What is the difference?
X/x2+3x+2 - 1/(X+ 2)(x+1)

ANSWER

The relation is not a function.

EXPLANATION

The relation is not a function because we have an x-coordinate mapping on to more than one y-coordinate.

This occurs at x=1.

The ordered pairs (1,1) and (1,3) disqualify the relation from being a function.

Hence the relation is not a function.

Answer:

The correct option is 2.

Step-by-step explanation:

Let A be the event that the college student belongs to a health club and B be the event that the college student lives off-campus.

The probability that a college student belongs to a health club is 0.3.

[tex]P(A)=0.3[/tex]

The probability that a college student lives off-campus is 0.4.

[tex]P(B)=0.4[/tex]

The probability that a college student belongs to a health club and lives off-campus is 0.12.

[tex]P(A\cap B)=0.12[/tex]

The probability that a college student belongs to a health club OR lives off-campus is

[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]

[tex]P(A\cup B)=0.3+0.4-0.12[/tex]

[tex]P(A\cup B)=0.58[/tex]

The probability that a college student belongs to a health club OR lives off-campus is 0.58. Therefore the correct option is 2.

Answer:

[tex]2.39\%[/tex]  

Step-by-step explanation:

we know that    

The compound interest formula is equal to  

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest  in decimal

t is Number of Time Periods  

n is the number of times interest is compounded per year

in this problem we have  

[tex]t=4\ years\\ P=\$10,000\\A=\$11,000\\ r=?\\n=4[/tex]  

substitute in the formula above  

[tex]11,000=10,000(1+\frac{r}{4})^{4*4}[/tex]  

[tex]1.1=(1+\frac{r}{4})^{16}[/tex]  

Elevated both sides to (1/16)

[tex]1.005975=(1+\frac{r}{4})[/tex]  

[tex]0.005975=\frac{r}{4}[/tex]  

[tex]r=0.005975*4=0.0239[/tex]  

Convert to percent

[tex]0.0239*100=2.39\%[/tex]  

Answer: [tex]\dfrac{1}{6}[/tex]

Step-by-step explanation:

The given event : A randomly drawn marble is blue.

The number of blue marbles in the bag = 2

The total number of marbles in the bag = [tex]2+7+3=12[/tex]

Now, the probability of drawing a blue marble is given by  :-

[tex]\text{P(Blue)}=\dfrac{\text{Number of blue marbles}}{\text{Total number of marbles}}\\\\\Rightarrow\text{P(Blue)}=\dfrac{2}{12}=\dfrac{1}{6}[/tex]

Hence, the probability of the given event event = [tex]\dfrac{1}{6}[/tex]

The answer is A, 71°.

180-109=71

Since m and n are parallel, angles 4 and 6 will add up to 180 degrees - just like angles 4 and 2. Remember that 180 degrees is a straight line: if angles 4 and 6 are put together, they will make a straight line.

Answer:

[tex]\large\boxed{(f-g)(x)=-3x+11}[/tex]

Step-by-step explanation:

[tex](f-g)(x)=f(x)-g(x)\\\\f(x)=2x-1+3=2x+2\\g(x)=5x-9\\\\\text{Substitute:}\\\\(f-g)(x)=(2x+2)-(5x-9)\\\\=2x+2-5x-(-9)\\\\=2x+2-5x+9\qquad\text{combine like terms}\\\\=(2x-5x)+(2+9)\\\\=-3x+11[/tex]

Answer:

[tex]\bullet\ \ 0.\overline{15}[/tex]

Step-by-step explanation:

  5/33 = (5·3)/(33·3) = 15/99 = 0.151515151515...

_____

You may recall that 1/9 = 0.11111...(repeating indefinitely). That is, a multiple of 1/9 is a single-digit repeating decimal.

Likewise, 1/99 = 0.01010101...(repeating indefinitely). This means when a 2-digit numerator has 99 as the denominator, the decimal equivalent is that number repeated indefinitely. Any fraction with 999 as the denominator is a 3-digit repeat in decimal; 9999 as the denominator gives a 4-digit repeat, and so on.

Answer:

27 SUVs

Step-by-step explanation:

Let number of ordinary cars be x and SUVs be y

We can write 2 equations and use substitution to solve for the number of SUVs.

