Mrs Wright spent 2/9 of her paycheck on food and 1/3 on rent. She spent 1/4 of the remainder on transportation. She had $210 left. How much was Mrs. Wright's paycheck ?

Answer:

Confidence interval = [ 0.4432, 0.4968]

Step-by-step explanation:

Registered voters approved of the job Barack Obama was doing as president,

p = 47% = 0.47

Total sample size, n = 2,294

Confidence level = 99%

now,

confidence interval for the proportion of all registered voters who approved of the job Barack Obama was doing as president

= p ± [tex]z\times\sqrt{\frac{p\times(1-p)}{n}}[/tex]

for 99% confidence level, z value is 2.58

= 0.47 ± [tex]2.58\times\sqrt{\frac{0.47\times(1-0.47)}{2294}}[/tex]

= 0.47 ± 0.0268

or

Confidence interval = [0.47 - 0.0268, 0.47 + 0.0268 ]

or

Confidence interval = [ 0.4432, 0.4968]

Answer:

The probability that the maximum number of draws is required is 0.2286

Step-by-step explanation:

The probability that the maximum number of draws happens when you pick different colors in the first four pick.

Assume you picked one sock in the first draw. Its probability is 1, since you can draw any sock.

In the second draw, 7 socks left and you can draw all but the one which is the pair of the first draw. Then the probability is [tex]\frac{6}{7}[/tex]

In the third draw, 6 socks left and you can draw one of the two pair colors which are not drawn yet. Its probability is [tex]\frac{4}{6}[/tex]

In the forth draw, 5 socks left and only one pair color, which is not drawn. The probability of drawing one of this pair is [tex]\frac{2}{5}[/tex]

In the fifth draw, whatever you draw, you would have one matching pair.

The probability combined is 1×[tex]\frac{6}{7}[/tex] ×[tex]\frac{4}{6}[/tex]× [tex]\frac{2}{5}[/tex] ≈ 0.2286

Final answer:

The probability that the maximum number of draws is required to get a matching pair of socks from four pairs is 1/7, or about 14.29%.

Explanation:

The question is asking for the probability that the maximum number of draws (7) from 8 socks (4 pairs) is needed before obtaining a matching pair. To calculate this probability, consider that with each draw, you are less likely to pick a sock that matches the ones already drawn, until the last pair remains.

The first sock can be any of the 8, so there's no chance of failure here. For the second sock, to avoid a match, we have 6 out of 7 chance, since one would match the first. For the third sock, to continue avoiding a match, there is a 4 out of 6 chance. This pattern continues until we are left with just two socks, and we must draw the one that matches the last sock drawn.

Therefore, the probability that the maximum number of draws is required is:

(6/7) * (4/6) * (2/4) * (1/2) = 1/7 or about 14.29%.

Answer: he has 65/20 meters of string left

Step-by-step explanation:

Montel bought a spool of string for making kites. It contained 10 3/20 meters of string. Let us first convert this length of string to improper fraction, it becomes 203/20 meters

He used 6 9/10 meters of string for kites. Changing 6 9/10 to improper fraction, it becomes 69/10 meters

To determine how many meters of string that Montel have left, we will subtract the length of string used from the total length of string. It becomes

203/20 - 69/10 = 203 - 138/20 = 65/20 meters of string left

Answer:

40 hours will it take for the pool to fill.

Step-by-step explanation:

A pump can fill a swimming pool in 8 hours.

Work done by pump to fill in 1 hour is  [tex]\frac{1}{8}[/tex]

The pool also has a drain that can empty the pool in 10 hours.

Work done by pump to drain in 1 hour is  [tex]\frac{1}{10}[/tex]

If someone turns on the pump to fill the pool, but forgets to shut the drain.

