when an automatic press is a manufacturing process is operaing properly, the lengths of the component it produces are normally distributed with a mean of 8 inches and a standard deviation of 1.5 inches. what is the probability thata randomly selected component is shorter than 7 inches long? (report your answer to 4 decimal places.)

Answer:

-----------------------------------

The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0.

To find the y-intercept, substitute 0 for x in the equation y = 4x + 2:

The y-intercept is 2 and matching choice is C.

Answer:

2

Step-by-step explanation:

We need to find out the y intercept of the given equation which is y = 4x + 2 .

y intercept is a point where the graph of the equation cuts the y axis. So at y intercept x coordinate will be 0 , so plug in x = 0 , in the given equation as ;

y = 4x + 2

y = 4(0) + 2

y = 0 + 2

y = 2

Therefore the y intercept of the line is 2 . ( option C )

The number of litres (Volume) of oil that is present in the tank of given dimension is calculated to be 806 litres (approximately).

The volume of any cylinder can be calculated using the formula,

V = πr²h

(Here V is the volume, r is the radius of the base, and h is the height of the cylinder)

As, the cylinder is one-fifth full of oil, which means that it is four-fifths empty. Therefore, the volume of oil in the tank is:

Volume of oil = (1/5) x Total Volume

Substituting the given values, we have:

Total Volume =  π(80cm)²(200cm) = 4,031,240 cm³

Volume of oil = (1/5) x 4,031,240 cm³ = 806,248 cm³

Converting cm³ to litres, we have:

1 litre = 1000 cm³

Volume of oil = 806,248 cm³ ÷ 1000 = 806.248 litres

Therefore, after rounding of the final volume (806.248 litres) to the nearest litre, the final answer is found to be 806 litres of oil.

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The surface area of the triangular prism is 2016 [tex]feet ^2[/tex], we get to the answer by adding area of all faces in this prism.

What is surface area ?

Surface area is the sum of the areas of all the faces or surfaces of a 3D object, measured in square units.

To find the surface area of the triangular prism, we need to calculate the area of each of its faces and then add them up.

First, let's find the area of the triangular base.

Area of triangle = (1/2) x base x height

Area of triangle = (1/2) x 18 x 24

Area of triangle = 216 [tex]feet ^2[/tex]

Next, let's find the area of each rectangular face.

Area of rectangle = length x width

Area of rectangle 1 = 24 x 22 = 528 [tex]feet ^2[/tex]

Area of rectangle 2 = 18 x 22 = 396 [tex]feet ^2[/tex]

Area of rectangle 3 =  22 x 30 = 660 [tex]feet ^2[/tex]

Now, we can add up the areas of all three faces to get the total surface area of the prism:

Surface area = area of rectangle (1+2+3) + 2 x area of triangle

Surface area = 528 + 396 + 660 + 2(216)

Surface area = 2016 [tex]feet ^2[/tex]

Therefore, the surface area of the triangular prism is 2016 [tex]feet ^2[/tex].

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Answer:

y = 1/7x + 29/7

Step-by-step explanation:

Slope intercept form is y = mx + b

m = the slope

b = y-intercept

m = 1/7

Y-intercept is located at (0, 29/7)

So, the equation is y = 1/7x + 29/7

Answer: 3/5

hope this helped you. Please brainliest! :D

Step-by-step explanation: If I am wrong tell me :D

U is a binomial random variable with n trials and probability of success given by 1 - p.

As Y is a binomial random variable with n trials and probability of success given by p. Using the moment-generating functions method, it can be shown that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p. The binomial distribution is described by two parameters: n, which is the number of trials, and p, which is the probability of success in any given trial. If a binomial random variable is denoted by Y, then:[tex]P(Y = k) = \binom{n}{k}p^{k}(1 - p)^{n-k}[/tex]

The method of generating moments can be used to show that U = n - Y is a binomial random variable with n trials and probability of success given by 1 - p. The moment-generating function of a binomial random variable is given by: [tex]M_{y}(t) = [1 - p + pe^{t}]^{n}[/tex]

The moment-generating function for U is: [tex]M_{u}(t) = E(e^{tu}) = E(e^{t(n-y)})[/tex]

