Use Laplace transforms to solve the initial value problem. x" + 9x = 1;x (0) = 0 = x' (0)

Answer:

[tex](a-2b)^{5}=-32b^{5}+80ab^{4}-80a^{2}b^{3}+40a^{3}b^{2}-10a^{4}b+a^{5}[/tex]

Step-by-step explanation:

The binomial expansion is given by:

[tex](x+y)^{n}=_{0}^{n}\textrm{C}x^{^{0}}y^{n}+_{1}^{n-1}\textrm{C}x^{1}y^{n-1}+...+_{n}^{n}\textrm{C}x^{n}y^{0}[/tex]

In our case we have

[tex]x=a\\y=-2b\\n=5[/tex]

Thus using the given terms in the binomial expansion we get

[tex](a-2b)^{5}=_{0}^{5}\textrm{C}a^{0}(-2b)^{5}+_{1}^{5}\textrm{C}a^{^{1}}(-2b)^{4}+{_{2}^{5}\textrm{C}}a^{2}(-2b)^{3}+_{3}^{5}\textrm{C}a^{3}(-2b)^{2}+_{4}^{5}\textrm{C}a^{4}(-2b)^{1}+_{5}^{5}\textrm{C}a^{5}(-2b)^{0}[/tex]

Upon solving we get

[tex](a-2b)^{5}=-32b^{5}+5\times a\times16b^{4}+10\times a^{2} \times (-8b^{3})+10\times a^{3}\times 4b^{2}+5\times a^{4}\times (-2b)+a^{5}\\\\(a-2b)^{5}=-32b^{5}+80ab^{4}-80a^{2}b^{3}+40a^{3}b^{2}-10a^{4}b+a^{5}[/tex]

Answer:

7257600

Step-by-step explanation:

Number of freshmen in the student council= 4

Number of sophomores in the student council= 5

Number of juniors in the student council= 6

Number of seniors in the student council= 7

Ways of choosing council members

⁴C₂×⁵C₂×⁶C₂×⁷C₂

[tex]^4C_2=\frac{4!}{(4-2)!2!}\\=\frac{24}{4}=6\\\\^5C_2=\frac{5!}{(5-2)!2!}\\=\frac{120}{12}=10\\\\^6C_2=\frac{6!}{(6-2)!2!}\\=\frac{720}{48}=15\\\\^7C_2=\frac{7!}{(7-2)!2!}\\=\frac{5040}{240}=21[/tex]

⁴C₂×⁵C₂×⁶C₂×⁷C₂=6×10×15×21=18900

Ways of lining up the four classes=4!=1×2×3×4=24

Ways of lining up members of each class=2⁴=2×2×2×2=16

Pictures are possible if each group of classmates stands​ together

⁴C₂×⁵C₂×⁶C₂×⁷C₂×4!×2⁴

=18900×24×16

=7257600

There are 453600 possible different pictures

The given parameters are:

Freshmen = 4

Sophomores = 5

Juniors = 6

Seniors = 7

Two council members are to be selected from each group.

So, the number of ways this can be done is:

n = ⁴C₂×⁵C₂×⁶C₂×⁷C₂

Apply combination formula, and evaluate the product

n = 18900

Each group are to stand together.

There are 4! ways to arrange the 4 groups.

So, the total number of pictures is:

Total = 4! * 18900

Evaluate the product

Total = 453600

Hence, there are 453600 possible different pictures

Read more about combination at:

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Answer: 0.1061

Step-by-step explanation:

Given : An insurance company found that 9% of drivers were involved in a car accident last year.

Thus, the probability of drivers involved in car accident last year = 0.09

The formula of binomial distribution :-

[tex]P(X=x)^nC_xp^x(1-p)^{n-x}[/tex]

If seven (n=7) drivers are randomly selected then , the probability that exactly two (x=2) of them were involved in a car accident last year is given by :-

[tex]P(X=2)=^7C_2(0.09)^2(1-0.09)^{7-2}\\\\=\dfrac{7!}{2!5!}(0.09)^2(0.91)^{5}=0.106147867882\approx0.1061[/tex]

Hence, the required probability :-0.1061

Answer:

The rent should be $ 390 or $ 410.

Step-by-step explanation:

Given,

The original monthly rent of an apartment = $350,

Also, the original number of apartment that could be filled = 90,

Let the rent is increased by x times of $ 10,

That is, the new monthly rent of an apartment =( 350 + 10x ) dollars

Since, for each $10 per month increase there will be two vacancies with no possibility of filling them.

