A researcher wants to know if the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. If we assume the standard deviation is 3 years, a 95% confidence interval for the average time served is:

Answer: 0.2205

Step-by-step explanation:

Given : Technology Services department at Lahey Electronics revealed company employees receive an average of "2.7" non-work-related e-mails per hour.

i.e. [tex]\lambda = 2.7[/tex]

If the arrival of these e-mails is approximated by the Poisson distribution.

Then , the required probability is given by :-

[tex]P(X=x)=\dfrac{\lambda^xe^{-\lambda}}{x!}\\\\P(X=3)=\dfrac{(2.7)^3e^{-2.7}}{3!}\\\\=0.22046768454\approx0.2205[/tex]

Hence, the probability Linda Lahey, company president, received exactly 3 non-work-related e-mails between 4 P.M. and 5 P.M. yesterday =0.2205

To determine the probability that Linda Lahey received exactly 3 non-work-related e-mails in one hour based on a Poisson distribution with an average rate of 2.7 e-mails per hour, we apply the Poisson formula. This calculation offers a precise way to understand the likelihood of such an event occurring within a set timeframe.

Given that, on average, company employees receive 2.7 non-work-related e-mails per hour, we can use the Poisson formula to calculate this probability.

To find the probability of receiving exactly k events in a fixed interval of time, we use the formula:
P(X = k) = (λ^k * e^-λ) / k!
where λ is the average rate (2.7 emails per hour in this case), k is the number of events (3 emails), and e is the base of the natural logarithm (approximately 2.71828).

Plugging in the values, we calculate the probability as follows:
P(X = 3) = (2.7^3 * e^-2.7) / 3!
This calculation gives us the specific probability that Linda Lahey received exactly 3 non-work-related e-mails in one hour.

Answer:

7

Step-by-step explanation:

[tex]x^2-y^2[/tex] is a difference of squares.

When factoring a difference of squares, you can use this formula [tex]u^2-v^2=(u-v)(u+v)[/tex].

So [tex]x^2-y^2[/tex] can be factored as [tex](x-y)(x+y)[/tex].

So back to the problem:

[tex]x^2-y^2=56[/tex]

Rewriting with a factored left hand side:

[tex](x-y)(x+y)=56[/tex]

We are given x-y=4 so rewriting again with this substitution:

[tex]4(x+y)=56[/tex]

Dividing both sides by 4:

[tex](x+y)=14[/tex]

So we have x+y equals 14.

We are asked to find the average of x and y which is (x+y)/2.

So since x+y=14 , then (x+y)/2=14/2=7.

To find the net worth of the Crabby Apple restaurant at the beginning of January before it lost $2500, we add back the loss to the net worth at the end of the month, resulting in an initial net worth of $2900.

The student is asking about calculating the net worth of a restaurant before a financial loss. To find the net worth at the beginning of the month, we would add the loss sustained during that month back to the net worth at the end of the month. Since the restaurant lost $2500 in January, and had a net worth of $400 at the end of January, we can calculate its net worth at the beginning of January by the following calculation:

Net Worth at Beginning of Month = Net Worth at End of Month + Loss During Month

Net Worth at Beginning of Month = $400 + $2500

Net Worth at Beginning of Month = $2900

Therefore, the net worth of the Crabby Apple restaurant at the beginning of January was $2900.

The net worth of the Crabby Apple restaurant at the beginning of January was $2900.

To determine the net worth of the Crabby Apple restaurant at the beginning of January, we need to account for the loss incurred during the month. Given that the net worth at the end of January is $400 and the restaurant lost $2500 during the month, we can set up the following equation:

[tex]\[\text{Net worth at the beginning of January} =[/tex][tex]\text{Net worth at the end of January} + \text{Loss during January}\][/tex]

Substitute the given values into the equation:

[tex]\[\text{Net worth at the beginning of January} = \$400 + \$2500\][/tex]

[tex]\[\text{Net worth at the beginning of January} = \$2900\][/tex]

