Prove the following
[tex] {sin}^{2} ( \frac{\pi}{4} - \alpha ) = \frac{1}{2} (1 - sin2 \alpha )[/tex]
​

The company should survey at least 601 customers to be 95% confident that the estimated proportion is within 4 percentage points of the true population proportion.

To determine the sample size needed, we can use the formula:

n = (z^2 * p * (1-p)) / E^2

where:

n is the sample size needed

z is the z-score for the desired confidence level (in this case, 1.96 for 95% confidence)

p is the estimated population proportion (we don't have an estimate, so we'll use 0.5, which gives the largest possible sample size)

E is the maximum margin of error (4 percentage points in this case, or 0.04)

Plugging in the values, we get:

n = (1.96^2 * 0.5 * 0.5) / 0.04^2

n = 600.25

We round up to the nearest whole number to get a sample size of 601.

This means that if the company surveys 601 randomly selected customers and finds that, for example, 60% of them keep up with regular vehicle maintenance, we can be 95% confident that the true proportion of all customers who keep up with regular vehicle maintenance is between 56% and 64%.

Learn more about proportion    here: brainly.com/question/31548894

#SPJ11

The coordinates where the fountain will be located is (2, 1/2)

From the question, we have the following parameters that can be used in our computation:

The center town forms a parallelogram with the vertices

A = (0, 4); B = (6, 1); C = (4, -3); D = (-2, 0)

The corner points when placed on a coordinate grid can be calculated using

Mid-point = 1/2(x1 + y1, x2 + y2)

So, we have

Center = 1/2 * (0 + 4, 4 - 3) or

Center = 1/2 * (6 - 2, 1 + 0)

Evaluate

Center = (2, 1/2) or Center = (2, 1/2)

Hence, the coordinates where the fountain will be located is (2, 1/2)

Read more about midpoint at

https://brainly.com/question/28462165

#SPJ1

Our quotient is 1482 R 4. We can also write this as 1482 and 4/5 or 1482.8.

To find the quotient of a division problem, we must divide the dividend by the divisor. In this case, our dividend is 7415 R 112 R 412 R 214 R 4 and our divisor is 5.

First, we divide 5 into 74 and get a quotient of 14 with a remainder of 4. We bring down the next digit, 1, and get a new dividend of 41. We divide 5 into 41 and get a quotient of 8 with a remainder of 1. We bring down the next digit, 2, and get a new dividend of 12.

We divide 5 into 12 and get a quotient of 2 with a remainder of 2. We bring down the next digit, 1, and get a new dividend of 21. We divide 5 into 21 and get a quotient of 4 with a remainder of 1.

Finally, we bring down the last digit, 4, and get a new dividend of 14. We divide 5 into 14 and get a quotient of 2 with a remainder of 4.

Therefore, our quotient is 1482 R 4. We can also write this as 1482 and 4/5 or 1482.8.

For more such questions on Our quotient

https://brainly.com/question/11418015

#SPJ11

The value of b such that p(70 ≤ x ≤ b) = 0.3 is 67.4 (rounded to one decimal place). The deviation from the mean for this value of b is about -2.12 times the standard deviation of 5.

To find b, we can use the z-score formula, where z = (x - μ) / σ. In this case, we want to find the value of b that corresponds to a probability of 0.3, which means that the area under the normal distribution curve between 70 and b is 0.3.
First, we need to find the z-score for x = 70. Using the formula, we get:
z = (70 - 70) / 5 =
Next, we need to find the z-score for the value of b that corresponds to a probability of 0.3. We can use a standard normal distribution table or a calculator to find this value. For example, using a calculator, we can input:
invNorm(0.3) = -0.5244
This means that the z-score for the value of b is -0.5244. We can use the z-score formula again to find the actual value of b:
-0.5244 = (b - 70) / 5
Solving for b, we get:
b = 67.38
Therefore, the value of b such that p(70 ≤ x ≤ b) = 0.3 is 67.4 (rounded to one decimal place).
In terms of deviation, we can see that the value of b is about 2.12 standard deviations below the mean (z = -0.5244 corresponds to an area of 0.3 under the normal distribution curve). This tells us that the value of b is relatively low compared to the mean of 70. The deviation from the mean for this value of b is about -2.12 times the standard deviation of 5.