Using the definition of moment-generating functions, we can write: [tex]M_{u}(t) = E(e^{t(n-y)})$$$$= \sum_{y=0}^{n} e^{t(n-y)} \binom{n}{y} p^{y} (1-p)^{n-y}[/tex]

Taking the summation of the above expression: [tex]= \sum_{y=0}^{n} e^{tn} e^{-ty} \binom{n}{y} p^{y} (1-p)^{n-y}$$$$= e^{tn} \sum_{y=0}^{n} \binom{n}{y} (pe^{-t})^{y} [(1-p)^{n-y}]^{1}$$$$= e^{tn} (pe^{-t} + 1 - p)^{n}[/tex]

Comparing this expression with the moment-generating function for a binomial random variable, we can say that U is a binomial random variable with n trials and probability of success given by 1 - p.

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The equation to show the relationship between x and y is y = 0.08x, where y is the amount of sales tax in dollars for the item and x is the cost of the item in dollars before tax.

Tax is a compulsory financial charge or some other type of levy imposed upon a taxpayer by a governmental organization in order to fund various public expenditures. A failure to pay, along with evasion of or resistance to taxation, is punishable by law. Taxes consist of direct or indirect taxes and may be paid in money or as its labour equivalent. Most countries have a tax system in place to pay for public, common or agreed national needs and government functions. Some levy a flat percentage rate of taxation on personal annual income, but most scale taxes based on annual income amounts.

This equation reflects the fact that the amount of sales tax for an item is 8% of its cost before tax.

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Sales tax: The equation to show the relationship between x and y is x + y = C,

An equation is a mathematical statement that expresses two mathematical expressions to be equal. It usually consists of two numbers, variables, or a combination of the two. An equation may be used to calculate a desired result, or to determine the relationship between two variables. Equations can also be used to test whether a statement is true or false. Most equations are written in the form of "a = b," where a and b are two numbers or variables.

where C is the cost of the item plus the sales tax. This equation can be explained mathematically as follows: the cost of the item (x) is added to the amount of sales tax (y) to give the total cost of the item with tax (C).

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Answer:

first one

Step-by-step explanation:

The equation that models the future cost of gasoline is y=1.19(1.003)^(x), where "y" represents the future cost of gasoline per gallon and "x" represents the number of months since January 1, 1999.

In this equation, the initial cost of gasoline is $1.19 per gallon, and the cost increases by 0.3% per month, which is represented by the factor of (1.003)^(x).

Using this equation, you can calculate the future cost of gasoline for any number of months after January 1, 1999. For example, if you want to calculate the cost of gasoline 24 months after January 1, 1999, you can plug in x=24 and calculate y as follows:

y = 1.19(1.003)^(24)

y = 1.19(1.08357)

y = 1.288 per gallon

Therefore, the predicted cost of gasoline 24 months after January 1, 1999 is $1.288 per gallon.

The values of x such that the graph of f has a horizontal tangent are;x = 2π/3 + 2πn, 4π/3 + 2πn, where n is an integer.

1. The given derivative can be found by finding the first few derivatives and observing the pattern that occurs as shown below;Differentiating sin x with respect to x gives the derivative cos x. Continuing this process, the pattern that emerges is that sin x changes sign for every odd derivative, and stays the same for every even derivative. Therefore the 115th derivative of sin x can be expressed as follows;(d115/dx115)(sin x) = sin x, for n = 58 (where n is an even number)2. To find the values of x such that the graph of f has a horizontal tangent, we differentiate f with respect to x, and then solve for x such that the derivative equals zero. We have;f(x) = x + 2sin xDifferentiating f(x) with respect to x gives;f'(x) = 1 + 2cos xFor a horizontal tangent, f'(x) = 0, thus;1 + 2cos x = 02cos x = -1cos x = -1/2The solutions of the equation cos x = -1/2 are;x = 2π/3 + 2πn or x = 4π/3 + 2πnwhere n is an integer. Therefore the values of x such that the graph of f has a horizontal tangent are;x = 2π/3 + 2πn, 4π/3 + 2πn, where n is an integer.