Thus, the new number of filled apartments = 90 - 2x,

Hence, the total revenue of the firm = ( 90 - 2x )(350 + 10x ) dollars,

According to the question,

( 90 - 2x )(350 + 10x ) = 31,980

[tex]90(350)+90(10x)-2x(350)-2x(10x)=31980[/tex]

[tex]31500+900x-700x-20x^2=31980[/tex]

[tex]-20x^2+200x+31500-31980=0[/tex]

[tex]20x^2-200x+480=0[/tex]

By the quadratic formula,

[tex]x=\frac{200\pm \sqrt{(-200)^2-4\times 20\times 480}}{40}[/tex]

[tex]x=\frac{200\pm \sqrt{1600}}{40}[/tex]

[tex]x=\frac{200\pm 40}{40}[/tex]

[tex]\implies x=\frac{200+40}{40}\text{ or }x=\frac{200-40}{40}[/tex]

[tex]\implies x=6\text{ or } x =4[/tex]

Hence, the new rent of each apartment, if x = 6, is $ 410,

While, if x = 4, is $ 390

Answer:

[tex]a=2 \quad \text{and} \quad b=-2[/tex]

Step-by-step explanation:

Take [tex]a=2 \quad \text{and} \quad b=-2[/tex], note that

[tex]2=(-1)\cdot(-2)[/tex]

hence b divides a. On the other hand, we have that

[tex]-2=(-1)\cdot2[/tex]

which tells us that a divides b. Moreover, [tex]a=2 \neq -2=b[/tex].

The number of  Degrees of Freedom is the answer.

The term 'degrees of freedom' refers to the number of sample values that can vary after specific restrictions have been placed on all data values. It is a critical concept in statistics, playing a role in areas such as hypothesis testing and confidence intervals.

The blank should be filled with 'degrees of freedom'. The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.

For instance, in a set of sample data with a fixed mean, if you know the values of all but one data point, you can calculate the value of the remaining one due to the restriction of the fixed mean. Therefore, in this case, the degrees of freedom would be n-1 (with 'n' representing the total number of sample data points).

The concept of degrees of freedom is an important aspect in various areas of statistics, including hypothesis testing and estimating confidence intervals.

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Answer:  (16%, 24%)

Step-by-step explanation:

The confidence interval for proportion is given by :-

[tex]p\pm E[/tex]

Given : Significance level : [tex]\alpha=1-0.95=0.05[/tex]

The proportion of the respondents answered "crime and violence." : p=0.20

Margin of sampling error : [tex]E=\pm0.04[/tex]

Now, the 95% confidence interval for the proportion of the population that would respond "crime and violence" to the question asked by the pollsters is given by :-

[tex]0.20\pm 0.04\\\\=0.20-0.04,\ 0.20+0.04\approx(0.16,0.24)[/tex]

In percentage, [tex](16\%,24\%)[/tex]

Hence, the 95% confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters =(16%, 24%)

The 95% confidence interval for the population that would answer "crime and violence" to the poll question, given a sample proportion of 20% and a margin of error of 4%, is between 16% and 24%.

This question is about constructing a confidence interval based on poll data, a mathematical and statistical concept. The poll indicates that 20% of respondents believe the most pressing issue in the country today is "crime and violence", with a margin of error of ±4%. A 95% confidence interval for the population proportion can be constructed by adding and subtracting the margin of error from the sample proportion.

So in this case, we can calculate the low and high end of the interval as follows:

So, we can be 95% confident that the true population proportion that would respond "crime and violence" to the question lies somewhere between 16% and 24% based on this poll.

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Answer with Step-by-step explanation:

We are given that tr(FG)=tr(GF) for any two matrix of order [tex]n\times n[/tex]

We have to show that if A and B are similar then

tr upper A=tr upper B

Trace of a square matrix A is the sum of diagonal entries in A and denoted by tr A

We are given that A and B are similar matrix  then there exist a inverse matrix P such that

Then [tex]B=P^{-1}AP[/tex]

Let [tex] G=P^{-1} [/tex] and F=AP

Then[tex] FG= APP^{-1}[/tex]=A

GF=[tex]P^{-1}AP=B[/tex]

We are given that tr(FG)=tr(GF)

Therefore, tr upper A=trB

Hence, proved

Answer:

12 cm²

Step-by-step explanation:

