Suppose in a theoretical experiment there is one favorable outcome. If two other outcomes are removed, the theoretical probability

To find the percentage of all students who like science but not humanities, we need to subtract the percentage of students who chose both science and humanities from the percentage of students who chose science only. 30% of all students like Science but not Humanities.

So, the percentage of students who like science but not humanities is:

41% (students who chose science) - 11% (students who chose both science and humanities) = 30%

Therefore, 30% of all students like science but not humanities.

To find the percentage of students who like Science but not Humanities, we need to subtract the percentage of students who chose both from the percentage who chose Science. Here's the step-by-step explanation:

1. The percentage of students who chose Science: 41%
2. The percentage of students who chose both Science and Humanities: 11%
3. Subtract the percentage of students who chose both from the percentage who chose Science: 41% - 11% = 30%

So, 30% of all students like Science but not Humanities.

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Answer: x=8

Step-by-step explanation: to get the answer you try to isolate the x you do this by multiplying both side by 10 which you get x= 40/5 which you can simplify so 8

Answer:

The answer to your problem is,

Step-by-step explanation:

So in this problem we will solve for ‘ x ‘

We will make equivalent fraction on the way.

The problem; [tex]\frac{4}{5}[/tex] = [tex]\frac{?}{10}[/tex]

How to find equivalent fractions:

First in order to find equivalent fractions do as I follow:

1. Multiply by 2 for the denominator and numerator

Make it into this fraction, [tex]\frac{4*2}{5*2}[/tex].

2. Solve:

4 × 2 = 8[tex]\frac{8}{10}[/tex]

5 × 2 = 10

3. Make into fraction, [tex]\frac{8}{10}[/tex]

[tex]\frac{8}{10}[/tex] is the answer

Thus the answer to your problem is, [tex]\frac{8}{10}[/tex]

The distance of the losing shot from the center is also uniformly distributed from 0 to 1.  To find the PDF of the distance of the losing shot from the center, we need to first determine the probability of one shot being closer to the center than the other.

Let X be the distance of the first shot from the center, and Y be the distance of the second shot from the center. Then, the probability that X is closer to the center than Y is given by the area of the region where X < Y, which is a triangular region with base 1 and height 1/2 (since the probability of X being closer to the center than Y is the same as the probability of Y being closer to the center than X). Therefore, the probability of X being closer to the center than Y is 1/4.

Now, let Z be the distance of the losing shot from the center. We know that Z is equal to the distance of the second shot from the center if the second shot is closer to the center than the first shot, and it is equal to the distance of the first shot from the center if the first shot is closer to the center than the second shot. Therefore, the PDF of Z is given by:

fZ(z) = (1/4)fY(z) + (3/4)fX(z)

where fX(x) and fY(y) are the PDFs of X and Y, respectively. Since X and Y are uniformly distributed from 0 to 1, their PDFs are both equal to 1 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Therefore, we have:

fZ(z) = (1/4) + (3/4) = 1

for 0 ≤ z ≤ 1. This means that the distance of the losing shot from the center is also uniformly distributed from 0 to 1.

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The set of values that satisfy the inequality is (-2, 0, 2, 4, 6).

Option A is the correct answer.

We have,

We can solve the inequality as follows:

(1/2)x + 3 ≥ 0

Subtracting 3 from both sides:

(1/2)x ≥ -3

Multiplying both sides by 2

(note that since we are multiplying by a positive number, we do not need to reverse the inequality):

x ≥ -6

We see that,

Out of the given sets of values, only (-2, 0, 2, 4, 6) contains values that satisfy this inequality.

Therefore,

The set of values that satisfy the inequality is (-2, 0, 2, 4, 6).

