Assessment: To help assess student learning in her developmental math courses, a mathematics professor at a community college implemented pre- and posttests for her students. A knowledge-gained score was obtained by taking the difference of the two test scores.

(a) What type of experimental design is this?

Answer:

the cost function is Cost=7000 m*$ /R + 50.265 $/m² * R²

Step-by-step explanation:

then the cost function is

Cost= cost of side area+ cost of top + cost of bottom = 2*π*R*L * 5$/m² +

π*R² * 8$/m² +  π*R² * 8$/m²

since the volume V is

V=π*R²*L → V/(π*R²)=L

then

Cost=2*π*R*V/(π*R²) * 5$/m² +  π*R² * 8$/m² +  π*R² * 8$/m²

replacing values

Cost=2*700 m³ /R * 5$/m² +  π*R² * 16$/m² = 7000 m*$ /R + 50.265 $/m² * R²

thus the cost function is

Cost=7000 m*$ /R + 50.265 $/m² * R²

Answer: 50.24r² + 7000/r

Step-by-step explanation:

The formula for determining the volume of a cylinder is expressed as

Volume = πr²h

Since the volume of the can is 700cm³, then

πr²h = 700

h = 700/πr²

The formula for determining the total surface area of a cylinder is expressed as

Total surface area = 2πr² + 2πrh

The surface area of the top and bottom of the can is 2πr².

Since top and bottom are made up of a material that costs $8 per square centimeter, then the cost is

2πr² × 8 = 16πr²

Since π = 3.14, the surface area of the top and bottom of the cylindrical can is

16 × 3.14 × r² = 50.24r²

The surface area of the side of the can is

2πrh = 2πr × 700/πr²

= 1400/r

Since the the sides are made of material that costs $5 per square centimeter, then the cost is

1400/r × 5 = 7000/r

The total cost of the material as a function of the radius, r of the cylinder is

50.24r² + 7000/r

Answer:

The probability that a child with a rash does not have a fever is 22%

Step-by-step explanation:

1. Probability of having fever:

[tex]P(fever)=0.70[/tex]

2. Probability of not having fever:

[tex]P(not fever)=1-P(fever)\\P(not fever)=1-0.70\\P(not fever)=0.30[/tex]

3. Probability of fave fevers and a rash:

[tex]P(fever and rash)=(0.70)(0.30)=0.21[/tex]

4. Probability of having a rash but not a fever:

[tex]P(rash and not fever)=(0.30)(0.20)=0.60[/tex]

5. Probability of having a rash:

[tex]P(rash)=P(rash and fever)+P(rash and no fever)\\P(rash)=0.21+0.06=0.27[/tex]

6. Probability a child with a rash does not have a fever

[tex]P=\frac{P(rash and not fever)}{P(rash)} =\frac{0.06}{0.27} =0.22[/tex]

22% of the child at the doctor's office with a rash does not have a fever.

Final answer:

To find the probability that a child with a rash does not have a fever, we analyze the given percentages, calculate how many children have a rash with and without fever, and then find the ratio of those without a fever to the total number with a rash, resulting in a probability of approximately 22.22%.

Explanation:

The question asks: What is the probability that a child at the doctor's office with a rash does not have a fever? To solve this, let's start by analyzing the given percentages.

70% of children who go to the doctor have fevers.

Of those with fevers, 30% also have a rash.

Of those without fevers, 20% have a rash.

To find the probability that a child with a rash does not have a fever, we need to calculate the proportion of children with a rash who are fever-free compared to all children with a rash.

Step-by-step Calculation:

Assuming 100 children visit the doctor: 70 will have fevers, and 30 will not.

Of the 70 with fevers, 21 (30% of 70) have a rash.

Of the 30 without fevers, 6 (20% of 30) have a rash.

In total, 27 children have a rash (21 with fever + 6 without).

The probability a child with a rash does not have a fever is the number of children with a rash but no fever divided by the total number of children with a rash: 6/27.

This calculation shows that the probability of a child having a rash but no fever is 6/27 or approximately 22.22%.

To pay off your credit card debt of $3,500 at 13% interest rate in 4 years, you will have to calculate the monthly payments by using a certain formula. After defining the parameters (monthly payment, monthly interest rate, present value or current loan amount, and time), all you have to do is put the values in it and calculate the monthly payment.

This question refers to a financial problem in the subject of Mathematics, particularly pertaining to the concept of

simple interest

. To find out how much you need to pay each month to clear out your credit card debt, you would need to understand how the 13% interest is applied to your debt of $3,500. The formula for finding monthly payments is

P = [r*PV(1 + r)^t]/[(1 + r)^t – 1]

, where:

After defining the parameters, all you have to do is put the values in the formula and calculate the monthly payment.

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To calculate the monthly payments on a credit card debt of $3,500 with a 13% interest rate over 4 years, you can use the formula for calculating the monthly payment on a fixed-rate loan. The monthly payment comes out to be approximately $94.78.