To know more about decimal place visit :

https://brainly.com/question/29770338

#SPJ11

Answer:

It is translated left 6 units.

The probability that the randomly selected sample will have less than 97.5% of the units that can hold at least 400 lbs of weight before breaking is 2.4249.

What is the binomial distribution?

In probability theory and statistics, the discrete probability distribution of the number of successes in a series of n separate experiments, each asking a yes-or-no question and each with its own Boolean-valued outcome: success or failure, is known as the binomial distribution with parameters n and p.

Here, we have

Given: For one product, 98% of all the units made at a particular factory can hold at least 400 lbs. of weight before breaking to check the product, quality control selects a random sample of 300 units made at the factory and determines whether or not the product can hold at least 400 lbs.

p = 0.98

q = 1 - p = 1 - 0.98 = 0.02

n = 300

Using the binomial distribution,

Standard deviation = σ = √npq = √300 × 0.98 × 0.02 = 2.4249

Standard deviation = σ = 2.4249

Hence, the probability that the randomly selected sample will have less than 97.5% of the units that can hold at least 400 lbs of weight before breaking is 2.4249.

To learn more about the binomial distribution from the given link

https://brainly.com/question/9325204

#SPJ4

The Number of tenths in 37.6 is 37.6 or 37 6/10 or 37 3/5 so the correct answer was given by Student 2

A decimal is simply another way of representing a fraction with a denominator of 10, 100, 1000, or any power of 10. In other words, the decimal point separates the whole number part from the fractional part, with each digit to the right of the decimal point representing a different power of 10.

The teacher asked how many tenths are equivalent to 37.6, which means we're looking for a fraction with a denominator of 10. To figure out the answer, we need to convert 37.6 into a fraction with a denominator of 10.

To do this, we look at the digit in the tenths place, which is 6. This tells us that 37.6 is equivalent to 37 and 6 tenths,

= 37 6/10.

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which in this case is 2.

= 37 3/5.

So, which student is correct? Student 2 answered 37.6 tenths, which is equivalent to 37 and 6 tenths, or 37 6/10. This means that Student 2's answer is correct.

To now more about Decimal here

https://brainly.com/question/9543292

#SPJ4

Answer:50

Step-by-step explanation:

50 , middle school students preferred espresso-flavored frozen yogurt.

The circle was created so the percentage of the total circle is 100%

Therefore, the sum of percentages of frozen yogurt flavors is 100

Dutch chocolate + country vanilla + sweet coconut + espresso + cake batter =100

Given,

Dutch chocolate = 21.5 percent,

country vanilla = 28.5 percent,

sweet coconut = 13 percent,

espresso = x (say),

cake batter = 27 percent

substituting in above we get x

21.5+28.5+13+x+27=100

90+x=100

x=100-90

x=10

Therefore, espresso = 10 percent

Total students = 10% of 500

=50

Read more about percentage:

https://brainly.com/question/24877689

Since we need to round up to the next integer, the required sample size for this experiment is 284 people for the confidence level.

To find the required sample size for NASA's experiment, we can use the following formula for the sample size estimation in a proportion experiment:

[tex]n = (Z^2 * p * (1-p)) / E^2[/tex]

where:
- n is the sample size
- Z is the z-score corresponding to the desired confidence level (85% in this case)
- p is the estimated population proportion (0.33)
- E is the margin of error (0.04)

First, we need to find the z-score for an 85% confidence level. We can look this up in a z-table, or use an online calculator. The z-score for an 85% confidence level is approximately 1.44.

