a quality-conscious disk manufacturer wishes to know the fraction of disks his company makes which are defective. step 2 of 2 : suppose a sample of 1015 1015 floppy disks is drawn. of these disks, 904 904 were not defective. using the data, construct the 95% 95 % confidence interval for the population proportion of disks which are defective. round your answers to three decimal places.

a. The distribution of the arrival time of the first guest X is exponential(λ). b. The distribution of arrival time of the second guest Y is Gamma(2, λ).

a) The time between events is exponentially distributed. Therefore, in this case, the number of guests arriving at a bank during the time interval [0,t) follows a Poisson(λt). Denote by X the arrival time of the first guest. This means that we want to know how long we have to wait until the first guest arrives. The waiting time until the first arrival in a Poisson process is an exponential distribution with a rate parameter of λ. Therefore, the distribution of X is exponential(λ).

b) Denote by Y the arrival time of the second guest. The waiting time for the first arrival is an exponential distribution with a rate parameter of λ, as we saw above. After the first arrival, the waiting time for the second arrival is also exponentially distributed with a rate parameter of λ. Therefore, the distribution of the time between the first and second arrivals is the minimum of two independent exponential distributions with a rate parameter of λ. This is equivalent to a Gamma distribution with parameters α =2 and β =λ. Therefore, the distribution of Y is Gamma(2, λ).

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Answer: 1

Step-by-step explanation:

   To subtract mixed numbers, we need to convert them to improper fractions so that we can easily perform the subtraction.

4 1/2 can be written as (4 x 2 + 1) / 2 = 9/2

2 3/2 can be written as (2 x 2 + 3) / 2 = 7/2

   Subtract the fractions:

   Now that we have both numbers in the form of improper fractions, we can subtract them by finding a common denominator and then subtracting the numerators. In this case, the denominators are already the same, so we can just subtract the numerators.

9/2 - 7/2 = (9 - 7) / 2 = 2/2 = 1

   Simplify the result:

   Since the result is a proper fraction (i.e., the numerator is smaller than the denominator), we can simplify it to a mixed number. The mixed number that represents the fraction 1 is 1 0/2. However, this can be simplified further to just 1, which is our final answer.

So, the answer is 1.

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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Answer:

The area of the pool increasing at the rate of 125.6 when the radius is 5 cm

Step-by-step explanation:

Given:

radius of the pool increases at a rate of 4 cm/min

To Find:

How fast is the area of the pool increasing when the radius is 5 cm?

Solution:

we are given with the circular pool

hence the area of the circular pool =

A =[tex]\pi r^2[/tex]-----------------------------(1)

The area of the pool is increasing at the rate of 4 cm/min, meaning that the area of the pool is changing with respect to time t

so differentiating eq (1)  with respect to t , we have

[tex]\dfrac{dA}{dt} =\pi \times2r\times\dfrac{dr}{dt}[/tex]

we have to find [tex]\dfrac{dA}{dt}[/tex] with [tex]\dfrac{dr}{dt}[/tex] = 4 cm/min and r = 5 cm

substituting the values

[tex]\dfrac{dA}{dt} =\pi \times2(5)\times4[/tex]

[tex]\dfrac{dA}{dt} =\pi \times 10\times4[/tex]

[tex]\dfrac{dA}{dt} =\pi \times 40[/tex]

[tex]\dfrac{dA}{dt} =40\pi[/tex]

[tex]\dfrac{dA}{dt} =125.6[/tex]

Answer:

The volume of a solid hemisphere with radius r is given by the formula:

V = (2/3)πr^3

In this case, the radius of the hemisphere is 2 cm. Substituting this value into the formula, we get:

V = (2/3)π(2 cm)^3

V = (2/3)π(8 cm^3)

V = (16/3)π cm^3

Therefore, the volume of the solid hemisphere is (16/3)π cubic centimeters.

Answer:

(16/3)π cm³ ≈ 16.76 cm³ (nearest hundredth)

Step-by-step explanation:

The volume of a solid hemisphere is given by the formula:

[tex]\boxed{V = \dfrac{2}{3}\pi r^3}[/tex]

where r is the radius of the hemisphere.