To calculate the monthly payments on a credit card debt, you can use the formula for calculating the monthly payment on a fixed-rate loan. The formula is:

Monthly Payment = (Principal * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Months))

In this case, the principal is $3,500, the monthly interest rate is 13% divided by 12 (0.13/12), and the number of months is 4 years multiplied by 12 months (4 * 12). Plugging in these values, we get:

Monthly Payment = (3500 * 0.13/12) / (1 - (1 + 0.13/12)^(-4 * 12))

Solving this equation will give us the monthly payment amount. Using a calculator, the monthly payment comes out to be approximately $94.78.

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initial volume of balloon =

[tex] \frac{4}{3} \times \frac{22}{7} \times r^{3} \\ \frac{4}{3} \times \frac{22}{7} \times {3}^{3} \\ 113.14[/tex]

so intial volume is 113.14 cubic inches

volume after one hour will be 113.14 - 2 = 111.14 inches

new radius

[tex]111.14 = \frac{4}{3} \times \frac{22}{7} \times {x}^{3} \\ \frac{2333.94}{88} = {x}^{3} \\ 26.52 = {x}^{3} \\ x = 2.98[/tex]

Rate change of radius is

[tex] \frac{2.98}{3} \times 100 \\ \frac{298}{3} \\ 99.93\%[/tex]

Rate change is 0.7% per hour the radius is decreasing

Answer:

200,000

Step-by-step explanation:

add 128,000+42,000+30,000

The implicit cost of her business is $56000

Implicit costs are a specific type of opportunity cost, the cost of resources already owned by the firm that could have been put to some other use.

Given that, the annual revenue of Aimee is $128,000, the explicit costs of her business are $42,000, and the opportunity costs of her business are $30,000.

We need to find the implicit costs of her business,

Therefore,

Economic profit = total revenues – explicit costs – implicit costs.

30000 = 128000-42000-implicit costs.

Implicit costs = 56000

Hence, the implicit cost of her business is $56000

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Answer:

Step-by-step explanation:

13. The set M4,6 of all 4x6 matrices is a vector space as it is closed under vector addition as also scalar multiplication. Also the 4x6 zero matrix is in M4,6.

14. The set M1,1 is a singleton, i.e. a 1x1 matrix. It is a vector space for the same reasons as in 13. above.

15. The degree of the zero polynomial is undefined and it is usually treated as a constant (of degree 0). If we treat 0 as a polynomial of degree 0, then the set of all 3rd degree polynomials is not a vector space despite being closed under vector addition and scalar multiplication as it does not contain the zero polynomial.

16. The set of all 5th degree polynomials is not a vector space despite being closed under vector addition and scalar multiplication as it does not contain the zero polynomial.

17. The set of all first degree polynomials ax (a≠0) is not a vector space. If p(x) = ax and q(x) = -ax , then p(x)+q(x) = 0*x which is not in the given set. Hence the set is not closed under vector addition and, therefore, it is not a vector space.

18. The set of all first degree polynomials ax +b (a,b ≠0) is not a vector space. If p(x) = ax +b and q(x) = -ax -b , then p(x)+q(x) = 0*x +0 which is not in the given set. Hence the set is not closed under vector addition and, therefore, it is not a vector space.

19. The set P4 of all polynomials of degree 4 or less isa vector space. If p(x) = a1x4+a2 x3+a3x2+a4x+a5 and q(x) = b1x4+b2 x3+b3x2+b4x+b5 are 2 arbitrary elements of P4, then p(x)+q(x)is in P4. Similarly, αp(x) is in P4 for any arbitrat scalar α. Hence, P4 is closed under vector addition and scalar multiplication. Also, the 0 polynomial is in P4. Hence P4 is a vector space.

A set forms a vector space if it meets all ten vector space axioms. For instance, a set of first-degree polynomial functions ax + b, where a, and b are non-zero and whose graphs do not pass through the origin, fails to meet vector space axiom for closure under addition. On the other hand, the set of all third-degree polynomials is a vector space as it satisfies all ten vector space axioms.

To determine whether a set combined with standard operations forms a vector space, it must satisfy ten vector space axioms. These axioms include requirements regarding the addition and scalar multiplication of vectors.

To illustrate, let's look at the set of all first-degree polynomial functions ax + b, with a, b ≠ 0, whose graphs do not pass through the origin. By definition, vectors in a vector space should be closed under addition, i.e., if you add any two vectors together, you should get another vector in the set. However, if we add two such functions together ax + b + cx + d = (a+c)x + (b+d), the resulting function will pass through the origin only if (b+d)=0. But given that b and d are non-zero, the sum will not yield a first-degree polynomial that passes through the origin. Therefore, this set fails to meet the vector space axiom for closure under addition.

On the other hand, the set of all third-degree polynomials will be a vector space as it adheres to all the ten axioms. For example, if two polynomials in the set are added together or multiplied by a scalar, the resultant is still a third-degree polynomial which means it still belongs to the set.

