The 8 leaders of the G8 nations convene in Rome and stand in a row as they get ready to have some pictures of them taken by the press. What is the probability that the picture that the New York Times' editors will randomly select for publishing the next day is one in which Berlusconi is not standing next to Obama? (assuming that there are pictures of all possible standing arrangements).

A) State the chain rule for integration

Ans. The chain rule for integration is also known as "  Integration by substitution "

Integration by substitution is taken in order to make integration solve easily in few steps.

For, [tex]I = \int\limits (x+2)^{2} \,dx[/tex]

Instead of expanding term [tex](x+2)^{2}[/tex]

With substitution of  [tex]u = (x+2) [/tex] and [tex]du= 1 dx [/tex]

We simplified the integration as

[tex]I = \int\limits (u)^{2} \, du[/tex]

[tex]I = \frac{(u)^{3}}{3}+C[/tex]

By replacaing value of u=x+2

[tex]I = \frac{(x+2)^{3}}{3}+C[/tex]

B) State the rule of differentiation for the sine function.

Ans. We know that [tex]\frac{d}{dx}Sinx dx = Cosx [/tex]

C) Find the indefinite integral using substitution.

Ans.

Given, [tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]

Take y = Sin14x

Differentiating both side

[tex]dy=14Cos14x dx [/tex]

[tex]\frac{dy}{14} = Cos14x\, dx[/tex]

Substituting values in integration,

[tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]

[tex]I = \int\limits {\frac{1}{y} } \,\frac{dy}{14} [/tex]

[tex]I = \frac{1}{14}\int\limits {\frac{1}{y} } \,dy [/tex]

[tex]I = \frac{1}{14} lny + C [/tex]

Replacing values in the integration

[tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]

D)Check your work by taking a derivative of your answer from part C.

Ans.

Answer for Part C is [tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]

Differentiating the answer

we get,

[tex]=\frac{1}{14}\frac{d}{dx}[ ln(Sin14x) + C]\\=\frac{1}{14}\frac{1}{Sin14x} \frac{d}{dx}(Sin14x)+ \frac{d}{dx}C\\=\frac{1}{14}\frac{1}{Cos14x}(14Cos14x)\\=\frac{Cos14x}{Sin14x} \\ =I[/tex]

Answer:

a. [tex]x-y > 0[/tex]

b. [tex]x+y \leq 50[/tex]

c. x=26; y=24 or (24,26)

Step-by-step explanation:

Let x = number of teachers hired.

Let y = number of tutors hired.

Now solving for part a we get

a. Write an inequality that represents the statement that the number of teachers hired must exceed the number of tutors hired.

[tex]x-y > 0[/tex]

solving for part b we get;

b. Write an inequality that represents the statement that the maximum number of teachers and tutors is 50.

[tex]x+y \leq 50[/tex]

solving for part c we get;

c. Choose a point that satisfies the situation, and explain why you chose that number of tutors and teachers.

Now we know that [tex]x-y > 0[/tex] also [tex]x+y \leq 50[/tex]

x=26; y=24

(24,26)

Explanation: To make number of teacher more than number of tutors this is the maximum value we can achieve for the requirement.

a. The inequality x > y represents the statement that the number of teachers hired must exceed the number of tutors hired. b. The inequality x + y ≤ 50 represents the statement that the maximum number of teachers and tutors is 50. c. The point (25, 10) satisfies both inequalities and represents hiring 25 teachers and 10 tutors.

a. To represent the statement that the number of teachers hired must exceed the number of tutors hired, you can write the inequality x > y. This means that the number of teachers, represented by x, must be greater than the number of tutors, represented by y.

b. To represent the statement that the maximum number of teachers and tutors is 50, you can write the inequality x + y ≤ 50. This means that the total number of teachers and tutors hired, represented by x + y, cannot exceed 50.

c. Let's choose the point (25, 10) which represents hiring 25 teachers and 10 tutors. This point satisfies both inequalities: 25 > 10 and 25 + 10 ≤ 50.