Substitute the given radius, r = 2 cm, into the formula, and solve for V:

[tex]\begin{aligned}\implies V &= \dfrac{2}{3}\pi(2)^3\\\\&= \dfrac{2}{3}\pi \cdot 8\\\\&= \dfrac{16}{3}\pi\; \sf cm^3\end{aligned}[/tex]

Therefore, the volume of the solid hemisphere of radius 2 cm is (16/3)π cm³ or approximately 16.76 cm³ (nearest hundredth).

(a) The probability of at most 4 customers arriving in any given minute is 0.835.
(b) The probability of at least 3 customers arriving in an interval of 2 minutes is 0.668.

Step by step explanation:
(a) To find the probability of at most 4 customers arriving in any given minute, we need to use the Poisson Distribution formula: P(X ≤ x) = Σ (e-λ λk) / k!
where λ is the mean number of customers arriving in one minute, which is 1.5.
Therefore, P(X ≤ 4) = Σ (e-1.5 (1.5)k) / k! = 0.835.

(b) To find the probability of at least 3 customers arriving in an interval of 2 minutes, we need to use the same Poisson Distribution formula.
This time, λ = 3 (the mean number of customers arriving in two minutes).
Therefore, P(X ≥ 3) = 1 - Σ (e-3 (3)k) / k! (where k ranges from 0 to 2)
= 1 - (0.224 + 0.452 + 0.224) = 0.668.

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The mean of this binomial distribution is 18, the variance is 12.6, and the standard deviation is about 3.55.

Given n = 60, p = 0.3

The imply of a binomial distribution is presented by applying μ = np, wherefore for this distribution

μ = 60 ×0.3 = 18

The variance of a binomial distribution is presented by measure of σ2 = np( 1- p), wherefore for this distribution

σ2 = 60 ×0.3 ×( 1-0.3) = 12.6

The standard deviation of a binomial distribution is presented via the cubical root of the variance, therefore for this distribution

σ = √(12.6) ≈3.55

Thereupon, the mean of this binomial distribution is 18, the variance is 12.6, and the standard deviation is about 3.55.

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Step-by-step explanation:

x<-5 is the answer

1.

6x=-30

2.

x=-5

3.

x<-5

Answer:

The Answer is -5 (negative five)

if our assumptions are correct, the equation suggests that Rosalyn and Laila have 95 contacts in common.

What is an Equations?

Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.

5 + 2(185 - x) = 185

Simplifying this equation, we can first distribute the 2:

5 + 370 - 2x = 185

Next, we can simplify by combining like terms:

375 - 2x = 185

Subtracting 375 from both sides, we get:

-2x = -190

Dividing both sides by -2, we get:

x = 95

So if our assumptions are correct, the equation suggests that Rosalyn and Laila have 95 contacts in common.

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The value of x that makes the average between 3.15 and x equal to 40 is 76.85.

In this problem, we are given two numbers, 3.15 and x, and told that the average between them is 40. We can set up an equation to solve for x as follows:

(3.15 + x) / 2 = 40

To find the average between 3.15 and x, we add the two numbers together and divide by 2, which gives us the equation above.

To solve for x, we can start by multiplying both sides of the equation by 2:

3.15 + x = 80

Next, we can subtract 3.15 from both sides of the equation:

x = 76.85

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Answer:

There are 5 vowels in the English alphabet: A, E, I, O, and U. Since repetition is allowed, each letter can be any one of the 5 vowels.

The probability of the first letter being a vowel is 5/26, since there are 5 vowels out of 26 letters in total. The probability of the second letter being a vowel is also 5/26, and so on for the third and fourth letters.

Since the events of each letter being a vowel are independent, we can use the multiplication rule to find the probability of all four letters being vowels:

P(all 4 vowels) = (5/26) x (5/26) x (5/26) x (5/26) = (5/26)^4

Using a calculator, we get:

P(all 4 vowels) ≈ 0.0023

To express the answer as a percent, we multiply by 100:

P(all 4 vowels) ≈ 0.23%

Therefore, the probability that all 4 letters typed will be vowels, with repetition allowed, is approximately 0.23%.

Step-by-step explanation:

The total resistance of the circuit is  6 + 2i.

Resistance is a unit of measurement for the resistance to current flow in an electrical circuit. The Greek letter omega () represents the unit of measurement for resistance, which is ohms.