"The number of ordinary cars is larger than the number of sport utility vehicles by 59.3%"-

This means that 1.593 times more is ordinary cars (x) than SUVs (y), so we can write:

x  = 1.593y

"The difference between the number of cars and the number of SUVs is 16" -

Since we know ordinary cars are "more", we can say x - y = 16

We can now plug in 1.593 y into x of the 2nd equation and solve for y:

x - y = 16

1.593y - y = 16

0.593y = 16

y = 27 (rounded)

Hence, there are 27 SUVs

The residual for the newborn is -0.9148 kg, indicating he is lighter than what the model predicts for his length.

To calculate the residual for the newborn's weight, we first use the least squares regression line equation, which is weight = -5.33 + 0.1926 * length. We then input the newborn's length of 48 cm into the equation to predict the weight.

Predicted weight = -5.33 + (0.1926 * 48) = -5.33 + 9.2448 = 3.9148 kg

The residual is the difference between the actual weight and the predicted weight, so for this newborn, the residual = actual weight - predicted weight = 3 kg - 3.9148 kg = -0.9148 kg.

The negative residual indicates that the newborn weighs less than what the regression model predicts for a boy of 48 cm in length. This could suggest that the child is lighter than average for his length

By Stokes' theorem,

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]

where [tex]S[/tex] is the surface with [tex]C[/tex] as its boundary. The curl is

[tex]\nabla\times\vec F(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]

Parameterize [tex]S[/tex] by

[tex]\vec\sigma(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(8-u\cos v)\,\vec k[/tex]

with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. Then take the normal vector to [tex]S[/tex] to be

[tex]\vec\sigma_u\times\vec\sigma_v=u\,\vec\imath+u\,\vec k[/tex]

Then the line integral is equal to the surface integral,

[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^9(2\,\vec\imath-u\cos v\,\vec k)\cdot(u\,\vec\imath+u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]

[tex]\displaystyle=\int_0^{2\pi}\int_0^9(2u-u^2\cos v)\,\mathrm du\,\mathrm dv=\boxed{162\pi}[/tex]

Answer:

The equation of the circle is (x + 10)² + (y + 5)² = 125 in standard form

Step-by-step explanation:

* lets study the standard form of the equation of a circle

- If the coordinates of the center of the circle are(h , k) and its radius

 is r, then the standard equation of the circle is:

 (x - h)² + (y - k)² = r²

* Now lets solve the problem

∵ The coordinates of the center of the circle are (-10 , -5)

∵ The standard form of the equation is (x - h)² + (y - k)² = r²

∵ h , k are the coordinates of the center

∴ h = -10 , k = -5

∴ The equation of the circle = (x - -10)² + (y - -5)² = r²

∴ The equation of the circle = (x + 10)² + (y + 5)² = r²

- To find the value of the radius lets use the point (-5 , 5) to

  substitute their coordinate instead of x and y in the equation

∵ The circle passes through point (-5 , 5)

∵ (x + 10)² + (y + 5)² = r²

- Use x = -5 and y = 5

∴ (-5 + 10)² + (5 + 5)² = r² ⇒ simplify

∴ (5)² + (10)² = r²

∴ 25 + 100 = r²

∴ r² = 125

* Now lets write the equation in standard form

∴ (x + 10)² + (y + 5)² = 125

* The equation of the circle is (x + 10)² + (y + 5)² = 125 in standard form

Answer: The required proportion is [tex]\dfrac{172}{523}[/tex] in fraction and [tex]0.329[/tex] in decimals.

Step-by-step explanation:

Since we have given that

Number of suspected criminals is drawn = 523

Number of criminals were captured = 172

We need to find the proportion of people who were caught after being on the 10 Most wanted list.

So, Proportion of people who were caught is given by

[tex]\dfrac{172}{523}\\\\=0.3288\\\\\approx 0.329[/tex]

Hence, the required proportion is [tex]\dfrac{172}{523}[/tex] in fraction and [tex]0.329[/tex] in decimals.

The estimated proportion of suspected criminals caught after being on the FBI's 10 Most Wanted list is 0.329, or 32.9%, based on a sample where 172 out of 523 individuals were captured.

