A football game in 2?011 had an attendance of 213,070 fans. The headline in the newspaper the next day read, About 200,000 people attend the big game! Is the newspapers estimate reasonable

When two fair dice are rolled,

(a) P(F) = 1/9

(b) P(F|E) = 1/5

a. To compute P(F), we need to find the probability that the total of two dice is five. There are four ways to obtain a total of five: (1,4), (2,3), (3,2), and (4,1). Since each die has six possible outcomes, there are 6x6=36 possible outcomes when two dice are rolled. Therefore, P(F) = 4/36 = 1/9.

b. To compute P(F|E), we need to find the probability that the total of two dice is five given that the total is odd. Since the sum of two odd numbers is always even, we know that if an odd total shows on the dice, then the sum must be either 3, 5, 7, 9, or 11. Out of these possibilities, only one yields a total of 5, which is (2,3). Therefore, P(F|E) = 1/5.

We would expect the probability of F to change when we condition on E because the occurrence of E affects the sample space. When we know that an odd total shows on the dice, we can eliminate some of the possible outcomes and reduce the sample space. This makes it more likely that the remaining outcomes will satisfy the condition for F, which increases the probability of F. Therefore, P(F|E) is greater than P(F) because E provides additional information that makes F more likely.

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For example, if the events were "you earn an A" and "your friend, who always studies with you, earns an A", these events would be dependent because the probability of your friend earning an A would be affected by whether or not you earn an A.

In this case, the events are not mutually exclusive because both events can happen at the same time (i.e., both you and a randomly selected student can earn an A in the course).

The events can be considered independent if one event does not affect the probability of the other event occurring. In this case, whether you earn an A does not affect the probability of the randomly selected student also earning an A. Therefore, the events can be considered independent.

Note that if the events were dependent, it would mean that the probability of one event occurring would affect the probability of the other event occurring. For example, if the events were "you earn an A" and "your friend, who always studies with you, earns an A", these events would be dependent because the probability of your friend earning an A would be affected by whether or not you earn an A.

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The age of the man , his son, and his grandson is equal to 60 years, 30 years, and  6 years old.

Let x be the age of the son

2x be the age of the man since he is twice as old as his son.

let y be the age of the grandson .

The sum of their ages is 96.

x + 2x + y = 96

Simplifying this equation, we get

⇒3x + y = 96

The man is ten times as old as his grandson,

⇒2x = 10y

Simplifying this equation, we get,

⇒x = 5y

Now substitute x = 5y into the first equation,

⇒3x + y = 96

⇒3(5y) + y = 96

⇒15y + y = 96

⇒16y = 96

⇒y = 6

So the grandson is 6 years old.

Using x = 5y

⇒The son is 30 years old.

Finally, the man is 2x = 2(30)

                                  = 60 years old.

Therefore, the man is 60 years old, his son is 30 years old, and his grandson is 6 years old.

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Answer: No, it is not.

Step-by-step explanation:

To figure out if a shape is a right triangle, we need to use the pythagorean theorem, which states that a^2 + b^2 = c^2.

In this case, a is equal to 5, b is equal to 9.2, and c is equal to 14.

a^2 is equal to 25 and b^2 is equal to 84.64, we can add these two values together to get 109.64.

Now, we calculate 14^2, which is 196.

We now have something to determine, is 109.64 equal to 196?

Since these two numbers are not equal to each other, the answer is no, and that means this triangle is not a right triangle.

Answer:

Triangle ABC is not a right triangle, as the sum of the squares of the shortest two sides do not equal to the square of the longest side.

Step-by-step explanation:

Pythagoras Theorem explains the relationship between the three sides of a right triangle. The square of the hypotenuse (longest side) is equal to the sum of the squares of the legs of a right triangle:

[tex]\boxed{a^2+b^2=c^2}[/tex]

where:

As we have been given the measures of all three sides of triangle ABC (where AB and AC are the shortest sides, and BC is the longest side), we can use Pythagoras Theorem to determine if the triangle is a right triangle.