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Answer:

0.84 or 84%

Step-by-step explanation:

The probability of an apple tree being injured in this period, is the probability of it being injured by the first frost (0.6), added to the probability of it being injured by the second frost (0.6) minus the probability of it being injured by both frosts (0.6 x 0.6):

[tex]P = P(F_1)+P(F_2)-P(F_1\ and\ F_2)\\P=0.6+0.6-(0.6*0.6)\\P=0.84 = 84\%[/tex]

There is a 0.84 or 84% probability of an apple tree being injured in this period.

The value of probability can only be from 0 to 1. The probability of an apple tree being injured in this period is 0.84 or 84 %.

Probability means possibility. It deals with the occurrence of a random event. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.

Frost damage to apple blossoms can severely reduce apple yield in commercial orchards.

It has been determined that the probability of a late spring frost causing blossom damage to Empire apple trees in the Hudson Valley of New York State is 0.6.

The probability of it being injured by first frost (0.6), added to the probability of it being injured second frost (0.6) minus the probability of it being injured by both touches of frost (0.6 × 0.6). That can be written as mathematically,

P = P(F₁) + P(F₂) - P(F₁ and F₂)

P = 0.6 + 0.6 - (0.6 × 0.6)

P 0.84

P = 84%

Thus, the probability of an apple tree being injured in this period is 0.84 or 84 %.

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Answer:

99% confidence interval for the proportion of adults who professed to be baseball fans in November 2003 is (46.9%, 55.1%)

This interval means that the lower limit of the proportion of adults who professed to be baseball fans is 46.9% and the upper limit of the proportion is 55.1%

Step-by-step explanation:

Confidence interval = P' + or - t×sqrt[P'(1-P') ÷ n]

P' is sample proportion = 51% = 0.51

n = 1001

confidence level = 99%

t-value corresponding to 99% confidence interval and infinity degree of freedom is 2.576

t × sqrt[P'(1-P') ÷ n] = 2.576 × sqrt[0.51(1-0.51) ÷ 1001] = 2.576 × 0.0158 = 0.041

Lower limit = P' - 0.041 = 0.51 - 0.041 = 0.469 = 46.9%

Upper limit = P' + 0.041 = 0.51 + 0.041 = 0.551 = 55.1%

99% Confidence interval is (46.9%, 55.1%)

The interval means that the proportion of adults who professed to be baseball fans in November 2003 is between 46.9% and 55.1%

Answer:

a) Parameter of interest [tex] p[/tex] representing the true proportion of the plates have blistered.

b) Null hypothesis:[tex]p\leq 0.1[/tex]  

Alternative hypothesis:[tex]p > 0.1[/tex]  

c) [tex]z=\frac{0.14 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=1.33[/tex]  

d) For this case we need to find a value in the normal standard distribution that accumulates 0.1 of the area in the right tail and for this case is:

[tex] z_{critc}= 1.28[/tex]

e) For this case since our calculated value is higher than the critical value 1.33>1.28 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion is significantly higher than 0.1

f) [tex]p_v =P(z>1.33)=0.0917[/tex]  

If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the true proportion is higher than 0.1 or 10%

Step-by-step explanation:

Data given and notation

n=100 represent the random sample taken

Part a

Parameter of interest [tex] p[/tex] representing the true proportion of the plates have blistered.

X=14 represent the number of the plates have blistered.

[tex]\hat p=\frac{14}{100}=0.14[/tex] estimated proportion of the plates have blistered.

[tex]p_o=0.1[/tex] is the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level

Confidence=90% or 0.90

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Part b: Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that  more than 10% of all plates blister under such circumstances.:  

Null hypothesis:[tex]p\leq 0.1[/tex]  

Alternative hypothesis:[tex]p > 0.1[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

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

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Part c: Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.14 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=1.33[/tex]  

Part d: Rejection region

For this case we need to find a value in the normal standard distribution that accumulates 0.1 of the area in the right tail and for this case is:

[tex] z_{critc}= 1.28[/tex]

Part e

For this case since our calculated value is higher than the critical value 1.33>1.28 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion is significantly higher than 0.1

Part f

Since is a right taild test the p value would be:  

[tex]p_v =P(z>1.33)=0.0917[/tex]  

If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the true proportion is higher than 0.1 or 10%

In the given sample, the parameter  [tex]\mu[/tex] represents the average temperature of the coffee in the population whose value is not known.

In statistics, a parameter means a numerical attribute of a population. A population parameter describes a particular aspect of the entire population, which is the complete set of individuals, items, or units of interest. The mean, standard deviation, and variance are some of the population parameters.

A coffee company wants to make sure that their coffee is being served at the right temperature. If it is too hot, the customers could burn themselves. If it is too cold, the customers will be unsatisfied. The company has determined that they want the average coffee temperature to be 65 degrees C. They take a sample of 20 orders of coffee and find the sample mean to be equal to 70.2 C. [tex]\mu[/tex] represents the average temperature of the coffee in the population, and the value is not known for this problem.

