The mean hours of sleep that students get per night is 7 hours, the standard deviation of hours of sleep is 1.7 hours, and the distribution is approximately normal. Complete the following sentence. For about 95% of students, nightly amount of sleep is between ______.a. 5.3 and 8.7 hrsb. 5 and 9 hrsc. 3.6 and 10.4 hrsd. 1.9 and 12.1 hrs

Answer:

8.99 days elapsed. Option (C) is correct

Step-by-step explanation:

The distribution  has density function k/t+1 for a constant k and t between 0 and 6.5 . Since the distribution is symmetrical in 6.5, the area it forms between 0 and 6.5 should be 1/2, thus

[tex]\frac{1}{2} = \int\limits_0^{6.5} \frac{k}{t+1} \, dt = k *(ln(t+1) \, |_0^{6.5}) = k * (ln(7.5)-ln(1)) = k*ln(7.5)[/tex]

Hence k = 1/(2ln(7.5)), approx 1/4.

We need to find the percentil 0.6, since the integral of the random variable is 1/2 over the first half, we need to find t such that the integral of the random variable between o and 6.5 + t is 0.6. This is equivalent to find t such that the integral between 6.5 and 6.5+t is 0.1. Due to the  over 6.5, this t should satisfy that the integral between 6.5-t and 6.5 is also 0.1. Lets compute the integral and find t

[tex]\int\limits^{6.5}_{6.5-t} {\frac{k}{t+1}} \, dx = \frac{1}{2ln(7.5)}*(ln(t+1) \, |_{6.5-t}^{6.5} \, ) = \frac{1}{2ln(7.5)} * (ln(7.5)-ln(7.5-t)) = \\\frac{1}{2} - \frac{ln(7.5-t)}{2ln(7.5)} = 0.1[/tex]

Therefore,

[tex]\frac{ln(7.5-t)}{2ln(7.5)} = 0.4\\\\ln(7.5-t) = 0.8*ln(7.5)\\\\7.5-t = e^{0.8*ln(7.5)}\\\\t = 7.5-e^{0.8*ln(7.5)} = 2.49[/tex]

As a result, the student sould have registered 2.49 days after the day 6.5, thus it should have registeredd at day 8.99. Option (C) is correct.

Answer:

Option C: The line y = x

Step-by-step explanation:

Let us take an ordered pair (x, y) of a function. Then the ordered pair of its inverse function would be (y, x).

That is to say, when we reflect a point (x, y) across the line y = x we get the point (y, x).

Note that since, this function is invertible, it is both 'one - one' and 'onto'.

The line of reflection for two functions with an inverse relationship is the line y = x, as this line shows the symmetry of the original function and its inverse on a graph.

When two functions have an inverse relationship, the line of reflection across which their graphs are mirrored is the line y = x. This is because for an inverse relationship, each x value in the first function corresponds to a y value in the second function, and vice versa. Thus, when graphing the original function and its inverse, you will notice that they are symmetric with respect to the line y = x.

Answer:

[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]

Step-by-step explanation:

We have been an integral [tex]\int \frac{9+\sqrt{x}+x}{x}dx[/tex]. We are asked to find the general solution for the given indefinite integral.

We can rewrite our given integral as:

[tex]\int \frac{9}{x}+\frac{\sqrt{x}}{x}+\frac{x}{x}dx[/tex]

[tex]\int \frac{9}{x}+\frac{1}{\sqrt{x}}+1dx[/tex]

Now, we will apply the sum rule of integrals as:

[tex]\int \frac{9}{x}dx+\int \frac{1}{\sqrt{x}}dx+\int 1dx[/tex]

[tex]9\int \frac{1}{x}dx+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]

Using common integral [tex]\int \frac{1}{x}dx=\text{ln}|x|[/tex], we will get:

[tex]9\text{ln}|x|+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]

Now, we will use power rule of integrals as:

[tex]9\text{ln}|x|+\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\int 1dx[/tex]

[tex]9\text{ln}|x|+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+\int 1dx[/tex]

[tex]9\text{ln}|x|+2x^{\frac{1}{2}}+\int 1dx[/tex]

[tex]9\text{ln}|x|+2\sqrt{x}+\int 1dx[/tex]

We know that integral of a constant is equal to constant times x, so integral of 1 would be x.

