16 < b - 8, if b = 22

   Andy Beth Tri Dale Ella.

To solve the problem using Polya's four-step method, assign variables to represent the runners, set up a system of equations, and solve for the unknowns.

To solve this problem using Polya's four-step method, we need to identify the given information and the unknowns. Let's assign variables to represent the runners: Andy (A), Beth (B), Dale (D), Ella (E), and Tri (T). From the given information, we know that Dale finished 5 seconds before Tri (D - T = 5), Tri finished 7 seconds after Beth (T - B = 7), Beth finished 7 seconds after Ella (B - E = 7), and Ella finished 4 seconds before Andy (E - A = 4). Now, we can set up a system of equations to solve for the order of the runners.
From the equations, we can solve for the values of the variables. Plugging the values back into the original equations, we find that the order in which the runners finished the race is Andy, Ella, Beth, Tri, and Dale.

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Answer:

185

Step-by-step explanation:

cause if you add up 80+65+40 right?

To find the overall probability of leaving on-time, we consider the probabilities of each airline and their respective on-time rates. The weighted average of the on-time rates is calculated using the percentages of scheduled flights for each airline. Summing up the weighted contributions gives us the overall probability of leaving on-time.

To find the overall probability of leaving on-time, we need to consider the probabilities of each airline and their respective on-time rates. First, we calculate the weighted average of the on-time rates using the percentages of scheduled flights for each airline.

Airline A contributes 50% of the flights with an 80% on-time rate, so its weighted contribution is 0.5 x 0.8 = 0.4.

Airline B contributes 30% of the flights with a 65% on-time rate, so its weighted contribution is 0.3 x 0.65 = 0.195.

Airline C contributes 20% of the flights with a 40% on-time rate, so its weighted contribution is 0.2 x 0.4 = 0.08.

To find the overall probability, we sum up the weighted contributions: 0.4 + 0.195 + 0.08 = 0.675.

Therefore, the overall probability of leaving on-time is 0.675 or 67.5%.

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Final answer:

To write the linear equation, let x represent the number of Beats Roses sold and y represent the number of Chocolate Hearts sold. The equation is 5x + 2.50y = 250.

Explanation:

To write a linear equation in standard form for this scenario, we need to define our variables. Let x represent the number of Beats Roses sold and y represent the number of Chocolate Hearts sold.

Based on the given information, we know that the price of Beats Roses is $5, and the price of Chocolate Hearts is $2.50.

The total amount of money Matt made at the end of the dance is $250. Using this information, we can write the equation:

5x + 2.50y = 250

Therefore, the linear equation in standard form for this scenario is 5x + 2.50y = 250.

Answer:

7) d)

standard error of the mean of one sample of 'n' observation = 0.20

8) a)

The margin of Error = 0.392

9) d

The 95% of confidence intervals are (8.61 , 9.39)

Step-by-step explanation:

7)

solution:-

The Given data sample size 'n' = 81

Given Population standard deviation 'σ' = 1.8 hours

The standard error of the mean of one sample of 'n' observation is

Standard error (SE)

                               = [tex]\frac{S.D}{\sqrt{n} }[/tex]  

                               = σ / √n

                               = [tex]\frac{1.8}{\sqrt{81} } =0.2[/tex]

standard error of the mean of one sample of 'n' observation = 0.20

8)

Solution:-

The Given data sample size 'n' = 81

Given Population standard deviation 'σ' = 1.8 hours

Given the probability is 0.95

The z- score = 1.96 at 0.05 level of significance.