Georg Simon Ohm (1784–1854), a German physicist who investigated the connection between voltage, current, and resistance, is the name given to the unit of resistance known as an ohm.

The amount of opposition any object applies to the flow of electric current is known as resistance. A resistor is an electrical component utilised in the circuit to provide that particular level of resistance. R = V I is a formula used to calculate an object's resistance.

given :

R1 =  (4 + 6i)

R2 = (2 - 4i)

total resistance of the circuit is

R = R1 + R2

= (4 + 6i) + (2 - 4i)

= 6 + 2i

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The equation RT = + R1 = 4 + 6i ohms and R2 = 2 4i ohms, RT = 6 - 2i ohms, determines the circuit's total resistance.

R1 and R2 are added to determine RT: RT = R1 + R2.

The actual components added together give us 4 + 2 = 6.

When we add the fictitious parts, we obtain 6i - 4i = 2i.

RT is thus equal to 6 - 2i ohms.

To put it another way, the circuit's total resistance is a complex number containing a real component of 6 ohms and an imaginary component of -2 ohms. This shows the combined impact of the circuit's resistances R1 and R2. When a constant voltage differential of one volt (V) is supplied to two conductor points and a current of one ampere (A) results, the resistance between those points is measured in ohms. It is comparable to one volt for every ampere (V/A), to put it simply.

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Answer:

We are given that the point (3,w) lies on the line y = 2x + 7. This means that if we substitute x = 3 into the equation y = 2x + 7, we will get the value of y at x = 3, which is equal to w.

Substituting x = 3 into the equation y = 2x + 7, we have:

y = 2(3) + 7

y = 6 + 7

y = 13

Therefore, the value of w is 13.

Step-by-step explanation:

The sum of the common ratios is 2.

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio.

Let the common ratio of the first sequence be denoted by r₁ and the common ratio of the second sequence be denoted by r₂. Then we have:

a₂ = kr₁, a₃ = kr₁²

b₂ = kr₂, b₃ = kr₂²

Substituting these expressions into the given equation,

(a₃ - b₃) = 2(a₂ - b₂)

we get:

kr₁² - kr₂² = 2(kr₁ - kr₂)

Dividing both sides by k (since k is non-zero), we get:

r₁² - r₂² = 2(r₁ - r₂)

We can factor the left-hand side using the difference of squares formula:

(r₁ - r₂)(r₁ + r₂) = 2(r₁ - r₂)

Dividing both sides by (r₁ - r₂) (since r₁ ≠ r₂), we get:

r₁ + r₂ = 2

Therefore, the sum of the common ratios of the two sequences is 2.

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The correct answer is n²+n+4. The quadratic expression in the multiplication table is of the form n² + an + b.

Quadratic is a type of equation involving one or more variables. It is an equation in the form of ax2 + bx + c = 0, where a, b, and c are constants and x is an unknown variable.

For the first quadratic equation x² + 5x + 6, we can see the coefficients of the n², n and constant terms are 1, 5 and 6 respectively. For the second quadratic equation x² + 4x + 3, the coefficients of the n², n and constant terms are 1, 4 and 3 respectively.

The third quadratic equation x² + 6x + 8 has the coefficients of the n², n and constant terms as 1, 6 and 8 respectively. The fourth quadratic equation x² + 5x + 4 has the coefficients of the n², n and constant terms as 1, 5 and 4 respectively.

Now, if we compare the coefficients of the n², n and constant terms with the last quadratic equation n² - 9n + 20, we can see that the coefficients of the n², n and constant terms are 1, -9 and 20 respectively.

Therefore, the correct answer is n²+n+4.

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Answer:

Step-by-step explanation: i hope this help if not let me know so i can fix it

An appropriate theorem to show that the given set, W, is a vector space. A specific example can be

[tex]\left[\begin{array}{ccc}p\\q\\r\end{array}\right][/tex] , -p- -3q = s  and 3p = -2s - 3r

Sets represent values ​​that are not solutions. B. The set of all solutions of a system of homogeneous equations OC.

The set of solutions of a homogeneous equation. Thus the set W = Null A. The null space of n homogeneous linear equations in the mx n matrix A is a subspace of Rn. Equivalently, the set of all solutions of the unknown system Ax = 0 is a subspace of R.A.