[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]

Therefore, our required integral would be [tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex].

The general indefinite integral of the expression 9 + sqrt(x) + x/x is obtained by separately integrating each term, resulting in the expression 9x + (2/3)x^(3/2) + x + C, where C is the sum of the constants of integration.

The question asks to compute the indefinite integral of an algebraic function. Firstly, it's crucial to understand that an indefinite integral is an antiderivative of a function, representing the reverse operation of differentiation. In the given expression, we essentially have three terms to integrate: 9, √x and x/x (which simplifies to 1). To find the integral of these expressions we can use the power rule of integration: ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration.

When we integrate 9, treating it as 9x^0, it becomes 9x + C1. The integral of the square root of x, which is x^1/2 in exponent form, becomes (2/3)x^(3/2) + C2 according to the power rule. Finally, x/x simplifies to 1, and its integral is simply x + C3.

So, the general indefinite integral of the original expression is 9x + (2/3)x^(3/2) + x + C, where 'C' represents the sum of the constants of integration: C1, C2, and C3.

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Answer:

Option D) There is sufficient evidence that the true population proportion is not equal to 70%.        

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 1165

p = 70% = 0.7

Alpha, α = 0.02

Number of people who agreed , x = 746

First, we design the null and the alternate hypothesis  

[tex]H_{0}: p = 0.7\\H_A: p \neq 0.7[/tex]

This is a two-tailed test.  

Formula:

[tex]\hat{p} = \dfrac{x}{n} = \dfrac{746}{1165} = 0.64[/tex]

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

Putting the values, we get,

[tex]z = \displaystyle\frac{0.64-0.7}{\sqrt{\frac{0.7(1-0.7)}{1165}}} = -4.46[/tex]

Now, [tex]z_{critical} \text{ at 0.02 level of significance } = \pm 2.33[/tex]

Since,  

Since, the calculated z statistic does not lie in the acceptance region, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.

Thus, there is enough evidence to support the claim that the true proportion differs from 70%.

Option D) There is sufficient evidence that the true population proportion is not equal to 70%.

Answer:

=19

Step-by-step explanation:

=6÷3+17

=2+17

=19

Answer:  The proofs are given below.

Step-by-step explanation:  We are given to prove that the following statements are tautologies using truth table :

(a) ¬r ∨ (¬r → p)                              b. ¬(p → q) → ¬q

We know that a statement is a TAUTOLOGY is its value is always TRUE.

(a) The truth table is as follows :

r                 p                 ¬r                       ¬r→p                     ¬r ∨ (¬r → p)

T                T                   F                         T                                T

T                F                   F                         T                                T

F                T                   T                         T                                T

F                F                   T                         F                                T  

So, the statement (a) is a  tautology.

(b) The truth table is as follows :

p                 q                 ¬q                       p→q             ¬(p→q)          ¬(p→q)→q

T                T                   F                         T                      F                    T

T                F                   T                         F                      T                    T

F                T                   F                         T                      F                    T

F                F                   T                         T                       F                   T

So, the statement (B) is a  tautology.              

Hence proved.

Final answer:

To prove that a statement is a tautology using propositional logic, we need to show that the statement is true under all possible truth values of its variables. By applying the laws of implication, disjunction, and contradiction, we can prove that the given statements are tautologies.

Explanation:

To prove that a statement is a tautology using the laws of propositional logic, we need to show that the statement is true under all possible truth value assignments of its variables. Let's consider each statement:

a. ¬r ∨ (¬r → p)

We can use the law of implication, which states that ¬p ∨ q is equivalent to p → q, to rewrite the statement as ¬r ∨ (r → p). By applying the law of disjunction, which states that p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r), we can further rewrite the statement as (¬r ∧ r) ∨ (r ∨ p). Using the law of contradiction, which states that p ∧ ¬p is always false, we can simplify the statement to r ∨ p, which is a tautology.

b. ¬(p → q) → ¬q

We can use the law of implication to rewrite the statement as ¬(¬p ∨ q) → ¬q. By applying De Morgan's law, which states that ¬(p ∨ q) is equivalent to ¬p ∧ ¬q, we can simplify the statement to (p ∧ ¬q) → ¬q. Using the law of contradiction, we know that p ∧ ¬p is always false, so the statement simplifies to false → ¬q, which is always true. Therefore, it is a tautology.

