A zoo has some new exhibits.
In a survey, 2400 zoo attendants are asked to choose their favorite.
The results are shown in the following table.
(a) find the percentage for each type of animal.

Answer:

3.1

4.2

5.2

42

Step-by-step explanation: They’re congruent so just copy what the other triangle has all you had to do was find the radius.

Answer:

the first option ...-4

Step-by-step explanation:

The solution is Option A.

The value of the equation is x = -4

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

A = Negative 2 x + (negative 8) = 2 x + 8

Substituting the values in the equation , we get

( -2x ) + ( -8 ) = 2x + 8   be equation (1)

On simplifying the equation , we get

-2x - 8 = 2x + 8

Adding 8 on both sides of the equation , we get

2x + 16 = -2x

Adding 2x on both sides of the equation , we get

4x + 16 = 0

Subtracting 16 on both sides of the equation , we get

4x = -16

Divide by 4 on both sides of the equation , we get

x = -4

Therefore , the value of x is -4

Hence , the value of the equation is x = -4

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Answer:

Time taken by A = 16 seconds

Time taken by B = 12 seconds

Step-by-step explanation:

Given:

Two elevator and their respective time to reach certain height.

Here time is function of height.

Time taken by elevator A = 0.8(h)+16

Time taken by elevator B = -0.8(h)+12

We have to find the time taken by the elevator to reach ground level.

Accordingly:

We know that ground level the height will be zero meaning that (h=0).

Plugging the h values in the equation we can find the time taken by both the elevators to reach zero height that is the ground level.

⇒ [tex]t_A=0.8h+16[/tex]                      ⇒ [tex]t_B=-0.8h+12[/tex]

⇒ [tex]t_A=0.8(0)+16[/tex]                   ⇒ [tex]t_B=-0.8(0)+12[/tex]

⇒ [tex]t_A=16[/tex] sec                          ⇒ [tex]t_B=12[/tex] sec

So,

Time taken by elevator A and elevator B to reach the ground is 16 seconds and 12 seconds respectively.

The option A mutual funds will be more effective.

Step-by-step explanation:

Option A:

Principal amount = $10000

Monthly deposit = $200

Time = 10 years

Rate of interest = 7%

Total deposit = (200 x 12 x 10) + 10000

= 24000 + 10000

= $34000

Interest = (34000 x 7 ) /100

= 340 x 7

= $2380

Total amount = 34000 + 2380

= $36380

Option B:

Principal amount = $10000

Monthly deposit = $200

Time = 5 years

Rate of interest = 9%

Total deposit = (200 x 12 x 5) + 10000

= 12000 + 10000

= $22000

Interest = (22000 x 9 ) /100

= $1980

Total amount = 22000+1980

= $23980

The option A mutual funds will be more effective.

Answer:

What is the difference in the final balances of the two mutual funds?

Step-by-step explanation:

The difference is $12,400.

Answer:

A = 68 unit^2

Step-by-step explanation:

Given:-

The piece-wise function f(x) is defined over an interval as follows:

                       f(x) =  { x^2+3x+4    , x < 3

                       f(x) =  { x^2+3x+4     , x≥3

                        Domain : [ -3 , 4 ]

Find:-

Find the area of the region enclosed by f(x) and the x axis

Solution:-

- The best way to tackle problems relating to piece-wise functions is to solve for each part individually and then combine the results.

- The first portion of function is valid over the interval [ -3 , 3 ]:      

                       [tex]f(x) = x^2+3x+4[/tex]

- The area "A1" bounded by f(x) is given as:

                      [tex]A1 = \int\limits^a_b {f(x)} \, dx[/tex]

Where,  The interval of the function { -3 , 3 ] = [ a , b ]:

                     [tex]A1 = \int\limits^a_b {x^2+3x+4} \, dx\\\\A1 = \frac{x^3}{3} + \frac{3x^2}{2} + 4x |\limits_-_3^3 \\\\A1 = \frac{3^3}{3} + \frac{3*3^2}{2} + 4*3 - \frac{-3^3}{3} - \frac{3(-3)^2}{2} - 4(-3)\\\\A1 = 9 + 13.5 +12 + 9-13.5+12\\\\A1 =42 unit^2[/tex]