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a. There are three independent variables in a 4 x 2 x 3 factorial design.

b. There are four possible main effects in a 4 x 2 x 3 factorial design, one for each independent variable.

c. There are 24 different conditions in a 4 x 2 x 3 factorial design, which is the product of the levels of each independent variable (4 x 2 x 3 = 24).

d. There are several possible interactions in a 4 x 2 x 3 factorial design, including two-way and three-way interactions between the independent variables.

a). In a factorial design, the independent variables are systematically manipulated in all possible combinations. In a 4 x 2 x 3 factorial design, there are three independent variables, each with a different number of levels.

The first independent variable has four levels, the second independent variable has two levels, and the third independent variable has three levels.

Therefore, there are three independent variables in this design.

b). A main effect is the effect of a single independent variable on the dependent variable, averaged across all levels of the other independent variables.

In a 4 x 2 x 3 factorial design, there are three independent variables, each of which can have a main effect. Therefore, there are three possible main effects in this design.

Additionally, there can be two-way and three-way interactions between the independent variables, which can also affect the dependent variable. However, a main effect is distinct from an interaction effect and is not dependent on any other independent variables.

c).The number of different conditions in a 4 x 2 x 3 factorial design is calculated by multiplying the number of levels for each independent variable.

In this case, the first independent variable has four levels, the second independent variable has two levels, and the third independent variable has three levels.

Therefore, the total number of different conditions is obtained by multiplying these three numbers (4 x 2 x 3 = 24). Each condition represents a unique combination of the levels of the three independent variables.

d). Interactions in factorial designs occur when the effect of one independent variable on the dependent variable depends on the level of another independent variable. In a 4 x 2 x 3 factorial design, there are several possible interactions between the independent variables.

For example, there may be a two-way interaction between the first and second independent variables, meaning that the effect of the first independent variable on the dependent variable depends on the level of the second independent variable.

Similarly, there may be a two-way interaction between the second and third independent variables, or between the first and third independent variables.

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The values of the sides and the angle in the triangle will be:

AB = 15

m∠A = 53.1°

m∠B = 36.9°

It should be noted that a triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees

From the picture attached,

By Applying Pythagoras theorem,

AB² = AC² + BC²

AB² = 9² + (12)²

AB² = 81 + 144

AB = √225

AB = 15

tan(B) = opposite / adjacent

         = 12/9

         = 1.3333

B will be inverse of town (1.333)

  = 53.13

  ≈ 53.1°

Since, m∠A + m∠B + m∠C = 180°

m∠A + 53.1° + 90° = 180°

m∠A = 180°- 143.1°

       = 36.9°

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This can be calculated by adding up the total number of passengers on all five flights (41+39+40+39+33 = 192) and dividing it by the total number of available seats on those flights (69 x 5 = 345). So, 192 divided by 345 equals 0.774 or 77.4%. The load factor for Jim Bob Airlines' last five flights was 55.65%.

To find the load factor for Jim Bob Airlines' last five flights, follow these steps:

1. Add the number of passengers on each flight:
  41 (MSY to DFW) + 39 (DFW to OKL) + 40 (OKL to TUL) + 39 (TUL to FWB) + 33 (FWB to MSY) = 192 passengers

2. Calculate the total number of available seats for the five flights:
  69 seats per flight × 5 flights = 345 available seats

3. Calculate the load factor by dividing the total passengers by the total available seats:
  Load factor = (192 passengers) / (345 available seats) = 0.5565 (rounded to four decimal places)

4. Convert the load factor to a percentage:
  Load factor percentage = 0.5565 × 100 = 55.65%

The load factor for Jim Bob Airlines' last five flights was 55.65%.

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Thus, the probability is 1/100 or 0.01, which means there is a 1% chance of getting a triple-digit number with all the same digits in this lottery game.

To calculate the probability of getting a triple-digit number where all digits are the same in a lottery game that consists of selecting a three-digit number, you should first determine the number of favorable outcomes and the total possible outcomes.

There are 9 favorable outcomes, as the triple-digit numbers with the same digits are: 111, 222, 333, 444, 555, 666, 777, 888, and 999. Note that we don't include 000 since it's not a triple-digit number.