Answer:

Step-by-step explanation:

The probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19,

It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.

It is given that:

The randomly chosen number between 1 and 100 is divisible by 3

Applying conditional probability:

Let A is the event: the numbers divisible by 3

Let B is the event: At least one digit equal to 5

P(A|B) = n(A∩B)/n(B)

P(A|B) = 6/19

Thus, the probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19.

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Final answer:

The described observational study is a cross-sectional study, as it involves collecting data once at a particular point in time, without any follow-up observations. So the correct option is B.

Explanation:

The type of observational study described in the scenario is a cross-sectional study. This classification is based on the researchers collecting data at a specific point in time by interviewing a sample of 1800 adults about their tendencies in handling emotions and measuring their blood pressure. Since the data on suppression of emotions and blood pressure is gathered just once from these individuals, without any follow-up at later dates, this defines the hallmark of a cross-sectional study. In contrast, a retrospective study would involve looking back at past behaviors or conditions, and a cohort study would imply following a group of individuals over a prolonged period of time to observe outcomes, neither of which applies to this particular research setup.

Final answer:

The study in the question is a cross-sectional study because it gathers data from subjects at a specific point in time to assess the association between emotional suppression and high blood pressure.

Explanation:

The study described in the question can be classified as a cross-sectional study because it involved collecting data from the individuals at a specific point in time. Participants in the study were asked about their response to emotions and their current tendency to express or hold in anger and other emotions. Additionally, their blood pressure was measured. This type of observational study is designed to analyze data at a single point in time to find any associations or correlations, such as the one being investigated between emotional suppression and high blood pressure.

Answer:

1) The solution of the system is

[tex]\left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]

2) The solution of the system is

[tex]\left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]

Step-by-step explanation:

1) To solve the system of equations

[tex]\left\begin{array}{ccccccc}&3x_2&-5x_3&=&89\\6x_1&&+x_3&=&17\\x_1&-x_2&+8x_3&=&-107\end{array}\right[/tex]

using the row reduction method you must:

Step 1: Write the augmented matrix of the system

[tex]\left[ \begin{array}{ccc|c} 0 & 3 & -5 & 89 \\\\ 6 & 0 & 1 & 17 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 2: Swap rows 1 and 2

[tex]\left[ \begin{array}{ccc|c} 6 & 0 & 1 & 17 \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 3:  [tex]\left(R_1=\frac{R_1}{6}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]

Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]

Step 5: [tex]\left(R_2=\frac{R_2}{3}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]

Step 6: [tex]\left(R_3=R_3+R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & \frac{37}{6} & - \frac{481}{6} \end{array} \right][/tex]

Step 7: [tex]\left(R_3=\left(\frac{6}{37}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 8: [tex]\left(R_1=R_1-\left(\frac{1}{6}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 9: [tex]\left(R_2=R_2+\left(\frac{5}{3}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]

Step 10: Rewrite the system using the row reduced matrix:

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right] \rightarrow \left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]

2) To solve the system of equations

[tex]\left\begin{array}{ccccccc}4x_1&-x_2&+3x_3&=&12\\2x_1&&+9x_3&=&-5\\x_1&+4x_2&+6x_3&=&-32\end{array}\right[/tex]

using the row reduction method you must:

Step 1:

[tex]\left[ \begin{array}{ccc|c} 4 & -1 & 3 & 12 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 2: [tex]\left(R_1=\frac{R_1}{4}\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 3: [tex]\left(R_2=R_2-\left(2\right)R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]

Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 5: [tex]\left(R_2=\left(2\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 6: [tex]\left(R_1=R_1+\left(\frac{1}{4}\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]