- Similarly for the other portion of piece-wise function covering the interval [3 , 4] :

                     [tex]f(x) = 4x+10[/tex]

- The area "A2" bounded by f(x) is given as:

                      [tex]A2 = \int\limits^a_b {f(x)} \, dx[/tex]

Where,  The interval of the function { 3 , 4 ] = [ a , b ]:

                     [tex]A2 = \int\limits^a_b {4x+10} \, dx\\\\A2 = 2x^2 + 10x |\limits_3^4 \\\\A2 = 2*(4)^2 + 10*4 - 2*(3)^2 - 10*3\\\\A2 = 32 + 40 - 18-30\\\\A2 =26 unit^2[/tex]

- The total area "A" bounded by the piece-wise function over the entire domain [ -3 , 4 ] is given:

                     A = A1 + A2

                     A = 42 + 26

                     A = 68 unit^2

To find the area enclosed by f(x) and the x-axis from x = -3 to x = 4 for the given piecewise function, calculate the integral of each piece separately and then sum the areas. The area is divided into two parts due to the function having different expressions before and after x = 3.

The student is asking to find the area under the curve of a given piecewise function f(x) on the interval from x = -3 to x = 4. Since the function is defined differently for x < 3 and x ≥ 3, the area calculation involves two parts:

The first part can be calculated using the integral of [tex]f(x) = x^2 + 3x + 4[/tex]from x = -3 to x = 3. The second part is the integral of [tex]f(x) = 4x + 10[/tex] from x = 3 to x = 4. The total area is the sum of these two areas. For this function, the areas are bounded above by the function and below by the x-axis, so all areas are considered positive.

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Answer:

42

Step-by-step explanation:

If the shortest side is 7 and it is enlarged to be 24.5.

It is enlarged by a scale of 3.5.

so if you scale 12 by 3.5

then you get 42

Answer:

(A) The additional information that is needed to confirm about the conditions for this test have been met is ‘Population is approximately normal.

(B) The test statistic = 1.9965

(C1) P-value = 0.0556

(C2) There is no significant difference between mean VOT for children and adults.

D1) It is possible to make type II error.

(D2) There should be a difference in the VOT of adults and children.

Step-by-step explanation:

Let na be the number of adults = 20

xa mean VOT for adults = 88.17

sa standard deviatiation of VOT for adults = 24.74

Let nc be the number of children = 10

xc mean VOT for children =60.67

sc standard deviatiation of VOT for children = 39.89

(A) From the information the population variances are unknown and the two sample are assumed to be independent and the sample the sample size are smaller that is (n<30).

This indicates that the additional information that is required for the conditions of the test to be satisfied is 'distribution of the population'. the addition assumption to be made is, that the population distribution is normal.

(b) Calculating the test statistics using the formula;

t = (xa -xc)/SE - d

where SE = standard deviation , d= hypothesized difference = 0

But SE = √sa²/na +sc²/nc

           = √24.74 ²/20 + 39.89²/10

           = √189.72559

           = 13.774

Substituting into test statistics equation, we have

t = (xa -xc)/SE - d

  = (88.17 - 60.67/13.774

  = 1.9965

Therefore the test statistic is 1.9965

(c) Calculating the p-value, we have;

Degree of freedom = na + nc -2

                                 = 10+ 20 -2

                                 = 28

The p-value for t=1.9965 at 28 degrees of freedom and 0.05 level of significance is 0.0556.

The p-value 0.0556 is greater than given level of significance 0.05 hence we fail to reject the null hypothesis and conclude that there is no significant difference between mean VOT for children and adults.

(D1) From the information in part (C) the null hypothesis is not rejected.

Since the null hypothesis is not rejected, there might be a chance that not rejecting null hypothesis would be wrong. In this type of situation the error that can occur would be type II error.

(D2) The type of error describe in the context of this study is obtained by the concept of the type II error which tells that the null hypothesis is not rejected when it is actually false.