Now, let's find the total possible outcomes. In a three-digit number, there are 10 possible digits (0-9) for each of the three positions. However, the first position cannot be 0, as that would not be a triple-digit number.

So, there are 9 possible digits for the first position and 10 for the other two. The total possible outcomes are 9 x 10 x 10, which equals 900.

Finally, to find the probability of getting a triple-digit number with all the same digits, divide the number of favorable outcomes by the total possible outcomes:

Probability = Favorable Outcomes / Total Possible Outcomes
Probability = 9 / 900

The probability is 1/100 or 0.01, which means there is a 1% chance of getting a triple-digit number with all the same digits in this lottery game.

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Normal random variable with mean 500 and standard deviation 100 (a) the probability that five randomly chosen seniors all score below 600 is approximately 0.4437. (b) a student must obtain a score of 766.5 or higher to be in the 99th percentile.

(a) To find the probability that five randomly chosen seniors all score below 600, we need to use the normal distribution formula. We know that the mean (μ) is 500 and the standard deviation (σ) is 100. To find the probability of a score below 600, we need to find the z-score for 600.

z = (x - μ) / σ

z = (600 - 500) / 100 = 1

Using a standard normal distribution table or calculator, we can find that the probability of a score below 600 is 0.8413. To find the probability of five randomly chosen seniors all scoring below 600, we need to multiply this probability by itself five times.

P(X < 600)^5 = 0.8413^5 = 0.4437

Therefore, the probability that five randomly chosen seniors all score below 600 is approximately 0.4437.

(b) To be in the 99th percentile, a student must score higher than 99% of the population. We can use the normal distribution table or calculator to find the z-score for the 99th percentile.

z = 2.33

Using the same formula as before, we can solve for x (the score needed to be in the 99th percentile).

2.33 = (x - 500) / 100

x = 500 + 2.33(100)

x = 766.5

Therefore, a student must obtain a score of 766.5 or higher to be in the 99th percentile.

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Answer:

To solve for x in the equation 8x + 7 = 6x + 15, you need to isolate the variable (x) on one side of the equation.

First, you can start by subtracting 6x from both sides of the equation to get:

8x + 7 - 6x = 15

Simplifying this gives: 2x + 7 = 15

Next, you can subtract 7 from both sides of the equation to get: 2x = 8

Finally, divide both sides of the equation by 2 to solve for x: x = 4

Therefore, the solution for x in the equation 8x + 7 = 6x + 15 is x = 4

Answer:

x = 4

Step-by-step explanation:

To solve the equation 8x + 7 = 6x + 15, you can start by isolating the variable on one side of the equation. To do this, you can subtract 6x from both sides of the equation to get 2x + 7 = 15. Then, you can subtract 7 from both sides of the equation to get 2x = 8. Finally, you can divide both sides of the equation by 2 to get x = 4 1.

I hope that helps!

The three-inch cube would be worth [tex]9\text{ lb}\times $20/\text{lb} = $180$.[/tex]

Since the cube is made of silver, we know that the ratio of volume to weight is constant, meaning that the density of silver is the same throughout the cube. Therefore, if a two-inch cube of silver weighs 3 pounds, then its volume is [tex]$(2\text{ in})^3=8\text{ in}^3$[/tex] and its density is [tex]3\text{ lb}/8\text{ in}^3 = 0.375\text{ lb/in}^3$.[/tex]

Now let's consider a three-inch cube. Its volume is [tex]$(3\text{ in})^3=27\text{ in}^3$[/tex], which is 3 times the volume of the two-inch cube. Since the density is the same, the weight of the three-inch cube will be 3 times the weight of the two-inch cube, or [tex]3\times 3\text{ lb}=9\text{ lb}$.[/tex]

To find the value of the three-inch cube, we need to know the price of silver per pound. Let's assume it's [tex]$$20$[/tex] per pound (this is just an example). Then the three-inch cube would be worth [tex]9\text{ lb}\times $20/\text{lb} = $180$.[/tex]

In general, the value of the three-inch cube would be proportional to the weight and the price of silver per pound.