Step 7: [tex]\left(R_3=R_3-\left(\frac{17}{4}\right)R_2\right)[/tex]

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & - \frac{117}{2} & \frac{117}{2} \end{array} \right][/tex]

Step 8: [tex]\left(R_3=\left(- \frac{2}{117}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 9: [tex]\left(R_1=R_1-\left(\frac{9}{2}\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 10: [tex]\left(R_2=R_2-\left(15\right)R_3\right)[/tex]

[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]

Step 11:

[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right]\rightarrow \left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]

Answer:

a) Quantitative

b) Quantitative

c) Qualitative

d) Qualitative

Step-by-step explanation:

a)

Percent of freshmen that will eventually graduate is a quantitative variable because it can be presented numerically for example 84% or 78% etc.

b)

GPA of incoming freshman is a quantitative variable because it can be presented by numerical quantities.

c)

Require SAT or ACT for admission is a categorical variable because it is divided into categories such as required, recommended and not used.

d)

College type is a categorical variable because it is divided into categories such as liberal arts college and national university etc.

The variables 'percent of freshmen who eventually graduate' and 'G.P.A of incoming freshmen' are quantitative because they can be measured numerically. The variables 'Require SAT or ACT for admission' and 'College type' are categorical because they are classified into specific categories.

In regards to the variables that contribute to the ranking of colleges, (a) The 'percent of freshmen who eventually graduate' can be identified as a QUANTITATIVE variable since it is a number that can be measured. For instance, an outcome could be '85% of freshmen graduate'.

(b) The 'GPAs of incoming freshmen' is also a QUANTITATIVE variable as it can also be measured numerically. An example could be 'The average GPA of incoming freshmen is 3.7 out of 4.0'.

(c) Whether or not a college 'requires SAT or ACT for admission' is a CATEGORICAL variable as it describes a category or characteristic, for example, the admission requirement can be one of the following: required, recommended or not used.

(d) Lastly, 'college type (liberal arts college, national university, etc.)' is also a CATEGORICAL variable, because it represents different types of educational institutions which are distinguished by categories.

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Answer:

the explanation is given below.

Step-by-step explanation:

from this, the lowest number is 1 and the highest number is 100

Answer:

a. mixed longitudinal study

True, by definition a mixed-longitudinal study is when "have defined some cohorts and these are followed for a shorter period and we can compare the precision, bias due to age and cohort effects" on the entire study. So that represent the perfect mix between longitudinal and cross sectional study.

Step-by-step explanation:

a. mixed longitudinal study

True, by definition a mixed-longitudinal study is when "have defined some cohorts and these are followed for a shorter period and we can compare the precision, bias due to age and cohort effects" on the entire study. So that represent the perfect mix between longitudinal and cross sectional study.

b. limited longitudinal study

This definition is not appropiate and is not usually used in the experimental designs.

c. longitudinal study

False, a longitudinal study is a design on which have "repeated observations of the same variables over short or long periods of time" and for this case we need cross sectional conditions, so for this case not applies.

d. cohort study

False, by definition a cohort study is an extesion of the longitudinal design but on this case " the samples are obtained from a cohort using cross-section intervals through time" and for this reason not applies for our case since we need a longitudinal design combined with the cross sectional design

Answer:

1. First equation is option B

2. Second equation is option C

3. Third equation is option E

4. Fourth equation no best option.

explanation:

Check the attachment for solution

Following are the calculation to the differential equation:

For point 1)

[tex]y'' + 8 y' + 15 y = 0\\[/tex]

B

[tex]Y = e^{-3x}[/tex] be the solution of this equation

[tex]Y' = -3 e^{-3x}\\\\ y''= 9 e^{-3x} \\\\\therefore \\\\y'' +8y' + 15 y= 9e^{-3x} + 8(-3e^{-3x})+ 15 e^{-3x} \\\\e^{-3x}( 9-24+15)=0[/tex]

For point 2)  

[tex]y'' + y = 0[/tex]

C

[tex]y = \sin x[/tex] be the solution of above equation  

[tex]y'= -\cos x \\\\y''= -\sin x = -y \\\\y''+y=0\\\\[/tex]