If triangle ABC is a right triangle, then AB and AC will be the legs, and BC will be the hypotenuse.

Substitute the values into the formula:

[tex]\implies AB^2+AC^2=BC^2[/tex]

[tex]\implies 5^2+9.2^2=14^2[/tex]

[tex]\implies 25+84.64=196[/tex]

[tex]\implies 109.64=196[/tex]

As 109.64 does not equal 196, triangle ABC is not a right triangle.

Answer:

When data has a narrow range and we want to see the shape of the data, the best way to display the data is through a line plot. A line plot is a graphical display of data that uses dots placed above a number line to show the frequency of values in a set of data. It is useful for showing the distribution of a small set of data, especially when the data has a narrow range of values. The line plot allows us to see how many times each value occurs and to identify the mode(s) of the data.

When data has a narrow range and we want to see its shape, a stem-and-leaf plot is the best way to display it. In a stem-and-leaf plot, the data is divided into two parts: the stem and the leaf. The stem is the leftmost digit(s) of each data point, and the leaf is the rightmost digit. This plot allows us to quickly see the distribution of the data and identify any outliers or patterns.

The perimeter of the figure in this problem is given as follows:

P = 36.6.

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

For this problem, three of the lengths are quite straightforward, as follows:

10, 4 and 10.

The fourth length is half the circumference of a circle of diameter 4 = radius 2, hence it is given as follows:

C = 2πr

C = 4π.

C = 12.6.

Hence the perimeter of the figure is given as follows:

P = 10 + 4 + 10 + 12.6

P = 36.6.

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Answer:

The answer is 56.52 .

Step-by-step explanation:

18 multiple 3.14

Answer:

x = 10.1, AXY = 71.7 degrees

Step-by-step explanation:

There are two ways to solve this problem. You could either do 7x+1+108.3=180 or 180-108.3, then take that number and set it equal to 7x+1.

I will be using the latter. (You can do this because angle YXB is a linear pair with angle AXY. This means they add up to 180. So to find angle AXY, you subtract 180 from 108.3)

[tex]180-108.3=71.7\\\\7x+1=71.7\\\\[/tex]

Subtract one from each side to move variables to the left and constants to the right.

[tex]7x+1-1=71.7-1\\\\7x=70.7[/tex]

Divide seven by both sides to isolate the variable.

[tex]\frac{7x}{7}=\frac{70.7}{7} \\\\x=10.1[/tex]

So now we know what x is. So to find AXY, you substitute it back into the equation.

[tex]7(10.1)+1=71.7\\\\70.1+1=71.7?\\\\71.1=71.7?[/tex]

Answer:

30 oranges

Step-by-step explanation:

Divide 3,000 by 100 and you get the number of 30 so which means they can put 30 oranges each box if they wanted to.

Step-by-step explanation:

Answer: 30

Step-by-step explanation:

divide 3000 by 100 and then you git your answer

The statement "Given the logistic function 3 x(t) = e-1.08 t +0.09 The time needed to reach x(t)= 98 is t=3." is :

(b) False

The logistic function is a mathematical function that is used to model growth processes that are limited by saturation. It is often used in the field of biology to model population growth, as well as in economics to model the growth of markets and the adoption of new technologies.

Given the logistic function x(t) = 3e^(-1.08t) + 0.09, you want to determine if x(t) = 98 when t = 3.

Step 1: Plug in t = 3 into the function
x(3) = 3e^(-1.08*3) + 0.09

Step 2: Calculate the result
x(3) ≈ 3e^(-3.24) + 0.09 ≈ 0.0705

Since x(3) ≈ 0.0705 and not 98, the statement "The time needed to reach x(t) = 98 is t = 3" is false. Therefore, the correct answer is:

b. False

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The percentage difference in the percent errors of the estimates is approximately 35.6%.

Let's start with the drywall estimate. The estimated value is 19 sheets, while the actual value is 16 sheets. Using the formula, we get:

Percent Error = (|19 - 16| / 16) x 100%

Percent Error = (3 / 16) x 100%

Percent Error = 18.75%

Therefore, the percent error in the drywall estimate is 18.75%.