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Jim and Mary, both from Berkeley and of the same age, take separate trips to Los Angeles. Jim travels at a velocity of 65 miles per hour and reaches his destination, where he waits for Mary. The next day, Mary begins her journey, driving at a faster velocity of 70 miles per hour.

Upon arriving in Los Angeles, both Jim and Mary are still the same age. This is because age is a function of time, and both individuals have aged the same amount of time since the start of their journeys. While their respective velocities and travel times may differ, their ages remain equal as they are both progressing through time at the same rate.

In conclusion, Jim and Mary will be the exact same age when they meet in Los Angeles, regardless of their differing travel velocities. The difference in speed does not affect their aging process, as time progresses at the same rate for both of them.

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The type of study described in the question is an observational study.

Specifically, it is a cross-sectional study because data was collected at a single point in time from a representative sample of the population of interest. In this study, the researcher used existing data from the 2014 U.S. National Health Interview Study to explore the relationship between smoking and depression.

The study did not involve manipulating any variables or assigning participants to different groups, which are characteristics of experimental studies. Instead, the researcher examined data on smoking and depression status to see if there was a correlation between the two variables.

Since the study was observational, it is not possible to establish causality or determine the direction of the relationship between smoking and depression. However, the results may be useful for generating hypotheses for further research or informing public health policies related to smoking and mental health.

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Answer: 240 cubic feet

The length of the trout that represents the 45th percentile is 32 centimeters.

The trout with a length of 34 centimeters is at the 70th percentile.

From the data:

26, 28, 28, 29, 30, 31, 31, 31, 32, 33, 33, 33, 33, 34, 36, 36, 37, 38, 38, 40

Using this formula, we can calculate the index of the observation that represents the 45th percentile:

= (45/100) × 20 = 9

For the second part,

= [(14/20) × 100] = 70

This means that the trout with a length of 34 centimeters is at the 70th percentile.

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(a) To find E(Ti2), we can use the fact that the interarrival times of a Poisson process are exponentially distributed. Since T2 is the time of the second arrival,

We can write T2 = T1 + X, where X is the time between the first and second arrivals. Thus, we have:

E(T2) = E(T1 + X) = E(T1) + E(X)

Since the Poisson process has rate 3, the interarrival times are exponentially distributed with parameter λ = 3. Therefore, we have E(X) = 1/λ = 1/3. Also, the time of the first arrival is distributed as an exponential random variable with parameter λ, so we have E(T1) = 1/λ = 1/3. Putting it all together, we get:

E(T2) = E(T1) + E(X) = 1/3 + 1/3 = 2/3

Therefore, E(Ti2) = 2/3.

(b) To find E(Tiz/N(2) = 5), we need to condition on the value of N(2). We have:

E(Ti2/N(2) = 5) = ∑k≥2 E(Ti2/N(2) = 5, N(2) = k) P(N(2) = k)

Since the Poisson process has independent and stationary increments, we know that the distribution of N(2) is Poisson with parameter 6. Therefore, we have:

P(N(2) = k) = e^(-6) 6^k / k!

For a fixed value of N(2) = k, we can think of the process up to time T2 as a Poisson process with rate 3, and condition on the times of the first k arrivals. The time of the ith arrival, given the times of the first i-1 arrivals, is distributed as an exponential random variable with parameter λ = 3. Therefore, we have:

E(Ti2/N(2) = 5, N(2) = k) = E(Ti2 | T1, T2, ..., Tk)

Using the memoryless property of the exponential distribution, we can write:

E(Ti2 | T1, T2, ..., Ti-1) = Ti + E(T2 | T1, T2, ..., Ti-1) = Ti + 2/3

Therefore, we have:

E(Ti2/N(2) = 5, N(2) = k) = Ti + 2/3

Putting it all together, we get:

E(Tiz/N(2) = 5) = ∑k≥2 ∑i≥1 (Ti + 2/3) e^(-6) 6^k / k!