For point 3)

 [tex]y' = 5 y[/tex]

[tex]y'=e^{5x}[/tex] be the solution of equation 3

[tex]y'= 5 e^{5y}= 5y =y'=5y[/tex]

For point 4)

[tex]2x^2 y'' + 3xy' = y[/tex]

Let [tex]y=\sqrt{x}[/tex] be the solution of equation  (4)

 [tex]y'=\frac{1}{2 \sqrt{x} }\\\\y''=- \frac{1}{2} \times \frac{1}{2} \times {x^{- \frac{3}{2}}} ==- \frac{1}{4} \times {x^{- \frac{3}{2}}} \\\\-2x^2 \times =- \frac{1}{4} {x^{- \frac{3}{2}}}+ 3x \times =- \frac{1}{2 \sqrt{x}}\\\\- \frac{1}{2} {x^{ \frac{1}{2}}}+ \frac{3}{2} x^{\frac{1}{2}} =\sqrt{x} =y\\\\[/tex]

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Answer:  0.1

Step-by-step explanation:

WE know that the total probability in an experiments = 1.

i.e. Sum of the probabilities of occurring each event is 1.

i.e. If there are n outcomes in any experiment., then the total probability will be:

[tex]P(E_1)+P(E_2)+P(E_3)+...........+P(E_n)=1[/tex]

Given : An experiment consists of four outcomes ,with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4.

Then , [tex]P(E_1)+P(E_2)+P(E_3)+P(E_4)=1[/tex]

Substitute corresponding values , we get

[tex]0.2+0.3+0.4+P(E_4)=1[/tex]

[tex]0.9+P(E_4)=1[/tex]

[tex]P(E_4)=1-0.9=0.1[/tex]

Hence , the probability of outcome [tex]E_4[/tex] is 0.1.

The probability of event E4 in the given experiment is 0.1, because the sum of probabilities of all outcomes should equal 1.

The problem falls under the subject of

Probability Theory

within Mathematics. In probability, the total probability of all possible outcomes is always 1. So, for the given problem where you have four events E1, E2, E3, E4, the total probability P(E1)+P(E2)+P(E3)+P(E4) should equal 1. Given that P(E1) = 0.2, P(E2) = 0.3 and P(E3) = 0.4, we can find P(E4) using the equation

1 - P(E1) - P(E2) - P(E3) = P(E4)

. By substituting the given values into this equation, we find that P(E4) = 1 - 0.2 - 0.3 - 0.4 =

0.1

. Therefore, the probability of outcome E4 is 0.1.

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Answer:

Stratified

Step-by-step explanation:

Stratified sampling is a type of sampling method in which the observer divides the overall population into different groups.

These groups are known as strata.

Within the each group, a probability sample is selected.

stratified sampling helps in reducing the sample size required to attain a required precision.

Answer:

See explanation below.

Step-by-step explanation:

Part a

For this case we can use the moment generating function for the bernoulli distribution for n trials, given by:

[tex] p(s)^n = (q+ps)^n =\sum_{r=0}^n (nCr) (ps)^r q^{n-r}[/tex]

Wehre p is the probability of success and [tex] P(s_ = q+ps[/tex]

Using this property we see that;

If we multiply the two generating functions we got:

[tex] p(s)^N = (q+ps)^N =\sum_{r=0}^N (NCr) (ps)^r q^{N-r}[/tex]

[tex] p(s)^M = (q+ps)^M =\sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]

[tex] p(s)^M p(s)^N = \sum_{r=0}^N (NCr) (ps)^r q^{N-r} \sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]

And the mass function would be given by:

[tex]= (N+M C r) p^{r} (1-p)^{N+M-r}[/tex]

So we see that follows a binomial random variable with parameters (N+M, p)

Part b

For this case we are assuming that [tex] X \sim Bin (N,p) , Y\sim Bin (M,p)[/tex] and for this case we can assume that [tex] 0 \leq k \leq N+M[/tex] for the proof.