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[tex]\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =42\\ r=2.8 \end{cases}\implies s=\cfrac{(42)\pi (2.8)}{180}\implies s=\cfrac{49\pi }{75}\implies s\approx 2.05 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{Perimeter of the sector} }{2.8~~ + ~~2.8~~ + ~~2.05} ~~ \approx ~~ \text{\LARGE 7.65}[/tex]

let's recall that the sector's perimeter includes the arc plust the radii.

The height that the T-short lands on the bleachers is x = 0.65.

The quadratic equation may be solved by adding and subtracting the same number within the parenthesis to get a perfect square trinomial, which is how the quadratic formula is obtained. A version of the equation that can be solved using the square root function is the result of this method.

Given that it can be used to solve any quadratic equation regardless of the values of the variables a, b, and c, the quadratic formula is an effective tool for solving quadratic equations.

The equation of the path of T-shirt is y = -18x² + 4x and that of the height of bleachers is 3y=2x−14 or y = 2/3x - 14/3

To find the height of the T-shirt landing we find the intersection of the two equations as follows:

-18x² + 4x = (2x - 14)/3

-54x² + 12x = 2x - 14

54x² - 10x - 14 = 0

27x² - 5x - 7 = 0

Using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Substitute the values;

x = (-(-5) ± √((-5)² - 4(27)(-7))) / 2(27)

x = (5 ± √(925)) / 54

x = (5 ± 5√37) / 54

x = (5 + 5√37) / 54 = 0.65 and

x = (5 - 5√37) / 54 = -0.47

Hence, the height that the T-short lands on the bleachers is x = 0.65.

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From the given graph, CE is longer than CD.

The length of the line segment bridging two locations in a plane is known as the distance between the points. d=√((x₂ - x₁)²+ (y₂ - y₁)²) is a common formula to calculate the distance between two points. This equation can be used to calculate the separation between any two locations on an x-y plane or coordinate plane.

Coordinates of E(8,6)

Coordinates of C(6,1)

Coordinates of D(3,-3)

x=8, y=6

x=6, y=1

x=3, y=-3

Distance CE=√{(8-6)² +(6-1)²} = √29

Distance CD=√{(6-3)² +(1+3)²}= √25=5

Therefore, CE is longer than CD.

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The estimated annual income can be calculated using the equation above. If the data point is used to calculate the slope and y-intercept of the regression line, then the annual income can be estimated using the equation y = m(14) + b.

If the education level is 14 years,

To create a simple linear regression model using education level as the independent variable, you will need to input data for education level and annual income.

This data can be used to estimate the annual income for any given education level. The equation for this linear regression model is y = mx + b, where y represents the annual income, m is the slope of the line, x is the education level, and b is the y-intercept.

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The expression that shows the measurement of the length of the land is L = 3x - 10a.

Let L be the length of the rectangular plot.

We know that the perimeter of a rectangle is given by the sum of the lengths of all its sides. In this case, the perimeter is 10x - 28a, so we can write:

Perimeter = 2L + 2W

10x - 28a = 2L + 2(2x - 4a)

10x - 28a = 2L + 4x - 8a

2L = 6x - 20a

L = 3x - 10a

In other words, the length of the land is equal to three times the width minus ten times the value of 'a'.

To explain this equation in a little more detail, we can say that the length of the rectangular plot is dependent on both the width of the land and the value of 'a'.

The expression tells us that as the value of 'a' increases, the length of the land decreases, and as the width of the land increases, the length of the land also increases. This equation is useful because it allows us to calculate the length of the land without having to measure it directly, as long as we know the value of 'a' and the width of the land.

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(a)  n = 10, p = 1/4, and x = 5. Using the formula of binomial probability function,P(5) = 10C5 * (1/4)^5 * (3/4)^5≈ 0.0267 (rounded to three decimal places)

(b) P(More than 3) = P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)≈ 0.2784 (rounded to three decimal places)

Here n = 10, p = 1/4, and x = 5.Using the formula of binomial probability function,P(5) = 10C5 * (1/4)^5 * (3/4)^5≈ 0.0267 (rounded to three decimal places)