[tex]\bf cos^2(x)+2cos(x)+1=0\implies \stackrel{\textit{let's notice, this is simply }ax^2+bx+c=0}{[cos(x)]^2+2cos(x)+1=0} \\[2em] [cos(x)+1][cos(x)+1]=0 \\\\[-0.35em] ~\dotfill\\\\ cos(x)+1=0\implies cos(x)=-1\implies x=cos^{-1}(-1)\implies \stackrel{\textit{for the range }[0,2\pi ]}{x=\pi } \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{all solutions}}{x=\pi +2\pi n~~,~~ n \in \mathbb{Z}}~\hfill[/tex]

Answer:

The correct option is D. a = -2 and b = 4.

Step-by-step explanation:

Consider the provided equation:

[tex]y=a(x-2)^2+b\ \text{and}\ y=5[/tex]

The vertex form of a quadratic is:

[tex]y= a(x-h)^2+k[/tex]

Where, (h,k) is the vertex and the quadratic opens up if 'a' is positive and opens down if 'a' is negative.

Now consider the provided option A. a = 1 and b = -4.

Since the value of a is positive the graph opens up and having vertex (2,-4). Thus graph will intersect the line y = 5.

Refer the figure 1:

Now consider the option B. a = 2 and b = 5.

Since the value of a is positive the graph opens up and having vertex (2,5). Thus graph will intersect the line y = 5.

Refer the figure 2:

Now consider the option C. a = -1 and b = 6.

Since the value of a is negative the graph opens down and having vertex (2,6). Thus graph will intersect the line y = 5.

Refer the figure 3:

Now consider the option D. a = -2 and b = 4.

Since the value of a is negative the graph opens down and having vertex (2,4). Thus graph will not intersect the line y = 5.

Refer the figure 4:

Hence, the correct option is D. a = -2 and b = 4.

Answer:

The number of combinations are made when one person taken at a time out of four person=4.

Step-by-step explanation:

We are given that a six person committee composed of Alice,Ben,Connie, Dolph,Egbert, and Francisco.

We have to select three persons out of six persons one is chairperson,secretary and treasurer.

We have to find the number of combinations of different officer are made when two persons Dolph and Francisco must hold office.

Now, if two persons Dolph and Francisco must hold the office then we have to select only one member out of 4 persons.

Therefore ,using combination formula

[tex]\binom{n}{r}[/tex]=[tex]\frac{n!}{r!(n-r)!}[/tex]

We have n=4 and r=1 then

The number of combination of different officer are made =[tex]\binom{4}{1}[/tex]

The number of combination of different officer are made=[tex]\frac{4!}{1!(4-1)!}[/tex]

The number of combination of different officer are made=[tex]\frac{4\times 3!}{3!}[/tex]

The number of combination of different officer are made=4

Hence, the number of combinations are made when one person taken at a time out of four person=4.

Answer: 4

Answer:

Step-by-step explanation:

Using linear differential equation method:

\frac{\mathrm{d} y}{\mathrm{d} x}+3y=e^5^x

I.F.= [tex]e^{\int {Q} \, dx }[/tex]

I.F.=[tex]e^{\int {3} \, dx }[/tex]

I.F.=[tex]e^{3x}[/tex]

y(x)=[tex]\frac{1}{e^{3x}}[\int {e^{5x}} \, dx+c][/tex]

y(x)=[tex]\frac{e^{2x}}{5}+e^{-3x}\times c[/tex]

substituting x=2

c=[tex]\frac{25-e^4}{5e^{-6}}[/tex]

Now

y=[tex]\frac{e^{2x}}{5}+e^{-3x}\times \frac{25-e^4}{5e^{-6}}[/tex]

Answer:

(0.23, 0.30)

Step-by-step explanation:

Number of green peas = 428

Number of yellow peas = 155

Total number of peas = n = 583

Since we have to establish the confidence interval for yellow peas, the sample proportion of yellow peas would be considered as success i.e. p = [tex]\frac{155}{583}[/tex]

q = 1 - p = [tex]\frac{428}{583}[/tex]

