A data set consists of the values 2, 6, 3, and 1. If we consider this a population (all the values available), the variance isA. 12
B. 14
C. the square root of 3.5
D. the square root of 14
E. none of the above

Answer:

[tex] n_A = \frac{6.45^2}{(\frac{1.02}{1.96})^2}=153.61 \approx 154[/tex]

[tex] n_B = \frac{7.84^2}{(\frac{0.72}{1.96})^2}=455.49 \approx 456[/tex]

For this case as we can see we have a larger sample size for sample B, so then the best option for this case would be:

(B) The sample size of A is less than the sample size of B.

Step-by-step explanation:

For this case we have the following data given:

[tex] \bar X_A= 45[/tex] represent the sample mean for A

[tex] s_A= 6.45[/tex] represent the sample deviation for A

[tex] ME_A = 1.02[/tex] represent the margin of error for A

[tex] \bar X_B= 43[/tex] represent the sample mean for B

[tex] s_B= 7.84[/tex] represent the sample deviation for B

[tex] ME_B= 0.72[/tex] represent the margin of error for B

And for this case we are assuming that we have the same confidence level of 95%

For this case we an use the fact that the sample deviation is an unbiased estimator for the population deviation [tex]\hat \sigma = \hat s[/tex] and we can use the following formula for the margin of error of the sample mean the following formula:

[tex] ME= z_{\alpha/2} \frac{\hat s}{\sqrt{n}}[/tex]

For this case the value of the significance is given by [tex] \alpha =1-0.95 =0.05[/tex] and the value for [tex]\alpha/2 =0.025[/tex] , so then the value for [tex] z_{\alpha/2}[/tex] represent a quantile of the normal standard distribution that accumulates 0.025 of the area on each tail of the normal standard distribution and for this case is [tex] z_{\alpha/2}=\pm 1.96[/tex].

So then since we have the value for z if we solve for n from the margin of error formula we got:

[tex] n = \frac{\hat s^2}{(\frac{ME}{z})^2}[/tex]

And for the case A we can find the sample size and we got:

[tex] n_A = \frac{6.45^2}{(\frac{1.02}{1.96})^2}=153.61 \approx 154[/tex]

And for the case B we can find the sample size and we got:

[tex] n_B = \frac{7.84^2}{(\frac{0.72}{1.96})^2}=455.49 \approx 456[/tex]

For this case as we can see we have a larger sample size for sample B, so then the best option for this case would be:

(B) The sample size of A is less than the sample size of B.

The sample size of B is larger than the sample size of A and this can be determined by using the formula of margin of error.

Given :

To determine the sample size for both cases A and B, the formula of Margin of Error can be used:

[tex]\rm ME =z_{\frac{\alpha }{2}} \dfrac{\hat{s}}{\sqrt{n} }[/tex]

[tex]\rm n =\left(\dfrac{\hat{s}}{\dfrac{ME}{z}}\right)^2[/tex]

Now, for case A:

[tex]\rm n_A =\left(\dfrac{6.45}{\dfrac{1.02}{1.96}}\right)^2[/tex]

[tex]\rm n_A\approx 154[/tex]

Now, for case B:

[tex]\rm n_B =\left(\dfrac{7.84}{\dfrac{0.72}{1.96}}\right)^2[/tex]

[tex]\rm n_B \approx 456[/tex]

So, the sample size of B is larger than the sample size of A.

Therefore, the correct option is B) The sample size of A is less than the sample size of B.

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To show AB and CD are independent events, we used the fact that A, B, C, and D are mutually independent. The calculations show that P(AB AND CD) equals P(AB)P(CD), satisfying the condition for independence of events AB and CD.

To demonstrate that events AB and CD are independent, we need to use the definition of mutual independence. By this definition, being mutually independent, events A, B, C, and D satisfy the condition that for any two distinct events, say A and B, P(A AND B) = P(A)P(B). Similarly, this extends to any three events and all four events together, giving us P(A AND B AND C) = P(A)P(B)P(C), and P(A AND B AND C AND D) = P(A)P(B)P(C)P(D).