Also, [tex]\bar{X}[/tex] represents the sample mean here which is known and equals 70.2 C.

b

Therefore, [tex]\mu[/tex] represents the population mean and is not known in this problem.

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The symbol mu in the coffee company's problem represents the population mean, the desired average coffee temperature, which should be 65 degrees C.

In the context of the coffee company's quality control problem, the symbol mu represents the population mean, which is the average temperature of all cups of coffee served by the company. The sample mean, denoted as x-bar, is 70.2°C, which is the average temperature calculated from the sample of 20 orders of coffee. The goal of the company is to have mu equal to 65°C, as they want this to be the average serving temperature to ensure customer satisfaction and safety. If they find that x-bar is significantly different from mu, it may suggest that corrective actions are needed to reach the desired temperature.

Answer:

Step-by-step explanation:

a)

To construct a confidence interval ,the histogram should be bell shaped.It means that the it should be normally distributed with no skewness.To eliminate the skewness a large sample size is required so that the sample is normally distribute about the mean Also ,in order to construct a t-interval the sample data must come from the population that is normally distributed or the sample size is larger than 30.Since the question stated that the population distribution is skewed to the right \bar{x} is guaranteed to be normally distributed if [tex]n\geq 30[/tex]

b)The sample satisfies the normal distribution because the sample sixe is greater than 30 which is 1001.

c) [[tex]\bar{x}=1.22 \,s=.65\, n=1001[/tex]

Area in the right tail =2.5% or .025

degree of freedom =1001-1 =1000

[tex]t_{\alpha /2}=1.96[/tex]

95% confidence interval is :[tex]\bar{x}\pm t_{\alpha /2}s/\sqrt{n} [/tex]

lower bound is :[tex]1.22-1.96*.65/\sqrt{1001} =1.18 [/tex]

upper bound is :[tex]-1.22+1.96*.65/\sqrt{1001}=1.26 [/tex]

d)No because the sample data was obtined from Americans age 15 or older which cannot be applied to a different age group

Answer:

interest income in amount of $4000 will be accrued

Step-by-step explanation:

given data

principal = $100,000

rate = 8 percent = 0.08  

time period = 6 month ( July - December ) = [tex]\frac{1}{2}[/tex] year

solution

we get here interest at December 31, 2015 that is express as

interest = principal × rate × time ..................1

put here value and we will get

interest =  $100,000 × 0.08 × [tex]\frac{1}{2}[/tex]    

interest = $100,000 × 0.08 × 0.5

interest = $4000

so interest income in amount of $4000 will be accrued

Answer:

Answer is explained in the attached document

Step-by-step explanation:

Hessenberg matrix- it a special type of square matrix,there there are two subtypes of hessenberg matrix that is upper Hessenberg matrix and lower Hessenberg matrix.

upper Hessenberg matrix:- in this type of matrix  zero entries below the first subdiagonal or in another words square matrix of n\times n is said to be in upper Hessenberg form  if ai,j=0

for all i,j with i>j+1.and upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero

lower Hessenberg matrix:-  in this type of matrix  zero entries upper the first subdiagonal,square matrix of n\times n is said to be in lower Hessenberg form  if ai,j=0  for all i,j with j>i+1.and lower Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero.

Answer:

Mean  = 14.347 and Standard Deviation = 18.89

Step-by-step explanation:

Mean = Sum of all the numbers/total numbers = 272.6/19 = 14.347

For standard deviation, at first we will find variance which is

variance = The sum of the squared differences between each data point and the mean, divided by the number of data points (n) - 1:

x= data points

n= total number of data points

= ∑(x_i - Mean (x))^2/(n-1)

variance = 357

Standard deviation = [tex]\sqrt{variance}[/tex]

SD = [tex]\sqrt{357}[/tex]

SD = 18.89

Answer:

The mean= 14.347

The standard deviation =18.390

Step-by-step explanation:

The mean or average of a set of scores = the summation of the set of scores divided by the summation of the frequency of the respective scores.

Adding up the scores we have:

0.13+0.68+0.91+1.36+2.74+3.08+4.07+4.71+4.96+6.56+7.29+7.91+8.37+12.11+31.61+32.65+33.78+36.72+72.96 = 272.60

The frequency or number of scores is 19, so dividing this figure by the number of items in the set, we have:

272/19 =14.347

Therefore the mean of the scores (time, in minutes) is 14.347(to 3 decimal places)

In order to calculate the standard deviation of the set, we need to find the summation of the squares of the different deviations from the mean.

Example, 0.13 is the first score in the data. We subtract the mean (14.35) from 0.13 and we'll have 14.22 which is the score's deviation from the mean. the next step is to find the square of the deviation,14.22 which will be 202.208.