Confidence Level = 95%

Z value associated with this confidence level = z = 1.96

Confidence interval for the population proportion is calculated as:

[tex](p-z\sqrt{\frac{pq}{n}} ,p+z\sqrt{\frac{pq}{n}})[/tex]

Using the values, we get:

[tex](\frac{155}{583}-1.96\sqrt{\frac{\frac{155}{583} \times\frac{428}{583}}{583} },\frac{155}{583}+1.96\sqrt{\frac{\frac{155}{583} \times\frac{428}{583}}{583} })\\\\ =(0.23,0.30)[/tex]

Conclusion:

We are 95% confident that true value of population proportion of yellow peas lie between 0.23 and 0.30

Answer:

[tex]f (x) = x (x + 5) (x-9)[/tex]

Step-by-step explanation:

The zeros of the polynomial are all the values of x for which the function [tex]f (x) = 0[/tex]

In this case we know that the zeros are:

[tex]x = 9,\  x-9 =0[/tex]

[tex]x = 0[/tex]

[tex]x = -5[/tex], [tex]x + 5 = 0[/tex]

Now we can write the polynomial as a product of its factors

[tex]f (x) = x (x + 5) (x-9)[/tex]

Note that the polynomial is of degree 3 because the greatest exponent of the variable x that results from multiplying the factors of f (x) is 3

The polynomial f(x) of degree 3 that has the zeros 9, 0, and -5 can be found by setting up and multiplying the factors (x-9), (x-0), and (x+5). The resulting polynomial f(x) is therefore x(x - 9)(x + 5).

To find a polynomial f(x) of degree 3 that has the given zeros, you use the fact that the zeros (or roots) of a polynomial are the values that make the polynomial equal to zero. In this case, the zeros are 9, 0, and -5. Consequently, the factors of the polynomial are (x-9), (x-0), and (x+5).

Now multiply these factors together to get the polynomial. The result is:

f(x) = x(x - 9)(x + 5).

This is the polynomial of degree 3 with the given zeros.

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Answer:

a) bijective

b) neither surjective or injective

c) injective

Step-by-step explanation:

Since we are looking from real numbers to real numbers, we want the following things

1) We want every real number y to get it. (surjective)

2) We want every y that gets hit to be hit only once. (injective).

---

If we have both things then the function is bijective.

a) f(x)=5x+4 this is a line with a positive slope.

That means it is increasing left to right.  Every increasing or even decreasing line is going to hit every real number y.  This is a bijection.

b) g(x)=2x^2-2 is a parabola. Parabola functions always have y's that get hit more than once and not all y's get hit because the parabola is either open up or down starting from the vertex.  This function is neither injective or surjective.

c) h(x)=1+(2/x)  x is not 0.

1+(2/x) is never 1 because 2/x is never 0 for any x. This means the real number y=1 will never be hit and is therefore not surjective. This function is injective because every 1 that is hit is only hit once. If you want use the horizontal line test to see this.

Answer:

The population would be 781.

The population is growing with the rate of 12.50 persons/yr.

Step-by-step explanation:

Since, the formula for calculating the population, increasing with a rate per period,

[tex]A=P(1+r)^{n}[/tex]

Where, P is the initial population,

r is the rate per period,

n is the number of period,

t is the total years,

Here, P = 500, r = 25 % = 0.25, n = 2 ( the number of '10 year period' of in 20 years is 2 )

Hence, the population in 20 years would be,

[tex]A=500(1+0.25)^2=500(1.25)^2=781.25\approx 781[/tex]

Now, the rate of increasing per 10 year is 25 %,

⇒ The rate of increasing per year is 2.5 %,

Thus, the growing people per year = 2.5 % of 500 = 0.025 × 500 = 12.50

Hence, the population is growing at 12.50 person per year.

Answer:

Height of the mountain is 5108.80 feet.

Step-by-step explanation:

From the figure attached, h is the height of a mountain AB.