Sine AB and CD are composed of mutually independent events, we can deduce the following:

Now, to show that AB and CD are independent, we need to verify if P(AB AND CD) = P(AB)P(CD).

Because A, B, C, and D are mutually independent, we can expand P(AB AND CD) as:

And since we already know the individual probabilities of AB and CD as shown above, we then have:

Observing that both P(AB AND CD) and P(AB)P(CD) result in the same product P(A)P(B)P(C)P(D), we conclude that AB and CD are indeed independent events.

Answer:

2.94 rads

Step-by-step explanation:

The number of radians that Jose sweeps out equals to the ratio of the chord length, which is the distance he travels in km, to the radius of the circular track, which is 0.9 km

= 2.65 / 0.9 = 2.94 rads

So Jose sweeps an angle of 2.94 radians

Final answer:

Using the formula for arc length in a circle, Jose sweeps out approximately 2.944 radians when he travels 2.65 km along a circular ski trail with a radius of 0.9 km.

Explanation:

The student is asking about how to convert a distance traveled along a circular path into radian measure. Specifically, Jose has skied 2.65 kilometers around a circular ski trail with a radius of 0.9 kilometers. To calculate the number of radians swept out by Jose, we can use the formula s = rθ, where s is the arc length (distance traveled), r is the radius, and θ is the angle in radians.

To find the angle θ, we rearrange the formula to θ = s / r. Using the given values, we get θ = 2.65 km / 0.9 km, which simplifies to approximately 2.944 radians.

So, Jose sweeps out roughly 2.944 radians on his ski trip along the circular trail.

Answer:

Step-by-step explanation:

The Pythagorean theorem is expressed as

Hypotenuse² = opposite side² + adjacent side²

If the distances of the routes given are Pythagorean triples, then they obey the Pythagorean theorem hence, they would form a right angle triangle.

1) for the bus routes between stop A, B and C,

13² = 12² + 5²

169 = 144 + 25 = 169

A Pythagorean triple is formed hence, stop A, B and C form a right angle triangle.

2) for the bus routes between stop C, E and E,

22² = 14² + 18²

484 = 196 + 324 = 520

A Pythagorean triple is not formed hence, stop C, E and E do not form a right angle triangle.

3) 25² = 9² + HJ²

625 = 81 + HJ²

HJ² = 625 - 81 = 544

HJ = √544 = 23.32

4) EG² = 15² + 8²

EG² = 225 + 64

EG² = 289

EG = √289 = 17

The third quartile (Q3) of the customs data set is the 27th value in the ordered set, which is 28.

The third quartile (Q3), also known as the upper quartile, is typically the 75th percentile of a data set. This means it splits off the highest 25% of data from the rest. In this data set with 35 values, we would find the position of the third quartile using the formula (3*(N+1))/4, where N represents the number of data points. Here, it would be (3*(35+1))/4 = 27. So, we look to the 27th value in the ordered data set, which is 28. Therefore, the third quartile of the customs data is 28.

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Answer: D. none of the above

Step-by-step explanation:

Let x = a random variable that denotes the hourly income of midwives.

As per given , we have

[tex]\mu=\$55[/tex]

[tex]\sigma=\$15[/tex]

Also, the distribution is normal.

Then, the probability that NJ midwives earn more than $60 per hour will be :_

[tex]P(x>60)=1-P(x<60)=1-P(\dfrac{x-\mu}{\sigma}<\dfrac{60-55}{15})\\\\=1-P(z<0.33)\ \ [\because\ z=\dfrac{x-\mu}{\sigma}]\\\\=1-0.6293\ \ [\text{By z-table}]\\\\ =0.3707=37.07\% [/tex]

Hence, the percentage of NJ midwives earn more than $60 per hour is  approximately 37.07%.

