of this set is 3, so Q1 is 3.

The second quartile (Q2) is the median of the entire data set, which is 6.

The third quartile (Q3) is the value that divides the data set into quarters, with 75% of the data falling below this value. To find Q3, we need to locate the median of the second half of the data set. The second half of the data set consists of (7, 9, 100). The median of this set is 9, so Q3 is 9.

Therefore, the first, second, and third quartiles of the given data set (1, 3, 5, 6, 7, 9, 100) are 3, 6, and 9 respectively. The correct answer is option C) 3, 6, and 9.

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The probability that a basketball player with a 71% free throw shooting accuracy does not make his next free throw shot is 29%.

The probability that a basketball player with a 71% free throw shooting accuracy does not make his next free throw shot.
To calculate the probability of missing a free throw shot, we need to subtract the shooting accuracy percentage from 100%.

In this case, the probability of making a free throw is 71%, which means the probability of missing the free throw is 29%.

Therefore, the probability that the basketball player does not make his next free throw shot is 29%.
It is important to note that free throw shooting accuracy can vary depending on the player's physical and mental condition, as well as external factors such as the audience's noise, the game's pressure, and the distance from the basket.

Thus, it is crucial for basketball players to train and practice regularly to improve their shooting skills and increase their chances of making free throw shots.
To answer this, we need to consider the complement of the success probability.

Since the player has a 71% chance of making the free throw, it means there is a 29% chance that he will not make it (100% - 71% = 29%).

The probability can also be expressed as a decimal, which is 0.29 (29/100 = 0.29).

Therefore, the probability that your favorite basketball player does not make his next free throw shot is 29% or 0.29.
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On flipping a coin, the number of total possibility for obtaining least one head if we flip the coin seven times are equal to 127.

We have an experiment for flip a coin and keep a record of the results.

Number of total possible outcomes on flipping a coin = 2 = { H, T}

Number of trials or a coin is flipped = 7

We have to determine the number of ways that at least one head if we flip the coin seven times. When we flip a two-sided coin n times, there are [tex]2^ n[/tex] possible outcomes. So, when a coin is flipped 7 times then total possible number of outcomes = 2⁷ = 128

Let's consider an Event A : getting at least one head in seven flips

Complement of this event is getting no head and it is denoted by [tex]A^ c =[/tex]{ T T T T T T T }

and this event will be occur in one way only. So, Number of ways of obtain at least one head if you flip the coin five times = 128 - 1 = 127 ways.

Hence, required value is 127 ways.

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There are 1,602,562,500 different passwords that can be made up using the given pattern and character choices.

We need to make a password with eight characters, where the first seven characters follow the pattern CVCVCVC, and the eighth character is a digit. There are 21 consonants and 5 vowels available, and there are 10 digits (0-9) available for the eighth character.

To find the number of possible passwords, we can count the number of choices for each character in the password, and then multiply the choices together.

For the first character, we can choose from 21 consonants. For the second character, we can choose from 5 vowels. For the third character, we can choose from 21 consonants again, and so on, alternating between consonants and vowels. For the eighth character, we can choose from 10 digits.

Therefore, the total number of possible passwords is:

21 x 5 x 21 x 5 x 21 x 5 x 21 x 10 = 1,602,562,500

So there are 1,602,562,500 different passwords that can be made up using the given pattern and character choices.

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Thus, the length of a side of the regular hexagon is 8 centimeters using the Pythagorean theorem.

The key to solving this problem is to realize that the perpendiculars drawn from the interior point to the sides of the hexagon form a right triangle with one leg being the perpendicular and the other leg being a side of the hexagon. We also know that the hexagon is regular, meaning all sides have the same length.

Let's label the length of the side of the hexagon as "x". We can use the Pythagorean theorem to find the length of each perpendicular as follows:

- For the perpendicular that is 4 cm long, we have x^2 = 4^2 + (x/2)^2
- For the perpendicular that is 5 cm long, we have x^2 = 5^2 + (x/2)^2
- For the perpendicular that is 6 cm long, we have x^2 = 6^2 + (x/2)^2
- For the perpendicular that is 8 cm long, we have x^2 = 8^2 + (x/2)^2
- For the perpendicular that is 9 cm long, we have x^2 = 9^2 + (x/2)^2
- For the perpendicular that is 10 cm long, we have x^2 = 10^2 + (x/2)^2

Simplifying each equation and using a bit of algebra, we get:

- 3x^2 = 16^2
- 7x^2 = 25^2
- 12x^2 = 36^2
- 24x^2 = 64^2
- 33x^2 = 81^2
- 40x^2 = 100^2

Solving for x in each equation, we find that x = 8 cm. Therefore, the length of a side of the regular hexagon is 8 centimeters.

In summary, we used the fact that the perpendiculars from an interior point to the sides of a regular hexagon form right triangles to set up equations using the Pythagorean theorem. Solving for the length of a side of the hexagon in each equation, we found that it is 8 cm long.

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Let G be a triangulation with n vertices, m edges and f faces. Since G is a triangulation, we know that each face has three edges and each edge is incident to two faces. Therefore, we have:

3f = 2m (1)

Now, let us consider the sum of the degrees of all the vertices in G. Each vertex v is incident to some number of edges, say d(v). Since each edge is incident to exactly two vertices, the sum of the degrees of all the vertices is given by:

2m = Σv∈G d(v) (2)

Using the handshake lemma, we know that the sum of degrees of all vertices in a graph is twice the number of edges, which gives us equation (2).

Now, we can find the average degree of a vertex in G by dividing the sum of degrees of all vertices by the number of vertices:

average degree = (Σv∈G d(v))/n

Substituting equation (2) into the above expression, we get:

average degree = 2m/n

Now, we can use Euler's formula to relate the number of vertices, edges, and faces in a planar graph:

n - m + f = 2

Substituting equation (1) into the above expression, we get:

n - (3f/2) + f = 2

n/2 = f + 2

Substituting the above expression into equation (2), we get:

2m = Σv∈G d(v) <= Σv∈G 6 = 6n

Dividing by n on both sides, we obtain:

2m/n <= 6

Substituting this expression into the expression for the average degree, we have:

average degree = 2m/n <= 6

Therefore, the average degree of a vertex in a triangulation is strictly less than 6.

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