At a point C angle of elevation of the mountain is 24°

Now survey team gets closer to the mountain by 1000 feet then angle of elevation is 26°.

Now from ΔABC,

tan24 = [tex]\frac{h}{x+1000}[/tex]

0.445 = [tex]\frac{h}{x+1000}[/tex]

h = 0.445(x + 1000)------(1)

From ΔABD,

tan26 = [tex]\frac{h}{x}[/tex]

0.4877 = [tex]\frac{h}{x}[/tex]

h = 0.4877x -----(2)

Now we equation 1 and equation 2

0.4452(x + 1000) = 0.4877x

0.4877x - 0.4452x = 1000(0.4452)

0.0425x = 445.20

x = [tex]\frac{445.20}{0.0425}[/tex]

x = 10475.29 feet

Now we plug in the value of x in equation 2.

h = (10475.29)×(0.4877)

h = 5108.80 feet

Therefore, height of the mountain is 5108.80 feet

Answer:  (A U B) n (B n C') = {5, 8, 11}.

Step-by-step explanation:  We are given the following sets :

U = {4, 5, 6, 7, 8, 9, 10, 11},

A = {5, 7, 9},

B = {4, 5, 8, 11}

and

C = {4, 6, 10}.

We are to find the following :

(A U B) n (B n C')

We know that for any two sets A and B,

A ∪ B contains all the elements present in set A or set B or both,

A ∩ B contains all the elements present in both A and B,

A - B contains all those elements which are present in A but not B

and

A' contains all the elements present in the universal set U but not A.

We will be suing the following rule of set of theory :

A ∩ B' = A - B.

Therefore, we have

[tex](A\cup B)\cap(B\cap C')\\\\=(A\cup B)\cap (B-C)\\\\=(\{5,7,9\}\cup\{4,5,8,11\})\cap (\{4,5,8,11\}-\{4,6,10\})\\\\=\{4,5,7,8,9,11\}\cap\{5,8,11\}\\\\=\{5,8,11\}.[/tex]

Thus,  (A U B) n (B n C') = {5, 8, 11}.

Final answer:

To solve the set operation (A U B) n (B n C'), we first find the union of A and B, then the complement of C, and the intersection of B and C'. The final step is to intersect the results of (A U B) and (B n C'), which gives us {4, 5, 8, 11}.

Explanation:

The question involves operations on sets, specifically union, intersection, and complement. We have a universal set U and subsets A, B, and C. The objective is to find the result of (A U B) n (B n C'), which involves set union (U), set intersection (n), and the complement of a set (').

First, let's find the union of sets A and B: A U B = {s, 7, 9, 4, 5, 8, 11}.

Next, we need to find the complement of set C, which is C' = {4, 5, 7, 8, 9, 11} as these are the elements of U that are not in C.

Then, identify the intersection of sets B and C': B n C' = {4, 5, 8, 11}, because these elements are common to both B and C'.

Finally, we find the intersection of the two results: (A U B) n (B n C') = {4, 5, 8, 11}.

Answer:

Step-by-step explanation:

∠C and ∠F are vertical angles, so are congruent. Then any angle complementary to one of those will also be complementary to the other.

Likewise, ∠B and ∠E are vertical angles and congruent. Any angle complementary to one of them will also be complementary to the other. Here, ∠E and ∠F are listed as complementary, so we know ∠B and ∠F will be also.

Answer:2 and 4

Step-by-step explanation:

^B and ^F

^F and ^E

Answer:

[tex]6ft[/tex] length on the east and west sides

[tex]12ft[/tex] length on the north and south sides

Step-by-step explanation:

Using x for the length of the east side (and is equal to the length  of the west side) and y for the length of the north side (and is equal to the length  of the south side), the equation that gives the total price equalized to 480 is:

[tex]20x+20x+10y+10y=480[/tex]

[tex]40x+20y=480[/tex]

Solving for y

[tex]y=\frac{-40x+480}{20}[/tex]

[tex]y=-2x+24[/tex]