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Answer:

A. 18x8 is 144 then divided by 2 is 72

Answer:

Step-by-step explanation:

Answer:

option (b) 12.5 V

Step-by-step explanation:

Given:

Number of turns, N = 500

Area of the coil, A = 0.050 m²

Angular speed, ω = 20 rad/s

Magnetic field, B = 0.050 T

Angle to the field, θ = 30°

Now,

EMF induced, ε = NBAωsinθ

on substituting the values, we get

ε = 500 × 0.050 × 0.050 × 20 × sin30°

or

ε  = 25 × 0.5

or

ε  = 12.5 V

Hence,

option (b) 12.5 V

To estimate the proportion of cyanobacteria in algae samples with a 99% confidence level and a margin of error ≤ 0.01, Ian would need a sample size of at least 12111.

To determine the required sample size (n) for estimating the proportion of cyanobacteria in the algae samples with a margin of error of at most (0.01) and a (99%) confidence level, you can use the formula for the margin of error in estimating a population proportion:

[tex]\[ E = Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]

Given:

- Margin of error (E) = 0.01

- Confidence level = (99%), which corresponds to a Z-score of approximately 2.576

- Estimated proportion [tex](\(\hat{p}\)) = \(\frac{38}{50} = 0.76\)[/tex] (from the given sample)

Now, the formula can be rearranged to solve for (n):

[tex]\[ n = \frac{\hat{p}(1-\hat{p})}{\left(\frac{E}{Z}\right)^2} \][/tex]

Substitute the given values:

[tex]\[ n = \frac{0.76 \times (1 - 0.76)}{\left(\frac{0.01}{2.576}\right)^2} \][/tex]

Now calculate:

[tex]\[ n \approx \frac{0.76 \times 0.24}{\left(\frac{0.01}{2.576}\right)^2} \][/tex]

n ≈  12111

Therefore, Ian would need a sample size of at least 12111 algae samples to estimate the proportion of cyanobacteria with a margin of error of at most (0.01) in a (99%) confidence interval.

Ian would require a sample size of approximately 12,102 algae to estimate the proportion of cyanobacteria with a margin of error of at most 0.01 in a 99% confidence interval.

To determine the required sample size for estimating the proportion of algae samples that are cyanobacteria with a margin of error of at most 0.01 and a 99% confidence level, we can use the formula for the confidence interval for a proportion:

[tex]\[ n = \left(\frac{z_{\alpha/2} \cdot \sqrt{p(1-p)}}{E}\right)^2 \][/tex]

where:

-  n is the sample size,

- [tex]\( z_{\alpha/2} \)[/tex] is the z-score corresponding to the desired confidence level (for 99%, [tex]\( z_{\alpha/2} = 2.576 \),[/tex]

- p is the estimated proportion of the population that has the characteristic of interest (in this case, being cyanobacteria),

-  E  is the margin of error.

The estimated proportion  p  can be taken from the initial sample of 50 algae, where 38 were cyanobacteria. Thus,  pis estimated as:

[tex]\[ p = \frac{\text{Number of cyanobacteria}}{\text{Total sample size}} = \frac{38}{50} = 0.76 \][/tex]

Now, we plug in the values into the formula:

[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.76(1-0.76)}}{0.01}\right)^2 \][/tex]

[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.76 \cdot 0.24}}{0.01}\right)^2 \][/tex]

[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.1824}}{0.01}\right)^2 \][/tex]

[tex]\[ n = \left(\frac{2.576 \cdot 0.427}{0.01}\right)^2 \][/tex]

[tex]\[ n = \left(\frac{1.101}{0.01}\right)^2 \][/tex]

[tex]\[ n = (110.1)^2 \][/tex]

[tex]\[ n \ = 12101.01 \][/tex]

Since we cannot have a fraction of an algae sample, we round up to the nearest whole number:

[tex]\[ n \ = 12102 \][/tex]

Answer:4 students will be in each group.

Step-by-step explanation:

28 and 36. The 2 numbers are divisible by 4 without remainder

Answer:

4 students will be in each group.

Step-by-step explanation:

28 and 36. The 2 numbers are divisible by 4 without remainder

brainilest pl.z? i've never got brainilest.