Now, let's calculate the percent error in the tile estimate. The estimated value is 85 square feet, while the actual value is 67 square feet. Using the formula, we get:

Percent Error = (|85 - 67| / 67) x 100%

Percent Error = (18 / 67) x 100%

Percent Error = 26.87%

Therefore, the percent error in the tile estimate is 26.87%.

To find the difference in the percent errors, we need to subtract the percent error in the drywall estimate from the percent error in the tile estimate and take the absolute value. We then divide the result by the average of the percent errors and multiply by 100 to get the percentage difference. The formula is as follows:

Percentage Difference = |(Percent Error Tile - Percent Error Drywall) / ((Percent Error Tile + Percent Error Drywall) / 2)| x 100%

Plugging in the values, we get:

Percentage Difference = |(26.87% - 18.75%) / ((26.87% + 18.75%) / 2)| x 100%

Percentage Difference = |8.12% / 22.81%| x 100%

Percentage Difference = 35.6%

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Answer: ''pencil veinya''    (Pennsylvania)

Step-by-step explanation:

The probability of a point chosen uniformly at random in the region [tex]-1\leq x\leq1[/tex] and [tex]-1\leq y\leq1[/tex]  to be classified as '+' is [tex](\frac{4 - A_{minus}}{4})[/tex].

To draw the Voronoi tile for the given data set and find the probability of a point being classified as '+', follow these steps:

1. Plot the data points: Plot the points (-1,-1), (-1,1), (1,-1), (1,1), and (0,0) on a graph. Label (0,0) as '-' and the other points as '+'.

2. Construct the Voronoi diagram:

For each pair of neighboring '+' points, draw a line that is equidistant from both points and bisects the line connecting them. These lines will divide the space into regions called Voronoi tiles, where each tile contains one data point, and any point within that tile is closer to the data point it contains than to any other data point.

3. Identify the tile containing the '-' point:

In this case, the Voronoi tile surrounding (0,0) will be the region that is closer to the '-' point than to any '+' point.

4. Calculate the area of the Voronoi tile containing the '-' point:

Since we are considering the region [tex]-1\leq x\leq1[/tex] and [tex]-1\leq y\leq1[/tex], find the area of the intersection of this region with the Voronoi tile containing the '-' point.

5. Calculate the total area of the considered region: The total area of the considered region is

(-1 to 1)(-1 to 1) = 2(2) = 4 square units.

6. Determine the probability of a point being classified as '+':

The probability of a point chosen uniformly at random in the considered region being classified as '+' is equal to the ratio of the area of the region not covered by the '-' Voronoi tile to the total area of the considered region.

Let's say the area of the '-' Voronoi tile is [tex]A_{minus}[/tex]. Then, the probability of a point being classified as '+' is [tex](\frac{4 - A_{minus}}{4})[/tex].

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The odds against the coin showing heads all 4 times its tossed is 15:1.

Probability of getting heads on one toss of a fair coin is 1/2

Since the coin is tossed four times, the probability of getting heads all four times is:

P(H) =  (1/2) x (1/2) x (1/2) x (1/2) = 1/16.

Recall that the odds against an event happening are the ratio of the number of ways it can't happen to the number of ways it can happen.

In this case, the number of ways the coin won't show heads all four times is:

P(T) = 15 (there are 16 possible outcomes and only one of them is all heads). Therefore, the odds against the coin showing heads all four times are 15 to 1.

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A) The correct features is,

⇒ a = 9000

⇒ r = 15%

B) The equation correctly models the context of the problem is,

⇒ y = 9000 (0.85)ˣ

We have to given that;

Tiffany’s mother bought a car for $9000 five years ago.

And, She wants to sell it to Tiffany based on a 15% annual rate of depreciation.