Work done by both pipe in 1 hour is

[tex]W=\frac{1}{8}-\frac{1}{10}[/tex]

[tex]W=\frac{10-8}{80}[/tex]

[tex]W=\frac{2}{80}[/tex]

[tex]W=\frac{1}{40}[/tex]

Both pipe filled [tex]\frac{1}{40}[/tex] part of pool in hours = 1

Both pipe filled complete pool in hours = [tex]\frac{1}{\frac{1}{40}}=40[/tex]

Therefore, 40 hours will it take for the pool to fill.

Answer:

  24

Step-by-step explanation:

Counting Li Ana, there are 6 friends distributing fliers, so each one is distributing an amount calculated as ...

  (144 fliers)/(6 persons) = 24 fliers/person

Each person will distribute 24 fliers.

Answer:

Self selection sampling.

Step-by-step explanation:

A poll that does not attempt to generate a random sample, but instead invites people to volunteer to participate is called - self selection sampling.

Self-selection sampling is a sampling method where researchers allow the people or individuals, to choose to take part in research on their own accord.

Answer:

p*(3/10)+14

Step-by-step explanation:

There are 10 girls for every 3 boys. Therefore, if we have the number of girls, p, the number of boys is equal to p*(3/10). Then, adding the 14 boys that left, our answer is p*(3/10)+14

Answer:

The linear inequalities representing the given situation is -

x≥0 , y≥0

13x + 9y ≥ 368

x + y ≤ 40

Step-by-step explanation:

Let Rolando work 'x' hours a week at starbucks and 'y' hours a week at the youth soccer league.

∵ No. of hours cannot be negative -

x≥0 ,  y≥0

Given that Rolando earns $13 an hour at starucks -

Total earning at starbucks per week = 13x

Also Rolando earns $9 an hour at the youth soccer league -

Total earning at the youth soccer league per week = 9y

Total earnings for Rolando per week = 13x+9y

Since his total earnings needs to be atleast $368 -

13x+9y ≥ 368

Total number of hours for which Rolando can work per week is -

x+y

Since he cannot work for more than 40 hours a week -

x+y ≤ 40.

∴ x≥0 , y≥0

  13x + 9y ≥ 368

  x + y ≤ 40

Final answer:

Rolando's work situation can be modeled by the linear inequalities 13x + 9y ≥ 368 and x + y ≤ 40, where x is the number of hours at Starbucks and y is the number at the youth soccer league.

Explanation:

To represent Rolando’s work situation with linear inequalities, let us define two variables: let x be the number of hours Rolando works at Starbucks, and let y be the number of hours he works at the youth soccer league. Given that Rolando earns $13 per hour at Starbucks and $9 per hour at the youth soccer league, the inequality to ensure he makes at least $368 per week is 13x + 9y ≥ 368.

Furthermore, because Rolando can work a maximum of 40 hours per week, we have another inequality: x + y ≤ 40. Thus, the system of linear inequalities that models Rolando’s situation is:

13x + 9y ≥ 368

x + y ≤ 40

These inequalities define the range of hours Rolando can work at both jobs to meet his weekly earnings goal without exceeding the maximum number of allowed hours.

Answer:

Answer is reflection across line. When transformation maps X"Y"Z" .Then first transformation for this composition is , and the second transformation is 90° rotation about point X'.

Step-by-step explanation:

When you transform a point across a line y=x then x-coordinate and y-coordinate changes.

Similarly when you reflect a point across the line y=-x then signs are changed and both coordinate change places.

It is the first transformation across the line.

Answer:

d, a reflection across line m

Step-by-step explanation:

Just did it on edge 2020

(its right)

Answer:

1/4

Step-by-step explanation:

Take my word for it.

(Yes I know this answer is late)

To solve the equation 3(‒4x + 5) = 12, apply the distributive property, then use subtraction to isolate the variable, and lastly, divide to find the value of x, which is 0.25.

To solve the linear equation 3(−4x + 5) = 12, we start by applying the distributive property to eliminate the parentheses:

Next, we use the subtraction property of equality to isolate the variable:

Finally, we apply the division property of equality to solve for x:

Answer:

Step-by-step explanation:

number 2 is false

Answer:The length of the missing side can be calculated by the following formula;

Step-by-step explanation:

Answer:

The price per t-shirts is $8.5.