The margin of Error   =  [tex]\frac{z_{0.95} S.D}{\sqrt{n} }[/tex]

                                   = [tex]\frac{1.96 (S.D)}{\sqrt{n} }[/tex]

                                   = [tex]\frac{1.96 (1.8)}{\sqrt{81} }[/tex]

                                   = 0.392

The margin of Error = 0.392

9)

Solution:-

The 95% of confidence intervals are

[tex](x^{-} - 1.96\frac{S.D}{\sqrt{n} } , x^{-} + 1.96\frac{S.D}{\sqrt{n} } )[/tex]

[tex](9 - 1.96\frac{1.8}{\sqrt{81} } , 9+ 1.96\frac{1.8}{\sqrt{81} } )[/tex]

(9 - 0.392 , (9 + 0.392)

(8.609 , 9.392)

The 95% of confidence intervals are (8.61 , 9.39)

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Answer:

  1/5000

Step-by-step explanation:

The scale is ...

  map distance / ground distance = (0.095 m)(475 m) = 0.0002 = 1/5000

To find the map scale, convert the actual distance from meters to centimeters and then set up a ratio with the map distance. After simplifying, the scale of the map is determined to be 1:5000, meaning 1 centimeter on the map equals 5000 centimeters in reality.

To determine the scale of the map we'll convert the given distance from George's house to the school into the same units and then create a ratio. Since the map shows the distance as 9.5 centimeters and the actual distance is 475 meters, we first need to convert meters to centimeters because the scale needs to have the same units for both measurements.

1 meter = 100 centimeters, so 475 meters is equivalent to 475 x 100 = 47500 centimeters.

Now, we can set up the scale of the map as a ratio:
9.5 centimeters (map distance) / 47500 centimeters (actual distance).

We can simplify this ratio to find the scale of the map by dividing both the numerator and the denominator by 9.5:

9.5 cm / 47500 cm = 1 cm / (47500 / 9.5) = 1 cm / 5000 cm

This means that every centimeter on the map represents 5000 centimeters in real life. Therefore, the map scale is 1:5000.

Answer:

Step-by-step explanation:

the explanation is attached below

The conditional pdfs of Y given that X is 5 x and X given that Y is 5 y is 20< y<30.

Explain about the probability?

Calculating or estimating how likely something is to occur is what probability is all about. The likelihood of an event occurring can be expressed using words like "certain," "impossible," or "probable." Probabilities are always expressed in mathematics as fractions, decimals, or percentages with values ranging from 0 to 1.

The definition of probability, methods for calculating the probabilities of single and multiple random events, and the distinction between probabilities and odds of an event occurring are all covered in this article. Key conclusions: The probability that an event will occur is determined by probability: P(A) = f / N.

f y/x(y/x) = f(x, y)/f x(x)

 =K(x²+y²)/ 10Kx²+0.05

     0

20<y<30

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Answer: The value of y is 13

Step-by-step explanation: To find the value of y, we will use properties of equality.

Step 1: -7y = -91  (We want to find the value of y, or 1 y)

Step 2: (Use the division property of equality) -7y/-7 = -91/-7

Step 3: (Answer) y = 13

Answer:

y= 13

Step-by-step explanation:

-7y = -91

divide both sides by -7

y = 13

The total lifetime value for 5% of packages to exceed is approximately 103.94 hours.

To solve this problem, we first need to find the distribution of the total lifetime of all nine batteries in a package. Since each battery's lifetime follows a normal distribution with a mean of 11 hours and a standard deviation of 1 hour, the total lifetime of all nine batteries will also follow a normal distribution.

The mean of the total lifetime of all nine batteries is [tex]\( 9 \times 11 = 99 \)[/tex] hours.

The standard deviation of the total lifetime of all nine batteries is [tex]\( \sqrt{9} \times 1 = 3 \)[/tex] hours.

Now, we need to find the value such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages. This is equivalent to finding the 95th percentile of the distribution of the total lifetime.

We'll use the Z-score formula to find the Z-score corresponding to the 95th percentile, and then use that Z-score to find the corresponding value in terms of hours.

The Z-score corresponding to the 95th percentile is approximately 1.645 (you can find this value from standard normal distribution tables or calculators).

Now, we'll use the formula:

[tex]\[ \text{Value} = \text{Mean} + (\text{Z-score} \times \text{Standard deviation}) \][/tex]

[tex]\[ \text{Value} = 99 + (1.645 \times 3) \][/tex]

[tex]\[ \text{Value} = 99 + 4.935 \][/tex]

[tex]\[ \text{Value} \approx 103.94 \][/tex]

So, the total lifetime value such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages is approximately 103.94 hours.