We are interested on the random variable [tex] Z= X+Y[/tex] since the two random variables are independent we can write the probability mass function for Z like this:

[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i , Y=k-i)[/tex]

[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i) P(Y=k-i)[/tex]

And we can replace the mass function for X and Y

[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} \sum_{i=0}^k (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]

And we can rewrite this like that:

[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]

[We can take out the constant p:

[tex] P(Z = X+Y = k) = p^k (1-p)^{N+M-k} \sum_{i=0}^k (NCi)(M C k-i)[/tex]

And using properties of the binomial formula we can write this like that:

[tex] P(Z = X+Y = k) = (N+M Ck) p^k (1-p)^{N+M-k} [/tex]

So then we see that [tex] Z= X+Y \sim Bin(N+M ,p)[/tex]

To design a sine function with a period of 24 and minima and maxima at given times, we scale the function using a coefficient of 2π/24, shift the function to make the peak occur at x=16, and stretch it by a factor of 3 to make it go from 10 to 16.

To design a sine function with a period of 24 and minima and maxima at given times, we first need to understand a few concepts about sine functions. The standard sine function, sine(x), has a period of 2π. Therefore, to stretch it to a period of 24 hours, we would scale the function using a coefficient of 2π/24 or π/12. Thus, our function becomes sine((π/12) x).

Next, we want to shift the function so its maximum occurs at t=16. Normally, the sine function peaks at π/2, so we need to shift the function to the right by an amount that makes the peak occur at x=16. This would be 16 - π/2, which gives us the function sine(π/12 x - 16 + π/2).

Finally, to stretch the function vertically to accommodate the minimum and maximum values of 10 and 16, we note that the amplitude of the sine function is usually 1 (from -1 to 1), so we need to stretch it by a factor of (16-10)/2 = 3 to make it go from 10 to 16. This gives us the function y= 3sin((π/12)x - 16 + π/2)+13.

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Answer:

The length of the circular arc is 8.64 inches

Step-by-step explanation:

Length of circular arc (L) = central angle/360° × 2πr

central angle = pie/4 = 45°, r (radius) = 11 inches

L = 45°/360° × 2 × 3.142 × 11 = 8.64 inches (to two decimal places)

Answer:

Step-by-step explanation:

The formula for determining the length of an arc is expressed as

Length of arc = θ/360 × 2πr

Where

θ represents the central angle.

r represents the radius of the circle.

π is a constant whose value is 3.14

From the information given,

Radius, r = 11 inches

θ = pi/4

2π = 360 degrees

π = 360/2 = 180

Therefore,

θ = 180/4 = 45 degrees

Therefore,

Length of arc = 45/360 × 2 × 3.14 × 11

Length of arc = 8.64 inches rounded up to 2 decimal places

Answer:

a) H0: mean =5 and Ha: mean≠ 5

Step-by-step explanation:

In hypothesis testing procedure the trait of null hypothesis is that it always contain an equality sign. We are known that diameter of spindle is known to be 5mm. This our null value. Hence the null hypothesis is

H0:μ=5.

Now for alternative hypothesis we are given that the mean diameter has moved away from the target. This means that mean diameter could be increases or decreases from 5mm. Hence the alternative hypothesis is

Ha:μ≠5

From the information given, it is found that the correct option is:

(A) H0: Mean= 5 and Ha: Mean is not equal to 5.

At the null hypothesis, it is tested if the motor works properly, that is, the spindle has diameter significantly close to 5 mm, hence:

[tex]H_0: \mu = 5[/tex]

At the alternative hypothesis, it is tested if the motor does not work properly, that is, the spindle has diameter different from 5 mm, either too high or too low, hence:

[tex]H_a: \mu \neq 5[/tex]

Thus, a is the correct option.