Now, We have;

the exponential growth formula is,

⇒ y = a(1 − r)ˣ

Here, We have;

⇒ a = 9000

⇒ r = 15%

Thus, We get;

The equation correctly models the context of the problem is,

⇒ y = a(1 − r)ˣ

⇒ y = 9000 (1 - 0.15)ˣ

⇒ y = 9000 (0.85)ˣ

Therefore, We get;

A) The correct features is,

⇒ a = 9000

⇒ r = 15%

B) The equation correctly models the context of the problem is,

⇒ y = 9000 (0.85)ˣ

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Complete question is,

Tiffany’s mother bought a car for $9000 five years ago. She wants to sell it to Tiffany based on a 15% annual rate of depreciation.

Part A. Identify each feature of the problem as it relates to the context and the exponential growth formula: y=a(1−r)t

a=

r=

Part B. Which equation correctly models the context of the problem?

Choose : A. y=9000(0.15)t

or B. y=9000(0.85)t

Answer : The equation is

Part C.

Tiffany wants to buy the car from her mother now. (t = 5)

A fair price for the car will be about $

in 5 years.

The probability that a randomly selected Ping-Pong ball has a diameter of less than 2.7 cm is approximately 0.0668 or 6.68%.

To solve this problem, we need to standardize the value of 2.7 using the formula:

z = (x - μ) / σ

where:

x = 2.7 (the value we want to find the probability for)

μ = 3 (mean)

σ = 0.2 (standard deviation)

z = (2.7 - 3) / 0.2

z = -1.

Now, we need to find the probability that a randomly selected Ping-Pong ball has a diameter of less than 2.7 cm, which is the same as finding the area to the left of z = -1.5 on the standard normal distribution curve. We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table, we can find the area to the left of z = -1.5 is 0.0668.

Therefore, the probability that a randomly selected Ping-Pong ball has a diameter of less than 2.7 cm is approximately 0.0668 or 6.68%.

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The solution to the given quadratic equation is x= -3 -2 -1 0 1 2 3 y= 2 3 2 -1 -6 -13 -22.

This table of values contains a quadratic function that changes direction at the value of x = 0. This is different from the other tables of values which all have the quadratic function changing direction at the value of x = -2. The y values in this table of values can be described as an upside-down parabola. At x = 0, the y value is -1 and it decreases as x increases to positive values, and it increases as x decreases to negative values.

Therefore correct answer is C. x -3 -2 -1 0 1 2 3 y 2 3 2 -1 -6 -13 -22.

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The answer is option [tex](cx^2y^2).[/tex]

To find the least common multiple (LCM) of [tex]xy, x^2,[/tex] and xy^2, we need to factor each term into its prime factors and then take the highest power of each factor.

xy = (x) * (y)

x^2 = (x) * (x)

xy^2 = (x) * [tex](y^2)[/tex]

The prime factorization of the given terms are:

xy = (x) * (y)

x^2 = (x) * (x)

xy^2 = (x) * [tex](y^2)[/tex]

So, the LCM can be found by taking the highest power of each factor, which gives us:

LCM = [tex](x^2)[/tex] * [tex](y^2)[/tex] =[tex]x^2y^2[/tex]

Therefore, the answer is option [tex](cx^2y^2).[/tex]

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To find out how many students prefer Kiwi when there are 357 total students, we can use the equivalent ratios of 5:12 or 12:5.

The ratio of students who prefer pineapple to students who prefer kiwi is given as 12 to 5, which means that for every 12 students who prefer pineapple, 5 students prefer kiwi. We can represent this ratio as 12:5.

To find out how many students prefer kiwi, we need to determine the proportion of the total number of students that prefer kiwi. Since the total number of students is 357, we can set up a proportion with the ratio of students who prefer Kiwi to the total number of students. Using the equivalent ratio of 5:12, we can set up the proportion as follows:

5/12 = x/357

Here, x represents the number of students who prefer Kiwi. To solve for x, we can cross-multiply and simplify the proportion as follows:

5 * 357 = 12 * x
1785 = 12x
x = 1785/12
x = 148.75

Since we cannot have a fractional number of students, we need to round our answer to the nearest whole number. Therefore, we can conclude that approximately 149 students prefer Kiwi out of a total of 357 students.