Find P(More than 3)For this, we need to calculate P(4), P(5), P(6),...,P(10) and add them.Using the formula of binomial probability function,P(4) = 10C4 * (1/4)^4 * (3/4)^6 = 0.2503 (rounded to three decimal places)P(5) = 10C5 * (1/4)^5 * (3/4)^5≈ 0.0267 (rounded to three decimal places)P(6) = 10C6 * (1/4)^6 * (3/4)^4≈ 0.0014 (rounded to three decimal places)P(7) = 10C7 * (1/4)^7 * (3/4)^3≈ 0.0001 (rounded to three decimal places)P(8) = 10C8 * (1/4)^8 * (3/4)^2≈ 0.0000 (rounded to three decimal places)P(9) = 10C9 * (1/4)^9 * (3/4)^1≈ 0.0000 (rounded to three decimal places)P(10) = 10C10 * (1/4)^10 * (3/4)^0≈ 0.0000 (rounded to three decimal places)P(More than 3) = P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)≈ 0.2784 (rounded to three decimal places)

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Using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.

An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values.

As in 3x + 5 = 15, for example.

There are many different types of equations, including linear, quadratic, cubic, and others.

So, we have:

x + y = 100                        .........A

40 × x + 140 y = 85 × 100

40 x + 140 y = 8500         ........B

Solving (A) and (B) as follows:

(40 x + 140 y) - 40 × ( x + y ) = 8500 - 40 × 100

(40 x - 40 x) + (140 y - 40 y) = 8500 - 4000

0 + 100 y = 4500

y = 4500/100

Hence, the price per unit of grocery is $1.40 = y = 45 pounds.

Now, put the value of y in equation (A) as follows:

x + y = 100

x = 100 - y

x = 100 - 45

x = 55 pounds

The number of groceries at the 40-cent price is x = 55 pounds.

Therefore, using equations we know that 45 pounds are included in the $1.40 worth of groceries and 55 pounds worth of groceries are being purchased for 40 cents.

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Correct question:
A grocer has two kinds of candies, one selling for 40 cents a pound and the other for $1.40 per pound. How many pounds of each kind must he use to make 100 pounds worth 85 cents a pound?

Answer:

The second figure.

Step-by-step explanation:

The first figure's perimeter is:

70 in + 42 in + 56 in = 168 inches.

And the second figure's perimeter is:

42 in + 33 in + 33 in + 64 in = 172 inches.

Therefore, Figure 1 < Figure 2.

Answer:

We can use similar triangles to solve this problem. Let's call the length of the kite's line "x". Then, we can set up a proportion:

(length of kite) / (length of shadow) = (height of kite) / (length of shadow)

x / 9 = 10 / 9

To solve for x, we can cross-multiply and simplify:

x = 90 / 9

x = 10

Therefore, the length of the kite's line is 10 feet.

Step-by-step explanation:

Ramona's economic profit last year was $92,000 - $55,000 = $37,000. Therefore, the correct option is b. $37,000.

Ramona owns a small coffee shop, where she works full-time. Her total revenue last year was $200,000, and her rent was $5,000 per month. She pays her one employee $3,000 per month, and the cost of ingredients averages $1,000 per month. Ramona could earn $55,000 per year as the manager of a competing coffee shop nearby. Her economic profit last year was $37,000.An economic profit can be calculated by subtracting total costs from total revenue. Given that Ramona's total revenue is $200,000, her total cost is $5,000 + $3,000 + $1,000 = $9,000 per month. Multiplying this by 12 gives us her total cost for the year: $9,000 x 12 = $108,000. Ramona's economic profit last year was therefore $200,000 - $108,000 = $92,000. However, this figure doesn't take into account the opportunity cost of Ramona earning $55,000 as the manager of a competing coffee shop nearby. This needs to be subtracted from Ramona's economic profit.

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If there are 500 students in total, the number of students who made the honor roll is 10 students, given that the ratio of students who made the honor roll to the total number of students is 1:50.

The number of students who made the honor roll can be found using proportions. Here's how to do it:

Let X be the number of students who made the honor roll.

The proportion can be set up using the given ratio as follows:

1:50 = X:500

Cross-multiplying this equation and solving for X gives:

50X = 500

X = 10

Therefore, 10 students made the honor roll.

