Solve for the right triangle
Angle V 51

Answer:

Step-by-step explanation:

3. the diagram has 3 rectangles, 2 triangles

the surface area is the area of each shape

for the first rectangle = length x breadth = 8 x 6 = 48

for the second rectangle = length x breath = 6 x 6 = 36

for the third rectangle = length x breath = 6 x 6 = 36

for the triangles

(base x height )/2 = (8 x 4.5) /2 = 4 x 4.5 = 18

Surface Area =  48 + 18 + 36 + 36 = 138

4.  there are 4 identical rectangles and a base rectangle

4 x (5 x3 ) + ( 5 x5) = 4 x 15 + 25 = 60 + 25 = 85 ft

5. there are 2 triangles of 6 x 8 and a rectangle of 10 x 8

surface area = 2 x( (base x height)/2) + 10x8

2 x ((6 x8) /2 ) + 80 = 2 x (48/2)  + 80 = 48 + 80 = 128ft  

Answer:

26.76% probability that a randomly chosen golfer's score is above 70 strokes.

Step-by-step explanation:

Problems of normally distributed samples can be 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 = 68.2, \sigma = 2.91[/tex]

What is the probability that a randomly chosen golfer's score is above 70 strokes?

This is 1 subtracted by the pvalue of Z when X = 70. So

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

[tex]Z = \frac{70 - 68.2}{2.91}[/tex]

[tex]Z = 0.62[/tex]

[tex]Z = 0.62[/tex] has a pvalue of 0.7324.

So there is a 1-0.7324 = 0.2676 = 26.76% probability that a randomly chosen golfer's score is above 70 strokes.

Final answer:

To find the probability that a golfer's score is above 70, calculate the Z-score using the formula Z = (X - μ) / σ, where X is 70, the mean (μ) is 68.2, and the standard deviation (σ) is 2.91. The result, approximately 0.62, corresponds to a cumulative probability that must be subtracted from 1 to find the probability of scoring above 70. This process estimates there's about a 26.8% chance a golfer scores above 70.

Explanation:

The question asks about the probability that a randomly chosen golfer on the PGA tour has a score above 70 strokes, given that the scores are normally distributed with a mean of 68.2 and a standard deviation of 2.91. To find this probability, we use the Z-score formula, which is Z = (X - μ) / σ, where X is the score of interest, μ (mu) is the mean, and σ (sigma) is the standard deviation.

Calculating the Z-score for a score of 70:
Z = (70 - 68.2) / 2.91 ≈ 0.62.

Next, we consult a Z-table or use a calculator to find the probability corresponding to a Z-score of 0.62, which tells us the probability of a score being less than 70. To find the probability of a score being above 70, we subtract this value from 1.

Note: Specific values from the Z-table or calculator are not provided here. Generally, the process would involve looking up the cumulative probability for a Z-score of 0.62, which might be around 0.732. Therefore, the probability of a score above 70 would be 1 - 0.732 = 0.268. This means there's approximately a 26.8% chance that a randomly chosen golfer's score is above 70.

In order to find the percentage of bulbs with a certain lifespan, one must calculate the z-scores for the given values and find the probabilities using the standard normal distribution, converting these into percentages.

The student's question involves using the properties of the normal distribution to determine the probability of light bulb life spans within certain intervals. To solve these problems, the z-score formula is used, which is (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation.

Remember that the answer will be in the form of a percentage representing the likelihood that any given bulb falls within the specified hour range.

Answer:

Step-by-step explanation:

Yes, integers like 27,57,87,117,.... and so on gives a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6.

Yes, there is an integer that has a remainder of 2 when divided by 5 and a remainder of 3 when divided by 6. One example of such an integer is 23.

Let's use variables to rewrite the given statement. We can represent the integer as 'n', and the two remainders as 'r1' and 'r2'.

Therefore, the rewritten statement is: Is there an integer 'n' such that 'n' has a remainder of 2 when divided by 5 and a remainder of 3 when divided by 6?

Yes, such integers exist. One example is 23. When 23 is divided by 5, the remainder is 3, and when it is divided by 6, the remainder is also 3.

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Answer:

cos 38° = 17/c

Step-by-step explanation:

in the triangle shown

the sum of angles in a triangle is 180°

its a right angle triangle meaning one of the angles is 90°

the other part is 52°

the third part is described as x

90° + 52° + x = 180° ( sum of angles )

142° + x = 180°

x = 180 - 142 = 38°

cos 38° = adjacent/hypothenus = 17/c

cos 38° = 17/c

Answer:

a) 45% probability that the person is a male, given that he is a democrat.

b) 42.52% probability that the person is a republican given that he is male.

c) 41.67% that an Independent person is a female.