Complete Question:

The lifetime of a certain type of battery is normally distributed with mean value 11 hours and standard deviation 1 hour. There are nine batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages? (Round your answer to two decimal places.)

______ hours

Answer:

The dimensions should be length=width=height=74 cm.

Step-by-step explanation:

We have an optimization with restriction problem.

We have to maximize the volume, subject to restriction in the sum of the length, width and height.

Let x be the length and width, that are equal, and z be the height.

The restriction can be expressed as:

[tex]x+x+z\leq222\\\\2x+z\leq222[/tex]

We can express z in function of x as:

[tex]2x+z=222\\\\z=222-2x[/tex]

The volume, the function to be optimized, can be expressed as:

[tex]V=x^2z=x^2(222-2x)=222x^2-2x^3[/tex]

To optimize, we derive and equal to zero.

[tex]\dfrac{dV}{dx}=\dfrac{d}{dx}[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74[/tex]

We have the optimum length. We can now calculate the height z:

[tex]z=222-2(74)=222-148=74[/tex]

Answer:

  2x -3y = 12

Step-by-step explanation:

For some horizontal change Δx and some vertical change Δy between the two points, an equation of the line through points (x1, y1) and (x2, y2) can be written as ...

  Δy·x -Δx·y = Δy·(x1) -Δx·(y1)

Here, we have ...

  Δy = y2 -y1 = 2 -(-2) = 4

  Δx = x2 -x1 = 9 -3 = 6

So, our equation can be ...

  4x -6y = 4·3 -6·(-2) = 24

Factoring out a common factor of 2 makes the equation be ...

  2x -3y = 12 . . . . . . equation of the line in standard form

Solving for y gives the equation in slope-intercept form:

  y = 2/3x -4

_____

More conventional solution

Plotting the points and drawing the line, you see that the y-intercept is -4. You also see that there is a "rise" of 2 grid squares for each "run" of 3 grid squares. Thus the slope of the line is 2/3. With this information, you can write the equation directly in slope-intercept form:

  y = mx + b . . . . . . line with slope m and y-intercept b

  y = 2/3x -4 . . . . . . the line through the given points

Final answer:

The line passing through the points (3, -2) and (9, 2) has a slope of 2/3, and its equation is y = (2/3)x - 4, which can be graphed by plotting the given points and ensuring the slope is represented correctly.

Explanation:

To graph the line that passes through the points (3, -2) and (9, 2), we first find the slope of the line. The slope formula is (y2 - y1) / (x2 - x1). Plugging in our points, we get (2 - (-2)) / (9 - 3) which simplifies to 4 / 6, further reduced to 2 / 3. Therefore, the slope of the line is 2 / 3.

Next, we use one of the points and the slope to write the equation in point-slope form, y - y1 = m(x - x1). Using the point (3, -2), the equation becomes y + 2 = (2/3)(x - 3). After distributing the slope and moving -2 to the other side, we get the equation y = (2/3)x - 4.

Finally, we can graph the line by plotting the two given points and drawing a straight line through them, ensuring that the rise over the run matches the slope of 2 / 3. The equation of the line y = (2/3)x - 4 can be verified using various x-values to see if the resulting y-values fall on the line plotted.

Answer:

f(13)=2x-1, x=7

Step-by-step explanation:

7*2=14, 14-1=13

vise versa

f(13)=2(7)-1

Answer: 25

Step-by-step explanation: Easy. Just substitute 13 for x into the equation. f(13)=2(13)-1

= 26-1

= 25

Answer:

Renaming a fraction

Step-by-step explanation:

Answer:

1. renaming a fraction

2. de-c

Step-by-step explanation:

Answer:

[tex]P(X>34) = 0.9889[/tex]

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 50

Standard Deviation, σ = 7

We are given that the distribution of random variable X is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P(X greater than 34)

[tex]P( X > 34) = P( z > \displaystyle\frac{34 - 50}{7}) = P(z > -2.2857)[/tex]

[tex]= 1 - P(z \leq -2.2857)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(X>34) = 1 - 0.0111= 0.9889= 98.89\%[/tex]

The attached image shows the normal curve.