To estimate the proportion of people who were caught after being on the FBI's 10 Most Wanted list, we can use the sample data provided. In the sample, 523 suspected criminals were monitored and 172 were captured. The estimated proportion of individuals caught is calculated by dividing the number of people captured by the total number in the sample.

To find this proportion, we perform the following calculation:

Proportion = Number of people captured / Total number of suspected criminals

Proportion = 172 / 523

Proportion = 0.329 (rounded to three decimal places)

So, the estimated proportion of people who were caught after appearing on the list is approximately 0.329, or 32.9%.

Answer:

12m

Step-by-step explanation:

We are given the following expression where a fraction is divided by another fraction:

[tex]\frac{\frac{16m^2}{m^2+5} }{\frac{4m}{3m^2+15} }[/tex]

To change this division into multiplication, we will take reciprocal of the fraction in the denominator and then solve:

[tex] \frac { 1 6 m ^ 2 } { m^2+5} } \times \frac{3m^2+15}{4m}[/tex]

Factorizing the terms to simplify:

[tex] \frac { 4 m ( 4m ) } { m ^ 2 + 5 } \times \frac { 3 ( m ^ 2 + 5 ) } { 4 m } [/tex]

Cancelling the like terms to get:

12m

Answer: [tex]12m[/tex]

Step-by-step explanation:

Given the expression [tex]\frac{\frac{16m^2}{m+5}}{\frac{4m}{3m^2+15}}[/tex], we can rewrite it in this form:

[tex](\frac{16m^2}{m+5})(\frac{3m^2+15}{4m})[/tex]

Now we must multiply the numerator of the first fraction by the numerator of the second fraction and   the denominator of the first fraction by the denominator of the second fraction:

 [tex]=\frac{(16m^2)(3m^2+15)}{(m^2+5)(4m)}}[/tex]

According to the Quotient of powers property:

[tex]\frac{a^m}{a^n}=a^{(m-n)}[/tex]

And the Product of powers property states that:

[tex](a^m)(a^n)=a^{(m+n)}[/tex]

Then, simplifying, we get:

[tex]=\frac{3(m^2+5)(4m)(4m)}{(m^2+5)(4m)}}\\\\=3(4m)\\\\=12m[/tex]

[tex]|\Omega|={_{53}C_3}\cdot 45=\dfrac{53!}{3!50!}\cdot45=\dfrac{51\cdot52\cdot53}{2\cdot3}\cdot45=1054170\\|A|=1\\\\P(A)=\dfrac{1}{1054170}\approx0.00000095\%[/tex]

[tex]|\Omega|=52\cdot51\cdot50=132600[/tex]

a)

[tex]|A|=4\cdot3\cdot2=24\\P(A)=\dfrac{24}{132600}=\dfrac{1}{5525}[/tex]

b)

[tex]|A|=13\cdot12\cdot11=1716\\P(A)=\dfrac{1716}{132600}=\dfrac{11}{850}[/tex]

a. Probability of all 3 cards being queens:

b. Probability of all 3 cards being spades:

Answer:

A. -1.6; not significant

Step-by-step explanation:

The z-score of a data set that is normally distributed with a mean of [tex]\bar x[/tex] and a standard deviation of [tex]\sigma[/tex], is given by:

[tex]z=\frac{x-\bar x}{\sigma}[/tex].

From the question, the test score is: [tex]x=48.4[/tex], the mean is [tex]\bar x=66[/tex], and the standard deviation is [tex]\sigma =11[/tex].

We just have to plug these values into the above formula to obtain:

[tex]z=\frac{48.4-66}{11}[/tex].

This simplifies to:  [tex]z=\frac{-17.6}{11}[/tex].

[tex]z=-1.6[/tex].

We can see that the z-score falls within two standard deviations of the mean.

Since [tex]-2\le-1.6\le2[/tex] the value is not significant.

The correct answer is A. -1.6; not significant

Answer:

a) 1/2 = 50%

b) 3/4 = 75%

c)  1 / 52 or 1,9%

Step-by-step explanation:

In a standard deck of cards, there are 52 cards in total:

13 are hearts, 13 are diamonds, 13 are clubs and 13 are spades.

​(a) Compute the probability of randomly selecting a club or spade

How many cards are a club or a spade?