Step-by-step explanation:

A probability is the number of desired people(outcomes) divided by the total number of people(outcomes).

Example.

In a sample of 50 people, 30 are Buffalo Bills fans. The probability that a randomly selected person is a Buffalo Bills is 30/50 = 0.6 = 60%.

So

a) Male given that the person is a Democrat?

There are 100 Democrats. Of them, 45 are male and 55 are female.

So there is a 45/100 = 0.45 = 45% probability that the person is a male, given that he is a democrat.

b) Republican given that the person is Male?

There are 127 males. Of those, 54 are Republican.

So there is a 54/127 = 0.4252 = 42.52% probability that the person is a republican given that he is male.

c) Female given that the person is an Independent?

There are 48 independent people. Of those, 20 are female.

So there is a 20/48 = 0.4167 = 41.67% that an Independent person is a female.

The probabilities that a randomly selected individual is: a) a male Democrat is 0.45, b) a male Republican is approximately 0.425, and c) a female Independent is approximately 0.417.

The subject of your question is probability in mathematics. Given the data of a survey, where a person is randomly selected from a group of 246 people, we are asked to find the probability that the person is:

For a), the total number of Democrats is 100, and out of these, 45 are males. So the probability is 45/100 = 0.45.

For b), the total number of males is 127, and out of these, 54 are Republicans. So the probability is 54/127 ≈ 0.425.

For c), the total number of Independents is 48, and out of these, 20 are females. So, the probability is 20/48 ≈ 0.417.

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Answer:

Option (E) 648

Step-by-step explanation:

the 3 digit number can be represented by the blanks as " _ _ _  "

Now,

we have 10 choices ( i.e 0,1,2,3,4,5,6,7,8,9) available for each place in the blank if no condition is applied.

For the hundreds digit, using the conditions given in the question, we have 8 choices left

as 1 and 0 cannot be included in the hundreds place.

for the tens place

we will have 9 choices left out of 10 ( as 1 choice is less because we cannot have same number as on the hundred place )

similarly, for the ones place we have 9 choices left out of 10 ( as 1 choice is less because we cannot have same number as on the tens place )

Therefore,

Total possibilities = 8 × 9 × 9 = 648

Hence,

Option (E) 648

Answer:

y(t)= 11/3 e^(-t) - 5/2 e^(-2t) -1/6 e^(-4t)

Step-by-step explanation:

[tex] Y(s)=\frac{s+3}{(s^2+3s+2)(s+4)} + \frac{s+3}{s^2+3s+2} +\frac{1}{s^2+3s+2} [/tex]

We know that [tex] s^2+3s+2=(s+1)(s+2)[/tex], so we have

[tex] Y(s)=\frac{s+3+(s+3)(s+4)+s+4}{(s+1)(s+2)(s+4)}  [/tex]

By using the method of partial fraction we have:

[tex] Y(s)=\frac{11}{3(s+1)} - \frac{5}{2(s+2)} -\frac{1}{6(s+4)} [/tex]

Now we have:

[tex] y(t)=L^{-1}[Y(s)](t) [/tex]

Using linearity of inverse transform we get:

[tex] y(t)=L^{-1}[\frac{11}{3(s+1)}](t) -L^{-1}[\frac{5}{2(s+2)}](t) -L^{-1}[\frac{1}{6(s+4)}](t) [/tex]

Using the inverse transforms

[tex] L^{-1}[c\frac{1}{s-a}]=ce^{at} [/tex]

we have:

[tex] y(t)=11/3 e^{-t} - 5/2 e^{-2t} -1/6 e^(-4t) [/tex]

Answer:

Step-by-step explanation:

a) The game tree for k = 5 has been drawn in the uploaded picture below where C stands for continuing and S stands for stopping:

b) Say we were to use backward induction we can clearly observe that stopping is optimal decision for each player in every round. Starting from last round, if player 1 stops he gets $3 otherwise zero if continues. Hence strategy S is optimal there.

Given this, player 2’s payoff to C is $3, while stopping yields $4, so second player will also chooses to stop. To which, player 1’s payoff in k = 3 from C is $1 and her payoff from S is $2, so she stops.

Given that, player 2 would stop in k = 2, which means that player 1 would stop also in k = 1.