Since , 37.07% is not given in any option.

So the correct answer to this question is "D.none of the above"

Answer:

The average height of poodles in your kennel is 25.19 inches.

Step-by-step explanation:

This is a weighed average problem.

To find the weighed average, we sum each value of the set multiplied by it's weight, and then we divide by the sum of the weights.

In this problem, we have that:

Average 20.5 inches has weight 28.

Average 32.1 inches has weight 19.

What is the average height of poodles in your kennel?

[tex]A = \frac{32.1*19 + 20.5*28}{19+28} = \frac{1183.9}{47} = 25.19[/tex]

The average height of poodles in your kennel is 25.19 inches.

To start with, in finding the height of the poodle, we consider both groups. We then add the multiplication of the number of dogs by the average, and finally, we divide them all by the number dogs. Mathematically, i am saying that;

Finally,

[tex]\frac{574+609.9}{47}[/tex] = 25.19

Therefore, the average height of the dogs in the kennel is 25.19 inches

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Answer:

mass of the water removed =  66.67 kg

Step-by-step explanation:

given data

solids is blended with sugar = 10%

obtain  final product = 15% product

obtain  final product = 15% sugar solids

The initial product is = 200 kg

solution

we get here amount of sugar required that is

amount of sugar required = 200 × [tex]\frac{10}{100}[/tex]

amount of sugar required =  20 kg

and we know total solid  = product solids +  sugar solids    ...............1

and

initial product =  final product    

so

0.20 ×  200kg= 0.30 ×  mass of the final product

mass of final product = 133.33 kg

but here

final product = 15% product solids and 15% sugar solids

so that amount of product solid = sugar solid

so

mass of the water removed = 200 kg - 133.33 kg

mass of the water removed =  66.67 kg

The quantity of final product obtained from 200 kg of liquid product is 20 kg. No sugar is required, and the mass of water removed during concentration is 180 kg.

Let's break down the problem step-by-step:

Therefore, the quantity of final product obtained from 200 kg of liquid product is 20 kg. No sugar is required, and the mass of water removed during concentration is 180 kg.

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Answer:

Step-by-step explanation:

Hello!

To see if there is an association between the variables:

"Level of satisfaction of an individual"

"Risk of diabetes of an individual"

The researchers studied 1621 people over 5 years.

Observations recorded:

Interviews of daily lives and hassles and hypothetical situations that were studied to assess their emotions.

Determination, if in the course of these 5 years the individuals experienced any type of diabetes.

Conclusion "Satisfied individuals are less likely to have diabetes"

a) This is a prospective cohort study.

In this type of study, a group of individuals that share the same characteristics is observed over some time, recording the events of interest.

b) Considering that the experiment concluded that "satisfaction" reduces the "risk of diabetes", we can determine that the response variable is "Risk of diabetes of an individual" and the explanatory variable is "Level of satisfaction of an individual".

Remember, the explanatory variable is the one considered to have a direct effect over the response variable.

c) "the research team also hasn't ruled out that a common factor like genetics could be causing both the emotions and the lung cancer."

There is a new variable that may affect the experiment. Be "genetic factor" the new variable and it affects directly "emotions and lung cancer", we can say that if some of those individuals are genetically predisposed to have lung cancer, this affects their emotions (satisfaction) and therefore, modifying their risk of having diabetes.

If this is so, then the genetic factor could be a lurking variable affecting directly the result of the observational experiment. Then the correct answer is:

A. The researchers may be concerned with confounding that occurs when the effects of two or more explanatory variables are not separated or when some explanatory variables were not considered in a study, but that affects the value of the response variable.

I hope it helps!

Final answer:

This response explains the type of observational study conducted, identifies the response and explanatory variables, and clarifies the researchers' concern about confounding due to genetics.

Explanation:

(a) Type of observational study: This study is an observational study because the researchers are observing and analyzing individuals to determine a relationship between satisfaction levels and the risk of diabetes without intervening or manipulating variables.