Step-by-step explanation:

Let us assume the price per t- shirt = $m

The cost of 75 t-shirts  =   $637.50

Now, the total cost for 75 t-shirts = 75 x ( cost of 1 t-shirt)

⇒  $637.50 = 75 x ( m)

 ⇒ m =  $637.50 / 75

   or, m = 8.5

Hence, the price per t-shirts is $8.5.

Answer:

Step-by-step explanation:

1) Twelve more than the product of 5 and a number x. This is expressed as

y = 5x + 12

when x = 2,

y = 5×2 +12 = 22

and when y = 10

10 = 5x + 12

5x = - 2

x = -2/5

2) 15 decreased by the product of a number x and 4. This is expressed as

y = 15 - 4x

When x = 2

y = 15 - 4×2 = 7

When y = 10

10 = 15 - 4x

4x = 15 - 10 = 5

x = 5/4

3) You eat 5 slices of bread. Your friend eats 2 slices fewer than you eat.

Let y represent the number of slices that your friend ate. The expression will be

y = 5 - 3 = 2

y = 2

4) Your uncle is 2 years older than 3 times your age. A. You are x years old. Let your uncle be y years old. Therefore,

y = 3x + 2

If you are 12 years old, your uncle well be

3×12 + 2 = 38 years old

Answer:

A) 24

Step-by-step explanation:

The given geometric series is {12, 6, 3, 3 /2 , 3 /4 , 3/ 8 ,...}

Each term can be represented as a product of its previous term and \[\frac{1}{2}\]

The generic term of the series can be represented as the product of the first term 12 and \[\frac{1}{2}^{n-1}\] where n is the index of the term in the series.

The sum to infinity of such a series is given by the following formula:

\[\frac{term1}{1-ratio}\]

Substituting and calculating:

\[\frac{12}{1-\frac{1}{2}}\]

=\[\frac{12}{\frac{1}{2}}\]

=12*2 = 24

Answer:

5

Step-by-step explanation:

To answer this, we must figure out the minimum amount we can add 32 to itself to reach 150. This means that we must multiply 32 by our desired amount, x, to reach the amount of hours closest to 150 (but above/equal to it). To find x, we can divide 150 by 32, and if it gives us a number with no decimals or remainders, that is x, as it is the smallest amount of people to get above/equal to 150. If it gives us a remainder/decimal, we must round up, as we cannot have a part of a person and we must get the work done, so we must get at least 150 hours. 150/32 is 4.6 something, so we must round up. 5 is our answer

The number of people required to do the work of 150 hours in a week is 5.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the work in an office takes 150 hours to complete every week. Each person in the office works 32 hours a week.

The number of people will be calculated as,

N = ( 150 / 32 )

N = 4.7 or 5 people

Therefore, the number of people required to do the work of 150 hours in a week is 5.

To know more about an expression follow

https://brainly.com/question/17134322

#SPJ5

Answer:

  see below

Step-by-step explanation:

We presume the damping constant is the opposite of the multiplier of time in the exponential term. Then the equations are ...

(a)  y = a·e^(-ct)·sin(ωt)

(b)  y = a·e^(-ct)·cos(ωt)

__

These are the standard equations for simple harmonic motion assuming there is no driving function.

  a = initial amplitude*

  c = damping constant**

  ω = frequency of oscillation in radians per second

  t = time in seconds

_____

* Of course, when y(0) = 0, the motion never actually reaches this amplitude because it is subject to decay before it can.

__

** In electrical engineering, damping is often specified in terms of a time constant, the time it takes for amplitude to decay to 1/e (≈36.8%) of the original amplitude. If that time is represented by τ, then the exponential factor is e^(-t/τ).

In a damped harmonic motion, the equation for displacement y at time t would be y = ae-ct/2mcos(ωt+φ). At t = 0, if y = 0, the phase constant φ is π/2; if y = a, then φ = 0.