The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.

Place value is the basis of our entire number system. This is the system in which the position of a digit in a number determines its value.

The number 42,316 is different from 61,432 because the digits are in different positions.

here, we have,

given that,

6,582,390,417

so, the values of each of the digits in 6,582,390,417 in word form is:

6: six billon

5: five hundred

8: eighty two million

3: three hundred thousand

and ninety four hundred

seventeen.

Hence, The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.

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Answer:

The proportion of slots which meet these specifications is 0.16634 or 16.63%.

Step-by-step explanation:

We are given that the width in inches of slots cut by a milling machine follows approximately the N(0.72,0.0012) distribution.

Also, the specifications allow slot widths between 0.71975 and 0.72025.

Let X = width in inches of slots cut by a milling machine

The z-score probability distribution for normal distribution is given by;

                           Z = [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] ~ N(0,1)

where, [tex]\mu[/tex] = population mean width = 0.72

            [tex]\sigma[/tex] = standard deviation = 0.0012

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, Probability that the specifications allow slot widths between 0.71975 and 0.72025 is given by = P(0.71975 < X < 0.72025)

      P(0.71975 < X < 0.72025)  = P(X < 0.72025) - P(X [tex]\leq[/tex] 0.71975)

     P(X < 0.72025) = P( [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] < [tex]\frac{ 0.72025-0.72}{{0.0012}} }} }[/tex] ) = P(Z < 0.21) = 0.58317

     P(X [tex]\leq[/tex] 0.71975) = P( [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] [tex]\leq[/tex] [tex]\frac{ 0.71975-0.72}{{0.0012}} }} }[/tex] ) = P(Z [tex]\leq[/tex] -0.21) = 1 - P(Z < 0.21)

                                                                    = 1 - 0.58317 = 0.41683

So, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 0.21 in the z table which has an area of 0.58317.

Therefore, P(0.71975 < X < 0.72025)  = 0.58317 - 0.41683 = 0.16634

Hence, the proportion of slots who meet these specifications is 16.63%.

9514 1404 393

Answer:

  cut the pizza ±1.855 inches from the centerline

Step-by-step explanation:

Here's an interesting approach that actually gives amazing accuracy.

If the pizza were a 14" square, the cuts would be 14/6 = 2.33 inches from the centerline. If we draw a diagram of the pizza as a circle of radius 7 centered at the origin, we propose to create a running total of the areas of trapezoidal slices in the first quadrant parallel to the y-axis out to a distance of x=2.3. We want to find where the accumulated area is close to 1/12 of the area of the circle. In order to get sufficient resolution as to where the cut should be, we choose to use a total of 20 slices to cover that area. (n=20 for the Riemann sum)

Of course, the equation of the circle is x^2 +y^2 = 49. Then the value of y of interest is ...

  f(x) = √(49 -x^2)

where the value of x is some multiple of 2.3/20, the slice width for our Riemann sum. If we number the slices 0 to 19, the accumulated area up to slice k is ...

  [tex]\displaystyle A_k=\dfrac{2.3}{20}\sum_{n=0}^{k}\dfrac{f(x_n)+f(x_{n+1})}{2}[/tex]

As you can tell, this is using the trapezoidal method of computing the Riemann sum. We considered left, right, and midpoint integration methods but settled on this as having the kind of accuracy we wanted.

We want an accumulated area of 1/3 of our quarter circle, or (1/12)(π)(7^2) = 49π/12, so we want to find the zero of the difference Ak -49π/12.