Length of rectangle = 5.6 cm

Width of rectangle = 2.1 cm

Area of rectangle = Length of rectangle×Width of rectangle

⇒Area of rectangle = 5.6×2.1

⇒Area of rectangle = 11.76 cm²

11.76 has 4 significant figures in order to write this term in 2 significant terms we round of the term

The last digit in the decimal place is 6. Now, 6≥5 so we round the next digit to 8 we get

11.8

Now the last digit in the decimal place is 8. Now, 8≥5 so we round the next digit to 2 we get

12

∴ Hence the area of the rectangle when rounded to 2 significant figures is 12 cm²

Answer:78

Step-by-step explanation:

For N=13 candidates

For pairwise comparison to determine the winner in an election we need to use combination

a pair of distinct candidates can be chosen in [tex]^NC_{2}=\frac{N\left ( N-1 \right )}{2}[/tex]

Therefore no of comparison to be made =[tex]^{13}C_{2}=frac{13\left ( 13-1 \right )}{2}=78[/tex]

Thus a total of 78 comparison is needed

To determine the number of comparisons needed using the pairwise comparison method when there are 13 candidates, we can use the formula (n-1) + (n-2) + ... + 1, where n represents the number of candidates. By substituting n = 13 into the formula, we find that a total of 78 comparisons must be made.

To determine the number of comparisons needed using the pairwise comparison method, we can use the formula:

(n-1) + (n-2) + ... + 1

where n represents the number of candidates. Substituting n = 13, we get:

(13-1) + (13-2) + ... + 1

Simplifying the equation, we find:

12 + 11 + ... + 1 = 78

Therefore, a total of 78 comparisons must be made when there are 13 candidates.

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Answer: There was population of  22808 in 2000.

Step-by-step explanation:

Since we have given that

Population was in 2002 = 25,570

According to question, the population in 2002 was 10% more than 2001.

So, the population in 2001 was

[tex]25570=P_1(1+\dfrac{r}{100})\\\\25570=P_1(1+\dfrac{10}{100})\\\\25570=P_1(1+0.1)\\\\25570=P_1(1.01)\\\\\dfrac{25570}{1.01}=P_1\\\\25316.8\approx 25317=P_1[/tex]

Now, we have given that

In 2001, the population in a town was 11% more than it was in 2000.

So, population in 2000 was

[tex]25317=P_0(1+\dfrac{r}{100})\\\\25317=P_0(1+\dfrac{11}{100})\\\\\25317=P_0(1+0.11)\\\\25317=P_0(1.11)\\\\P_0=\dfrac{25317}{1.11}\\\\P_0=22808[/tex]

Hence, there was population of  22808 in 2000.

Answer:  The required number of ways is 46200.

Step-by-step explanation:  Given that a catering service offers 8 ​appetizers, 11 main​ courses, and 7 desserts.

A banquet committee is to select 7 ​appetizers, 8 main​ courses, and 4 desserts.

We are to find the number of ways in which this can be done.

We know that

From n different things, we can choose r things at a time in [tex]^nC_r[/tex] ways.

So,

the number of ways in which 7 appetizers can be chosen from 8 appetizers is

[tex]n_1=^8C_7=\dfrac{8!}{7!(8-7)!}=\dfrac{8\times7!}{7!\times1}=8,[/tex]

the number of ways in which 8 main courses can be chosen from 11 main courses is

[tex]n_2=^{11}C_8=\dfrac{11!}{8!(11-8)!}=\dfrac{11\times10\times9\times8!}{8!\times3\times2\times1}=165[/tex]

and the number of ways in which 4 desserts can be chosen from 7 desserts is

[tex]n_3=^7C_4=\dfrac{7!}{4!(7-4)!}=\dfrac{7\times6\times5\times4!}{4!\times3\times2\times1}=35.[/tex]

Therefore, the number of ways in which the banquet committee is to select 7 ​appetizers, 8 main​ courses, and 4 desserts is given by

[tex]n=n_1\times n_2\times n_3=8\times165\times35=46200.[/tex]

Thus, the required number of ways is 46200.

Answer:

  The probability of success is 1/2.

Step-by-step explanation:

The histogram of a binomial distribution has a mode of n×p. For that to be n/2, the value of p must be 1/2.

Answer:   0.1342

Step-by-step explanation:

The cumulative distribution function for exponential distribution is :-

[tex]P(x)=1-e^{\frac{-x}{\lambda}}[/tex], where [tex]\lambda [/tex] is the mean of the distribution.