Using the fact that the interarrival times are exponentially distributed, we can compute the sum over i as:

∑i≥1 (Ti + 2/3) = E(T2) + 2/3 = 8/3

Therefore, we have:

E(Tiz/N(2) = 5) = (8/3) ∑k≥2 e^(-6) 6^k / k! = (8/3) (1 - e^(-12))

Thus, E(Tiz/N(2) = 5) ≈ 1.81.

(c) To find E(N(S) N(2) = 5), we can use the fact that the number of arrivals in a Poisson process of rate λ in an interval of length t is a Poisson random variable with parameter λt. Therefore, we have:

E(N(S) N(2) = 5) = E(N(5) N(2) = 5) = E(N(5)^2 | N(2) = 5) P(N(2) = 5)

For a fixed value of N(2) = 5, we can think of the process up to time 5 as a Poisson process with rate 3, and condition on the times of the first 5 arrivals. Therefore, we have:

E(N(5)^2 | N(2) = 5) = E((N(5) - 5)^2 | N(2) = 5) + E(10 N(5) - 25 | N(2) = 5) + 25

Using the fact that the number of arrivals in an interval of length t is Poisson with parameter λt, we have:

E((N(5) - 5)^2 | N(2) = 5) = Var(N(3)) = 3

Also, we have:

E(10 N(5) - 25 | N(2) = 5) = 10 E(N(5) | N(2) = 5) - 25 = 10 (5 + 2) - 25 = 15

Putting it all together, we get:

E(N(S) N(2) = 5) = (3 + 15 + 25) P(N(2) = 5) = 43 e^(-6) 6^5 / 5!

Thus, E(N(S) N(2) = 5) ≈ 1.94.

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A graph of the given angle measures is shown in the image attached below.

In Mathematics and Geometry, a rotation refers to a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

In this scenario, we would use an online graphing to calculator to plot or draw each of the angle measures that are provided as shown in the graph attached below.

For instance, we would convert -π/2 to degrees;

θ = -π/2 × 180/π

θ = -180/2

θ = -90 degrees.

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The quantity of the substance left at any time t > 0 can be found using the formula q(t) = 20 * 2^(-t/3200).The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay. In this case, the half-life is 3200 years, which means that after 3200 years, half of the substance will have decayed.

After another 3200 years, half of the remaining substance will have decayed, and so on.

To find the quantity q(t) of the substance left at time t > 0, we can use the formula q(t) = q(0) * 2^(-t/h), where q(0) is the initial quantity of the substance, t is the time elapsed, and h is the half-life of the substance.

In this case, q(0) = 20 g and h = 3200 years. So, the formula becomes q(t) = 20 * 2^(-t/3200).

For example, if t = 3200 years, then q(t) = 20 * 2^(-3200/3200) = 10 g, which means that half of the substance has decayed. If t = 6400 years, then q(t) = 20 * 2^(-6400/3200) = 5 g, which means that three-quarters of the substance has decayed.

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Let's call the cost of the dryer "x".

We know from the problem that the washer costs $68 less than the dryer, so the cost of the washer would be "x - 68".

The problem also tells us that the combined cost of the washer and dryer is $782. So we can set up an equation:

x + (x - 68) = 782

Simplifying this equation, we get:

2x - 68 = 782

Adding 68 to both sides, we get:

2x = 850

Dividing both sides by 2, we get:

x = 425

So the dryer costs $425.

The probability that the sample mean would be less than 23,013 dollars if a sample of 134 persons is randomly selected is approximately 0.0000397.

We can use the central limit theorem to approximate the distribution of the sample mean as a normal distribution, with a mean of μ = 23,037 dollars and a standard deviation of σ/√n = √(149,769/134) = 33.23 dollars.