We'll repeat this same process for the remainder of the scores and then sum up the squares of the deviations.Doing this, the summation of the squares of all the deviations from the mean score will be = 6425.67

Once this is calculated we then solve to obtain the standard deviation of the scores by applying the formula:

√summation of (x - mean deviations)/total number of scores

=√(6425.67/19)

= √ (338.193)

= 18.390( to 3 decimal places)

Therefore the mean and the standard deviation of the set of scores are 14.347 and 18.390 respectively

Answer:

Test statistics = 1.87

Step-by-step explanation:

We are given that a political study took a sample of 1000 voters in the town and found that 56% of the residents favored annexation.

And, a political strategist wants to test the claim that the percentage of residents who favor annexation is more than 53%, i.e;

Null Hypothesis, [tex]H_0[/tex] : p = 0.53 {means that the percentage of residents who favor annexation is 53%}

Alternate Hypothesis, [tex]H_1[/tex] : p > 0.53 {means that the percentage of residents who favor annexation is more than 53%}

The test statistics we will use here is;

                T.S. = [tex]\frac{\hat p -p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }[/tex] ~ N(0,1)

where, p = actual percentage of residents who favor annexation = 0.53

            [tex]\hat p[/tex] = percentage of residents who favor annexation in a sample of

                  1000 voters = 0.56

            n = sample of voters = 1000

So, Test statistics = [tex]\frac{0.56 -0.53}{\sqrt{\frac{0.56(1- 0.56)}{1000} } }[/tex]

                              = 1.87

Therefore, the value of test statistics is 1.87 .

Answer:

mmmm

yesssirrreeee

Answer:children at home prefer hot breakfast than hot school breakfast because at home you could put it in the refrigerator but at school you have to throw it in the trash before you go to trash.

Step-by-step explanation:

Parental care in this context is considered the independent variable because it influences children's school performance.

In this scenario, parental care is considered an independent variable. The reason for this is because it is the variable that influences or predicts the outcome, which in this case, is children's performance in school. The independent variable is the one that is manipulated or controlled in a study to observe its effects on the dependent variable (children's school performance here). Examples of parental care might include ensuring the child eats a good breakfast, aiding with schoolwork, or providing emotional support.

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Answer:392.5

Step-by-step explanation:

50/2=25

25*25=625

625*3.14=1962.5

1962.5/5=392.5

Im right  look at pic for proof

Answer:

The probability distribution of [tex]\hat p[/tex] is, [tex]N(0.70, 0.032)[/tex].

Step-by-step explanation:

The random variable X is defined as the number of brand A external hard drives work in a satisfactory manner throughout the warranty period.

The sample selected is of size, n = 15.

The probability of selecting a hard drive that works in a satisfactory manner throughout is, p = 0.70.

Every hard drive works independently of the others.

The random variable X follows a Binomial distribution with parameters n = 15 and p = 0.70.

The probability mass function of X, the binomial random variable is:

[tex]P(X=x)={15\choose x}0.70^{x}(1-0.70)^{15-x};\ x=0,1,2,3...[/tex]

The probability distribution of X is shown below.

Now a statistic is defined as:

[tex]\hat p=\frac{X}{n}[/tex]

This statistic is known as the sample proportion.

A Normal approximation to Binomial can be used to approximate the distribution of sample proportion if the following conditions are satisfied:

Then the sampling distribution of [tex]\hat p[/tex] follows a Normal; distribution with mean [tex]p=0.70[/tex] and standard deviation [tex]\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.70(1-0.70)}{200}}=0.032[/tex].

Check the conditions:

[tex]np=15\times0.70=10.5\\n(1-p)=15\times (1-0.70)=4.5\approx5[/tex]

So, the probability distribution of [tex]\hat p[/tex] is, [tex]N(0.70, 0.032)[/tex].

The sampling distribution of X/n is approximately normally distributed with a mean of 0.7 and a standard deviation of 0.117.

To determine the sampling distribution of the statistic X/n, we need to consider the probability distribution of the number of successes (X) in the sample of n = 15 drives.

Since the probability of a success (a working hard drive) is p = 0.7 and the probability of a failure (a non-working hard drive) is q = 1 - p = 0.3, we can model X using the binomial distribution with parameters n = 15 and p = 0.7.

The probability of getting exactly x successes (working hard drives) in a sample of n = 15 drives is given by the binomial probability mass function (PMF):

P(X = x) = (nCx) * px * (1-p)^(n-x)

where:

nCx is the binomial coefficient, which represents the number of ways to choose x successes out of n trials.

px is the probability of success (working hard drive) on each trial.

(1-p)^(n-x) is the probability of failure (non-working hard drive) on each trial.

For x = 3 successes (working hard drives), we calculate the probability:

P(X = 3) = (15C3) * (0.7)^3 * (0.3)^12 ≈ 0.193

This means that the probability of observing 3 successes (working hard drives) in a sample of 15 drives is approximately 0.193.