Answer:

Step-by-step explanation:

Total cost of the two T-shirts that you purchased is $25. The sales tax on the T-shirts is 8%. This means that the additional amount that you will pay on the T shirt is

25/100 × 25 = 0.25 × 25 = $6.25

The new cost of the T shirt would be the sum of the original cost and the value of the sales tax. Therefore,

New cost = 25 + 6.25 = $31.25

You have a $30 gift card from Amazon. The new cost of the shirts is higher than that value of the gift card. So the gift card will not cover the cost of the T shirts

To determine if the gift card will cover the cost of the two T-shirts, calculate the sales tax and subtract it from the total cost. The gift card will cover the cost if the remaining balance is positive.

To determine if your gift card will cover the cost of the two T-shirts, you need to calculate the cost of the sales tax and subtract it from the total cost of the T-shirts. First, convert the sales tax percentage to decimal form by dividing it by 100 (8% becomes 0.08). Then, multiply the total cost of the T-shirts (which is $25) by the sales tax rate to find the amount of sales tax. Multiply $25 by 0.08 to get $2. Next, subtract the sales tax amount from the total cost of the T-shirts: $25 - $2 = $23. Finally, determine if the gift card covers the cost by comparing it to the final cost: $30 - $23 = $7.

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Answer: a) 12/51

b) 3/51

Step-by-step explanation:

we are assuming the cards are dealt without replacement

a) given that the first card is heart, we are left with 12 hearts and 51 cards in total

Therefore, the probability that the second card is heart is:

P2 = 12/51

b) probability that the two cards are hearts is given by:

P = 13/52 * 12/51

P = 3/51

(a) The chance that the second card is a heart given the first card is a heart is [tex]\(\frac{12}{51}\)[/tex].

(b) The chance that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].

(a) To find the probability that the second card is a heart given that the first card is a heart, we use conditional probability. There are 13 hearts in a standard deck of 52 cards. Once the first heart is drawn, there are 12 hearts left and the total number of cards left is 51. The probability of drawing a heart as the second card, given that the first card is a heart, is the number of remaining hearts divided by the total number of remaining cards.

[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{12}{51} \][/tex]

This simplifies to:

[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{4}{17} \][/tex]

(b) To find the probability that both the first and second cards are hearts, we multiply the probability of drawing a heart first by the probability of drawing a heart second given that the first card is a heart. The probability of drawing a heart first is [tex]\(\frac{13}{52}\)[/tex], which simplifies to [tex]\(\frac{1}{4}\)[/tex]. We already calculated the probability of drawing a heart second given a heart first as [tex]\(\frac{12}{51}\) or \(\frac{4}{17}\)[/tex].

[tex]\[ P(\text{first card is a heart and second card is a heart}) = P(\text{first card is a heart}) \times P(\text{second card is a heart | first card is a heart}) \][/tex]

[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{4} \times \frac{4}{17} \][/tex]

[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{17} \][/tex]

Thus, the probability that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].

Answer:

The angles formed on line b when cut by the transversal are congruent with ∠2 are [tex]\angle{6}\text{ and }\angle{7}[/tex]

Step-by-step explanation:

Consider the provided information.

If transversal line crossed by two parallel lines, then, the corresponding angles and alternate angles are equal .

The angles on the same corners are called corresponding angle.

Alternate Angles:  Angles that are in opposite positions relative to a transversal intersecting two lines.

∠2 and ∠6 are corresponding angles

Therefore, ∠2 = ∠6

∠2 and ∠7 are alternate exterior angles

Therefore, ∠2 = ∠7

Hence, the angles formed on line b when cut by the transversal are congruent with ∠2 are [tex]\angle{6}\text{ and }\angle{7}[/tex]

Answer:

Step-by-step explanation:

A function is increasing when it is rising to the right. Here, that is everywhere right of x=-1.

A function is decreasing when it is falling to the right. Here, that is everywhere left of x=-1.

A function is constant when its graph is horizontal. There are no places on this graph like that.

Answer:

a) Yes b) No

Step-by-step explanation:

Given that the p value for a two-sided test of the null hypothesis H0: μ = 10

is 0.06

a) [tex]p value = 0.06.\\Confidence level = 95%\\Significance level = 100-95 = 5%\\Alpha = 0.05\\\\p  \geq \alpha is true[/tex]

Hence we accept null hypothesis

This implies that 10 will be within the confidence interval of 95%

b) If confidence level = 90%

[tex]\alpha = 0.10\\p value = 0.06\\p \leq 0.10[/tex]

So we have to reject null hypothesis.