Answer: 32

Step-by-step explanation:

Use a fact family. 42 - 10 = 32.

You will get 32 as an answer.

So x = 32.

Answer:

32

Step-by-step explanation:

If x+10=42 then x=42-10 which makes x=32

Answer:

$9

Step-by-step explanation:

12 + 18 = $30 Total

30 - 21 = $9 Discount

30 - 9 = $21 After discount

Answer:

x= [tex]\frac{135}{7}[/tex]

Step-by-step explanation:

1. [tex]-\frac{7}{3} (x-3)=-52[/tex]

2. combine -7/3(x - 3)

  =    [tex]-\frac{7}{3}x-7 = -52[/tex]

3. Do combine like terms on both sides

   =   -52 + -7 = -45

4. left with, [tex]-\frac{7}{3}x = -45[/tex]

5. divide -45 by -7/3

6. left with, [tex]x = \frac{135}{7}[/tex]

Answer:

21

Step-by-step explanation:]

plz mark me the Brainliest

Answer:

q = 26

Step-by-step explanation:

Given that (3, q ) lies on the line, then the coordinates of the point make the equation true.

Substitute x = 3, y = q into the equation

q = 8(3) + 2 = 24 + 2 = 26

Final answer:

To determine the value of q for the point (3, q) on the line y=8x+2, substitute x with 3 to get y=26. Hence, q is 26, and the point is (3, 26).

Explanation:

To find the value of q for the point (3, q) that lies on the line with the equation y=8x+2, we substitute the x-coordinate of the point, which is 3, into the equation and solve for y. This gives us:

y = 8(3) + 2

y = 24 + 2

y = 26

Therefore, the value of q is 26, and the point is (3, 26). This demonstrates how to use a line's equation to find a specific point on the line. The question illustrates the algebraic relationship between the coordinates of a point on a line and the equation of the line itself.

Answer:

[tex]\frac{16}{28}[/tex]

Step-by-step explanation:

Question asked:

Select all of the ratios that are equivalent to 4 to 7.

1) 5/8

2) 12/10

3) 8.14

4) 5 to 10

5) 16/28

6) 9 to 16​

Solution:

To know the ratios are equivalent to [tex]\frac{4}{7}[/tex] or not, we have to check each by cross multiplication:-

1) [tex]\frac{5}{8}[/tex]

[tex]\frac{4}{7} =\frac{5}{8} \\ \\[/tex]

By cross multiplication:

[tex]4\times8=5\times7\\ \\ 32=35[/tex]

Not equivalent.

2) [tex]\frac{12}{10}[/tex]

[tex]\frac{4}{7} =\frac{12}{10} \\ \\[/tex]

By cross multiplication:

[tex]4\times10=12\times7\\ \\ 40=84[/tex]

Not equivalent.

3) 8.14 = [tex]\frac{814}{100} \ eliminating\ decimal[/tex]

[tex]\frac{4}{7} =\frac{814}{100} \\ \\[/tex]

By cross multiplication:

[tex]4\times100=814\times7\\ \\ 400=5698[/tex]

Not equivalent.

4) [tex]\frac{5}{10}[/tex]

[tex]\frac{4}{7} =\frac{5}{10} \\ \\[/tex]

By cross multiplication:

[tex]4\times10=5\times7\\ \\ 40=35[/tex]

Not equivalent.

5)[tex]\frac{16}{28}[/tex]

[tex]\frac{4}{7} =\frac{16}{28} \\ \\[/tex]

By cross multiplication:

[tex]4\times28=16\times7\\ \\ 112=112[/tex]

Yes, this is equivalent.

6) [tex]\frac{9}{16}[/tex]

[tex]\frac{4}{7} =\frac{9}{16} \\ \\[/tex]

By cross multiplication:

[tex]4\times16=9\times7\\ \\ 64=63[/tex]

Not equivalent.

Thus, only [tex]\frac{16}{28}[/tex] is equivalent.