The sample proportion (fraction) of successes, X/n, is a continuous random variable that can take values between 0 and 1. To determine the sampling distribution of X/n, we need to consider the distribution of X and how it transforms to X/n.

The distribution of X/n is approximately normal for large sample sizes (n ≥ 30), but for smaller sample sizes, it can be skewed. In this case, with n = 15, the distribution of X/n is slightly skewed, but it is still close to being normal.

The mean of the sampling distribution of X/n is equal to the population proportion, which is p = 0.7.

The standard deviation of the sampling distribution of X/n is given by:

σ(X/n) = √(p * q / n) ≈ √(0.7 * 0.3 / 15) ≈ 0.117

Therefore, the sampling distribution of X/n is approximately normally distributed with a mean of 0.7 and a standard deviation of 0.117.

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Answer: 771,243

Step-by-step explanation:

Answer:

D. All of these

Step-by-step explanation:

Requirements determination is the process of transforming a system's request into more detailed business statement that is clear and precise. It is the beginning sub phase of analysis, so all the given options in the question can be included.

Answer:

[tex]z=1.28<\frac{a-239}{4.6}[/tex]

And if we solve for a we got

[tex]a=239 +1.28*4.6=244.89[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 244.89.  

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 heights of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(239,23)[/tex]  

Where [tex]\mu=239[/tex] and [tex]\sigma=23[/tex]

Since the distribution for X is normal then we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

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

And we are interested on a value a such that:

[tex]P(\bar X>a)=0.10[/tex]   (a)

[tex]P(\bar X<a)=0.90[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.90 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1

If we use condition (b) from previous we have this:

[tex]P(\bar X<a)=P(\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{a-\mu}{\frac{\sigma}{\sqrt{n}}})=0.9[/tex]  

[tex]P(z<\frac{a-\mu}{\frac{\sigma}{\sqrt{n}}})=0.9[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=1.28<\frac{a-239}{4.6}[/tex]

And if we solve for a we got

[tex]a=239 +1.28*4.6=244.89[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 244.89.  

Answer:

a) 0.5 per minutes

b) 5 arrivals expected in 10 minutes

c) P ( x = 0 ) = 0.00673 , P ( x = 1 ) =  0.03368 , P ( x = 2 ) = 0.08422 ,P ( x = 3 ) = 0.14037                  

d)  P ( X >= 4 ) = 0.735                

Step-by-step explanation:

Given:

- The number of customer arriving at window is modeled by Poisson distribution. The distribution is given by:

                         P(x) = ( λ^x ) (e^-λ) / x!            x = 0 , 1 , 2 , 3 , ......

- Average rate λ = 30 / hr

Find:

a. How many customers arrive per minute, on average?

b. How many customers would you expect to arrive in a 10-minute interval?c. Use equation 13.1 to determine the probability of exactly 0, 1, 2, and 3 arrivals in a 10-minute interval.

d. What is the probability of more than three arrivals occurring in a 10-minute interval?

Solution:

- The average rate λ in number of customers that arrive in a minute is given by:

                              λ1 = 30 / 60 = 0.5 arrival per minutes

- The average number of customer that are expected to arrive in 10-minutes window is:

                              λ2 = 10*λ1 = 10*0.5 = 5 arrivals expected in 10 minutes

- The probability of exactly 0,1 , 2 , and 3 arrivals in 10 minute windows:

                   P ( x = 0 ) = ( 5^0 ) (e^-5) / 0! = 0.00673  

                   P ( x = 1 ) = ( 5^1 ) (e^-5) / 1! = 0.03368

                   P ( x = 2 ) = ( 5^2 ) (e^-5) / 2! = 0.08422

                   P ( x = 3 ) = ( 5^3 ) (e^-5) / 3! = 0.14037

- The probability of more than three arrivals occuring in 10-minute interval is:

                  P ( X >= 4 ) = 1 - P ( X =< 3 )

                  P ( X >= 4 ) = 1 - [ P ( x = 0 ) + P ( x = 1 ) +  P ( x = 2 ) + P ( x = 3 ) ]

                  P ( X >= 4 ) = 1 - [ 0.00673 + 0.03368 + 0.08422 + 0.14037 ]

                  P ( X >= 4 ) = 1 - [ 0.265 ]

                  P ( X >= 4 ) = 0.735

Using the Poisson distribution, it is found that:

a) 0.5 customers per minute.

b) 5 customers are expected to arrive.

c)

0.0068 = 0.68% probability of exactly 0 arrivals in a 10-minute interval.

0.0337 = 3.37% probability of exactly 1 arrivals in a 10-minute interval.

0.0842 = 8.42% probability of exactly 2 arrivals in a 10-minute interval.

0.1404 = 14.04% probability of exactly 3 arrivals in a 10-minute interval.

d) 0.7349 = 73.49% probability of more than three arrivals occurring in a 10-minute interval.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

The parameters are:

Item a:

30 in one-hour(60 minutes), hence 0.5 customers per minute.