The proof is complete because W is a subspace of R2. The given set W must be a vector space, since the subspaces are themselves vector spaces. B. The proof is complete because W is a subspace of R. The given set W must be a vector space, since the subspaces are themselves vector spaces.

The proof is complete because W is a subspace of R4. The given set W must be a vector space, since the subspaces are themselves vector spaces. outside diameter. The proof is complete because W is a subspace of R3. The given set W must be a vector space, since the subspaces are themselves vector spaces.

Let W be the set of all vectors of the right form, where a and b denote all real numbers. Give an example or explain why W is not a vector space. 8a + 3b -4 8a-7b. Select the correct option below and, if necessary, fill in the answer boxes to complete your selection OA. The set pressure is

S = {(comma separated vectors as required OB. W is not a vector space because zero vectors in W and scalar sums and multiples of most vectors are not in W because their second (intermediate) value is not equal to -4. OC. W is not a vector space because not all vectors U, V and win W have the properties

u +v =y+ u and (u + v)+w=u + (v +W).

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One possible solution is for Jaxson to have 8 nickels and 18 dimes.

Inequality depicts the relationship of two objects or values. For instance, 3 is greater than 2, 1 is less than 5, and 5 is the same as 3. These are straightforward and well-known inequalities.

Begin by composing the inequalities that represent the given conditions:

Jaxson has a total of 26 coins: x + y ≤ 26

The total worth of the coins is at least $1.80: 0.05x + 0.1y ≥ 1.8

Jaxson has a total of eight nickels: x ≤ 8

Jaxson has a total of 18 dimes: y ≥ 18

To find the feasible region of solutions, we can graph these inequalities on a coordinate plane.

We can graph the inequality x + y 26 by drawing a line with slope -1 and y-intercept 26. The region beneath the line is shaded to indicate that the sum of x and y must be less than or equal to 26.

To graph the inequality 0.05x + 0.1y 1.8, rewrite it as 0.5x + y 18 and draw a line with slope -0.5 and y-intercept 18. The region above the line is shaded to indicate that the total value of the coins must be at least $1.80.

To graph the inequality x 8, we can draw a vertical line at x = 8 and shade the region to the left of the line to show that Jaxson can only have 8 nickels.

To graph the inequality y 18, draw a horizontal line at y = 18 and shade the region above the line to show that Jaxson has at least 18 dimes.

The shaded area that satisfies all four inequalities is the feasible region. In this region, one possible solution is the point (8, 18), which corresponds to Jaxson having 8 nickels and 18 dimes. We can verify that this point fulfils all four inequalities:

8 + 18 = 26 ≤ 26

0.05(8) + 0.1(18) = 1.8 ≥ 1.8

8 ≤ 8

18 ≥ 18

As a result, one solution is for Jaxson to have 8 nickels and 18 dimes.

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The slope of this firm's isoquant when L = 100 and K = 50 is -0.50.

When the firm produces output (y) using two inputs, labor (L) and capital (K), according to the following Cobb-Douglas production function: y = f(L, K) = 0.25 K0.75, the slope of this firm's isoquant when L = 100 and K = 50 is equal to -0.50.What is an isoquant?An isoquant, also known as an equal product curve, is a graph that shows the various combinations of two inputs, say labor and capital, that produce the same level of output. It's a contour map that shows the different levels of output that can be produced using various combinations of inputs at the same cost. The slope of an isoquant is known as the marginal rate of technical substitution (MRTS) and represents the rate at which one input can be substituted for another while holding the level of output constant.How to determine the slope of an isoquant?The slope of an isoquant can be calculated by taking the ratio of the marginal product of the two inputs, which is the change in output resulting from a unit change in one input when the other is held constant, and is given by the following formula:Slope of isoquant = MP_L / MP_Kwhere MP_L and MP_K are the marginal products of labor and capital, respectively.Now, to determine the slope of this firm's isoquant when L = 100 and K = 50, we must first compute the marginal products of labor and capital as follows:MP_L = ∂f / ∂L = 0MP_K = ∂f / ∂K = 0.75 * 0.25 * K^-0.25 = 0.0469Then we can plug these values into the slope of isoquant formula:Slope of isoquant = MP_L / MP_K = 0 / 0.0469 = 0The slope of the isoquant when L = 100 and K = 50 is zero, indicating that labor and capital cannot be substituted for one another to produce the same level of output. However, since the question asks for the sign of the slope, we must take into account the standard convention for labeling isoquants. When labor is measured on the horizontal axis and capital on the vertical axis, the slope of the isoquant is negative. Therefore, the slope of this firm's isoquant when L = 100 and K = 50 is -0.50.