(b) Response and explanatory variables: The response variable in this study is the occurrence of diabetes, while the explanatory variable is the level of satisfaction of the individuals.

(c) Explanation of sentence: The sentence suggests that a common factor like genetics could be influencing both the emotions and the risk of diabetes, indicating that the researchers are considering the possibility of confounding where unaccounted variables may affect the relationship observed.The statement indicates concern about confounding variables, where genetics might influence both satisfaction and diabetes risk, suggesting they have not separated the effects of these intertwined factors fully. (Option A)

Answer:

Max 0.09B+0.2s  

 B>=0.6 Bond fund minimum

 0.06B+0.2S>=0.085 Minimum Return

 B+S=1 All funds invested

 B,S>=0

Step-by-step explanation:

(a) In linear programming,  the  mathematical model and the linear objective function set of linear constraints the variables are not negative.

B=% funds invested in the bond fund

S=% of funds invested in the stock fund

Max 0.09B+0.2s  

 B>=0.6 Bond fund minimum

 0.06B+0.2S>=0.085 Minimum Return

 B+S=1 All funds invested

 B,S>=0

Solve the above by using the graphical solution procedure?

steps to solve the graphs

1)draw the graphs, making sure the constraints are consider

2)consider all the constraints

3)draw the objective function line to the decision variables to

4) place the parallel lines of objective function towards larger objective function

5)  consider as optimal function  the feasible solution on the objective function line with the largest value ia

In this data set, the mean number of sibling pairs is 2.38, the median is 2, and the first and third quartiles are 2 and 3, respectively. The sample standard deviation would require a more complex calculation involving the mean and variance.

To calculate the relevant statistics for this data set, we need to use the formulas associated with each statistic.

1. The mean is the average number of sibling pairs in these families, calculated as the sum of all responses divided by the total number of responses. In this case, the mean is (1*13 + 2*22 + 3*15 + 4*6 + 5*3 + 7*1) / 60 = 2.38.

2. The median is the middle value in the ordered data set. Here, since we have 60 responses, the median is the average of the 30th and 31st values, which both fall within the '2 siblings' category. So, the median is 2.

3. To calculate the sample standard deviation, we first find the variance (the average of the squared differences from the mean). Then standard deviation is the square root of the variance. The exact calculation is quite lengthy, so you might want to use a statistical calculator for this.

4. The first quartile (Q1) is the value that separates the first 25% of the data. Because 25% of 60 equals 15, Q1 is also 2.

5. The third quartile (Q3) is the value that separates the first 75% of the data. Since 75% of 60 is 45, Q3 falls within the '3 siblings' category, so Q3 is 3.

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Answer:

An experimental research and particularly a quasi experimental study.

Step-by-step explanation:

An experimental research is one where the investigator study the effects from random samples he got and tested. the investigator manipulates parameters to considers some underlying factors in order to arrive at a conclusion. it is usually used to investigate relationships between variable and make comparison. In this case the investigator is conducting a research on how the consumption of oatmeal in adults reduces the level of bad cholesterol,  he measures and study the level of bad cholesterol for 6months and then compare their results.

for example, I can conduct an extensive experimental study on the long term effects of exhaust fumes on the passengers (particularly adults between the ages of 25-50) of public transport in Nigeria. this will be studied and their effects will be compared and a conclusion will be reached on the exposure and the non exposure as the case maybe. most underlying conditions of experimental study are under the direct control of the researcher of the investigator. there are three different types of experimental research ; Pre-experimental study, quasi-experimental study and true-experimental study.

The research being described is a cohort study, which follows two groups of people over time to measure the impact of their diet on cholesterol levels.

The study described in the question is a cohort study. In this type of study, a researcher follows a group over time to measure factors like diet and health outcomes. This particular study is comparing two cohorts: adults over the age of 40 who consume oatmeal regularly and those who do not, monitoring their levels of bad cholesterol over six months. While some other study designs, such as cross-sectional studies or observational studies, might look at data from different populations at a single point in time or look for correlations without affecting the participants' behavior, cohort studies are specifically designed to follow a group over time to see how specified factors affect their health.