The area of the garden is [tex]A=xy[/tex], to find the largest, substitute y in the formula of the area

[tex]A=x(-2x+24)=-2x^2+24x[/tex]

For the optimization, find the largest area, is needed the critical point. To find this point, derive A and equalize the derivative to zero:

[tex]A'=-4x+24=0[/tex]

Solve for x:

[tex]-4x=-24[/tex]

[tex]x=\frac{-24}{-4}[/tex]

[tex]x=6[/tex]

To see if x=6 is a maximum or a minimum, derive A' and substitute with x=6

[tex]A''=-4[/tex]

In this case, the second derivative of A doesn't depend on x, and it has a negative value, meaning the value found is a maximum. Using x=6 to find y

[tex]y=-2x+24[/tex]

[tex]y=-2(6)+24[/tex]

[tex]y=12[/tex]

The area is:

[tex]A=xy=6*12=72 ft^2[/tex]

Answer:

B

Step-by-step explanation:

Let x = acres of corn and

     y = acres of soybean

If the cost of corn is 60 per acre, we represent that as 60x.

If the cost of soybean is 80 per acre, we represent that as 80y.  

The addition of these 2 grains cannot go over 4800; that means that it can be 4800 on the dot, but it cannot be more.  So "less than or equal to" is our sign.  Putting that all together:

60x + 80y ≤ 4800

The scientist can determine the amount of Solution A and Solution B required by setting up and solving a system of two linear equations representing the total solution volume and the total salt amount.

Lets let the amount of Solution A the scientist will use be x and the amount of Solution B she will also use be y. We know that x + y = 110 ounces because her final mix should be 110 ounces. Also, we know that 0.4x + 0.65y = 0.55*(x+y) = 60.5 because the amount of salt from Solution A and Solution B should add up to the amount of salt in the final mixture. Solving this system of linear equations to obtain the values for x and y, gives the required amounts of Solution A and Solution B needed.

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Answer:

[tex]y'=\frac{5}{x} \cdot \cos(\ln(2x^5))[/tex]

Step-by-step explanation:

[tex]y=\sin(\ln(2x^5))[/tex]

We are going to use chain rule.

The most inside function is [tex]y=2x^5[/tex] which gives us [tex]y'=10x^4[/tex].

The next inside function going out is [tex]y=\ln(x)[/tex] which gives us [tex]y'=\frac{1}{x}[/tex].

The most outside function is [tex]y=\sin(x)[/tex] which gives us [tex]y'=\cos(x)[/tex].

[tex]y'=10x^4 \cdot \frac{1}{2x^5} \cdot \cos(\ln(2x^5))[/tex]

[tex]y'=\frac{5}{x} \cdot \cos(\ln(2x^5))[/tex]

Answer:

He invest for 2 years.

Step-by-step explanation:

Given : Suppose you have $1,950 in your savings account at the end of a certain period of time. You invested $1,700 at a 6.88% simple annual interest rate.

To find : How long, in years, did you invest your money?

Solution :

Applying simple interest formula,

[tex]A=P(1+r)^t[/tex]

Where, A is the amount A=$1950

P is the principal P=$1700

r is the interest rate r=6.88%=0.0688

t is the time

Substitute the values in the formula,

[tex]1950=1700(1+0.0688)^t[/tex]

[tex]\frac{1950}{1700}=(1.0688)^t[/tex]

[tex]1.147=(1.0688)^t[/tex]

Taking log both side,

[tex]\log(1.147)=\log ((1.0688)^t)[/tex]

Applying logarithmic formula, [tex]\log a^x=x\log a[/tex]

[tex]\log(1.147)=t\log (1.0688)[/tex]

[tex]t=\frac{\log(1.147)}{\log (1.0688)}[/tex]

[tex]t=2.06[/tex]

Approximately, He invest for 2 years.

Answer:

30 gallons of 20% acid solution should be mixed.

Step-by-step explanation:

Let x gallons of a 20% acid solution was mixed with 30 gallons of a 40% solution, to obtain a mixture of 30% acid solution.