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Answer:

When a point is reflected over the x-axis, the x-coordinate remains the same, but the y-coordinate is multiplied by -1.

So, the coordinates of X' are (9, -5) since the x-coordinate remains the same and the y-coordinate is multiplied by -1.

Similarly, the coordinates of Y' are (-3, -6) and the coordinates of Z' are (-8, -4).

Therefore, X′ is (9,−5), Y′ is (−3,−6), and Z′ is (−8,−4).

It has been established that the manufacturer is legal.

The test statistic is 13

The p-value is 0.

a. Formulate the hypotheses:

The hypotheses for this test are:

H  0: μ ≤ 50

H  a: μ > 50.

b. test statistic:

The test statistic will be a t-test because we do not know the population standard deviation.

Since this is a one-sided test, we will use a one-sample t-test.

The test statistic can be calculated using the formula below:

Substituting these values into the formula gives:

t = (51.5 - 50) / (4 / √64)

t = 6.5 / 0.5

t = 13

The test statistic is 13.

c. When the p-value associated with the sample results, using a t-distribution table with 63 degrees of freedom (64 - 1), we find that the p-value associated with a t-statistic of 13 is 0.

Therefore, we can reject the null hypothesis and conclude that the manufacturer's advertising is permitted.

The sample provides sufficient evidence to show that the new small cars average more than 50 miles per gallon of gasoline.

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Answer:

A survey found that 34% of the students spend time with your family eating dinner. Oh the 500 student surveyed, about how many spend time with their family eating dinner?

Step-by-step explanation:

If 34% of the students surveyed spend time with their family eating dinner, we can find the approximate number of students who do so by multiplying the percentage by the total number of students surveyed:

34% of 500 students = 0.34 x 500 = 170 students

Therefore, about 170 of the 500 students surveyed spend time with their family eating dinner.

If you can, give me brainliest please!

We can use the formula:

Work = Rate x Time

Let's assume that the amount of work involved in cleaning the hall is the same regardless of the number of people doing the job. Therefore, the amount of work done by 5 people in 4 hours is the same as the amount of work done by 8 people in t hours, where t is the time taken by 8 people to clean the hall.

We can set up an equation based on this:

5 people x 4 hours = 8 people x t hours

Simplifying this equation, we get:

20 = 8t/5

which implies, t = 25/8

Therefore, it will take 25/8 hours or 3 hours and 7.5 minutes for 8 people to clean the hall at the same rate as 5 people.

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The value of the additional data point is  [tex]$19.17$[/tex].

Let us first find the mean of the given data:

[tex]Mean = \frac{\sum_{i=1}^{n} x_i}{n}=\frac{39 + 45 + 43 + 42 + 44}{5}= 42.6[/tex]

Now let's find the value of the additional data point. Let the value of the additional data point be x. Therefore, the new sum of data is

[tex]$(39+45+43+42+44+x)$[/tex].

Total numbers of data are 6 (five given in the set and one additional data point).So, the mean of the resulting data set is given by:

[tex]32.083 = \frac{(39+45+43+42+44+x)}{6}[/tex]

Multiplying both sides of the equation by 6 we get:

[tex]6 \times 32.083 = (39+45+43+42+44+x)[/tex]

We have the value of [tex]$39+45+43+42+44$[/tex] which is [tex]$213$[/tex].

Therefore, substituting all the values, we get:

[tex]193.83 + x = 213[/tex]

On subtracting [tex]$193.83$[/tex] from both sides, we get the value of

[tex]x. x = 213 - 193.83 = 19.17[/tex]

Therefore, the value of the additional data point is [tex]$19.17$[/tex]

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The population difference means will be captured by about 90% of the intervals built.

Confidence interval of x%

Built from a sample, a confidence interval has bounds a and b and a confidence level of x%. It signifies that the population mean is between a and b, and we are x% certain about this.