Next, we can plug the values into the formula:
[tex]n = (1.44^2 * 0.33 * (1-0.33)) / 0.04^2[/tex]
n ≈ (2.0736 * 0.33 * 0.67) / 0.0016
n ≈ 283.66

Since we need to round up to the next integer, the required sample size for this experiment is 284 people.

Learn more about confidence level here:

https://brainly.com/question/22851322

#SPJ11

In a two-player game with one player having four available strategies and the other player having three available strategies, there can be a total of twelve outcomes. This is determined by the four strategies available to the first player multiplied by the three strategies available to the second player.

Solving a system of equations we can see that she has 10 dimes and 50 pennies.

Let's define the variables:

x = number of pennies.

y = number of dimes.

With the given information we can write a system of equations:

x = 5*y

x*0.01 + y*0.10 = 1.50

We can replace the first equation into the second one:

5*y*0.01 + y*0.10 = 1.50

y*0.15 = 1.50

y = 1.50/0.15 = 10

So there are 10 dimes and 50 pennies.

Learn more about systems of equations at:

https://brainly.com/question/13729904

#SPJ1

The power factor in a system is defined as the cosine of the angle between the voltage and current waveforms. The power factor in this system is 0.906, indicating a relatively efficient use of power.

In this case, the voltage waveform is given as V = 120V sin(377t + 20°) and the current waveform is given as I = 60A sin(377t + 45°). To find the power factor, we need to determine the angle between the voltage and current waveforms. First, let's convert the voltage and current waveforms to phasor form:
V = 120V ∠ 20°
I = 60A ∠ 45°
The angle between the voltage and current phasors is given by:
θ = θv - θi = 20° - 45° = -25°
The power factor is the cosine of this angle, so:
PF = cos(-25°) = 0.906

Learn more about voltage here:

https://brainly.com/question/13521443

#SPJ11

Answer:

Part A: 120000000 is the difference

Part B: A 8 minutes, 20 seconds

Step-by-step explanation:

I'm an expert

The probability of drawing exactly two pairs from a deck of cards containing four aces, four kings, four queens, and four jacks is approximately 0.3623 or about 36.23%.

To have exactly two pairs in a five-card hand, we need two cards of one rank, two cards of another rank, and one card of a different rank.

The number of ways to choose two ranks out of four for the pairs is (4 choose 2) = 6.

For each pair, we can choose two cards out of four in (4 choose 2) = 6 ways.

Finally, we can choose one card from the remaining 44 cards in (44 choose 1) ways.

Therefore, the number of ways to get exactly two pairs is:

6 x 6 x (44 choose 1) = 1584.

The total number of ways to draw five cards out of 16 is (16 choose 5) = 4368.

Therefore, the probability of drawing exactly two pairs is:

P(exactly two pairs) = (number of ways to get exactly two pairs) / (total number of ways to draw five cards)

= 1584 / 4368

= 0.3623 (rounded to four decimal places).

For similar question on probability.

https://brainly.com/question/2239055

#SPJ11

The change in radius is supposed to be 2.04 units to get 800 as volume.

To calculate the volume of a cylinder, we use the formula V = πr^2h, where V represents the volume, r represents the radius, and h represents the height.

Given that the height is fixed at 10 units, and the volume is desired to be 800 cubic units, we can rearrange the formula to solve for the radius:

V = πr^2h

800 = πr^2(10)

To isolate the radius, we divide both sides of the equation by π * h * 10:

800 / (π * 10) = r²

Simplifying further:

80 / π = r²

To find the value of the radius, we take the square root of both sides:

√(80 / π) = r

Using a calculator to approximate the square root of 80 divided by π, we find:

r ≈ 5.04

Therefore, to achieve a volume of 800 cubic units while keeping the height at 10 units, the radius would need to be approximately 5.04 units.

And change would be 5.04-3 = 2.04.

Learn more about Volume here:

https://brainly.com/question/1578538

#SPJ1

Answer:

answer below

Step-by-step explanation:

a) will be all of them

b)will be all of their unions, so the values they all have in common in this case 4, 6

c)will be the values in common with a and b and all of c,

d)will be all of the values of a and b and all of the values in common with c

sorry I csnnot give an actual answer at the moment, but i can explain what each question wants from you in literal word form.