Answer: The examples that use the stratified method are: (1).  A questionnaire is created to gauge student opinion on a new university cafeteria. A sample of 40 eshmen, 50 sophomores, 60 juniors, and 50 seniors is selected to fill out the questionnaire. (2). A potato field is believed to be infected with a plant disease. The field is divided into 10 equal areas, and 25 potatoes are selected from each area to be tested for the disease.

Step-by-step explanation: Stratified sampling technique is a type of sampling where the population under study has a number of distinct categories or sub-groups in which it is divided into. These categories or sub-groups are called strata and are defined by certain characteristics related to the variable or particular finding under interest. The sampling frame can be organized into separate mutually exclusive strata and then each ‘stratum’ is being sampled as an independent sub-population out of which individual elements can be randomly selected. In this case, each unit in a stratum, that is, each element in a group has a chance of being selected. With stratified sampling, the best result occurs when elements within strata are internally homogenous.

Answer:

Step-by-step explanation:

The relationship between predictor and response variables in scatterplots can be analyzed in terms of direction (positive or negative), form (linear or non-linear), and strength. Each plot is described based on this analysis.

To determine the relationship between the predictor and response variables in each of the provided scatterplots, we need to analyze the form, direction, and strength of the scatterplots.

For plot (1), if the points are following a downward path but not a straight line, we would classify this as Negative, non-linear.

For plot (2), if the points are following an upward path but not a straight line, we would classify this as Positive, non-linear.

For plot (3), if the points are increasing in a straight line, we would classify this as Positive, linear.

For plot (4), if the points are decreasing in a straight line, we would classify this as Negative, linear. However, if the points seem to be randomly scattered with no discernible pattern, then we would classify this as No association.

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Answer:

Reason is Matrix Multiplication Technique

Step-by-step explanation:

Matrix Multiplication: In matrix multiplication each element of a row is multiplied with each element of a column. So, row with all its zero entries is multiplied with all of the columns and making corresponding entries as zeros as well.

Therefore, A with a row having all zero entries also produces a row with all zero entries during multiplication with any matrix B.

Answer:

See explanation below.

Explanation:

If the statement is a tautology is true for all the possible combinations and we can check this with the table of truth for the statements

Part a

[tex] (p \land q) \Rightarrow (p \lor r)[/tex] lets call this condition (1)

[tex] (p \land q)[/tex] condition (2) and [tex](p \lor r)[/tex]  condition (3)

We can create a table like this one:

p       q     r      (2)       (3)     (1)  

T       T     T      T        T       T

T       T     F      T        T       T        

T       F     T      F        T       T

T       F     F      F        T       T

F       T     T      F        T       T

F       T     F      F        F       T

F       F     T      F        T       T

F       F     F      F        F       T

So as we can see we have a tautology since for all the possibilites we got true the final result.

Part b

[tex] p \Rightarrow (r \Rightarrow p)[/tex] let's call this condition (1)

And let [tex] (r \Rightarrow p)[/tex] condition (2)

We can create the following table:

p     r       (2)     (1)

T     T       T       T

T     F       T       T

F     T       F       T

F     F       T       T

So is also a tautology since the statement is true for all the possibilities or combinations.

The question involves computing parameters of a joint probability density function given a specific function and ranges. The process involves setting up and evaluating appropriate double integrals over the given ranges.

The subject of this problem is related to joint probability density functions (pdfs) and probability theory which comes under mathematics, specifically statistics.

(a) To find the value of k that makes this a valid probability density function, we use the property that the integral of a pdf over its range should equal 1:

(b) To find P(Y1 ≤ 3/4, Y2 ≥ 1/2), you integrate the joint pdf over the given intervals:

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Answer:

A pitcher can hold only 2 cup in his hands

The explicit solution to the initial-value problem is y = (-5x^2 + 2x^5 + 4x^2y^2 - 9)/(10(1 + 4y^2)).