Therefore, final volume of the solution will be (x + 30) gallons.

Now concept to solve this question is

20%.(x) + 40%.(30) = 30%.(x + 30)

0.20(x) + 0.40(30) = 0.30(x + 30)

0.20x + 12 = 0.30x + 9

0.30x - 0.20x = 12 - 9

.10x = 3

x = [tex]\frac{3}{0.1}[/tex]

x = 30 gallons

Therefore, 30 gallons of the 20% acid solution should be mixed.

Answer:

The standard form of required line is x-y=7.

Step-by-step explanation:

The standard form of a line is

[tex]ax+by=c[/tex]

Where, a,b,c are integers with no factor common to all three and a≥0.

The give equation of line is

[tex]x+y=2[/tex]

Here a=1 and b=1.

The slope of a standard line is

[tex]m=\frac{-a}{b}[/tex]

[tex]m_1=\frac{-1}{1}=-1[/tex]

The product of slops of two perpendicular lines is -1.

[tex]m_1\cdot m_2=-1[/tex]

[tex](-1)\cdot m_2=-1[/tex]

[tex]m_2=1[/tex]

The slope of required line is 1.

The point slope form of a line is

[tex]y-y_1=m(x-x_1)[/tex]

Where, m is slope.

The slope of required line is 1 and it passes through the point (1,-6). So, the equation of required line is

[tex]y-(-6)=1(x-1)[/tex]

[tex]y+6=x-1[/tex]

Add 1 on each side.

[tex]y+7=x[/tex]

Subtract y from both the sides.

[tex]7=x-y[/tex]

Therefore the standard form of required line is x-y=7.

Answer: There is 49.28% of cumulative markup for the department at the end of April.

Step-by-step explanation:

Since we have given that

Price of opening inventory = $170,000

Mark up rate = 48%

Amount of mark up is given by

[tex]\dfrac{48}{100}\times 170000\\\\=\$81600[/tex]

Price of additional merchandise = $80000

Mark up rate = 52%

Amount of mark up is given by

[tex]\dfrac{52}{100}\times 80000\\\\=\$41600[/tex]

So, total mark up would be

$81600 + $41600 = 123200

So, the cumulative markup percentage for the department at the end of April is given by

[tex]\dfrac{123200}{170000+80000}\times 100=\dfrac{123200}{250000}\times 100=49.28\%[/tex]

Hence, there is 49.28% of cumulative markup for the department at the end of April.

Answer:

=10km/h

Step-by-step explanation:

Let motor boat speed be represented by x and current y

The speed upstream = Motor boats speed - rate of current

=x-y

The net speed down stream = Motor boats speed + rate of current

=x+y

Let us find the speed upstream =distance/ time taken

=150km/5hrs

=30km/h

Speed down stream= 150km/3h

=50 km/h

The problem forms simultaneous equations.

x-y=30

x+y=50

Using elimination method we solve the equations.

Add the two equations to eliminate y.

2x=80

x=40

Current, y= 50-x

=10km/h

Answer:

1) [tex]40\ \frac{km}{h}[/tex]

2) [tex]10\ \frac{km}{h}[/tex]

Step-by-step explanation:

 Let' call "b" the speed of the motorboat and "c" the speed of the current.

We know that:

[tex]V=\frac{d}{t}[/tex]

Where "V" is the speed, "d" is distance and "t" is time.

Then:

[tex]d=V*t[/tex]

We know that distance traveled upstream is 150 km and the time is 5 hours. Then, we set up the folllowing equation:

 [tex]5(b-c)=150[/tex]  (Remember that in the trip upstream the speed of the river is opposite to the motorboat)

For the return trip:

 [tex]3(b+c)=150[/tex]  

 By solving the system of equations, we get:

- Make both equations equal to each other and solve for "c".