In the context of damped harmonic motion, the displacement y at time t of an object with damping constant c, period 2π/ω, and initial amplitude a generally would follow the equation: y = ae-ct/2mcos(ωt+φ) , where m is the mass of the object and φ is the phase constant.

(a) If y = 0 at time t = 0, then the phase constant φ is π/2.

(b) If y= a at time t= 0, then φ=0 because all the displacement is in the stretch or compression, not the oscillation.

https://brainly.com/question/34795407

#SPJ11

Answer:

(x + 6, y + 0), 180° rotation, reflection over the x‐axis

Step-by-step explanation:

The answer can be found out simply , a trapezoid has its horizontal sides usually parallel meanwhile the vertical sides are not parallel.

The horizontal parallel sides are on the x-axis.

Reflection over y- axis would leave the trapezoid in a vertical position such that the trapezoid ABCD won't be carried on the transformed trapezoid as shown in figure.

So option 1 and 2 are removed.

Now, a 90 degree rotation would leave the trapezoid in a vertical position again so its not suitable again.

In,The final option (x + 6, y + 0), 180° rotation, reflection over the x‐axis, x+6 would allow the parallel sides to increase in value hence the trapezoid would increase in size,

180 degree rotation would leave the trapezoid in an opposite position and reflection over x-axis would bring it below the Original trapezoid. Hence, transformed trapezoid A`B`C`D` would carry original trapezoid ABCD onto itself

So the scale used was

1 inch = 6 feet

Step-by-step explanation:

Given

Length of library on drawing = 13 inches

Actual length of library = 78 feet

In order to find the scale we divide the actual length by the length of library on drawing.

So,

Scale = [tex]Scale = \frac{Actual\ length}{Length\ on\ drawing}\\=\frac{78}{13}\\=6[/tex]

So the scale used was

1 inch = 6 feet

Keywords: Scale, Maps

Learn more about scaling at:

#Learnwithbrainly

Answer:

Step-by-step explanation:

Multiply the budget total by the percentage for each category:

__

Find "the rest" by subtracting the total of percentages from 100%, or by subtracting the total of budget categories from $2400. Here, everything (not including savings) adds to 90% × $2400 = $2160, so the amount to savings is 10%, or $240.

In a family's monthly budget of $2,400, $1,056 is spent on housing, $552 on food, $144 on clothing, $408 on transportation, and the remaining $240 is put into savings.

To calculate how much money is spent on each category for a family with a monthly budget of $2,400, we need to multiply each percentage by the total budget. For example:

  Transportation: 17% of $2,400 is $408.00.  

The rest of the money, which is 10% of the total budget (the remaining percentage when subtracting the other categories from 100%), is put into savings. Therefore, the amount put into savings is 10% of $2,400, which equals $240.00.

https://brainly.com/question/36705672

#SPJ3

Answer:

Step-by-step explanation:

Marcy wants to buy packages of hot dogs and hamburgers for her booth at the carnival and the total cost must not exceed her budget of $150

she found packages of hot dogs that cost $6 per one and packages of hamburger is that cost $20 per one .

Let x = number of packages of hot dogs that she can buy.

Let y = number of packages of hamburgers that she can buy.

The cost of x packages of hot dogs = 6×x = $6x

The cost of y packages of hamburgers = 20×y = $20y

The equation will be

Since 6x + 20y must not exceed 150

The equation will be

6x + 20y lesser than or equal to 150

Answer:

20/7 days (just less than 3 days)

Step-by-step explanation:

Recall that (1 job) = (rate)(time), so time = (1 job) / (rate).

Set up and solve the following equation:

  1 job

------------------------------- = time required for 2 pumps working together

1 job         1 job

---------- + -------------

4 days      10 days

This comes out to:

              1 job

------------------------------------------- = time required

10 job-days      4 job-days

------------------ + -----------------

   40 days           40 days

or:

    1 job

-------------------- = (40/14) days, or 20/7 days (just less than 3 days)

14 job·days

-----------------

   40 days  

The number of days required to complete the work together by the pump is 20 / 7 days.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that one pump can empty a pool in 4 days, whereas a second pump can empty the pool in 10 days.