To solve the given initial-value problem (1 + x^4)dy + x(1 + 4y^2)dx = 0, with y(1) = 0, the method of separation of variables is applied. Rearrange terms to isolate y and x:

(1 + x^4)dy = -x(1 + 4y^2)dx

Now, integrate both sides:

∫(1 + x^4)dy = -∫x(1 + 4y^2)dx

Integrating, we get:

y + (x^5)/5 = -(x^2)/2 - 2x^2y^2/2 + C

Solve for C using the initial condition y(1) = 0:

0 + 1/5 = -1/2 - 4/2 + C

C = 9/10

Substitute C back into the equation:

y + (x^5)/5 = -(x^2)/2 - 2x^2y^2/2 + 9/10

Now, simplify and solve for y:

y = (-5x^2 + 2x^5 + 4x^2y^2 - 9)/(10(1 + 4y^2))

This is the explicit solution to the initial-value problem. It is essential to note that the obtained solution is implicit and may not have a simple form due to the nature of the given differential equation.

Answer Choices:

Answer:

Observational

The study design of using cameras to monitor an intersection and count cars during green light intervals is considered an "C. Observational" study, as it involves systematic data collection without experimental manipulation.

The study design described, where cameras are set up to watch an intersection and determine how many cars are allowed through with each green light interval, would be considered an example of a "C. Observational" study design.

Observational studies involve the systematic collection and analysis of data without manipulating any variables. In this case, researchers are merely observing and recording the number of cars passing through the intersection when the traffic light is green. They are not actively intervening, controlling variables, or conducting experiments. Instead, they are passively gathering information from the real-world scenario without any interference.

This type of observational study can provide valuable insights into traffic patterns, efficiency, and safety at the intersection without introducing external biases that might occur in experimental designs. It allows researchers to collect data in a naturalistic setting, making it suitable for studying real-world phenomena where experimentation might be impractical or unethical, such as traffic flow analysis.

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Que. Cameras are set up to watch an intersection and determine how many cars are let through with each green light interval. This study design would be considered:

A. Simulation

B. Survey

C. Observational

D. Experimental

Answer:

Step-by-step explanation:

For first game PA = 1/3

For second game PA = 1/3 ( If A is not lost in first game )

 = 2/5 (If A is lost in first game )

For conclusion of game in three matches :

If A wins , the probability is

PA , PB , PA  = 1/3 X 2/3 X 3/5 = 6/45

PB , PA , PA = 2/3 X 3/5 X 1/3 = 6/45

If B wins

PA, PB, PB = 1/3 X 2/3 X 2/5 =  4/ 45

PB, PA , PB = 2/3 X 3/5 X 2/3 = 4 / 15

Total probability of conclusion of game in 3 matches

= 6/45 +6/45 + 4/45 +4/15 = 28/45

b )

For the game concluding in 2 matches , the probability are as follows

PA,PA = 1/3 X 1/3 = 1/9

PBPB = 2/3 X 2/5 = 4 / 15

Total probability

= 1/9 + 4/15 = 17/45

So PA = 1/9 / 17/45

= 5/17

Answer:

b. False

Step-by-step explanation:

De Morgan's law states that considering two statements A and B;

                      not (A or B) = not A and not B; and                  

                       not (A and B) = not A or not B

In set theory;

                      [tex]\overline{A u B} = \overline{A} n \overline{B}\\\overline{A n B} = \overline{A} u \overline{B}[/tex]

Applying De Morgan's law to the question,

A = Kwame will take a job in industry

B = go to graduate school

  not (A or B) = Kwame will not take a job in industry and not go to graduate school

Also;  

  not (A and B) = Kwame will not take a job in industry or not go to graduate school