Item b:

0.5 customers per minute, hence, in a 10 minute interval, 5 customers are expected to arrive.

Item c:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-5}(5)^{0}}{(0)!} = 0.0068[/tex]

[tex]P(X = 1) = \frac{e^{-5}(5)^{1}}{(1)!} = 0.0337[/tex]

[tex]P(X = 2) = \frac{e^{-5}(5)^{2}}{(2)!} = 0.0842[/tex]

[tex]P(X = 3) = \frac{e^{-5}(5)^{3}}{(3)!} = 0.1404[/tex]

0.0068 = 0.68% probability of exactly 0 arrivals in a 10-minute interval.

0.0337 = 3.37% probability of exactly 1 arrivals in a 10-minute interval.

0.0842 = 8.42% probability of exactly 2 arrivals in a 10-minute interval.

0.1404 = 14.04% probability of exactly 3 arrivals in a 10-minute interval.

Item d:

This probability is:

[tex]P(X > 3) = 1 - P(X \leq 3)[/tex]

In which:

[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]

From item c:

[tex]P(X \leq 3) = 0.0068 + 0.0337 + 0.0842 + 0.1404 = 0.2651[/tex]

Then:

[tex]P(X > 3) = 1 - P(X \leq 3) = 1 - 0.2651 = 0.7349[/tex]

0.7349 = 73.49% probability of more than three arrivals occurring in a 10-minute interval.

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Answer:

A) Estimated standard deviation = 148cm³

B) The estimated standard deviation of 148 cm³ is less than the true standard deviation of 183.2 cm³. This estimated standard deviation is not even within 15 cm³ of the true standard deviation and thus we can say it's not accurate.

Step-by-step explanation:

From the question, the given standard deviation is 183.2 cm³

Also that the range of values is between 902 cm³ to 1494 cm³.

Range is the difference between highest and lowest values in the data set.

Thus, Range = 1494 cm³ - 902 cm³ = 592 cm³

Now, The range rule tells us that the standard deviation of a sample is approximately equal to one-fourth of the range of the data. In other words s = (Maximum – Minimum)/4

Thus, estimated standard deviation = 592/4 = 148 cm³

Answer: B, C, D

Step-by-step explanation:

since it is rotated, the parallel sides stay parallel.

The trapezoid ABCD rotated to form A'B'C'D, A'B' ∥ D'C',

Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, reflection, dilation and translation.

Rotation is a rigid transformation, hence it preserves the shape and size. If a point A(x, y) is rotated on 180° about the origin, the new point is A'(-x, -y).

Hence for the trapezoid ABCD rotated to form A'B'C'D, A'B' ∥ D'C',

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The area of the cross section of the column is [tex]18 \pi \ m^2[/tex]

Explanation:

Given that a building engineer analyzes a concrete column with a circular cross section.

Also, given that the circumference of the column is [tex]18 \pi[/tex] meters.

We need to determine the area of the cross section of the column.

The area of the cross section of the column can be determined using the formula,

[tex]Area= \pi r^2[/tex]

First, we shall determine the value of the radius r.

Since, given that circumference is [tex]18 \pi[/tex] meters.

We have,

[tex]2 \pi r=18 \pi[/tex]

   [tex]r=9[/tex]

Thus, the radius is [tex]r=9[/tex]

Now, substituting the value [tex]r=9[/tex] in the formula [tex]Area= \pi r^2[/tex], we get,

[tex]Area = \pi (9)^2[/tex]

[tex]Area = 81 \pi[/tex]

Thus, the area of the cross section of the column is [tex]18 \pi \ m^2[/tex]

Answer:

a) 94.06% probability that a water specimen collected from the river has an alkalinity level exceeding 45 milligrams per liter.

b) 94.06% probability that a water specimen collected from the river has an alkalinity level below 55 milligrams per liter.

c) 50.98% probability that a water specimen collected from the river has an alkalinity level between 48 and 52 milligrams per liter.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 50, \sigma = 3.2[/tex]

a. exceeding 45 milligrams per liter.

This probability is 1 subtracted by the pvalue of Z when X = 45. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{45 - 50}{3.2}[/tex]

[tex]Z = -1.56[/tex]

[tex]Z = -1.56[/tex] has a pvalue of 0.0594.

1 - 0.0594 = 0.9406

94.06% probability that a water specimen collected from the river has an alkalinity level exceeding 45 milligrams per liter.

b. below 55 milligrams per liter.

This probability is the pvalue of Z when X = 55.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{55 - 50}{3.2}[/tex]

[tex]Z = 1.56[/tex]

[tex]Z = 1.56[/tex] has a pvalue of 0.9604.

94.06% probability that a water specimen collected from the river has an alkalinity level below 55 milligrams per liter.

c. between 48 and 52 milligrams per liter.