C = 13 clubs + 13 spades = 26 cards

Out of the 52 total, that means that:

P (club or spade) = 26/52 = 1/2 = 50%

​(b) Compute the probability of randomly selecting a club or spade or heart. ​

How many cards are a club or a spade?

C = 13 clubs + 13 spades  + 13 hearts = 39 cards

Out of the 52 total, that means that:

P (club or spade or heart) = 39/52 = 3/4 = 75%

(c) Compute the probability of randomly selecting a two or diamond.

There's only ONE two of diamond in  regular deck of cards, so...

P(2 of diamond) = 1 / 52 or 1,9%

Answer:

5.(3)

Step-by-step explanation:

5.(3)=16/3

Answer:

d

Step-by-step explanation:

Step-by-step explanation:

The average rate of change of a function f(x) over an interval [a, b] is:

(f(b) − f(a)) / (b − a)

(f(4) − f(0)) / (4 − 0)

(7/17 − -1/5) / 4

(52/85) / 4

13/85

Answer:

yes thank you so much i was struggling so much with this tysm

Step-by-step explanation:

Answer:

[tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]

Step-by-step explanation:

We need to find out the partial differential [tex]F_{yx}[/tex] of [tex]F(x,y)=x^{2}sin(xy)[/tex]

First, differentiate [tex]F(x,y)=x^{2}sin(xy)[/tex] both the sides with respect to 'y'

[tex]\frac{d}{dy}F(x,y)=\frac{d}{dy}x^{2}sin(xy)[/tex]

Since, [tex]\frac{d}{dt}\sin t =\cos t[/tex]

[tex]\frac{d}{dy}F(x,y)=x^{2}cos(xy)\times \frac{d}{dy}(xy)[/tex]

[tex]\frac{d}{dy}F(x,y)=x^{2}cos(xy)\times x[/tex]

[tex]\frac{d}{dy}F(x,y)=x^{3}cos(xy)[/tex]

so, [tex]F_y=x^{3}cos(xy)[/tex]

Now, differentiate above both the sides with respect to 'x'

[tex]F_{yx}=\frac{d}{dx}x^{3}cos(xy)[/tex]

Chain rule of differentiation: [tex]D(fg)=f'g + fg'[/tex]

[tex]F_{yx}=cos(xy) \frac{d}{dx}x^{3} + x^{3} \frac{d}{dx}cos(xy)[/tex]

Since, [tex] \frac{d}{dx}x^{m} =mx^{m-1}[/tex] and [tex] \frac{d}{dt} cost =-\sin t[/tex]

[tex]F_{yx}=cos(xy)\times 3x^{2} - x^{3} sin(xy)\times \frac{d}{dx}(xy)[/tex]

[tex]F_{yx}=cos(xy)\times 3x^{2} - x^{3} sin(xy)\times y[/tex]

[tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]

hence, [tex]F_{yx}=3x^{2} cos(xy)- yx^{3} sin(xy)[/tex]

Answer: Option 'A' is correct.

Step-by-step explanation:

Since we have given a situation that

If an increase in one variable causes a decrease in another​ variable,

Then, there is inverse relationship.

When one variable is increased whereas other variable falls.

There will be inverse relationship.

Since inverse relation has negative relation.

Then, there is a negative relationship.

Hence, Option 'A' is correct.

An increase in one variable causing a decrease in another indicates a negative relationship between the two variables, characterized by opposite directional movements and graphically represented by a line with a negative slope.

When discussing the correlation between two variables, it is important to consider the direction and type of relationship they share. If an increase in one variable causes a decrease in the other variable, this is defined as a negative relationship. In a negative relationship, the two variables move in opposite directions, meaning that as one variable increases, the other decreases and vice versa.

The relationship is depicted graphically as a line with a negative slope on a graph, where the line descends as it moves from left to right. This situation should not be confused with dependent, direct, or independent relationships, which describe different aspects of variable interaction.

Answer: Hence, a) 0,  b) 2, and  c) 0

Step-by-step explanation:

As we know that If the difference of sum of odd places values and sum of even places value is divisible by 11, then the number is itself divisible by 11.

(a) 362,375,__35

Sum of odd places values : 3+2+7+5+x=17+x

Sum of even places values : 6+3+5+3=17

Difference between them is 17+x-17=x

So, x should be 0 to get divisible by 11 as 0 is divisible by 11.