Now considering the question, answer provided "Kwame will not take a job in industry or will not go to graduate school." is FALSE

Answer:

c = 10.000cm

Step-by-step explanation:

If the 2 boards made the right angle, c would be the hypotenuse with 2 sides of 6 cm and 8 cm. We can then use Pythagorean formula to solve for c

[tex]c^2 = 6^2 + 8^2 = 36 + 64 = 100[/tex]

[tex]c = \sqrt{100} = 10.000 cm[/tex]

Answer:

Ö + θ ( (k/m) + (g/l)) = 0

Step-by-step explanation:

Use the FBD attached:

Apply Newtons 2 nd Law in tangential direction:

Sum ( Ft ) = m*a

Sum of all tangential forces is:

m*g*sin(θ) + k*l*sin(θ)*cos(θ) = - m*l*Ö

Using small angle approximations:

sin (θ) = θ

cos (θ) = 1

Ö = angular acceleration.

m*g*θ + k*l*θ = -m*l*Ö

Ö + θ ( (k/m) + (g/l)) = 0

Answer: A . A Sample

Step-by-step explanation:

In online class , there should be male students too.

Thus , the population : All students who take part in our online class

So the data of all female students who take part in our online class is just a sample of the entire population.

∴ All female students who take part in our online class can be described as a sample.

Hence, the correct answer is A. A Sample .

Answer:  10

Step-by-step explanation:

Given : Jason has five coins in his pocket: a penny, a nickel, a dime, a quarter, and a half-dollar.

Since each coin has different value from others.

So the combination of any 3 coin will give a different amount.

We know that the combination of r things out of n things = [tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]

Therefore , the combination of 3 coins out of 5 = [tex]^5C_3=\dfrac{5!}{3!(5-3)!}=\dfrac{5\times4\times3!}{3!\times2!}=10[/tex]

Hence, the number of different sums of money can be formed using exactly three of the coins = 10

Answer: [tex]a+b+c=\frac{n+1}{n}.[/tex]

Step-by-step explanation: The function [tex]f(x_1,x_2,\ldots)=x_1^2+x_2^2+\ldots[/tex] is always positive except at the origin where it is equal to zero. This means that the absolute minumum of this function must be [tex]a=0[/tex]. Absolute maximum is when all of the variables are equal to zero except [tex]x_1[/tex] which is equal to 1 (f evaluated at this point is equal to 1 do b=1). The function itself is then equal to 1. This is because when [tex]f(\cdots)=x_1^2+x_2^2+\ldots\leq x_1^2+2x_2^2+3x_3^2+\ldots\leq1[/tex] so it is at most equal to 1 and this happens exactly at the point [tex](x_1,x_2,x_3,\ldots)=(1,0,0,\ldots).[/tex]

The absolute minimum at the boundary of this function happens when all the variables are equal to 0 except [tex]x_n=\frac{1}{\sqrt{n}}[/tex] and this minimum is equal to c=1/n. To see this notice that

[tex]nf=nx_1^2+nx_2^2+\cdots nx_n^2\geq x_1^2+2x_2^2+\cdots nx_n^2=1[/tex]

(the equality sign is because now we are on the boundary). We notice that nf is greater than or equal to 1 and the minimum of nf=1 (this implies the minimum for f to be 1/n) is attained exactly when [tex](x_1,x_2,\ldots,x_n)=(0,0,\ldots,\frac{1}{\sqrt{n}})[/tex].

So, finally, [tex]a+b+c=0+1+\frac{1}{n}=\frac{n+1}{n}.[/tex]

Answer:

the probability is P=0.012 (1.2%)

Step-by-step explanation:

for the random variable X= weight of checked-in luggage, then if X is approximately normal . then the random variable X₂ = weight of N checked-in luggage = ∑ Xi  , distributes normally according to the central limit theorem.