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This contradicts our assumption, so we conclude that 40 does not leave a remainder of 3 upon division by 7.

(a) To show that 45 leaves a remainder of 3 upon division by 7, we need to find an integer k such that 45 = 7k + 3. We can write 45 as 42 + 3, which gives us 45 = 7(6) + 3. Thus, n = 45 satisfies the definition and leaves a remainder of 3 upon division by 7.

(b) To show that -32 leaves a remainder of 3 upon division by 7, we need to find an integer k such that -32 = 7k + 3. We can write -32 as -35 + 3, which gives us -32 = 7(-5) + 3. Thus, n = -32 satisfies the definition and leaves a remainder of 3 upon division by 7.

(c) To show that 3 leaves a remainder of 3 upon division by 7, we need to find an integer k such that 3 = 7k + 3. We can write 3 as 0 + 3, which gives us 3 = 7(0) + 3. Thus, n = 3 satisfies the definition and leaves a remainder of 3 upon division by 7.

(d) To show that -4 leaves a remainder of 3 upon division by 7, we need to find an integer k such that -4 = 7k + 3. We can write -4 as -7 + 3, which gives us -4 = 7(-1) + 3. Thus, n = -4 satisfies the definition and leaves a remainder of 3 upon division by 7.

(e) To prove that 40 does not leave a remainder of 3 upon division by 7, we assume the opposite, that is, we assume that 40 does leave a remainder of 3 upon division by 7. This means that there exists an integer k such that 40 = 7k + 3. Rearranging this equation gives us 37 = 7k, which means that k is not an integer, since 37 is not divisible by 7. This contradicts our assumption, so we conclude that 40 does not leave a remainder of 3 upon division by 7.

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To know chi-square goodness-of-fit test conducted to determine whether the data provide convincing evidence that the distribution has changed. The test statistic was 10.13 with a p-value of 0.0175.

To ascertain if the observed data adheres to a predetermined distribution, the chi-square goodness-of-fit test is utilised.

The test statistic is determined using the following formula: 2 = [(O - E)2 / E]where 2 is the test statistic, is the sum of all the categories, and O and E are the observed and predicted frequencies.

If the null hypothesis is true, the p-value is the likelihood that a test statistic will be equally extreme or more extreme than the observed one.

The null hypothesis in this situation is that the distribution has not altered.

If the p-value is less than 0.05, we reject the null hypothesis and come to the conclusion that there is a statistically significant difference between the observed and predicted frequencies, indicating that the distribution has really changed. This is because the generally used significance level is 0.05.

The test statistic in this instance is 10.13, and the p-value is 0.0175.

We reject the null hypothesis since the p-value is less than 0.05 and come to the conclusion that the data is strong evidence that the distribution has altered.

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Han's retirement account earns $247,163.25 in simple interest.

To find the simplest interest, we need to use the formula:

Simple Interest = Principal × Rate × Time

In this case, the Principal is $410,000 and the Rate is $15,785 per year. We don't know the time period, but we can solve for it using the formula:

Time = Simple Interest ÷ (Principal × Rate)

Plugging in the values, we get:

Time = $15,785 ÷ ($410,000 × 1) = 0.0385 years

Therefore, the simplest interest is:

Simple Interest = $410,000 × $15,785 × 0.0385 = $247,163.25

So Han's retirement account earns $247,163.25 in simple interest.

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We can show that ∆ABC is congruent to ∆A′B′C′ by a translation of (x-2, y) unit(s) and a across the x-axis.

The coordinates of the triangle are A(8, 8), B(10, 4), C(2, 6), while the triangle A'B'C' is at A'(6, -8), B'(8, -4), C'(0, -6).

If a point O(x, y) is translated a units on the x axis and b units on the y axis, the new coordinate is O'(x+a, y+b).

If a point O(x, y) is reflected across the x axis, the new coordinate is O'(x, -y)

Hence if triangle ABC is translated -2 units on the x axis (2 units left), the new coordinates are A*(6, 8), B*(8, 4), C*(0, 6). If a reflection across the x axis is then done, the new coordinates are A'(6, -8), B'(8, -4), C'(0, -6).