Given : [tex]\lambda =20[/tex]

Then , the probability that the battery will last between 10 and 15 hours is given by :-

[tex]P(10<x<15)=P(15)-P(10)\\\\=1-e^{\frac{-15}{20}}-(1-e^{\frac{-10}{20}})\\\\=-e^{-0.75}+e^{-0.5}=0.13416410697\approx0.1342[/tex]

Hence, the probability that the battery will last between 10 and 15 hours = 0.1342

The probability that the battery will last between 10 and 15 hours is 18.13%.

In order to find the probability that the battery will last between 10 and 15 hours, we need to use the exponential distribution. The exponential distribution is defined by the formula P(X

For this problem, the mean is 20, so λ = 1/20. Plugging in the values, we get P(10

Therefore, the probability that the battery will last between 10 and 15 hours is 18.13%.

Answer:

Because they have never had to express one quantity in terms of another. The idea of such a relationship is completely new, as is the vocabulary for expressing such relationships.

Step-by-step explanation:

"Function" is a simple concept that says you can relate two quantities, and you can express that relationship in a number of ways. (ordered pairs, table, graph)

The closest experience most students have with functions is purchasing things at a restaurant or store, where the amount paid is a function of the various quantities ordered and the tax. Most students have never added or checked a bill by hand, so the final price is "magic", determined solely by the electronic cash register. The relationship between item prices and final price is completely lost. Hence the one really great opportunity to consider functions is lost.

Students rarely play board games or counting games (Monopoly, jump rope, jacks, hide&seek) that would give familiarity with number relationships. They likely have little or no experience with the business of running a lemonade stand or making and selling items. Without these experiences, they are at a significant disadvantage when it comes to applying math to their world.

Answer:

n = 601

Step-by-step explanation:

Since we know nothing about the percentage of computers with new operating system, we assume than 50% of the computers have new operating system.

So, p = 50% = 0.5

q = 1 - p = 1 - 0.5 = 0.5

Margin of error = E = 4 percentage points = 0.04

Confidence Level = 95%

z value associated with this confidence level = z = 1.96

We need to find the minimum sample size i.e. n

The formula for margin of error for the population proportion is:

[tex]E=z\sqrt{\frac{pq}{n}}[/tex]

Re-arranging the equation for n, and using the values we get:

[tex]n=(\frac{z}{E} )^{2} \times pq\\\\ n=(\frac{1.96}{0.04})^{2} \times 0.5 \times 0.5\\\\ n = 601[/tex]

Thus, the minimum number of computers that must be surveyed is 601

Answer:

He drove approximately 478.78 km

Step-by-step explanation:

We know that,

Distance = Speed × time,

Given,

Time taken by john in driving = 3.5 hours,

His average speed = 85 mph,

So, the total distance he drove = 3.5 × 85 = 297.5 miles,

Since, 1 miles = 1.60934 km,

Thus, the total distance he drove = 1.60934 × 297.5 = 478.77865 km ≈ 478.78 km

Answer:

120 minutes

Step-by-step explanation:

Total plants are 16 and only 5 can be placed in one go. so total number of rounds for the plants will be: 16/5 = 3.2 rounds ≅ 4 rounds

As there are four rounds to go in 8 hours, so the time for 1 round will be: 8/4 = 2 hours.

Therefore, 2 hours or 120 minutes of sunlight are possible for one plant ..

Answer:

The answer is actually 150 minutes.

Step-by-step explanation:

Answer:

I'm going to write both of these because maybe you have a fill in the blank question. I don't know.

[tex](x-4)^2+(y-5)^2=9^2[/tex]

Simplify:

[tex](x-4)^2+(y-5)^2=81[/tex]

Step-by-step explanation:

[tex](x-h)^2+(y-k)^2=r^2[/tex] is the equation of a circle with center (h,k) and radius r.

You are given (h,k)=(4,5) because that is the center.

You are given r=9 because it says radius 9.

Let's plug this in.

[tex](x-4)^2+(y-5)^2=9^2[/tex]

Simplify:

[tex](x-4)^2+(y-5)^2=81[/tex]

Answer:

(x-4)^2 + (y-5)^2 = 9^2

or

(x-4)^2 + (y-5)^2 =81

Step-by-step explanation:

The equation for a circle is (x-h)^2 + (y-k)^2 = r^2

Where (h,k) is the center and r is the radius

(x-4)^2 + (y-5)^2 = 9^2

or

(x-4)^2 + (y-5)^2 =81

Answer:

The answer is 105.5 kcal/slice of bread.