As it turns out, the desired area is bracketed by k=16 and k=17. By linearly interpolating between the area values for these numbers of slices, we find that x=1.8546 is the location we need to cut the pizza. The problem is symmetrical, so the other cut is at x = -1.8546 inches from center.

_____

When evaluating functions multiple times, it is convenient to use a graphing calculator or spreadsheet. With a spreadsheet, you could list function values in one column, the trapezoidal method area in another column, and the accumulated area in yet another column. As here, the interpolation required for a final answer is also easily handled in a spreadsheet.

__

Additional comment

The arc α that encloses a sector equal to 1/3 the area of the pizza will satisfy the equation α -sin(α) -2π/3 = 0. That is about 2.6053256746 radians. The cut distance we're trying to find is the 7cos(α/2) ≈ 1.85452459222 inches, which differs by about 0.005% from the value we found.

Answer:

Step-by-step explanation:

Considering the central limit theorem, the distribution is normal since the number of samples is large. Also, the population standard

deviation is known. We would determine the z score.

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.90 = 0.1

α/2 = 0.1/2 = 0.05

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.05 = 0.95

The z score corresponding to the area on the z table is 2.05. Thus, confidence level of 90% is 1.645

Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Confidence interval = mean ± z × σ/√n

Where

σ = population standard Deviation

Confidence interval = x ± z × σ/√n

x = 22.8 hours

σ = 6.4 hours

n = 175

i) Confidence interval = 22.8 ± 1.645 × 6.4/√175

= 22.8 ± 0.80

The lower end of the confidence interval is

22.8 - 0.80 = 22

The upper end of the confidence interval is

22.8 + 0.80 = 23.6

ii) error bound is the same as the margin of error

Error bound = 0.8

Answer:

Loans

Step-by-step explanation:

I don´t know how to explain it,and I hope my answer is correct though.

Answer:

banks create money by issuing loans and opening checking accounts  

Step-by-step explanation:

Answer:

RANDOM

Step-by-step explanation:

Answer:

A box designer has been charged with the task of determining the surface area of various open boxes (no lid) that can be constructed by cutting four equal-sized surface corners from an 8-inch by 11.5 inch sheet of cardboard and folding up the sides.

1. Determine a function that relates the total surface area, s, (measured in square inches) of the open box to the size of the square cutout x (measured in inches).

2. What is the domain and range of the function s?

3. What is the surface area when a 1" x 1" square is cut out?

4. What size square cutout will result in a surface area of 20 in?

5. What is the surface area of the box when the volume is maximized? (Calculator)

Step-by-step explanation:

Answer:

[tex]z=-2.33<\frac{a-8.9}{1.9}[/tex]

And if we solve for a we got

[tex]a=8.9 -2.33*1.9=4.47[/tex]

And the best answer for this case would be:

A) 4.47 lb

Step-by-step explanation:

Let X the random variable that represent the weights of juvenile turkeys, and for this case we know the distribution for X is given by:

[tex]X \sim N(8.9,1.9)[/tex]  

Where [tex]\mu=8.9[/tex] and [tex]\sigma=1.9[/tex]

The z score formula very useful for this case is given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

For this part we want to find a value a, such that we satisfy this condition:

[tex]P(X>a)=0.99[/tex]   (a)

[tex]P(X<a)=0.01[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.01 of the area on the left and 0.99 of the area on the right it's z=-2.33. On this case P(Z<-2.33)=0.01 and P(z>-2.33)=0.99

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.01[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.01[/tex]

So we have this relation

[tex]z=-2.33<\frac{a-8.9}{1.9}[/tex]

And if we solve for a we got

[tex]a=8.9 -2.33*1.9=4.47[/tex]

And the best answer for this case would be:

A) 4.47 lb

Answer:

The answer is option 1.