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In this particular question, we are being asked to find the number of normal subgroups of order 7 in a group of order 14. Before we begin, let's try to understand the different types of subgroups.Subgroups can be of two types: Proper Subgroups and Improper SubgroupsProper Subgroups are defined as subgroups which have more than one element and not equal to the entire group. Improper Subgroups are the subgroups which contain every element of the group.To determine the number of normal subgroups of order 7 in a group of order 14,

we need to use the following formula: `n_p = n/H`, where `n` represents the total number of subgroups of the group, `p` is the order of the subgroup we are looking for, and `H` is the normalizer of the subgroup in question.Using this formula, we can find that the order of the group is 14, and since 7 is a prime number, any subgroup of order 7 will be cyclic. Now, let's look at the different subgroups of the group of order 14:As 7 is a prime number and 7 divides 14, there will be one unique subgroup of order 7, and it will be cyclic.Since the group of order 14 is not a cyclic group, there are no other subgroups of order 7 besides the cyclic subgroup. Thus, the number of normal subgroups of order 7 in a group of order 14 is 1.

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No, the statements are not equivalent as the statement "a > b" simply means that "a" is greater than "b" and there exists a number "c" such that when added to "b", it equals "a". This does not necessarily mean that "a" is greater than "b".

The statements "a>b" and "there is the number c, so that a=b+c" are not equivalent. The statement "a>b" simply means that "a" is greater than "b," while the statement "there is the number c, so that a=b+c" means that "a" can be expressed as the sum of "b" and another number "c."

These statements are not equivalent because even if "a" is greater than "b," it may not be possible to express "a" as the sum of "b" and another number "c."

Additionally, even if "a" can be expressed as the sum of "b" and another number "c," it may not necessarily be true that "a" is greater than "b."

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The standard deviation of a normal distribution of exam scores is 8. A score that is 12 points above the mean would have a z-score of 1.5, and a score that is 20 points below the mean would have a z-score of -2.5.

The z-score can be calculated by dividing the difference between a data value and the mean of the data set by the standard deviation of the data set.

The z-score of a score that is 12 points above the mean in a normal distribution of exam scores with a standard deviation of 8.

z = (x−μ)/σ = (x−μ)/σ = (12−0)/8 = 1.5

The z-score of a score that is 12 points above the mean in a normal distribution of exam scores with a standard deviation of 8 is 1.5.

The z-score of a score that is 20 points below the mean in a normal distribution of exam scores with a standard deviation of 8.

z = {x-μ}/{σ}  = {-20-0}/{8}  = −2.5

The z-score of a score that is 20 points below the mean in a normal distribution of exam scores with a standard deviation of 8 is -2.5.

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Answer:

  138°

Step-by-step explanation:

You want the measure of the angle at two tangents when they intercept an arc of 42°.

The short answer is that the exterior angle x is the supplement of the measure of the arc:

  x = 180° -42°

  x = 138°

An exterior angle where secants meet is half the difference of the arcs of the circle they intercept. Here, the secants have been located so the corresponding chord length between the near and far circle intercept points have degenerated to zero. That is, they are tangents.

The angle relation still holds:

  x = (long arc - short arc)/2 = ((360° -42°) -42°)/2 = (360° -2·42°)/2

  x = 180° -42° = 138°

The tangents, together with their associated radii form a quadrilateral. The angles at the tangents are 90°, and the total of all angles is 360°. This gives us the relation ...

  x + 90° +42° +90° = 360°

  x +42° = 180° . . . . . . . . . . . . . subtract 180°

  x = 180° -42° = 138°

(We solved this with an extra step, so you could see the same "supplementary angles" relationship between x and 42°.)

If the average of three numbers is 16 and one of the numbers is 18, then the sum of the other two numbers is option (d) 30

Let's use algebra to solve this problem. Let x and y be the other two numbers we are looking for. We know that the average of the three numbers is 16, so we can write:

(18 + x + y) / 3 = 16

Multiplying both sides by 3, we get,

[(18 + x + y) / 3] × 3 = 16 ×3

18 + x + y = 48

Subtracting 18 from both sides, we get,

18 + x + y - 18 = 48

x + y = 30

Therefore, the correct option is (d) 30

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Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

Therefore, the smallest positive integer with at least 8 odd factors and at least 16 even factors is N = 1800.

In mathematics, combination is a way to count the number of possible selections of k objects from a set of n distinct objects, without regard to the order in which they are selected.