So we do not have 10 in the confidence interval

The reason is the lower the confidence level, the narrower the confidence interval.  In this case, 10 has gone outside the narrower interval hence we get this.

Answer:

a. See attachments

B. 163630.2ft

C. 147056.70ft

D. 49205.4ft

Step-by-step explanation:

Check attachments for details

Angle a.. Angle of dep to beginning of runway

Angle b... Angle of del to end of runway

AD... Altitude of plane

DB..... Ground distance before touch down

AC... Air distance travelled until touch down on the near end of runway

Answer:

[tex]\displaystyle \boxed{66 \times 7}\:7(60 + 6)[/tex]

[tex]\displaystyle \boxed{97 \times 4}\:4(100 - 3)[/tex]

[tex]\displaystyle \boxed{8(4 + 2)}\:32 + 16 = 48[/tex]

[tex]\displaystyle \boxed{5(9 - 6)}\:(5 \times 9) - (5 \times 6)[/tex]

[tex]\displaystyle \boxed{3(4 + 7)}\:(3 \times 4) + (3 \times 7)[/tex]

Step-by-step explanation:

According to the Order of Operations [GEMS\BOMDAS\PEMDAS etc.], you evaluate everything in parentheses first before preceding with your Division & Multiplication and Subtraction & Addition. When you do this, you will know exactly which expression corresponds with its Distributive Property expression.

I am joyous to assist you anytime.

Answer:

she keeps her earnings the same

Answer:

$8,000.00 /yr

Step-by-step explanation:

The original fuel cost was 30% of 400,000 or $120,000.

If the previous cost was 1.50/L, then 120,000/1.5 = 80,000L/yr

The extra $0.10/L thus adds .1(80,000) = $8,000.00 /yr

Answer:

  x < -7

Step-by-step explanation:

Eliminate parentheses:

  3x -12 > 5x +2

  -14 > 2x . . . . . . . . . add -2-3x

  -7 > x . . . . . . . . . . . divide by 2

Any value of x less than -7 is in the solution set.

Answer: the number of students that were surveyed is 40

Step-by-step explanation:

Let x = total number of students that were surveyed about their favorite colors

1/4 of the students preferred red.

This means that the number of students that preferred red is 1/4 × x = x/4

1/8 of the students preferred blue.

This means that the number of students that preferred blue is 1/8 × x = x/8

The remaining number of students will be the total number of students - the sum of the number of students that preferred red and the number of students that preferred blue. It becomes

x - (x/4 + x/8) = x - 3x/8 = 5x/8

3/5 of the remaining students were for green. This means that the number of students that preferred green is 3/5 × 5x/8 = 3x/8

if 15 students prefer green, then

3x/8 = 15

3x = 120

x = 120/3 = 40 students

Answer: Length = 11 feet

Width = 6 feet

Step-by-step explanation:

A rectangle is a four sided shape that has 2 equal lengths,L and 2 equal widths,W

The perimeter of the rectangular piece of land is the total distance around it. It is expressed as

Perimeter = 2L + 2W = 2(L+W)

The perimeter is given as 34ft. It means that

34 = 2(L+W) - - - - - - - -1

The length of the rectangular piece of land is 5ft more than two times its width. It means that

L = W + 5

Substituting L = W + 5 into equation 1, it becomes

34 = 2(W + 5 +W) = 2(2W + 5)

2W + 5 = 34/2 = 17

2W = 17 - 5 = 12

W = 12/2 = 6 feet

L = W + 5 = 6+5

L = 11 feet

Answer:

  (E)  xy/(y-x)

Step-by-step explanation:

The sum of the individual rates is the total rate. Each machine's rate is nails per hour is the inverse of its rate in hours per nail.

  total rate = (800 nails)/(x hours) = 800/x nails/hour

  A rate = (800 nails)/(y hours) = 800/y nails/hour

We want to find the time "b" such that ...