Answer:2 identical side rectangle

2 identical end triangle

1 bottom rectangle

area of trialge=1/2bh

aera of rectangle=legnth tiems width

2 side rectangles are 5 by 10=50

times 2 since 2 of them 50*2=100

end triangles

1/2 times 8 times 3=12

2 of them

12 times 2=24

bottom

8 by 10

80

add everybody

100+24+80=204 in^2 i think this is how

Step-by-step explanation:

Answer:

2 identical side rectangle

2 identical end triangle

1 bottom rectangle

area of trialge=1/2bh

aera of rectangle=legnth tiems width

2 side rectangles are 5 by 10=50

times 2 since 2 of them 50*2=100

end triangles

1/2 times 8 times 3=12

2 of them

12 times 2=24

bottom

8 by 10

80

add everybody

100+24+80=204 in^2 i think this is how

Step-by-step explanation:

Answer:

C: 107

Step-by-step explanation:

135-28 = 107

Answer:

it would have to be 107 bro

Step-by-step explanation:

Answer:

24, -48

Step-by-step explanation:

In this geometric sequence, the previous number is being multiplyed by -2 each time

This said the next number is 24.

-12*-2=24

and the next one is -48.

24*-2=-48

Final answer:

The next two terms of the geometric sequence -3, 6, -12 are 24 and -48, found by repeatedly multiplying with the common ratio of -2.

Explanation:

The first three terms of the given geometric sequence are -3, 6, and -12. To find the next two terms in this sequence, we need to determine the common ratio between the terms. The common ratio (r) is the factor by which we multiply one term to get the next term. In this case, 6 divided by -3 equals -2, and similarly, -12 divided by 6 also equals -2. This confirms that our common ratio is -2.

Now, to find the fourth term of the sequence, we multiply the third term (-12) by the common ratio (-2):

-12 × -2 = 24

To find the fifth term, we multiply the fourth term (24) by the common ratio (-2):

24 × -2 = -48

Therefore, the fourth and fifth terms of the geometric sequence are 24 and -48, respectively.

Answer:

2

Step-by-step explanation:

24÷ 6÷ 2=2

Answer:

3x-2

Step-by-step explanation:

Answer:

3x-2

Step-by-step explanation:

"Less than" makes the whole expression changed.

If the less than was not there, it would have been 2-3x.

"Product" is multiplication. So it would be 3x.

The "Less than" also has less in it which is subtraction.

Answer:

a = 3

d = 4

Step-by-step explanation:

6x^2+14x+4 = (ax+1)(2x+d)

6x^2+2x+12x+4

2x(3x+1)+4(3x+1)

(3x+1)(2x+4)

3 = a

4 = d

Answer:

Step-by-step explanation:

a=3, d=4

Answer:

[tex]x^2+8x+15[/tex]

Step-by-step explanation:

[tex]f(x)=x+4\\g(f(x))=g(x+4)\\g(x+4)=(x+4)^2-1\\(x+4)^2-1=x^2+8x+15[/tex]

Answer:

The volume of the large box is 384 cubic units

Step-by-step explanation:

Here, we have;

The number of cubes along the base of the front of the large box (width) = 8 cubes

The number of cubes along the base of the side of the large box (length) = 8 cubes

The number of cubes along the right  side of the front of the large box (the box height)  = 6 cubes

The volume of the large box = Length × Width × Height

∴ The volume of the large box = 8 cubes × 8 cubes × 6 cubes = 384 cubic units.

The volume of the large box = 384 cubic units.

Answer:

2.061, 2.34, 2.6, 2.601, 2.7

Step-by-step explanation:

Hello!

This is a bit hard to explain! If you want me to try, just comment.

I have arranged the sequence in ascending order below:

[tex]2.061,\:2.34,\:2.6,\:2.601,\:2.7[/tex].

Hope this helps!