To find f(t) given ℒ−1{1/(s^2-6s+10)}, we can use partial fraction decomposition and complete the square to rewrite the denominator:

1/(s^2-6s+10) = A/(s-3+I) + B/(s-3-I)

where A and B are constants to be determined, and I is the imaginary unit.

Multiplying both sides by (s-3+I)(s-3-I) = s^2 - 6s + 10, we get:

1 = A(s-3-I) + B(s-3+I)

Expanding and collecting like terms, we get:

1 = (A+B)s - 6A + 10B - AI - BI

Since this equation must hold for all values of s, we can equate the coefficients of each power of s:

A + B = 0 (coefficient of s^1)
-6A + 10B - AI - BI = 1 (coefficient of s^0)

Solving these equations simultaneously, we get:

A = -1/2 + I/2, B = -1/2 - I/2

Substituting these values back into the partial fraction decomposition, we get:

1/(s^2-6s+10) = (-1/2 + I/2)/(s-3+I) + (-1/2 - I/2)/(s-3-I)

Taking the inverse Laplace transform of each term separately using standard formulas, we get:

f(t) = e^(3t/2) sin(t/2)

Therefore, the function f(t) is e^(3t/2) sin(t/2).

Learn more about fraction here:brainly.com/question/10354322

#SPJ11

The possible value of angle G are given as follows:

g = 10º.

Suppose we have a triangle in which:

The lengths and the sine of the angles are related as follows:

[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]

The relation for this problem is given as follows:

sin(126º)/83 = sin(g)/17

Hence the measure of angle g is obtained as follows:

sin(g) = 17 x sine of 126 degrees/83

sin(g) = 0.1657

g = arcsin(0.1657)

g = 10º. -> rounded to the nearest degree.

More can be learned about the law of sines at https://brainly.com/question/4372174

#SPJ1

The probability of Z being less than 0.618 is approximately 0.7314. the probability of P(0.7 < X < 0.75) is approximately 0.7314.

a. The mean of X1 can be found by taking the expected value of the distribution:

E(X1) = ∫0^10 x f(x) dx

= ∫0^10 x[(3x^2)/2 + x] dx

= 78.75/4

= 19.6875

Therefore, the mean of X1 is 19.6875.

b. The variance of X1 can be found using the formula:

Var(X1) = E(X1^2) - [E(X1)]^2 E(X1^2) can be found by taking the second moment of the distribution:

E(X1^2) = ∫0^10 x^2 f(x) dx

= ∫0^10 x^2 [(3x^2)/2 + x] dx

= 1095/8

Therefore,

Var(X1) = 1095/8 - (78.75/4)^2

= 16.3203125

c. Using the central limit theorem, we can approximate the distribution of the sample mean with a normal distribution.

The mean of the sample mean is the same as the population mean, which we found to be 19.6875 in part a. The variance of the sample mean can be found by dividing the population variance by the sample size:

Var(X) = Var(X1)/n

= 16.3203125/100

= 0.163203125

Then, we can standardize the sample mean using the formula:

Z = (X - μ)/(σ/√n)

where μ is the population mean, σ is the population standard deviation (which we found to be √Var(X1) ≈ 4.0407), and n is the sample size.

Plugging in the values, we get:

Z = (0.725 - 0.7)/(4.0407/√100)

= 0.618

Using a standard normal distribution table or calculator, we can find that the probability of Z being less than 0.618 is approximately 0.7314. Therefore, the probability of P(0.7 < X < 0.75) is approximately 0.7314.

To know more about probability click here

brainly.com/question/15124899

#SPJ11

Part A: The best graphical representation to display the data is histogram.

Part B : The histogram is represented below.

Part A

A histogram would be the ideal type of chart to display the provided data. The distribution and frequency of data can be shown using a histogram in various intervals or bins. It enables us to see how the grades are distributed and see any patterns or trends.