  B rate = (800 nails)/(b hours) = 800/b nails/hour

__

As we said, the total rate is the sum of the individual rates:

  800/x = 800/y + 800/b

Multiplying by xyb/800, we get

  yb = xb + xy

Solving for b, we have ...

  yb -xb = xy

  b(y -x) = xy

  b = xy/(y-x) . . . . . matches choice E

It takes Machine B xy/(y-x) hours to produce 800 nails.

Answer:

D. 5x^2-6/2

Step-by-step explanation:

5x^2-18/6 , 18/6 is 6/2

5x^2-6/2 ,

After simplification, the equation obtained is 5x² - 6/2. Hence, option D is correct.

A phrase must be simplified and made shorter using a variety of methods. The steps required to decrease something are carried out according to a specified order known as BOD MAS.

Make a fraction simpler by reducing it to the base form. If both the numerator and denominator of a fraction only include the integer, the fraction is considered to be in its basic form.

As per the given information in the question,

The given equation in the question is,

15x² - 18/6

Divide the equation by 3,

= 5x² - 6/2.  

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Answer:

Omar can fill 28 containers

Step-by-step explanation:

Omar have 7 pounds of cherries, each pound need 4 containers because each one only hold of [tex]\frac{1}{4}\\[/tex] pound, now we multiplicate 7 pounds with 4 container for each one and we get 28 containers.

[tex]Number containers = \frac{7 pounds}{\frac{1}{4}pounds } = 28[/tex]

Answer: it will take her 56.67 seconds or 0.945 minutes to race 2 laps

Step-by-step explanation:

It takes Ella 1 minute and 25 seconds to complete the cheep beach course.

We can express this time in seconds or minutes. Expressing it in seconds,

1 minute = 60 seconds

Therefore,

1 minute and 25 seconds = 60 +25 = 85 seconds

If the course is 3 laps long, that means she completed 3 laps in 85 seconds. The time it will take her to race 2 laps would be

(2 × 85)/3

= 56.67 seconds

Converting to minutes, it will be

56.7/60 = 0.945 minutes

Answer:

Q=95, P= 52.5

Step-by-step explanation:

The profit maximizing level of output in monopolies is reached when the marginal cost is equal to the marginal revenue. This is also the profit maximizing rule in perfect competition, the difference between both is that is perfect competition the marginal revenue is equal to the price while in monopolies, the demand curve is often above the marginal revenue curve, then the actual price (defined by the demand curve) is often higher than the marginal revenue price.

For this problem the profit maximizing level of output is:

MC=MR

5=100-Q

Q=95

Because monopolies decide the selling price based on the demand curve, you should replace this quantity in the demand curve equation:

95=200-2P

95-200/2=-P

P= 52.5

When output is set so that marginal revenue and marginal cost are equal in this economics dilemma, the monopolist will make the most money. That would be at a manufacturing level of 95 units in this instance.

When establishing output, a monopolist in economics attempts to maximize profit by ensuring that Marginal Cost (MC) and Marginal Revenue (MR) are equal. We must assign MR to MC in this case given that MR = 100 - Q and MC = 5. Therefore, 100 - Q = 5, giving Q=95. In order to maximize its profit, the monopolist should create 95 units of the good.

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In the given arithmetic sequence, the ratio of the first term to the second term is 1:2.

The problem is based on the properties of the arithmetic sequence. To solve it, we use the formula for the sum of an arithmetic sequence: S = n/2*(a + l), where n is the number of terms, a is the first term, and l is the last term.

From the question, we know that 4 times the sum of the first five terms equals the sum of first ten terms. Therefore, 4 * (5/2 * (a + a + 4d)) = 10/2 * (a + a + 9d), where d is the common difference. Simplifying, we find that the ratio of the first term a to the second term (a + d) is 1:2.

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Answer:

your answer is 1/4

Step-by-step explanation:

part 1

7/8 + -2/3 = 5/24

part 2

5/24÷5/6 = 1/4

Answer:

First you would translate triangle ABC to the right . next you would then translate triangle ABC up . Last you would rotate triangle ABC clockwise and matched angle A with angle D.

Step-by-step explanation:

Answer:

18.63

Step-by-step explanation:

45*115%= 45*1.15 = 51.75

51.75*0.36 = 18.63

Answer:

18.63 dollars.

Step-by-step explanation:

The selling price was  45 + 15% or 45

= 45 + 0.15*45

= 51.75 dollars.