Then we can standardize the sample mean using the z-score formula:

z = ([tex]\bar{X}[/tex] - μ) / (σ/√n)

where [tex]\bar{X}[/tex] is the sample mean.

Plugging in the given values, we get:

z = (23,013 - 23,037) / 33.23 ≈ -0.722

Using a calculator, we can find that the probability of getting a z-score less than -0.722 is approximately 0.0000397.

Therefore, the probability that the sample mean would be less than 23,013 dollars if a sample of 134 persons is randomly selected is approximately 0.0000397.

This is a very small probability, indicating that it is unlikely to obtain a sample mean this low if the true population mean is 23,037 dollars.

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The factored version of the equation would be (b + 9) (b - 7) = 0.

We are given the equation:

b ^ 2 + 2b - 72 + 9 = 0

b ^ 2 + 2b - 63 = 0

We can rewrite this equation, based on two numbers that multiply to - 63 and add up to 2. These numbers would be 9 and 7 so the equation becomes :

(b + 9) (b - 7) = 0

The answers after the simplification would be:

b + 9 = 0

b = -9

b - 7 = 0

b = 7

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It's important to consider the overall nutrient profile of a cereal, rather than just focusing on a few specific ingredients.

To determine which ingredients are positively correlated with the nutritional rating, we can examine the scatter plots and look for a general trend between each ingredient and the rating.

In general, the higher the nutritional rating, the lower the values for fat, sugar, and sodium, and the higher the values for protein and fiber.

Based on this observation, we can conclude that protein and fiber are positively correlated with the nutritional rating, while fat, sugar, and sodium are negatively correlated with the nutritional rating.

It's worth noting that correlation does not imply causation, and there may be other factors that contribute to the nutritional value of a cereal beyond these five ingredients.

Therefore, it's important to consider the overall nutrient profile of a cereal, rather than just focusing on a few specific ingredients.

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Dilbert's boss was wrong to use the number 40% to assume that the employees were cheating because it only represents a percentage and not the actual number of days that the employees were absent.

In reality, if the workweek is 5 days, then 40% would only be 2 days. Therefore, assuming that the employees were cheating based on this percentage alone is not a fair or accurate assessment. It's important to consider the actual number of days missed by each employee and investigate the reasons for their absence before jumping to conclusions.

While 40% might seem like nearly half of the days in a workweek, it doesn't necessarily imply that employees are cheating. To make a fair assessment, one should consider other factors such as workload, productivity, and employee performance, rather than solely focusing on a percentage.

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To indicate the direction in which the curve is traced as the parameter increases, we can draw an arrow that points from lower left to upper right, since the curve moves from the point (1, 1) at t = 0 to the right and upward as t increases.

To eliminate the parameter, we can use the fact that x = e^t and y = e^(-3t) to solve for t in terms of x and y. Taking the natural logarithm of both sides of x = e^t, we get ln(x) = t. Similarly, taking the natural logarithm of both sides of y = e^(-3t), we get ln(y) = -3t, or t = (-1/3)ln(y). Substituting this expression for t into the equation we found for ln(x), we get ln(x) = (-1/3)ln(y), which simplifies to ln(x^3) = ln(y^(-1)), or x^3 = 1/y. This is the Cartesian equation of the curve.

To sketch the curve, we can start by noting that both x and y are positive for all values of t, since e^t and e^(-3t) are always positive. As t increases, x and y both increase, but y increases much more slowly than x since e^(-3t) decreases rapidly as t increases. This means that the curve starts out very steep (since the slope of the tangent line at t = 0 is dx/dt = 1 and dy/dt = -3), but becomes flatter and flatter as t increases. The curve approaches the x-axis as t approaches infinity, but never touches it.