(b) 82,919,__21

Sum of odd places values : 8+9+9+2=28

Sum of even places values : 2+1+x+1=4+x

Difference between them is 28-(4-x)=24-x

So, x should be 2 so, that it becomes 24-2=22 which is divisible by 11.

(c) 57,13__,473

Sum of odd places values : 5+1+x+7=13+x

Sum of even places values : 7+3+4+3=17

Difference between them is 17-(13+x)=4-x

So, x should be 4 so that it becomes 4-4=0 which is divisible by 11.

Hence, a) 0,  b) 2, and  c) 0

For this case we must factor the following expression:

[tex]4x ^ 2 + 23x-72[/tex]

We rewrite the middle term as a sum of two terms whose product is [tex]4 * (- 72) = - 288[/tex] and whose sum is 23. These numbers are -9 and +32. So:

[tex]4x ^ 2 + (- 9 + 32) x-72\\4x ^ 2-9x + 32x-72[/tex]

We factor the highest common denominator of each group.

[tex]x (4x-9) +8 (4x-9)[/tex]

We factor taking into account the common term [tex](4x-9):[/tex]

[tex](4x-9) (x + 8)[/tex]

Finally, the factored expression is:

[tex](4x-9) (x + 8)[/tex]

Answer:

Option D

Answer:

The correct answer option is D. (4x – 9)(x + 8).

Step-by-step explanation:

We are given the following polynomial and we are to find its factored form:

[tex]4x^2+23x-72[/tex]

Finding factors of (-72 * 4 = ) -288 such that when added they give a result of 23 and when multiplied it gives a product of -288.

[tex] 4 x ^ 2 + 3 2 x - 9 x - 7 2[/tex]

[tex] 4 x ( x + 8 ) - 9 ( x + 8 ) [/tex]

[tex] ( 4 x - 9 ) ( x + 8 )[/tex]

Answer:

1.406×[tex]10^{[tex]10^{18}km cubed

Step-by-step explanation:

The volume of a sphere is

[tex]V=\frac{4}{3}\pi r^3[/tex]

Filling in our formula:

[tex]V=\frac{4}{3}\pi (695,000)^3[/tex]

Cubing first gives us:

[tex]V=\frac{4}{3}\pi (3.35702[/tex]×[tex]10^{17}[/tex]

Do the multiplication and division of those numbers, multiply in the value of pi on your calculator, and you'll get 1.406×[tex]10^{18}[/tex]

To determine the volume of the Sun with a radius of about 695,000 km, we first convert the radius to centimeters and then apply the formula V = (4/3)πr³. After performing the calculations, the volume of the Sun is approximately 1.401 x 10³³ cm³in scientific notation with three decimal places in the mantissa.

The student has asked what the volume of the Sun is, given its radius of about 695,000 km. To find the volume of a sphere, the formula to use is V = (4/3)πr³, where V represents the volume and r is the radius.

First, we need to convert the radius from kilometers to centimeters because the standard unit for volume in scientific notation often involves cubic centimeters. There are 100,000 centimeters in a kilometer, so the radius in centimeters is 695,000 km × 100,000 cm/km = 6.95 x 10¹⁰cm.

Now, we can calculate the volume using the formula:

V = (4/3)π(6.95 x 10¹⁰ cm)³

V = (4/3)π(6.95^3 x 10³⁰) cm³

V = (4/3)π(334.14 x 10³⁰) cm³

V = (4/3)π(3.3414 x 10³²) cm³

V ≈ 4.1888 x 3.3414 x 10³² cm³

V ≈ 1.401 x 10^33 cm³

Therefore, the volume of the Sun in scientific notation, using three decimal places in the mantissa, is approximately 1.401 x 10³³ cm³.

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Answer:

  3x -4y = 2

Step-by-step explanation:

A plot of the points makes it clear that the longest diagonal is BD. The 2-point form of the line through those points can be found by filling in ...

  y = (y2 -y1)/(x2 -x1)(x -x1) +y1

  y = (4 -(-2))/(6 -(-2))(x -(-2)) +(-2) . . . . . fill in points B and D

  y = (6/8)(x +2) -2

  4y = 3(x +2) -8 . . . . . .  multiply by 4

  3x -4y = 2 . . . . . . . . . . add 2-4y

Answer:

Quincy

Step-by-step explanation:

Each student is assigned a two digit number, so let's split the random number line into two digit numbers:

59, 78, 44, 43, 12, 15, 95, 40, 92, 33, 00, 04, 67, 43, 18, 02, 61, 05, 73, 96, 16, 84, 33, 84, 54

Now let's identify the numbers between 00 and 29.