36% of 51.75

=  0.36 * 51.75

=  18.63 dollars  which he had to spend on books.

Answer:

Part 1) The perimeter of triangle ABC is 24 units

Part 2) [tex]y=97\°[/tex]

Step-by-step explanation:

Part 1) we know that

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side

see the attached figure to better understand the problem

so

Applying the midpoint theorem

step 1

Find the value of BC

[tex]BC=\frac{1}{2}XY[/tex]

[tex]XY=2AY[/tex] ---> because A is the midpoint

substitute the given value of AY

[tex]XY=2(7)=14\ units[/tex]

[tex]BC=\frac{1}{2}(14)=7\ units[/tex]

step 2

Find the value of AC

[tex]AC=\frac{1}{2}YZ[/tex]

[tex]YZ=2BZ[/tex] ---> because B is the midpoint

substitute the given value of BZ

[tex]YZ=2(8)=16\ units[/tex]

[tex]AC=\frac{1}{2}(16)=8\ units[/tex]

step 3

Find the value of AB

[tex]AB=\frac{1}{2}XZ[/tex]

substitute the given value of XZ

[tex]AB=\frac{1}{2}(18)=9\ units[/tex]

step 4

Find the perimeter of triangle ABC

[tex]P=AB+BC+AC[/tex]

substitute

[tex]P=9+7+8=24\ units[/tex]

Part 2) Find the measure of angle y

step 1

Find the measure of angle z

we know that

The sum of the interior angles in a triangle must be equal to 180 degrees

so

[tex]55\°+42\°+z=180\°[/tex]

solve for z

[tex]97\°+z=180\°[/tex]

[tex]z=180\°-97\°[/tex]

[tex]z=83\°[/tex]

step 2

Find the measure of angle y

we know that

[tex]z+y=180\°[/tex] ----> by supplementary angles (form a linear pair)

we have

[tex]z=83\°[/tex]

substitute

[tex]83\°+y=180\°[/tex]

solve for y

[tex]y=180\°-83\°[/tex]

[tex]y=97\°[/tex]

Number of calories in 6 ounces of pumpkin is 61.5 calories.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that, 8 ounces of canned pumpkin has 82 calories

Now, number of calories in 1 ounce = Total number of calories ÷ Number of ounces

= 82/8

= 10.25 calories

Number of calories in 6 ounces of pumpkin

= 6×10.25

= 61.5 calories

Therefore, number of calories in 6 ounces of pumpkin is 61.5 calories.

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Answer:

a) mean= 158.4 , standard deviation = 3.394

b)  Best option : B. Yes, since the original population is normal, the sampling distribution of the sum will also be approximately normal.

c)  P(X>160) = P(Z>0.471) = 1-P(Z<0.471) = 0.3188

Step-by-step explanation:

1) Notation

n = sample size = 8

[tex] \mu [/tex] = population mean = 19.8

[tex] \sigma [/tex] = population standard deviation = 1.2

2) Definition of the variable of interest

Part a

The variable that we are interested is [tex] \sum x_i  [/tex] and the mean and the deviation for this variable are given by :

E([tex] \sum x_i [/tex]) =  [tex] \sum E(x_i) [/tex] = n [tex] \mu [/tex] = 8x19.8 = 158.4

Var([tex] \sum x_i [/tex]) =  [tex] \sum Var(x_i) [/tex] = n [tex] \sigma^2 [/tex]

Sd([tex] \sum x_i [/tex]) =   [tex] \sqrt{n \sigma^2} [/tex] = [tex] \sqrt(8) [/tex]  x 1.2 = 3.394

Part b

For this case the populations are normal, then the distribution for the sample ([tex] \sum x_i [/tex])  is normal too.

Based on this the distribution for the variable X would be normal, so the best option should be:

B. Yes, since the original population is normal, the sampling distribution of the sum will also be approximately normal.

Part c

From part a we know that the mean = 158.4 and the deviation = 3.394

The z score is defined as

Z = (X -mean)/ deviation = (160-158.4)/ 3.394 = 0.471

Then we can find the probability P(X>160) = P(Z>0.471) = 1-P(Z<0.471) = 0.3188