[tex]5(b-c)=3(b+c)\\\\5b-5c=3b+3c\\\\5b-3b=3c+5c\\\\2b=8c\\\\c=\frac{b}{4}[/tex]

- Substitute "c" into the any original equation and solve for "b":

[tex]5(b-\frac{b}{4})=150\\\\\frac{3}{4}b=30\\\\b=40\ \frac{km}{h}[/tex]

- Substitute "b" into [tex]c=\frac{b}{4}[/tex]:

[tex]c=\frac{40}{4}\\\\c=10\ \frac{km}{h}[/tex]

Answer: 0.125

Step-by-step explanation:

Given: A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed in interval (50,52).

∴ The probability density function of X will be :-

[tex]f(x)=\dfrac{1}{b-a}=\dfrac{1}{52-50}=\dfrac{1}{2}[/tex]

The required probability will be:-

[tex]P(51.25<x<51.5)=\int^{51.5}_{51.25}f(x)\ dx\\\\=\dfrac{1}{2}\int^{51.5}_{51.25}\ dx\\\\=\dfrac{1}{2}[x]^{51.5}_{51.25}\\\\=\dfrac{1}{2}(51.5-51.25)=\dfrac{0.25}{2}=0.125[/tex]

Hence, the probability that a given class period runs between 51.25 and 51.5 minutes =0.125

Answer:

The correct option is C. Fredo

Step-by-step explanation:

In a Mario Puzo's fictional novel named The Godfather, Frederico Corleone or Fredo is a fictional character. In the novel, Fredo's father is killed by the assasins. Witnissing his father being shot, Fredo goes into a shock.

To protect and aid Fredo's recovery, his elder brother Sonny, sends him to Las Vegas.

Therefore, Fredo is sent to Las Vegas

Step-by-step explanation:

The probability of getting tails each time is 1/2.  Seven times in a row, the probability is:

P = (1/2)^7

P = 0.0078125

the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

The probability of obtaining seven tails in a row when flipping a coin can be calculated using the principle of independent events in probability. Each flip of the coin is an independent event with two possible outcomes: heads or tails.

Probability can be calculated by using the formula:

[tex]P = \frac{number\ of\ desired\ outcomes}{total\ outcomes}[/tex]

here the desired outcome is 1 as we only need tails so we can say that:

[tex]P(tails) = \frac{1}{2} = 0.5[/tex]

To find the probability of obtaining seven tails in a row, you need to multiply the probability of getting tails on each individual flip:

[tex]Probability (7\ tails\ in\ a\ row) = 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5\\\\Probability (7\ tails\ in\ a\ row) = (0.5)^7 \approx 0.0078125[/tex]

Therefore, the probability of obtaining seven tails in a row when flipping a coin is approximately 0.78%.

Answer: The value of b = 0.99

The probability that a day requires more water than is stored in city reservoirs is 0.161.

Step-by-step explanation:

Given : Water use in the summer is normally distributed with

[tex]\mu=310.4\text{ million gallons per day}[/tex]

Standard deviation : [tex]\sigma=40 \text{ million gallons per day}[/tex]

Let x be the combined storage capacity requires by the reservoir on a random day.

Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{350-310.4}{40}=0.99[/tex]

The probability that a day requires more water than is stored in city reservoirs is  :

[tex]P(x>350)=P(z>0.99)=1-P(z<0.99)\\\\=1-0.8389129=0.1610871\approx0.161[/tex]    

Hence, the probability that a day requires more water than is stored in city reservoirs is 0.161

Answer:

Step-by-step explanation:

Let us calculate std dev and std error of two samples

Sample     N     Mean    Std dev   Std error

  1              65     52.8      9.9         1.22279

  2             111      54.1       11.09      1.0526

Assuming equal variances

df =111+65-2=174

Pooled std deviation = combined std deviation = 10.6677

Pooled std error = 10.6677/sqrt 174 = 0.808

difference in means =52.8-54.1 = -1.3

Margin of error = 0.5923

( t critical 1.6660)

Confidence interval

=(-4.4923, 1.8923)

Since 0 lies in this interval we can accept null hypothesis that 10 year old boys and girls have the same average height