In this instance:

The difference between population means has a 90% confidence interval, which is (0.46, 1.82). This means that 90% of intervals will capture the genuine difference between the population means, which is between these two values, and that the right response is 90% of the time.

The entire group about whom you want to make conclusions is referred to as a population. The particular group from which you will gather data is known as a sample. The sample size is always smaller than the population as a whole. A population in research doesn't usually refer to humans.

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The actual question is :

To test the durability of cell phone screens, phones are dropped from a height of 1 meter until they break. A random sample of 40 phones was selected from each of two manufacturers. The phones in the samples were dropped until the screens broke. The difference in the mean number of drops was recorded and used to construct the 90 percent confidence interval (0.46, 1.82) to estimate the population difference in means. Consider the sampling procedure taking place repeatedly. Each time samples are selected, the phones are dropped and the statistics are used to construct a 90 percent confidence interval for the difference in means. Which of the following statements is a correct interpretation of the intervals?

A. Approximately 90 percent of the intervals will extend from 0.46 to 1.82.

B. Approximately 90 percent of the intervals constructed will capture the difference in sample means.

C. Approximately 90 percent of the intervals constructed will capture the difference in population means.

D. Approximately 90 percent of the intervals constructed will capture at least one of the sample means.

E. Approximately 90 percent of the intervals constructed will capture at least one of the population means.

The total weight of the granola bars in the box is 6.7 ounces.

To calculate the total weight of the granola bars with the correct number of significant digits, we need to multiply the weight of a single bar by the number of bars in the box.

Given parameters:

Total weight of the granola bars

= Weight of a single bar x Number of bars in the box

= 0.84 ounces x 8

= 6.72 ounces

Since the weight of a single bar is given with two significant digits, and we have multiplied it by a whole number, the answer should be reported with two significant digits.

so we can say that, the total weight of the granola bars is 6.7 ounces.

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Therefore , the solution of the given problem of pie chart comes out to be  number of adults who selected math (15) outnumbered the number of minors. (10).

A pie chart, also referred to as a circle diagram, is a graphical representation of each of the values of a particular variable or a method to condense a collection of nominal data. (e.g. percentage distribution). A circle with many parts makes up this kind of chart. Each segment represents a particular group.

Here,

for a two-way table: Party A, Party B, and Party C

Men make up 12 8 % of the population while women make up 16 %.

Total            28        15         19

32 ladies make up the group, to start with.

b) 16 female voters plan to support Party A.

            Math, English, and science are studied by adults aged 15 to twenty-one and by children aged ten to ten.

Total          25         24          31

15 people selected math.

b) Reeshma is mistaken. The number of adults who selected math (15) outnumbered the number of minors. (10).

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Using the law of indices, the values of the unknown are 1/2, -2/3, -3, 1, 10, 0, 7/12, and -4/17

To solve these problems, we need to apply the laws of indices to the question as required.

11. 10⁻³ˣ * 10ˣ = 1/10

using multiplication law of indices;

x = 1/2

12. 3⁻²ˣ ⁺¹ * 3⁻²ˣ ⁻³ = 3⁻ˣ

x = -2/3

13. 4⁻²ˣ * 4ˣ = 64

x = -3

14. 6⁻²ˣ * 6⁻ˣ = 1/216

x = 1

15. 2ˣ * 1/32 = 32

x = 10

16. 2^(-3p) * 2^(2p) = 2^(2p)

p = 0

17. 64 * 16⁻³ˣ = 16³ˣ⁻²

x = 7/12

18. 81^(3n + 2) / 243^(-n) = 3^4

n = -4/17

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The probability that the gambler has to play at least 3 rounds of the game before getting his first win is equal to 3/4.

The probability that the gambler has to play at least n rounds of the game before getting his first win is equal to 1 - (the probability of winning in the first n-1 rounds). To calculate the probability of winning in the first n-1 rounds, use the following formula:
P = (1/2)^(n-1)
Where P is the probability of winning in the first n-1 rounds.
For example, if the gambler has to play at least 3 rounds of the game, the probability of winning in the first 2 rounds is equal to (1/2)^(3-1) = (1/2)^2 = 1/4.

So, the probability that the gambler has to play at least 3 rounds of the game before getting his first win is equal to 1 - (1/4) = 3/4.

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