Answer:

2.061, 2.34, 2.6, 2.601, 2.7

Step-by-step explanation:

Because the numbers all have a 2 in the ones place, you need to evaluate the numbers in the tenths place and the numbers with the lowest go first and so on :)

Answer:

37

Step-by-step explanation:

The first thing is to calculate critical z factor

the alpha and the critical z score for a confidence level of 90% is calculated as follows:

two sided alpha = (100% - 90%) / 200 = 0.05

critical z factor for two sided alpha of .05 is calculated as follows:

critical z factor = z factor for (1 - .05) = z factor for (.95) which through the attached graph becomes:

critical z factor = 2.58

Now we have the following formula:

ME = z * (sd / sqrt (N) ^ (1/2))

where ME is the margin of error and is equal to 6, sd is the standard deviation which is 14 and the value of z is 2.58

N the sample size and we want to know it, replacing:

6 = 2.58 * (14 / (N) ^ (1/2))

solving for N we have:

N = (2.58 * 14/6) ^ 2

N = 36.24

Which means that the sample size was 37.

Answer:

x = +2, x = -2

Step-by-step explanation:

The equation to solve in this problem is

[tex]10x^2-5=35[/tex]

The first step we do is to subtract 35 on both sides of the equation, so we get:

[tex]10x^2-5-35=0\\10x^2-40=0[/tex]

Now we simplify the equation by dividing both terms by 10:

[tex]\frac{10x^2-40}{10}=0\\x^2-4=0[/tex]

Now we observe that the term on the left is the difference between two squares, so it can be rewritten using the property:

[tex]a^2-b^2=(a+b)(a-b)[/tex]

Where here,

a = x

b = 2

So we can rewrite the equation as:

[tex]x^2-4=0\\(x+2)(x-2)=0[/tex]

And this equation is zero when either one of the two factors is zero, so the two solutions are:

[tex]x+2=0\rightarrow x=-2\\x-2=0 \rightarrow x=+2[/tex]

Answer:

a -pound bag of cat food can enough to feed her cat for 7 days

Step-by-step explanation:

Given:

Number of cat food used in 7 days (in ounces): 7*6 = 42 ounces

As we know,1 pound (lb) is equal to 16 Ounces (oz)

<=> 42 ounces = 42 /16 = 2.625 pounds < 3-pound bag

So a -pound bag of cat food can enough to feed her cat for 7 days

Hope it will find you well.

Answer:

Yes it will

Step-by-step explanation:

Let's first work with a single unit so that our work will be a lot easier.

We are going to convert pounds to ounces and since 1 pound = 16 ounces, 3 pounds = 16 × 3 = 48 ounces of cat food.

And she feeds her cat 6 ounces of food each day,that means that in 7 days the cat will only be able to consume just 7 × 6 = 42 ounces,but remember that the amount of cat food bought is equal to 48 pounds.

This here means that the cat food will be enough to feed the cat for 7 days

To make F the subject of the formula in the Celsius to Fahrenheit conversion, multiply by 9, divide by 5, and then add 32, resulting in F = (9/5)C + 32.

To make F the subject of the formula when given the Celsius to Fahrenheit conversion formula c = 5(f - 32)/9, we start by isolating Fahrenheit on one side of the equation. Here's a step-by-step process:

Now the formula for F in terms of C is in the form aC + b with a = 9/5, b = 32, and there's no c as in the denominator since the conversion is direct.

[tex]\( F = \frac{9c + 160}{5} \).[/tex] In form [tex]\( aC + \frac{b}{c} \), \( a = 9 \), \( b = 160 \),[/tex] and [tex]\( c = 5 \).[/tex]

Let's break down the process of rearranging the formula step by step.

Given formula: [tex]\( c = \frac{5(F - 32)}{9} \)[/tex]

We want to isolate [tex]\( F \)[/tex] on one side of the equation.

1. Multiply both sides by [tex]\( \frac{9}{5} \):[/tex]

[tex]\[ \frac{9}{5} \cdot c = \frac{9}{5} \cdot \frac{5(F - 32)}{9} \][/tex]

  This cancels out the fraction on the right side.

  [tex]\[ \frac{9}{5} \cdot c = F - 32 \][/tex]

2. Add 32 to both sides to isolate [tex]\( F \):[/tex]

[tex]\[ \frac{9}{5} \cdot c + 32 = F \][/tex]

Now, [tex]\( F \)[/tex] is isolated on the right side of the equation.