The number of days will be calculated as,
1 / N  = ( 1 / 10 ) + ( 1 / 4 )

1 / N = (10 + 4 ) / 40

1 / N = 14 / 40

1 / N = 7 / 20

N = 20 / 7 days

Therefore, the number of days required to complete the work together by the pump is 20 / 7 days.

To know more about an expression follow

https://brainly.com/question/16818901

#SPJ5

Answer:

Converse: If the circumference of the two circle is same then they have same diameter.

Inverse: If two circle do not have same diameter then they do not have same circumference.

Contrapositive: If two circle do not have same circumference then they do not have same diameter.

Step-by-step explanation:

Let the statement's be,

p : Two circles have same diameter.

q : Two circles have same circumference.

The given statement is a conditional statement logically is given as

p → q

It means p implies q, that is if p then q

where p is called antecedent or hypothesis and

           q is called consequent or conclusion.

The converse of this statement logically is given as

q → p

It means q implies p, that is If q then p.

Converse: If the circumference of the two circle is same then they have same diameter.

The Inverse of this statement logically is given as

~p → ~q

It means negation p implies negation q, that is If not p then not q.

Inverse: If two circle do not have same diameter then they do not have same circumference.

The Contrapositive of this statement logically is given as

~q → ~p

It means negation q implies negation p, that is If not q then not p.

Contrapositive: If two circle do not have same circumference then they do not have same diameter.

Final answer:

The converse, inverse, and contrapositive statements of the original mathematical statement concerning circles and their diameters and circumferences are explained in relation to the formula C = πd.

Explanation:

The original statement given is: If two circles have the same diameter, then they have the same circumference.

The converse of this statement would be: If two circles have the same circumference, then they have the same diameter.

The inverse of the original statement is: If two circles do not have the same diameter, then they do not have the same circumference.

The contrapositive is: If two circles do not have the same circumference, then they do not have the same diameter.

In mathematics, particularly in geometry, a circle's circumference (C) is related to its diameter (d) by the formula C = πd, where π is a constant approximately equal to 3.14159. Hence, two circles with the same diameter will indeed have the same circumference.

The probability that the sum of two spins on this spinner will be five is 7/32, which can be calculated by determining individual probabilities for each combination that gives a sum of 5 and then adding these probabilities together.

The question is asking for the probability that the sum of two spins on a specific spinner will be equal to five. First, we need to look into the different ways we can achieve a sum of 5. These possible combinations include (1,4), (2,3), and (3,2), where the values in the parentheses represent the results of the first and second spins respectively.

Now we calculate the probability of each combination happening. The probability of getting a 1 on a spin is 3/8 because there are three sections numbered 1 out of eight total sections. The probability of getting a 2 on a spin is 1/8 because there's only one section numbered 2. Likewise, the probability of getting a 3 is 4/8 (simplified to 1/2) because there are four sections numbered 3.

The combined probability of each scenario would be (3/8*1/2) for (1,4), (1/8*1/2) for (2,3) and (1/2*1/8) for (3, 2). To find the total probability, we simply add these probabilities together. That gives us (3/16) + (1/32) + (1/32) = 7/32. Therefore, the probability that the sum of two spins will be five is 7/32.

https://brainly.com/question/33594301

#SPJ12

(a) The formula for linear approximation: Δg ≈ g × (Δr / r). (b) The acceleration will be negative. (c) The percentage change in acceleration due to gravity when moving from sea level to the top of Mount Elbert is approximately 0.06703%.

(a) To find the linear approximation to the change in acceleration due to gravity, Δg, when increasing the distance from the centre of the Earth by Δr = x, use the formula for linear approximation:

Δg ≈ g × (Δr / r)

Here, g is the original acceleration due to gravity, Δr is the change in distance (x), and r is the original distance from the centre of the Earth.