59, 78, 44, 43, 12, 15, 95, 40, 92, 33, 00, 04, 67, 43, 18, 02, 61, 05, 73, 96, 16, 84, 33, 84, 54

So the fifth student in the list is #18, or Quincy.

Answer:

Quincy is the answer

Step-by-step explanation:

Answer:

[tex]\large\boxed{7.\ B.\ 8s^2+16s+1}\\\\\boxed{8.\ D.\ -15t^9u^{11}}\\\\\boxed{11.\ C.\ 3,\ -2}[/tex]

Step-by-step explanation:

[tex]7.\\(3s^2+7s+2)+(5s^2+9s-1)=3s^2+7s+2+5s^2+9s-1\\\\\text{combine like terms}\\\\=(3s^2+5s^2)+(7s+9s)+(2-1)\\\\=8s^2+16s+1[/tex]

[tex]8.\\(-3t^2u^3)(5t^7u^8)=(-3\cdot5)(t^2t^7)(u^3u^8)\qquad\text{use}\ a^na^m=a^{n+m}\\\\=-15t^{2+7}u^{3+8}=-15t^9u^{11}[/tex]

[tex]11.\\n-the\ number\\\\n^2=n+6\qquad\text{subtract}\ n\ \text{and}\ 6\ \text{from both sides}\\\\n^2-n-6=0\\\\n^2+2n-3n-6=0\\\\n(n+2)-3(n+2)=0\\\\(n+2)(n-3)=0\iff n+2=0\ \vee\ n-3=0\\\\n+2=0\qquad\text{subtract 2 from both sides}\\n=-2\\\\n-3=0\qquad\text{add 3 to both sides}\\n=3[/tex]

Answer:

Step-by-step explanation:

Since we want twice as much Mocha as Decaf, we can create a "bag" that contains 2 lbs of Mocha (at 11.50 each) and 1 lb of Decaf (at 10). The value of this "bag" is then 2×11.50 +10.00 = 33.00. For 181.50, we can buy ...

  181.50/33.00 = 5.5

"bags". This amount is ...

  11 lbs of Arabian Mocha and 5.5 lbs of Columbian Decaf

Answer: 0.7619

Step-by-step explanation:

Given : Mean : [tex]\mu=45.0 [/tex]

Standard deviation : [tex]\sigma =8.0[/tex]

Sample size : [tex]n=9[/tex]

We assume that the variable is normally distributed.

The value of z-score is given by :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

a) For x= 46.9 years

[tex]z=\dfrac{46.9-45.0}{\dfrac{8}{\sqrt{9}}}=0.7125[/tex]

The p-value : [tex]P(z<0.7125)=0.7619224\approx0.7619[/tex]

Hence, the  probability that the average age of those doctors is less than 46.9 years =0.7619

The question relates to probability in a normally distributed population. We calculated the standard error and z-score, then used the z-table to find that there is approximately a 76.11% chance that the average age of 9 randomly chosen doctors from this hospital will be less than 46.9 years.

The subject of this question pertains to Probability and Statistics, specifically the application of the Normal Distribution in the context of calculating the probability of a particular outcome in a real-world scenario. We'll apply the rule for the Central Limit Theorem (CLT) since the sample size is reasonably large (n = 9).

The first step is to calculate the standard error (SE). The SE of the mean can be calculated by dividing the standard deviation by the square root of the number of doctors:

SE = 8.0/sqrt(9) = 8.0/3 = 2.67.

Next, you would calculate the z-score. The z-score of 46.9 is obtained by subtracting the population mean from 46.9 and then dividing by the SE:

Z = (46.9 - 45.0)/2.67 = 0.71.

To determine the probability that the average age is less than 46.9 years, you will want to look up the z-score of 0.71 in a z-table, which gives a value of 0.7611, or 76.11%. So there is approximately a 76.11% chance that the mean age of the 9 doctors chosen will be less than 46.9 years old.

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