3. Rewrite [tex]\( F \)[/tex] in the required form [tex]\( aC + \frac{b}{c} \):[/tex]

 [tex]\[ F = \frac{9}{5}c + 32 \][/tex]

  To express [tex]\( F \)[/tex] in the required form, we can rewrite [tex]\( \frac{9}{5}c \) as \( \frac{9c}{5} \),[/tex]so the form becomes [tex]\( aC + \frac{b}{c} \).[/tex]

  So, [tex]\( a = 9 \), \( b = 32 \), and \( c = 5 \).[/tex]

4. Final Form:

 [tex]\[ F = \frac{9c + 160}{5} \][/tex]

  So, in the form [tex]\( aC + \frac{b}{c} \),[/tex] we have [tex]\( a = 9 \), \( b = 160 \), and \( c = 5 \).[/tex]

Answer:

[tex]\frac{225}{16} cm/s[/tex]

Step-by-step explanation:

We are given that

[tex]\frac{dV}{dt}=25cm^3/s[/tex]

Side of base=4 cm

l=w=4 cm

Height,h=12 cm

We have to find the rate at which the water level rising when the water level is 4 cm.

Volume of pyramid=[tex]\frac{1}{3}lwh=\frac{1}{3}l^2h[/tex]

[tex]\frac{l}{h}=\frac{4}{12}=\frac{1}{3}[/tex]

[tex]l=\frac{1}{3}h[/tex]

Substitute the value

[tex]V=\frac{1}{27}h^3[/tex]

Differentiate w.r.t t

[tex]\frac{dV}{dt}=\frac{3}{27}h^2\frac{dh}{dt}[/tex]

Substitute the values

[tex]25=\frac{1}{9}(4^2)\frac{dh}{dt}[/tex]

[tex]\frac{dh}{dt}=\frac{25\times 9}{16}=\frac{225}{16} cm/s[/tex]

Final answer:

Using the volume of a pyramid and the concept of similar triangles, we set up a proportion to find the changing base area at a specific water level. Then, applying the product rule for differentiation, we relate the rate of volume change to the rate of height change, which allows us to solve for the water level rising rate when it is 4 cm.

Explanation:

To find the rate at which the water level is rising when the water level is 4 cm in an inverted pyramid being filled at a constant rate of 25 cubic centimeters per second, we can use the concept of similar triangles and the volume of a pyramid.

The volume V of a pyramid is given by V = (1/3)Bh, where B is the base area and h is the height. As the pyramid fills, the water forms a smaller, similar pyramid whose volume increases at a rate of 25 cm3/s.

Since the sides of the smaller pyramid are proportional to the height, we can set up a proportion using the side length s of the water level: s/4 = 4/12. Solving for s gives us s = 4 * (4/12) = 4/3 cm. The base area B of the water at this level is B = s2 = (4/3)2 cm2.

To find the rate of the rise of water dh/dt, we use the relation dV/dt = (1/3) * d(Bh)/dt. Since B is also changing with h, we have to use the product rule for differentiation: dV/dt = (1/3)(B(dh/dt) + h(dB/dt)). However, because B is a function of h2, dB/dt can be expressed as a function of dh/dt. This allows us to solve for dh/dt.

The required students will need to make 10 tiles to create a row that is 4 feet 2 inches long.

Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression. The goal is to obtain an expression that is easier to work with, manipulate, or solve.

Here,

To solve this problem, we first need to convert the length of the row from feet and inches to inches.

4 feet 2 inches is equal to (4 x 12) + 2 = 50 inches.

Next, we can divide the length of the row by the length of each tile to find the number of tiles needed:

Number of tiles = Length of row / Length of each tile

Number of tiles = 50 inches / 5 inches

Number of tiles = 10 tiles

Therefore, the students will need to make 10 tiles to create a row that is 4 feet 2 inches long.

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Answer:

C

Step-by-step explanation:

Answer:

The answer should be C

Step-by-step explanation:

we know that

the triangle AOB is congruent with triangle AOC

because

AB=AC

OB=OC-----> the radius of the circle

The OB side is common

but

there is no additional information that allows me to calculate the OBA angle to determine if it is a right angle

therefore

the answer is the option

C.There is no indication that AB and AC are perpendicular to the radii at the points of intersection with the circle.