Its expected value will be:

μ₂ = ∑ E(Xi) = N*E(Xi) = 121 seats * 68 lbs/seat = 8228 lbs

for N= 121 seats and E(Xi) = 68 lbs/person* 1 person/seat = 68 lbs/seat

the variance will be

σ₂² = ∑ σ² (Xi)= N*σ²(Xi) → σ₂ = σ *√N = 11 lbs/seat *√121 seats = 121 Lbs

then the standard random variable Z

Z= (X₂- μ₂)/σ₂ =

Zlimit= (8500 Lbs - 8228 lbs)/121 Lbs = 2.248

P(Z > 2.248) = 1- P(Z ≤ 2.248) = 1 - 0.988 = 0.012

P(Z > 2.248)= 0.012

then the probability that on a randomly selected full flight, the checked-in luggage capacity will be exceeded is P(Z > 2.248)= 0.012 (1.2%)

To assess if there's a linear relationship between the Gesell score and the age at first word, one would create a scatterplot with the Gesell score as (y) and age as (x). The presence of a linear trend could be indicated by a clustering of points near a line, while the strength of the relationship would be further analyzed using the least-squares regression line and correlation coefficient.

To determine whether there is a linear relationship between the Gesell adaptive score test (Gesell score) and the age at which children speak their first word, you would start by plotting a scatterplot with the Gesell score as the response variable (y) and the age at first word as the explanatory variable (x). In the scatterplot, each point represents one child's data with their corresponding age at first word on the x-axis and Gesell score on the y-axis.

After plotting the data, you would examine the scatterplot to see if the points suggest a linear trend. If the points cluster around a line that slopes upwards or downwards, this could indicate a positive or negative linear relationship respectively. Conversely, if the points are widely scattered without any discernible pattern, it might suggest that there is no significant relationship between the variables.

If there seems to be a potential linear relationship, you might proceed to calculate the least-squares regression line to find the best-fitting line through the data and the correlation coefficient to measure the strength and direction of the relationship between the variables. Significant correlation coefficients (typically those near -1 or 1) would support the presence of a linear relationship, while coefficients near zero would suggest little to no linear relationship.

Final answer:

To analyze the relationship between the Gesell adaptive score test and the age of first word spoken by children, one would plot a scatterplot with Gesell score on the y-axis and age at first word on the x-axis. This visualization aids in identifying any potential linear or non-linear patterns.

Explanation:

To investigate if there is a relationship between the Gesell adaptive score test and the age at which a child speaks their first word, we would create a scatterplot with the age at first word (in months) as the x-axis (explanatory variable) and the Gesell score as the y-axis (response variable). By analyzing the pattern of the dots on the scatterplot, we could determine if there is any apparent linear relationship or if the data suggests a more complex relationship, such as an inverted U-shaped relationship.

It's important to note that an absence of linear correlation from a statistical test doesn't necessarily deny the existence of any relationship between two measures; the relationship could be non-linear or might change over time. An example of this would be a change in problem-solving strategies in children causing an inverted U-shaped relationship in cognitive ability over different stages. Therefore, a scatterplot is a crucial tool for visually identifying patterns and potential relationships in data.

Answer:

Step-by-step explanation:

consider B in matrix form

We have

[tex]\left[\begin{array}{ccc}1&1&1\\1&1&0\\0&1&1\end{array}\right][/tex]

Reduce this to row echelon form

by R1= R1-R3

we get

[tex]\left[\begin{array}{ccc}1&0&0\\1&1&0\\0&1&1\end{array}\right][/tex]

Now R2-R1 gives Identity matrix in row echelon form.  So rank =3 hence this is a basis for R cube

to find (1,2,3) as linear combination of B

Let a, b, and c be the scalars such that

a(1,1,1)+b(1,1,0)+c(0,1,1) = (1,2,3)

Equate corresponding terms as

a+b= 1:   a+b+c =2:  a+c =3

Solving b = -1, c = 1 and a = 2

(1,2,3) = 2(1,1,1)-1(1,1,0)+1(0,1,1)