This is the pvalue of Z when X = 52 subtracted by the pvalue of Z when X = 48. So

X = 52

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{52 - 50}{3.2}[/tex]

[tex]Z = 0.69[/tex]

[tex]Z = 0.69[/tex] has a pvalue of 0.7549

X = 48

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{48 - 50}{3.2}[/tex]

[tex]Z = -0.69[/tex]

[tex]Z = -0.69[/tex] has a pvalue of 0.2451

0.7549 - 0.2451 = 0.5098

50.98% probability that a water specimen collected from the river has an alkalinity level between 48 and 52 milligrams per liter.

The probabilities are calculated using the normal distribution properties: exceeding 45 mg/L is approximately 0.9406, below 55 mg/L is approximately 0.9406, and between 48 and 52 mg/L is approximately 0.4678.

a) To find the probability that the alkalinity level exceeds 45 milligrams per liter, we can use the standard normal distribution with the given mean (50 mg/L) and standard deviation (3.2 mg/L). First, we need to calculate the z-score for 45 mg/L:

z = 45 - 50/3.2 = -1.5625

Then, we find the probability of the alkalinity level exceeding 45 mg/L by finding the area to the right of this z-score in the standard normal distribution. Using a standard normal table or calculator, we find this probability to be approximately 0.9406.

b) Similarly, to find the probability that the alkalinity level is below 55 milligrams per liter, we calculate the z-score for 55 mg/L:

[tex]\[ z = \frac{55 - 50}{3.2} = 1.5625 \][/tex]

Then, we find the probability of the alkalinity level being below 55 mg/L by finding the area to the left of this z-score in the standard normal distribution. Using a standard normal table or calculator, we find this probability to be approximately 0.9406.

c) To find the probability that the alkalinity level is between 48 and 52 milligrams per liter, we first calculate the z-scores for these values:

z₁ = 48-50/3.2 = -0.625

z₂ = 52-50/3.2 = 0.625

Then, we find the area between these two z-scores in the standard normal distribution, which represents the probability of the alkalinity level being between 48 and 52 mg/L. Using a standard normal table or calculator, we find this probability to be approximately 0.3146.

In conclusion:

Answer:

Let X the random variable who represent the number of occurences in a period of time for the calls.

For this case we have the following parameter [tex] \lambda = 1 \frac{call}{2 minutes}[/tex]

And we are interested in the expected number of calls in one hour.

We know that 1 hr = 60 mins so then the expected number of calls that arrive in one hour are:

[tex] \lambda = \frac{1 call}{2 minutes} * \frac{60 minutes}{1 hour} = 30 calls per hour[/tex]

Step-by-step explanation:

Definitions and concepts

The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:

[tex]P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}[/tex]

And the parameter [tex]\lambda[/tex] represent the average ocurrence rate per unit of time.

Solution to the problem

Let X the random variable who represent the number of occurences in a period of time for the calls.

For this case we have the following parameter [tex] \lambda = 1 \frac{call}{2 minutes}[/tex]

And we are interested in the expected number of calls in one hour.

We know that 1 hr = 60 mins so then the expected number of calls that arrive in one hour are:

[tex] \lambda = \frac{1 call}{2 minutes} * \frac{60 minutes}{1 hour} = 30 calls per hour[/tex]

Answer:

The probability of rolling a brown on the next toss is 0.2303.

Step-by-step explanation:

The data recorded is:

Brown = 35

Purple = 50

Green = 44

Yellow = 23

TOTAL = 152 tosses

The probability of an event E is computed by dividing the favorable number of outcomes by the total number of outcomes.

[tex]P(E)=\frac{Favorable\ outcomes}{Total\ no.\ of\ outcomes}[/tex]

Using this formula compute the probability of rolling a brown on the next toss as follows:

[tex]P(Brown)=\frac{35}{152}= 0.2303[/tex]

Thus, the probability of rolling a brown on the next toss is 0.2303.

Answer:

(a)2/3

(b)2/3

(c)9/13

Step-by-step explanation:

Total Number of Cards in new Deck=39

Hearts(Red)=13

Diamonds(Red)=13

Spades(Black)=13

(a)P(drawing a red card)

Total number of red cards = 13+13=26

P(drawing a red card)=26/39=2/3

(b)Drawing a heart or a red card

Number of Hearts=13

Number of red cards=26

Number of Red Hearts = 13

Since the two events are not mutually exclusive

P(Hearts or Red) = P(Hearts) + P(Red) - P( Hearts and Red)

P(H∪R)=P(H)+P(R)-P(H∩R)

=13/39 + 26/39 - 13/39

=26/39 =2/3

(c)Drawing a jack or a red card.

Number of Jacks=3

Number of red cards=26

Number of Red Jacks = 2

Since the two events are not mutually exclusive

P(Jack or Red) = P(Jacks) + P(Red) - P( Jacks and Red)

P(J∪R)=P(J)+P(R)-P(J∩R)

=3/39 + 26/39 - 2/39

=27/39 =9/13