Thanks (4)

Answer:

a) For this case we select a sample size of n=9. And we know that the distribution of X is normal so then the distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

b) [tex] SE = \frac{\sigma}{\sqrt{n}} =\frac{5.8}{\sqrt{9}} =1.9333[/tex]

c) [tex] P\bar X >39.5)[/tex]

And we can use the z score given by:

[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we find the z score for 39.5 we got:

[tex] z = \frac{39.5-38}{\frac{5.8}{\sqrt{9}}}= 0.78[/tex]

And using the complement rule we got:

[tex] P(z >0.78) =1-P(Z<0.78) = 1-0.7823= 0.2177[/tex]

d) [tex] P\bar X >37.5)[/tex]

And we can use the z score given by:

[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we find the z score for 37.5 we got:

[tex] z = \frac{37.5-38}{\frac{5.8}{\sqrt{9}}}= -0.26[/tex]

And using the complement rule we got:

[tex] P(z >-0.26) =1-P(Z<-0.26) = 1-0.3974= 0.6026[/tex]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the life of batteries of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(38,5.8)[/tex]  

Where [tex]\mu=38[/tex] and [tex]\sigma=5.8[/tex]

Part a

For this case we select a sample size of n=9. And we know that the distribution of X is normal so then the distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

Part b

The standard error is given by:

[tex] SE = \frac{\sigma}{\sqrt{n}} =\frac{5.8}{\sqrt{9}} =1.9333[/tex]

Part c

We want this probability:

[tex] P\bar X >39.5)[/tex]

And we can use the z score given by:

[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we find the z score for 39.5 we got:

[tex] z = \frac{39.5-38}{\frac{5.8}{\sqrt{9}}}= 0.78[/tex]

And using the complement rule we got:

[tex] P(z >0.78) =1-P(Z<0.78) = 1-0.7823= 0.2177[/tex]

Part d

We want this probability:

[tex] P\bar X >37.5)[/tex]

And we can use the z score given by:

[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we find the z score for 37.5 we got:

[tex] z = \frac{37.5-38}{\frac{5.8}{\sqrt{9}}}= -0.26[/tex]

And using the complement rule we got:

[tex] P(z >-0.26) =1-P(Z<-0.26) = 1-0.3974= 0.6026[/tex]

The distribution of the mean life of Batteries produced by Power+, Inc. follows Normal distribution. The standard error is calculated to be 1.9333. Using Z scores, it's discovered that approximately 21.77% of samples will have a mean life more than 39.5 hours while around 60.26% samples will have a mean useful life more than 37.5 hours.

Understanding the Distribution of the Mean Life of Batteries

a. The distribution of the sample mean should approximate a normal distribution because we know the distribution of the population (life of the batteries) is normal. The expectation is that the sample mean should also follow normal distribution, based on the Central Limit Theorem.

b. The standard error of the distribution of the sample mean, is calculated as the standard deviation divided by the square root of the number of samples. Therefore, the standard error is 5.8 / sqrt(9), that is approximately 1.9333.

c. To find the proportion of samples with a mean useful life of more than 39.5 hours, we first find the Z score for 39.5. The Z score is calculated by (sample mean - population mean) / standard error. Therefore, Z = (39.5 - 38) / 1.9333 = approximately 0.78 (rounded to 2 decimal places). Looking this up on a Z table gives us 0.7823. However, because we want the proportion where it is more than 39.5 hours, we need to subtract this from 1. So, 1-0.7823 = 0.2177 (i.e., 21.77% samples will have a mean useful life more than 39.5 hours).

d. Following the same procedure, the Z score for a sample mean of 37.5 is approx negative -0.26 (rounded to 2 decimal places) using the same calculation as above. Looking this up on a Z table gives us 0.3974. But, because we want the proportion greater than 37.5 hours, we need to subtract this from 1. So, 1-0.3974 = 0.6026 (i.e., 60.26% samples will have a mean useful life more than 37.5 hours).

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