Step-by-step explanation:

It is given that the area of sector is, A = θ/360 × π × r² where r is the radius of circle. Using the formula, you are able to find the shaded sector :

θ = 90°

r = 6 yd

A = 90/360 × π × 6²

= 1/4 × π × 36

= 9π

= 28.3 yd² (near. tenth)

Answer:

The answer is 7.1yd

Step-by-step explanation:

got the wrong answer so I hope this helps yall and god bless️

Answer:

Step-by-step explanation:

The first mathematician orders 1 beer, the second orders 1/2 a beer, the third orders 1/4 a beer, the fourth orders 1/8 a beer, the fifth orders 1/16 a beer. . .

Therefore, if the first mathematician orders 6 beers, the second orders 2 beer, the third orders 2/3 a beer, and so on

To find out how many beers the bartender should pour for the infinite series of mathematicians ordering in a pattern starting with 6 beers and each following ordering two-thirds of the previous, we use the sum formula for a geometric series. The sum is 18 beers.

The student is asking a question related to an infinite series in mathematics. In the scenario described, the first mathematician orders 6 beers, the second orders 2 beers, and each mathematician that follows orders a fraction of the previous mathematician's order, specifically two-thirds of the previous amount. To find out how many beers the bartender should pour, we need to find the sum of the geometric series.

The general form of a geometric series is a + ar + ar² + ar³ + ..., where 'a' is the first term and 'r' is the common ratio between terms. In this case, the first term 'a' is 6 (the first order) and the common ratio 'r' is 2/3 (each subsequent mathematician orders two-thirds of the previous one's amount).

To find the sum of this infinite series, we can use the formula S = a / (1 - r), when |r| < 1. Substituting the values from our question, we get S = 6 / (1 - 2/3) = 6 / (1/3) = 6 * 3 = 18 beers. Therefore, the bartender should pour 18 beers.

Given Information:

Cost price of lettuce = $10

Selling price of lettuce = $21

Salvage value = $2

Average demand of lettuce = μ = 262 boxes

Standard deviation of lettuce = σ = 43 boxes

Required Information:

How many boxes of lettuce should the supermarket purchase = ?

Answer:

n = 271 boxes

Step-by-step explanation:

The required number of lettuce boxes that supermarket should purchase is given by

n = μ + (z-score)σ

Where μ is the average demand of lettuce boxes, σ is the standard deviation, and z-score can be calculated by

p = C_us/(C_us + C_os)

Where the cost of under stocking is given by

C_us = Selling price of lettuce - Cost price of lettuce

C_us = $21 - $10

C_us = $11

The cost of over stocking is given by

C_os = Cost price of lettuce - Salvage value

C_os = $10 - $2

C_os = $8

p = C_us/(C_us + C_os)

p = 11/(11 + 8)

p ≈ 58%

The z-score corresponding to 58% is 0.202

n = 262 + (0.202)43

n = 270.68

n = 271 boxes

Therefore, the supermarket should purchase 271 boxes of lettuce tomorrow.

Answer:

x = 92

Step-by-step explanation:

Supplementary angles add to 180 degrees,

x+y = 180

We know y =88

x+88 = 180

Subtract 88 from each side

x+88-88=180-88

x =92

Answer:

1.19% probability that fewer than 101 commuters drive to work alone

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].

In this problem, we have that:

[tex]p = 0.75, n = 150[/tex]

[tex]\mu = E(X) = 150*0.75 = 112.5[/tex]

[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{150*0.75*0.25} = 5.3[/tex]

(a) What is the probability that fewer than 101 commuters drive to work alone

Using continuity corretion, this is [tex]P(X < 101-0.5) = P(X < 100.5)[/tex], which is the pvalue of Z when X = 100.5. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{100.5 - 112.5}{5.3}[/tex]

[tex]Z = -2.26[/tex]

[tex]Z = -2.26[/tex] has a pvalue of 0.0119

1.19% probability that fewer than 101 commuters drive to work alone

The probability that fewer than 101 U.S. commuters drive to work alone, based on a 75% solo driving rate, is calculated using the binomial probability formula, resulting in the answer.

To solve this problem, we can use the binomial probability formula, as this is a binomial distribution (success/failure) with a known probability of success.