(b) The change in acceleration, Δg, will be negative. This is because as you move farther away from the centre of the Earth, the gravitational acceleration decreases, which means Δg is a negative value.

(c) Given the height of Mount Elbert, Δr = 4.29 km, and assuming the radius of the Earth is r = 6400 km, we can use the linear approximation formula from part (a) to find the percentage change in acceleration due to gravity:

Δg ≈ g × (Δr / r)

Δg ≈ g × (4.29 / 6400)

Take the absolute value to find the percentage change:

Percentage Change = |(Δg / g)| × 100

Percentage Change = (Δg / g) × 100

Percentage Change = [(g × 4.29 / 6400) / g] × 100

Percentage Change = (4.29 / 6400) × 100

Calculate the percentage change:

Percentage Change ≈ (0.0006703125) × 100

Percentage Change ≈ 0.06703%

So,  the formula for linear approximation: Δg ≈ g × (Δr / r), the acceleration will be negative, and the percentage change in acceleration due to gravity when moving from sea level to the top of Mount Elbert is approximately 0.06703%.

Learn more about Acceleration here:

https://brainly.com/question/12550364

#SPJ12

The gravitational acceleration decreases as you move away from the Earth, with the rate of change proportional to the amount you increase your distance away from the Earth. The decrease is negative and is very minuscule even when moving from sea level to the top of Mount Elbert in Colorado.

The acceleration due to gravity, g, is inversely proportional to the square of the distance from the center of the Earth, r. Therefore, as we move away from the Earth, the value of g decreases.

For part (a), the change in acceleration due to an increase in distance (Δg) would be the derivative of g concerning r, times Δr=x. Therefore, Δg is approximately equal to -GM/r3x, where G is the gravitational constant and M is the mass of the Earth. When divided by g (which is GM/r2), this gives Δg/g = -x/r.

For part (b), this means that as you increase your distance away from the Earth, the gravitational acceleration decreases, therefore Δg is negative.

For part (c), the percentage change in g is given by the change in distance (4.29 km in this case), divided by the original radius of the Earth (6400 km), times 100%. Therefore, the percentage change is approximately 0.067% or an extremely small change.

https://brainly.com/question/31321801

#SPJ11

Answer:

0 < t < [tex]\frac{5}{3}[/tex]

After 1.67 days the stocks would be sold out.

Step-by-step explanation:

The price of a certain computer stock after t days is modeled by

p(t) = 100 + 20t - 6t²

Now we will take the derivative of the given function and equate it to zero to find the critical points,

p'(t) = 20 - 12t = 0

t = [tex]\frac{20}{12}[/tex]

t = [tex]\frac{5}{3}[/tex] days

Therefore, there are two intervals in which the given function is defined

(0, [tex]\frac{5}{3}[/tex]) and ([tex]\frac{5}{3}[/tex], ∞)

For the interval (0, [tex]\frac{5}{3}[/tex]),

p'(1) = 20 - 12(1) = 20

For the interval ([tex]\frac{5}{3}[/tex], ∞),

p'(2) = 20 - 12(2) = -4

Positive value of p'(t) in the interval (0, [tex]\frac{5}{3}[/tex]) indicates that the function is increasing.

0 < t < [tex]\frac{5}{3}[/tex]

Since at the point t = 1.67 days curve is showing the maximum, so the stocks should be sold after 1.67 days.

The price of the computer stock rises for the first 5/3 days after it is issued. Ideally, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

The price of the stock becomes maximum when the rate of change of the price is 0. To find when that is, we need to take the derivative of the price function, which is p'(t) = 20 - 12t, and set it equal to 0. Solving for t, we get t = 20/12 = 5/3. So, the price of the stock rises for t in (0, 5/3) days. Therefore, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

https://brainly.com/question/31545805

#SPJ11

Answer: 5:42:2 am

Step-by-step explanation:

He bought a 16 1/2 pound turkey. Converting to improper fraction, it becomes 33/2 pounds of turkey.