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Answer:

7. Dr. Agoncillo completed a total of 14 years of post-secondary education. (4 years undergrad + 4 years medical school + 5 years residency + 1 year fellowship = 14 years)

8. Dan consumed a total of 124 fluid ounces. (2 x 20 + 3 x 16 + 1 x 32 = 40 + 48 + 32 = 120 fluid ounces)

9. There are 175 ACE wraps in stock at this physical therapy clinic. (27 x 2 + 43 x 1 + 93 x 1 + 12 x 1 = 54 + 43 + 93 + 12 = 175 ACE wraps)

10. Karen consumed a total of 67 grams of sugar, 19 grams of fat, and 5 grams of protein in this snack. (Snickers: 20g sugar + 11g fat + 3g protein = 34g total; Doritos: 1g sugar + 8g fat + 2g protein = 11g total; Mt. Dew: 46g sugar + 0g fat + 0g protein = 46g total; 34g + 11g + 46g = 91g total sugar; 11g + 0g + 0g = 11g total fat; 3g + 2g + 0g = 5g total protein)

Step-by-step explanation:

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The statement that is true is that D. Patricia is not correct because both 3 – 4i  and 11+√2i  must be roots.

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc.

From the information, Patricia is studying a polynomial function f(x). Three given roots of f(x) are -11-√2i , 3 + 4i, and 10. Patricia concludes that f(x) must be a polynomial with degree 4.

In this case, the correct option is D.

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Patricia is studying a polynomial function f(x). Three given roots of f(x) are -11-√2i , 3 + 4i, and 10. Patricia concludes that f(x) must be a polynomial with degree 4. Which statement is true?

A. Patricia is correct because -11+√2i must be a root.

B. Patricia is correct because 3 – 4i must be a root.

C. Patricia is not correct because both 3 – 4i  and -11+√2i  must be roots.

D. Patricia is not correct because both 3 – 4i  and 11+√2i  must be roots

Answer: D

Step-by-step explanation:

edge 2023

Answer:

V = 58.7 m^3

Step-by-step explanation:

The formula is V = 1/3 *b*h

where b is the base area

h is the height

From the diagram,

h= 11m

Now we need to find the area of the base.

The base is made of an equilateral triangle, so we know that one of the sides is 8m.

The area of the base is found: 1/2 * a * l

Where a = 8m

l = a/2 = 8m/2 = 4m

So the area of the base is:

b = 1/2 * 8 m* 4m

b = 16m^2

Now plug this into the volume formula:

V = 1/3 b * h

V = 1/3 * 16m^2 * 11m

V = 58.7 m^3

The correlation of ranks is approximately 0.2.

Option A is the correct answer.

We have,

To compute the correlation of ranks, we first need to rank the scores in each subject:
Physics: 10, 17, 23, 28, 35, 43, 47
Rank: 1, 2, 3, 4, 5, 6, 7
Mathematics: 4, 8, 12, 23, 30, 31, 33, 45, 49
Rank: 1, 2, 3, 4, 5, 6, 7, 8, 9

Then, we can calculate the differences between the ranks for each student:
Physics ranks: 1-5, 2-3, 3-7, 4-6, 5-1, 6-4, 7-2
Differences: -4, -1, -4, -2, 4, 2, 5
Mathematics ranks: 1-8, 2-6, 3-7, 4-4, 5-1, 6-5, 7-2, 8-3, 9-9
Differences: -7, -4, -4, 0, 4, -1, 5, 5, 0

Next, we can calculate the sum of the products of the differences:
= Sum of products

= (-4)(-7) + (-1)(-4) + (-4)(-4) + (-2)(0) + (4)(4) + (2)(-1) + (5)(5)
= 28 + 4 + 16 + 0 + 16 - 2 + 25
= 87

Finally, we can use the formula for the correlation of ranks:
r = 1 - (6Σd²)/(n(n² - 1))
where d is the difference in ranks and n is the number of scores

Plugging in the values, we get:
r = 1 - (6(87))/(9(81-1))
= 1 - (522)/(648)
= 1 - 0.8056
= 0.1944

= 0.2

Therefore,

The correlation of ranks is approximately 0.2.