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Answer:

a) ∫_{-6}^{6} ∫_{0}^{36} ∫_{x²}^{36} (-y) dy dz dx

b)  ∫_{0}^{36} ∫_{-6}^{6} ∫_{x²}^{36} (-y) dy dx dz

c)  ∫_{0}^{36} ∫_{x²}^{36} ∫_{-6}^{6} (-y) dx dy dz

e) ∫_{x²}^{36} ∫_{-6}^{6} ∫_{0}^{36} (-y) dz dx dy

Step-by-step explanation:

We write the equivalent integrals for given integral,

we get:

a) ∫_{-6}^{6} ∫_{0}^{36} ∫_{x²}^{36} (-y) dy dz dx

b)  ∫_{0}^{36} ∫_{-6}^{6} ∫_{x²}^{36} (-y) dy dx dz

c)  ∫_{0}^{36} ∫_{x²}^{36} ∫_{-6}^{6} (-y) dx dy dz

e) ∫_{x²}^{36} ∫_{-6}^{6} ∫_{0}^{36} (-y) dz dx dy

We changed places of integration, and changed boundaries for certain integrals.

The expression 10∠30 + 10∠30 can be simplified by adding the magnitudes (10 + 10) and keeping the angle the same.

Given, that 10∠ 30 + 10∠ 30 .

In polar form, the magnitude is represented by the absolute value of a complex number and the angle is measured counterclockwise from the positive real axis.

To find the polar form of the sum, we first add the magnitudes: 10 + 10 = 20.

Next, keep the angle the same: 30 degrees.

Therefore, the polar form of 10∠30 + 10∠30 is 20∠30.

This means that the complex number is represented by a magnitude of 20 and an angle of 30 degrees.

Know more about polar form,

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The value of  A and B given that the function [tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex] has a minimum value of 54 at x = 81 is 243 and 3 respectively

Given the function

[tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex]

If y= 54 where x = 81, hence

[tex]54=\frac{A}{\sqrt{81} }+B\sqrt{81}\\54=\frac{A}{9}+9B\\486=A+81B\\ A+81B=486[/tex]

At the minimum point [tex]\frac{dy}{dx} = 0[/tex]

Differentiate the given function:

[tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}\\y'=\frac{-0.5A}{{x^{3/2}} }+\frac{B}{x^{1/2}} \\\frac{-0.5A}{{x^{3/2}} }+\frac{B}{x^{1/2}}=0[/tex]

Substitute x = 81 to hav:

[tex]\frac{-0.5A}{81^{2/3}} +\frac{B}{81^{1/2}}=0\\\frac{-A}{81} + B=0\\-A+81B=0\\A=81B ......................... 2[/tex]

Substitute equation 2 into 1:

[tex]81B+81B= 486\\162B=486\\B=\frac{486}{162} \\B=3[/tex]

Get the value of A:

[tex]A=81B\\A=81(3)\\A=243[/tex]

Hence the value of  A and B given that the function [tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex] has a minimum value of 54 at x = 81 is 243 and 3 respectively

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The correct values for A and B, given that the function y=Ax√+Bx√ has a minimum value of 54 at x = 81, would be A=243 and B=6. The sum of both divided by 81 equals 54, as stated in the equation.

The function provided in this Mathematics problem is y = Ax√ + Bx√:

We are told that function has a minimum value of 54 at x = 81. So if we insert 81 into x, we would get:

54 = 81A + 81B

Then simplify:

54 = 81(A + B)

To find the value for A and B, we need to know more about the relationship between A and B. Unfortunately, the problem doesn't supply enough information for us to determine exact figures of A and B. But from the options provided, we need A and B that sum up to 54/81. Of the choices provided, only (A = 243, B = 6) will give us that sum, so b.) is the correct choice.

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Answer:

im pretty sure you have to find out how far Joel walked and then double joels distance to get Brents

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer: ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )

Step-by-step explanation:

Given : A random sample of 11 days were selected from last year's records maintained by the maternity ward in a local hospital, and the number of babies born each day of the days is given below:

3  7   7   10    0    7    1    2    5   3    0

We first arrange them in increasing order , we get

0    0    1    2   3   3   5     7   7   7   10

Here , N= 11

Now , we can see that

Minimum value = 0

Maximum value = 10

First quartile [tex]Q_1[/tex]= [tex](\dfrac{N+1}{4})^{th}\ term=(\dfrac{12}{4})^{th}\ term = 3^{rd} term =1[/tex]

Second quartile [tex]Q_2[/tex]= Median = Middlemost number = 3

Third quartile [tex]Q_3[/tex] =  [tex](\dfrac{3(N+1)}{4})^{th}\ term=(\dfrac{36}{4})^{th}\ term[/tex]

[tex]= 9^{th} term =7[/tex]

∴ The required five number summary : ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )

Answer: Spatial correlation is defined as a specific relationship between spatial proximity among observational units and numeric similarities among values.