The cookbook says to cook the turkey for 20 minutes per pound and 25 minutes per pound if the turkey is stuffed. Since he wants to stuff the Turkey, he would have to cook for 25 minutes per pound. The total amount of time that he would have to cook the stuffed turkey will be 25 × 33/2 = 412.5 minutes

Let us converted to hours

If 60 minutes = 1 hour,

412.5 minutes = 412.5/60 = 6.875

He would spend 6.875 = 55/8 hours

in cooking the stuffed turkey.

The cookbook also says to let the turkey cool for 10 minutes after coming out the oven. Let us convert the 10 minutes to hours

60 minutes = 1 hour

10 minutes = 10/60 = 1/6

Total time it will take to get the Turkey ready for dinner will be

1/6 + 55/8 = 169/24 = 7.042 hours which means 7 hours, 2 minutes and 58 seconds

If he has to have dinner ready for 12:45 pm, he needs to put the turkey in the oven at 5:42:2 am at most

Answer: slope=-1, slope is -1/6, line a has a greater slope

Step-by-step explanation:

slope of line a is "m" in y=mx+b form

so slope=-1

slope for line b

first find two point, slope is change in y divided by change in x

1 down for every 6 units forward, so slope is -1/6

line a has a greater slope

Answer: (0.076, 0.140)

Step-by-step explanation:

Confidence interval for population proportion (p) is given by :-

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

, where [tex]\hat{p}[/tex] = sample proportion.

n= sample size.

[tex]\alpha[/tex] = significance level .

[tex]z_{\alpha/2}[/tex] = critical z-value (Two tailed)

As per given , we have

sample size : n= 500

The number of Independents.:  x= 54

Sample proportion of Independents[tex]\hat{p}=\dfrac{x}{n}=\dfrac{54}{500}=0.108[/tex]

Significance level 98% confidence level : [tex]\alpha=1-0.98=0.02[/tex]

By using z-table , Critical value : [tex]z_{\alpha/2}=z_{0.01}=2.33[/tex]

The 98% confidence interval for the true percentage of Independents among Haywards 50,000 registered voters will be :-

[tex]0.108\pm (2.33)\sqrt{\dfrac{0.108(1-0.108)}{500}}\\\\=0.108\pm2.33\times0.013880634\\\\=0.108\pm0.03234187722\\\\\approx0.108\pm0.032=(0.108-0.032,\ 0.108+0.032)=(0.076,\ 0.140)[/tex]

Hence, the 98% confidence interval for the true percentage of Independents among Haywards 50,000 registered voters.= (0.076, 0.140)

Final answer:

To calculate a 98% confidence interval for the percentage of Independents among registered voters in Hayward, we use the sample proportion and z-score to find that the true percentage is likely between 8% and 13.6%.

Explanation:

To calculate a 98% confidence interval for the true percentage of Independents among Hayward's 50,000 registered voters, based on a sample of 54 Independents out of 500 voters, we use the formula for the confidence interval for a proportion, which is p ± z*[tex]\sqrt{ ((p(1-p))/n)}[/tex], where p is the sample proportion, z is the z-score corresponding to the confidence level, and n is the sample size.

First, we find the sample proportion (p) of independents: p=54/500 = 0.108 or 10.8%.

Next, we look up the z-score for a 98% confidence level, which is approximately 2.33.

We then plug the values into the formula: 0.108 ± 2.33*[tex]\sqrt{((0.108(1-0.108))}[/tex]/500), and calculate the confidence interval.

Calculating the margin of error: 2.33*[tex]\sqrt{((0.108(1-0.108))}[/tex])/500) ≈ 0.028 or 2.8%.

Therefore, the 98% confidence interval is 10.8% ± 2.8%, which means it ranges from 8% to 13.6%.

The true percentage of Independents among Hayward's registered voters is likely between 8% and 13.6% with 98% confidence.