Part B

Follow these steps to make a histogram to represent the given data:

A visual representation of how the grades are distributed over the various intervals will be provided by the resulting histogram, making it simple to understand the class's overall performance.

Know more about histogram,

https://brainly.com/question/28164315

#SPJ1

The given parabola 8(y-2) = (x + 2)², the focus is (-2, 4), the directrix is y = 6, the vertex is (-2, 2), and the axis of symmetry is the vertical line x = -2.

To find the focus, directrix, vertex, and axis of symmetry of a parabola in standard form, we can rewrite the given equation as y = (1/8)(x + 2)² + 2. Comparing this equation with the standard form y = a(x - h)² + k, we can determine the values of h, k, and a. From the equation, we can see that the vertex is given by (h, k), which in this case is (-2, 2). The vertex represents the point where the parabola reaches its minimum or maximum value.

The axis of symmetry is a vertical line passing through the vertex. Therefore, the axis of symmetry for this parabola is x = -2.

The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the directrix. The distance between the focus and the vertex is given by the equation |1/(4a)|, where a is the coefficient of the x-term. In this case, a = 1/8, so the distance between the focus and the vertex is |1/(4(1/8))| = |2| = 2. Since the vertex is at (-2, 2), the focus is located at (-2, 2+2) = (-2, 4).

The directrix of a parabola is a line perpendicular to the axis of symmetry and is equidistant from the focus. Since the vertex is at (h, k) = (-2, 2) and the focus is at (-2, 4), the directrix is a horizontal line located at y = 2 + 2 = 6.

To learn more about parabola here brainly.com/question/11911877

#SPJ11

Answer:

x = 19

Step-by-step explanation:

Use the Intersecting Secants Theorem to solve this:

8(8 + x) = 9(9 + 15)

64 + 8x = 81 + 135

8x = 216 - 64 = 152

x = 152/8 = 19

The exact solutions for the equation tan(θ) = -1 in the interval 0≤θ≤2π are θ = (3π)/4 and θ = (7π)/4.

To find all exact solutions on the interval 0≤θ≤2π for the equation tan(θ) = -1, follow these steps:

Step 1: Identify the principal angles where tan(θ) = -1.
The tangent function is negative in the second and fourth quadrants.

Recall that tan(θ) = sin(θ) / cos(θ). In the second quadrant, sin(θ) is positive and cos(θ) is negative. In the fourth quadrant, sin(θ) is negative and cos(θ) is positive.

The principal angles where tan(θ) = -1 are θ = (3π)/4 and θ = (7π)/4, as these angles have equal magnitude for sin(θ) and cos(θ) but opposite signs.

Step 2: Check if the principal angles are within the given interval.
Both (3π)/4 and (7π)/4 lie within the interval 0≤θ≤2π.

Step 3: List the exact solutions.
The exact solutions for the equation tan(θ) = -1 in the interval 0≤θ≤2π are θ = (3π)/4 and θ = (7π)/4.

Know more about the equation here:

https://brainly.com/question/29174899

#SPJ11

The product (k-1) (6k+5) is C, [tex]6k^2 - k - 5.[/tex] therefore, option C, [tex]6k^2 - k - 5.[/tex] is correct.

To find the product of (k-1) and (6k+5), we can use the distributive property of multiplication.

We can multiply each term in the first expression (k-1) by each term in the second expression (6k+5), and then simplify:

[tex](k-1)(6k+5) = k(6k+5) - 1(6k+5)\\(k-1)(6k+5) = 6k^2 + 5k - 6k - 5\\(k-1)(6k+5) = 6k^2 - k - 5[/tex]

Therefore, the answer is C, [tex]6k^2 - k - 5.[/tex]

We can check our answer by multiplying it out using the distributive property, and we should get the original expressions (k-1) and (6k+5) back.

for such more question on

https://brainly.com/question/29667212

#SPJ11

The Central Limit Theorem allows us to use the sample mean to estimate the population mean even when the population distribution is unknown. The standard deviation of the population, sigma, is an important factor in determining the spread of the sample means.