Step-by-step explanation: Spatial analysis focuses on individual as units located in spatial oriented structures such as gangs. In crime mapping and behavioral studies, spatial autocorrelation is used to determine the adjacency between area of influence and individuals within an area.

Answer:

No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.

Step-by-step explanation:

It is important to know the concepts of units, tenths and cents.

For example

1 = 1 unit

10 = 1*10 + 0 = The tens digit is one the unit digit is 0

21 = 2*10 + 1 = The tens digit is two and the unit digit is 1.

120 = 1*100 + 2*10 + 0 = The cents digit is 1, the tens digit is two and the unit digit is 0.

So

Adding 1 is the same as the unit digit going up by 1.

Adding 10 is the same as the tens digit going up by 1.

Adding 100 is the same as the cents digit going up by 1.

In this problem, we have that:

864,865,866,867,868,869

Each value is the 1 added to the previous value, that is, the unit digit goes up by 1.

Mariel thinks the tens digit goes up by 1 in these numbers. Do you agree?

No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.

Answer:

disagree, its the unit number that goes up by 1

Step-by-step explanation:

Answer:

20

Step-by-step explanation:

You could find 5/⅓

by using 5 × 3

Knowing that:

i. Any number multiplied by 1, gives the number itself.

ii. Dividing any number by itself gives 1.

You would agree with me that

i. (5×3)/(5×3) = 1

ii. Writing 5/⅓ as 5/⅓ × 1 doesn't change the value.

Then I can write 5/⅓ as

5/⅓ × (5×3)/(5×3) = 1

This can become

[5×(5×3)] / [(⅓) × (5×3)]

= 75/(15/3)

= 75/5

= 15

In a similar way,

12/ 3/5

= [12/ (3/5)] × [(5×3)/(5×3)]

= 12×(5×3) / (3/5)×(5×3)

= (12×5×3) / [(3×5×3)/5]

= 180 / (45/5)

= 180 / 9

= 20

Answer:

x=-224/229,

y=296/229,

z=118/229

Step-by-step explanation:

-5x-7y-4z=-6

6x+2y+3z=-2

-x+2y-7z=0...........( multiple with (-5) and sum with 1st equation, mult with 6 and                    sum with 2nd equation)

______________

-x+2y-7z=0

-17y+31z=-6.....(mult with 14)

14y-39z=-2.....(mult with -17) then sum

___________

-x+2y-7z=0

-229z=-118, so here we have z=118/229.

14y-39*(118/229)=-2, from here we have y=296/229

-x+2*(296/229)-7*(118/229)=0, we get that x=-234/229

In the same way you can do this in the matrix form>>

Answer: individual

Step-by-step explanation:

An individual is a person or object that is a member of the population being studied. A population is defined as a group of individuals with a common characteristic living and interbreeding within a given area, in statistics, population is a collection of individuals to be studied. Individuals can also be referred to as the objects/person described by a set of data. For example: when studying the height of students in a school, the students attending that school are individuals.

The probability that a single randomly selected salary is at least $134,000 is approximately 0.5120 or 51.20%.

To find the probability that a single randomly selected salary is at least $134,000, we need to calculate the z-score and use the standard normal distribution.

1. Calculate the z-score:

z = (x - μ) / σ

where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

In this case, x = $134,000, μ = $133,000, and σ = $31,000.

z = (134000 - 133000) / 31000

z = 1 / 31000

z ≈ 0.0323

2. Find the probability associated with the z-score:

We can use a z-table or a calculator to find the probability.

From the z-table, we find that the probability corresponding to a z-score of 0.0323 is approximately 0.5120.

Therefore, the probability that a single randomly selected salary is at least $134,000 is 0.5120 or 51.20%.