The formula for the probability mass function of a binomial distribution is:

[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k} \][/tex]

where:

-[tex]\( n \)[/tex] is the number of trials (sample size),

- [tex]\( k \)[/tex]  is the number of successful outcomes,

- [tex]\( p \)[/tex]is the probability of success on a single trial.

In this case, [tex]\( n = 150 \)[/tex] (number of commuters),[tex]\( p = 0.75 \)[/tex] (probability of driving alone), and we want to find the probability that fewer than 101 commuters drive alone (\( k < 101 \)).

[tex]\[ P(X < 101) = P(X \leq 100) = \sum_{k=0}^{100} \binom{150}{k} \cdot 0.75^k \cdot (1 - 0.75)^{150 - k} \][/tex]

Now, we can use a calculator or statistical software to compute this probability. Keep in mind that the binomial coefficient[tex]\(\binom{n}{k}\)[/tex] is the number of ways to choose \(k\) successes from \(n\) trials and can be calculated as[tex]\(\frac{n!}{k! \cdot (n - k)!}\).[/tex]

Learn more about Binomial probability here:

https://brainly.com/question/39666605

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Final answer:

To compare the populations of the two bird species, we calculate the average rate of change for Bird A by dividing the change in population by time over 18 months, and for Bird B by evaluating the given exponential function at the endpoints of the time interval.

Explanation:

To compare the population changes of Bird A and Bird B and interpret the average rates of change over the interval [0, 18], we first need to calculate the average rate of change for Bird A. Given Bird A's population at different times, we can calculate the average rate of change by dividing the change in population by the change in time, over the interval [0, 18].

For Bird A, the population increases from 8.3 to 9.1 thousand over 18 months. The average rate of change for Bird A is thus (9.1 - 8.3) / (18 - 0) = 0.8 / 18 = 0.0444 thousand per month.

For Bird B, the population change is given by a function y= 3.6(1.06)ˣ, where y is the population in thousands and x is the time in months. To find the average rate of change over [0, 18], we evaluate the function at the endpoints of the interval: y(0) = 3.6 and y(18) = 3.6(1.06)¹⁸. After calculating y(18), we'd use the same average rate of change formula.

Interpreting the results, if Bird A's average rate of change is less than that of Bird B, it means Bird B's population is growing faster on average than Bird A's population over the 18 months.

Answer:

The confidence interval for the mean is given by the following formula:  

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

The margin of error is given by:

[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:

Of confidence intervals with this margin of error, 99% will contain the population proportion

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Solution to the problem

The confidence interval for the mean is given by the following formula:  

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

The margin of error is given by:

[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:

Of confidence intervals with this margin of error, 99% will contain the population proportion

Answer:

The dimensions of the rectangle that will allow for the most economical fence to be built are 30x15 feets, where two sides of 30 feets long cost $33 each one per foot, one side of 15 feets costs also $33 and the remaining side costs $99

Step-by-step explanation:

If x and y were the dimensions of the rectangle (in feets), then we have that x*y = 450. Therefore, y = 450/x.

Note that the rectangle as a result is formed by 2 sides with length x and 2 other sides with length 450/x. Lets suppose that x is the length of the 2 sides that costs both $33 and the other two sides, which have length 450/x, one costs also $33 and the other costs $99.

The cost, in $, function f,in terms of x, is given as follows

[tex] f(x) = 2 * 33 * x + 33*\frac{450}{x} + 99*\frac{450}{x} = 66x + \frac{59400}{x} [/tex]

We want to minimize f, so we will derivate it and equalize the derivate to 0:

[tex] f'(x) = 66 - \frac{59400}{x^2} [/tex]

[tex] f'(x) = 0 \leftrightarrow 66 = \frac{59400}{x^2} \leftrightarrow x^2 = \frac{59400}{66} = 900 \leftrightarrow x = \sqrt{900} = 30 [/tex]

(Note that x cant be negative, so in the equation we didnt count the opposite of the square root of 900)

We concluded that one dimension is 30 feets, and the other should be 450/30 = 15.