The number of combinations of k objects from a set of n objects is denoted by [tex]nCk[/tex] or [tex]C(n,k),[/tex] and is given by the formula:

[tex]nCk = n! / (k! *(n-k)!)[/tex]

where n! denotes the factorial of n, i.e., the product of all positive integers up to n.

by the question.

Now, let's consider the parity (evenness or oddness) of the factors of N. A factor of N is odd if and only if it has an odd number of factors of each odd prime factor of N. Similarly, a factor of N is even if and only if it has an even number of factors of each odd prime factor of N. Therefore, the condition that N has at least 8 odd factors and at least 16 even factors can be expressed as:

[tex](a_{1} +1) * (a_{2} +1) * ... * (an+1) = 8 * 2^{16}[/tex]

Let's consider the factor 2 separately. Since N has at least 16 even factors, it must have at least 16 factors of 2. Therefore, we have a_i >= 4 for at least one prime factor p_i=2. Let's assume without loss of generality that p[tex]1=2[/tex] and [tex]a1 > =4.[/tex]

Now, let's consider the remaining prime factors of N. Since N has at least 8 odd factors, it must have at least 8 factors that are not divisible by 2. Therefore, the product (a2+1) * ... * (an+1) must be at least 8. Let's assume without loss of generality that n>=2 (i.e., N has at least three distinct prime factors).

Since a_i >= 4 for i=1, we have:

[tex]N > = 2^4 * p2 * p3 > = 2^4 * 3 * 5 = 240[/tex]

Let's now try to find the smallest such N. To minimize N, we want to make the product (a2+1) * ... * (an+1) as small as possible. Since 8 = 2 * 2 * 2, we can try to distribute the factors 2, 2, 2 among the factors (a2+1), (a3+1), (a4+1) in such a way that their product is minimized. The only possibility is:

[tex](a2+1) = 2^2, (a3+1) = 2^1, (a4+1) = 2^1[/tex]

This gives us:

[tex]N = 2^4 * 3^2 * 5^2 = 1800[/tex]

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(a) Starting with the 75% estimate for Italians, the sample you must collect in order to estimate the proportion of PTC tasters within ± 0.1 with 90 % confidence is n = 51.

(b) The sample size required if you made no assumptions about the value of the proportion who could taste PTC is n = 68.

(a) To estimate the sample size needed to find the proportion of PTC tasters within ± 0.1 with 90% confidence, we will use the formula for sample size estimation in proportion problems:
n = (Z² * p * (1-p)) / E²

Where n is the sample size, Z is the Z-score corresponding to the desired confidence level (1.645 for 90% confidence), p is the proportion of PTC tasters (0.75), and E is the margin of error (0.1).

n = (1.645² * 0.75 * (1-0.75)) / 0.1²
n = (2.706 * 0.75 * 0.25) / 0.01
n ≈ 50.74

Since we need a whole number, we round up to the nearest whole number:
n = 51

(b) If no assumptions were made about the proportion of PTC tasters, we would use the worst-case scenario, which is p = 0.5 (maximum variance):

n = (1.645² * 0.5 * (1-0.5)) / 0.1²
n = (2.706 * 0.5 * 0.5) / 0.01
n ≈ 67.65

Again, rounding up to the nearest whole number:
n = 68

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There are, 6300 different possibilities for the researcher’s study.

Total number of class I railroads = 10Number of class I railroads selected = 3Total number of class II railroads = 6Number of class II railroads selected = 2Total number of class III railroads = 5Number of class III railroads selected = 1Number of different possibilities for selecting 3 class I railroads from 10 class I railroads = 10C3 = (10 x 9 x 8)/(3 x 2 x 1) = 120

Number of different possibilities for selecting 2 class II railroads from 6 class II railroads = 6C2 = (6 x 5)/(2 x 1) = 15Number of different possibilities for selecting 1 class III railroad from 5 class III railroads = 5C1 = 5Total number of different possibilities for selecting 3 class I railroads from 10 class I railroads, 2 class II railroads from 6 class II railroads, and 1 class III railroad from 5 class III railroads = 10C3 x 6C2 x 5C1= 120 x 15 x 5= 6300Therefore, there are 6300 different possibilities for the researcher’s study.

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