The statement you are referring to is known as the Central Limit Theorem. It states that when sampling from a population with an unknown distribution, if the sample size is sufficiently large (usually n>30), the sample mean will follow an approximately normal distribution regardless of the shape of the population distribution. This is particularly useful in statistics because it allows us to make inferences about the population mean based on the sample mean.

The standard deviation, sigma, plays an important role in the Central Limit Theorem because it determines how spread out the population is. If sigma is small, the sample means will be tightly clustered around the population mean, while if sigma is large, the sample means will be more spread out.

In conclusion, the Central Limit Theorem allows us to use the sample mean to estimate the population mean even when the population distribution is unknown. The standard deviation of the population, sigma, is an important factor in determining the spread of the sample means.

1. When sampling from a population with an unknown distribution, mean mu, and standard deviation sigma,

2. If the sample size (n) is sufficiently large,

3. The sample mean (x bar) will have approximately a normal distribution.

The CLT(central limit theorem) is a vital tool in many areas of statistical analysis, as it provides a foundation for making inferences about populations based on sample data, even when the original population distribution is unknown or non-normal.

Learn more about  standard deviation  here:

brainly.com/question/13498201

#SPJ11

Answer:

Mass of 1 gallon skim milk is 3.92 kg.

Step-by-step explanation:

Density is the ratio of mass to volume.

Density = (mass)/(volume)

0.264 gallons = 1 L

So, 1 gallon =  = 3.79 L

So, mass of 3.79 L of skim milk = (density of skim milk)(volume of skim milk)

                                                   =

                                                   = 3.92 kg

So, mass of 1 gallon skim milk is 3.92 kg.

A regression line is referred to as the line of best fit because it is a line that is drawn through a scatter plot of data points to show the general trend or pattern of the data. The line is "best" in the sense that it is the one that minimizes the distance between the line and the data points.

In other words, the regression line is the line that comes closest to passing through as many of the data points as possible while still maintaining a relatively small amount of distance between the line and the points.

Know more about the regression line

https://brainly.com/question/17004137

#SPJ11

Answer:

Step-by-step explanation:

hope this helps . Please mark my answer as best

The solution of the equation is x = -3 and x = 1

From the information given, we have that the quadratic equation is expressed as;

2x² + 4x-6=0

Using the factorization method , we have the multiply the coefficient of x squared by the constant value in the expression, we get;

2(-6) = -12

Now, find the pair factors of this product whose sum is 4, we have;

+6 - 2

Substitute the values

2x² + 6x - 2x - 6 = 0

Group in equation in pairs

(2x² + 6x) - (2x -6) = 0

factorize the terms

2x(x + 3)- 2(x + 3) = 0

Then, we have;

x = 2/2 = 1

x = -3

Learn about quadratic equations at: https://brainly.com/question/1214333

#SPJ1

The arc length traveled by the handle of the wrench is 0.825 meters.

When a wrench rotates, the handle moves along an arc due to its angular displacement.

To calculate the arc length traveled by the handle, we can use the formula:

Arc length = Radius [tex]\times[/tex] Angular displacement.

In this case, the length of the wrench is given as 0.33 meters, which acts as the radius.

The angular displacement is provided as 2.5 radians.

Plugging in these values, we have:

Arc length = 0.33 meters [tex]\times[/tex] 2.5 radians.

To calculate the product, we multiply the length of the wrench (0.33 meters) by the angular displacement (2.5 radians):

Arc length = 0.33 meters [tex]\times[/tex] 2.5 radians = 0.825 meters.

Therefore, the arc length traveled by the handle of the wrench is 0.825 meters.

This means that as the wrench rotates through an angle of 2.5 radians, the handle moves along an arc with a length of 0.825 meters.

For similar question on angular displacement.

https://brainly.com/question/29173215

#SPJ11