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The correct test statistic for Amanda's significance test would be option B:
z = (0.49 - 0.60) / sqrt(0.49(0.51)/300)

This is because option B includes the sample proportion and the sample size, which are necessary for calculating the test statistic for a significance test involving proportions. The formula for the test statistic for a two-tailed test of a population proportion is:

z = (p - P) / sqrt(P(1 - P) / n)

where p is the sample proportion, P is the hypothesized population proportion, and n is the sample size.

In this case, we are not given the hypothesized population proportion, so we use the sample proportion as an estimate. The formula becomes:

z = (p - P) / sqrt(P(1 - P) / n) = (p - 0.5) / sqrt(0.5(0.5) / n) = (0.49 - 0.5) / sqrt(0.5(0.5) / 300)

Simplifying this expression gives us the test statistic in option B.

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the biology professor cannot conclude that the percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

The biology professor can use hypothesis testing to determine if the percentage of children who develop asthma in the new country is significantly different from the percentage observed in the US study.

Here are the steps she can take:

1. Define the null and alternative hypotheses:
- Null hypothesis (H0): The percentage of children who develop asthma in the new country is the same as the percentage observed in the US study (i.e., 22.1%).
- Alternative hypothesis (Ha): The percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

2. Determine the test statistic to use:
- The appropriate test statistic for this scenario is the one-sample proportion z-test.

3. Set the significance level (alpha):
- Let's assume a significance level of 0.05.

4. Calculate the test statistic:
- The sample proportion of children who developed asthma in the new country is p = 70/336 = 0.2083.
- The standard error of the sample proportion is SE = sqrt[(p*(1-p))/n] = sqrt[(0.2083*(1-0.2083))/336] = 0.027.
- The test statistic is z = (p - P) / SE, where P is the proportion observed in the US study. So, z = (0.2083 - 0.221) / 0.027 = -0.463.

5. Determine the p-value and make a decision:
- The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. Using a standard normal distribution table or calculator, we find that the p-value is 0.3212.
- Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the percentage of children who develop asthma in the new country is significantly different from the percentage observed in the US study.

Therefore, the biology professor cannot conclude that the percentage of children who develop asthma in the new country is lower than the percentage observed in the US study.

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The linear approximation of F at (Pi, 0) is [tex]-Pi^2cos^2(Pi).[/tex]

To discover the linear approximation of the given function at (Pi, 0), we need to first discover the partial derivatives of the function with respect to x and y evaluated at (Pi, zero).

Partial derivative of F with recognize to x:

∂F/∂x = -2sin(x)cos(x)

evaluated at (Pi, 0):

∂F/∂x(Pi, 0) = -2sin(Pi)cos(Pi) = 0

Partial derivative of F with recognize to y:

∂F/∂y = 1/(2√y)

evaluated at (Pi, 0):

∂F/∂y(Pi, 0) = 1/(2√0) = undefined

For the reason that partial derivative of F with respect to y is undefined at (Pi, 0), we can't use the multivariable Taylor collection to discover the linear approximation. as an alternative, we will use the formula for the linear approximation:

[tex]L(x,y) = f(a,b) + ∂f/∂x(a,b)(x-a) + ∂f/∂y(a,b)(y-b)[/tex]

Wherein (a,b) is the factor at which we want to find the linear approximation.

In this case, a = Pi and b = 0. So, the linear approximation is:

[tex]L(x,y) = F(Pi, 0) + ∂F/∂x(Pi, 0)(x - Pi)[/tex]

[tex]L(x,y) = sqrt(0) + (cos(Pi))^2(0 - Pi)[/tex]

[tex]L(x,y) = -Pi^2cos^2(Pi)[/tex]

Consequently, the linear approximation of F at (Pi, 0) is [tex]-Pi^2cos^2(Pi).[/tex]

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