Step-by-step explanation:

The kcal per gram of:

Now, you have:

If we multiply the amount of components by his kcal:

Now gram/gram = 1, and we can cancel the grams in the equation:

Finally, the result of the kcal of the slice bread is:

Answer:

$15

Step-by-step explanation:

He had $20 and he spent [tex]\frac{3}{4}[/tex] of that in baseball cards, that is

[tex]20*\frac{3}{4}= \frac{20*3}{4} = \frac{60}{4} = 15.[/tex]

So, he spent $15 in baseball cards.

Answer:

A. 6.5

Step-by-step explanation:

First we find the average  [tex]\bar{x}[/tex] of the 9 data:

[tex]\bar{x} =\frac{\sum_{x=1}^{n}x_{i}}{n}[/tex]

Where n is the data number, that in this case is 9.

[tex]\bar{x} =\frac{1+ 2+ 11+ 8+ 16+ 16+20+ 16+ 18}{9}=12\\[/tex]

The formula of the standard deviation [tex]\sigma[/tex] is:

[tex]\sigma=\sqrt{\frac{\sum_{x=1}^{n}(x_{i}-\bar{x})^{2}}{n}}[/tex]

We replace the data and find the value of the standard deviation:

[tex]\sigma=\sqrt{\frac{(1-12)^{2}+(2-12)^{2}+(11-12)^{2}+(8-12)^{2}+(16-12)^{2}+(16-12)^{2}+(20-12)^{2}+(16-12)^{2}+(18-12)^{2}}{9}}[/tex]

[tex]\sigma=\sqrt{\frac{(-11)^{2}+(-10)^{2}+(-1)^{2}+(-4)^{2}+(4)^{2}+(4)^{2}+(8)^{2}+(4)^{2}+(6)^{2}}{9}}=6,54[/tex]

We approximate the number and the solution is 6,5

Answer:

This is a stem leaf data, in which the stem generally stands for the "tens" place value while the leaf stands for the "ones" place value.

Expand the data, and find the median by finding the middle number:

10, 13, 16, 20, 21, 23, 27, 27, 28, 29, 30, 32, 33, 33, 33, 33, 38, 39, 46, 46, 46, 48

There are 22 numbers in all. To find the Median when there is a even amount of numbers, Find the two middle numbers, and find the mean of the two numbers:

(32 + 30)/2 = (62)/2 = 31

31 is your answer.

~

Answer:

251.28 cubic feet

Step-by-step explanation:

The height of the cylinder is 9.7 ft.

The base length is 7 feet. So, the radius(R) = [tex]\frac{7}{2} = 3.5 feet[/tex]

The length of 4 feet cylinder cut out. So, the radius of the cut cylinder (r) = [tex]\frac{4}{2} = 2 feet[/tex]

We have to find the volume of solid cylinder figure without cutting part.

= Volume of the whole cylinder - Volume of the hole cut

We know that volume of a cylinder is [tex]\pi *r^2*h[/tex]

Using this formula,

= [tex]\pi *R^2*h - \pi *r^2*h[/tex]

= [tex]\pi h [R^2 - r^2][/tex]

Here π = 3.14, R = 3.5, r = 2 and h = 9.7

Plug in these values in the above, we get

= [tex]3.14*9.7 [3.5^2 - 2^2]\\= 30.458[12.25 - 4]\\= 30.458[8.25]\\= 251.2785 ft^3[/tex]

When round of to the nearest hundredths place, we get

So, the volume of solid cylinder figure not including cut out= 251.28 cubic feet

Answer:

No down payment = $73 267; 10 % down payment = $81 408

Step-by-step explanation:

1. With no down payment

The formula for a maximum affordable loan (A) is

A = (P/i)[1 − (1 + i)^-N]

where  

P = the amount of each equal payment

i = the interest rate per period

N = the total number of payments

Data:

     P = 500

APR = 2.83 % = 0.0283

      t = 15 yr

Calculations:

You are making monthly payments, so

i = 0.0283/12 = 0.002 358 333

The term of the loan is 15 yr, so

N = 15 × 12 = 180

A = (500/0.002 3583)[1 − (1 + 0.002 3583)^-180]

= 212 014(1 - 1.002 3583^-180)

= 212 014(1 - 0.654 424)

= 212 014 × 0.345 576

= 73 267

You can afford to spend $73 267 on a home.