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The US Crime Commission needs to take a sample of at least 1068 crimes to estimate the proportion of crimes in which firearms are used with a margin of error of 3% and a 95% confidence level, assuming no preliminary estimate is available.

To calculate the minimum required sample size, we can use the formula:

[tex]n = [(Z^2) \times p \times q] / E^2[/tex]

where:

n is the sample size

Z is the Z-score corresponding to the desired confidence level (95% in this case), which is 1.96

p is the estimated proportion of crimes in which firearms are used (since no preliminary estimate is available, we can use 0.5 as a conservative estimate to get the maximum sample size)

q is the complement of p, which is 1 - p

E is the margin of error, which is 3% or 0.03

Substituting the values, we get:

[tex]n = [(1.96^2) \times 0.5 \times 0.5] / 0.03^2[/tex]

n = 1067.11

Rounding up to the nearest integer, the minimum required sample size is 1068. Therefore, the US Crime Commission needs to take a sample of at least 1068 crimes to estimate the proportion of crimes in which firearms are used with a margin of error of 3% and a 95% confidence level, assuming no preliminary estimate is available.

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Answer:

25 x 0.01

Step-by-step explanation:

here you go . ..........................................

The , th only eigenvalue for the transformation that rotates points by 90 degrees about some axis through the origin is 1.

Let's assume that the transformation rotates points by 90 degrees about some axis through the origin. We can represent this transformation by a matrix A, and the eigenvectors of A will be the axis of rotation. Since the rotation is by 90 degrees, the eigenvectors will be orthogonal to the axis of rotation.

To find the eigenvalues of A, we can use the characteristic equation:

det(A - λI) = 0

where λ is the eigenvalue and I is the identity matrix. Since A is a rotation matrix, its determinant is equal to 1, and we can write:

det(A - λI) = det(A) - λ det(I) = 1 - λ

To find the eigenvalues, we need to solve the equation:

1 - λ = 0

which gives us λ = 1. Therefore, the only eigenvalue for the transformation that rotates points by 90 degrees about some axis through the origin is 1.

Note that the eigenvectors associated with this eigenvalue will be any two orthogonal vectors in the plane perpendicular to the axis of rotation.

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We will use mathematical induction to prove the given statement: Base Case: For n = 1, we have 8 = (7(1)(1) + 2(1))² = (7(1) - 1)² = 49 - 14 + 1 = 36, which is true.

Inductive Hypothesis: Assume that the statement is true for some positive integer k, i.e.,

8 + 15 + 22 + ... + (7k - 1) = (7k(k - 1) + 2k)²

Inductive Step: We need to prove that the statement is also true for k + 1, i.e.,

8 + 15 + 22 + ... + (7(k + 1) - 1) = (7(k + 1)(k) + 2(k + 1))²

Starting with the left-hand side, we can rewrite it as:

8 + 15 + 22 + ... + (7(k + 1) - 1) = (8 + 15 + 22 + ... + (7k - 1)) + (7(k + 1) - 1)

Using the inductive hypothesis, we can substitute the expression for the sum of the first k terms:

(7k(k - 1) + 2k)² + (7(k + 1) - 1)

Expanding the square and simplifying, we get:

49k² + 49k + 14 + 14k + 1

= 49k² + 63k + 15

= 7(k + 1)(7k + 9)

Now, we can rewrite the right-hand side of the statement for k + 1 as:

(7(k + 1)(k) + 2(k + 1))²

= (7k(k + 1) + 2(k + 1))²

= (7k² + 7k + 2k + 1)²

= (7k² + 9k + 1)²

= 49k⁴ + 98k³ + 62k² + 18k + 1

= 7(k + 1)(7k + 9)

This is exactly the same as the expression we obtained for the left-hand side, so we have shown that the statement is also true for k + 1.

Therefore, by mathematical induction, the statement is true for all positive integers n.

Hence, we have proved that 8 + 15 + 22 + ... + (7n - 1) = (7n(n - 1) + 2n)² for all positive integers n.

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