2. With a 10 % down payment

Without down payment, loan = 73 267

With 10 % down payment, you pay 0.90 × new loan

           0.90 × new loan = 73 267

New loan = 73267/0.90 = 81 408

With a 10 % down payment, you can afford to borrow $81 408 .

Here’s how it works:

Purchase price =  $81 408

Less 10 % down =    -8 141

                 Loan = $73 267

And that's just what you can afford.

[tex]A\cap B=\{x:x\in A \wedge x\in B\}[/tex]

[tex]\large\boxed{A\cap B=\{a,c\}}[/tex]

Answer:

Given an unguided connected graph, an extension tree of this graph is a subgraph which is a tree that connects all vertices. A single graph may have different extension trees. We can mark a weight at each edge, which is a number that represents how unfavorable it is, and assign a weight to the extension tree calculated by the sum of the weights of the edges that compose it. A minimum spanning tree is then an extension tree with a weight less than or equal to each of the other possible spanning trees. Generalizing more, any non-directional graph (not necessarily connected) has a minimal forest of trees, which is a union of minimal extension trees of each of its related components.

Answer:

  C.  (-4,10,7)

Step-by-step explanation:

Use the first equation to substitute for y in the last equation:

  x = (x +2z) -14

  14 = 2z . . . . . . add 14-x

  7 = z . . . . . . . . divide by 2

Now, find x:

  7 = -1 -2x . . . . substitute for z in the second equation

  8/-2 = x = -4 . . . . . add 1, divide by -2

Finally, find y:

  y = -4 +2(7) = 10 . . . . . substitute for x and z in the first equation

The solution is (x, y, z) = (-4, 10, 7).

Answer:

C

Step-by-step explanation:

EDGE

Answer:

  49 m/s

Step-by-step explanation:

We don't know what your model is, so we'll solve this based on the balance of forces. Air resistance exerts an upward force of ...

  (4 N/(m/s))v

Gravity exerts a downward force of ...

  (20 kg)(9.8 m/s²) = 196 N

These are balanced (no net acceleration) when ...

  (4 N/(m/s))v = 196 N

  v = (196 N)/(4 N/(m/s)) = 49 m/s

The terminal velocity is expected to be 49 m/s.

Answer:

Step-by-step explanation:

Given that n =30, x bar = 375 and sigma = 81

Normal distribution is assumed and population std dev is known

Hence z critical values can be used.

For 95% Z critical=1.96

Margin of error = [tex]1.96(\frac{81}{\sqrt{30} } )=29[/tex]

Confidence interval = 375±29

=(346,404)

B) 99% confidence

Margin of error = 2.59*Std error =38

Confidence interval = 375±38

=(337, 413)

C) For 90%

Margin of error = 20

Std error = 20/1.645 = 12.158

Sample size

[tex]n=(\frac{81}{12.158} )^2\\=44.38[/tex]

Atleast 44 people should be sample size.

To determine the confidence intervals for the average kilowatt-hours, a formula is used that includes the sample mean, Z-values, population standard deviation, and sample size. For a 95% confidence level, the interval is 324.95 to 425.05 kWh, and for a 99% confidence level, the interval is 311.01 to 438.99 kWh. To have a confidence level of 90% with a maximum error amount of 20kWh, the minimum sample size required is approximately 35 households.

A. Determine the interval of 95% confidence for the average kilowatt-hours for the population:

To determine the interval of 95% confidence, we can use the formula:

95% confidence interval = sample mean ± (Z-value) * (population standard deviation / √sample size)

Substituting the given values, we have:

95% confidence interval = 375 ± (1.96) * (81 / √30) = 324.95 to 425.05 kWh

B. Determine the 99% confidence interval:

Using the same formula, but with a Z-value of 2.57 (corresponding to 99% confidence), we have:

99% confidence interval = 375 ± (2.57) * (81 / √30) = 311.01 to 438.99 kWh

C. Minimum sample size for a confidence level of 90% and an error amount of 20kWh:

To determine the minimum sample size, we can rearrange the formula for the confidence interval and solve for the sample size:

Sample size = ((Z-value) * (population standard deviation / error amount))^2

Substituting the given values, we have:

Sample size = ((1.645) * (81 / 20))^2 = 34.64 or approximately 35 households

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