The sub game perfect equilibrium is therefore the profile of strategies where both players always stop: (S, S, S) for player 1, and (S, S) for player 2.

c) Irrespective of whether both players would be better off if they could play the game for several rounds, neither can credibly commit to not stopping when given a chance, and so they both end up with small payoffs.

i hope this helps, cheers

Final answer:

The subgame perfect equilibrium of the envelope game for any finite integer k is that both players will choose to stop in the final round (k) and each player will keep their own money.

Explanation:

Extensive-form tree for k = 5:

Backward Induction:

To find the subgame perfect equilibrium, we start from the last round (round 5) and work our way backwards: 1. In round 5, both players have the choice to stop or continue. Since both players want to maximize their earnings, they will both choose to stop, resulting in each player keeping their own money. 2. In round 4, player 2 knows that player 1 will choose to stop in round 5. Therefore, player 2 will choose to stop in round 4, resulting in each player keeping their money. 3. In round 3, player 1 knows that player 2 will choose to stop in round 4. Therefore, player 1 will choose to stop in round 3, resulting in each player keeping their money. 4. In rounds 2 and 1, both players have the choice to stop or continue. Since both players want to maximize their earnings and they know that the other player will choose to stop in the previous rounds, they will both choose to stop, resulting in each player keeping their money.

Backward Induction Outcome for Any Finite Integer k:

Based on the backward induction analysis, the outcome of the game for any finite integer k is that both players will choose to stop in the final round (k) and each player will keep their own money. This outcome is the subgame perfect equilibrium of the game, as it represents the strategy that maximizes the earnings for both players.

The data, the number of days traveled by randomly selected employees, is classified as quantitative, discrete data with a ratio level of measurement.

The data in question, namely, the number of days traveled last month by 100100 randomly selected employees, is considered quantitative data. This is because it deals with numbers that can be quantitatively analyzed. In terms of whether the data is discrete or continuous, it is discrete. The number of days traveled can be counted in whole numbers (you can't travel 2.5 days for example); thus, it is a countable set of data. Lastly, considering the level of measurement, the data falls under the ratio level as it not only makes sense to say that someone traveled more days than someone else (therefore an ordered relationship), but it also makes sense to say someone traveled twice as many days as someone else (giving us a proportion and a well-defined zero point).

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Answer:

We can conclude that the value of 1732 is a statistic who represent the sample selected. And the value of 1622 represent the population mean from all the customers with the previous data.

Step-by-step explanation:

Previous concepts

A statistic or sample statistic "is any quantity computed from values in a sample", for example the sample mean, sample proportion and standard deviation

A parameter is "any numerical quantity that characterizes a given population or some aspect of it".

Solution to the problem

For this case the analyst knows that from previous records that the mean for all the customers (that represent the population of interest) is [tex] \mu = 1622[/tex]

He have a random sample of n =500 customers from the previous month and he knows that 1732 represent the sample mean for the selected customers calculated from the following formula:

[tex] \bar X = \frac{\sum_{i=1}^{500} X_i}{500}= 1732[/tex]

So on this case we can conclude that the value of 1732 is a statistic who represent the sample selected. And the value of 1622 represent the population mean from all the customers with the previous data.

Final answer:

The number 1732 represents the sample mean of monthly credit card purchases for a sample of 500 customers, distinct from the population mean of $1622.

Explanation:

The number 1732 in the scenario given refers to the sample mean, which is the average amount of money spent on purchases last month by the 500 customers in the sample. This contrasts with the population mean, which the analyst knows to be $1622 for all customers holding the credit card. The difference between the sample mean and the population mean can be a subject of further statistical analysis to understand customer behavior and spending patterns better.

Answer:

P = 100 lbs in tension

F = 125 lbs in shear force downward direction

Moments are: 50 lb-ft and 125 lb-ft both in counter clockwise direction

Step-by-step explanation:

Forces

If we consider the entire structure as a system with two external forces P and F acting on it.

Translating these forces at point E,

Hence point E experiences the following forces:

P = 100 lbs in tension

F = 125 lbs in shear force

Moments

The bending moments are caused by the forces acting at respective offsets from point E

P = 100 lbs causes a bending moment of Mp = 100 lbs * (6/12) ft = 50 lb-ft

F = 125 lbs causes a bending moment of Mf = 125 lbs*(12/12)ft = 125 lb-ft

Moments are: 50 lb-ft and 125 lb-ft

Answer:

See explanation below.

Step-by-step explanation:

Definition

The cross product is a binary operation between two vectors defined as following:

Let two vectors [tex] a = (a_1 ,a_2,a_3) , b=(b_1, b_2, b_3)[/tex]

The cross product is defined as:

[tex] a x b = (a_2 b_3 -a_3 b_2, a_3 b_1 -a_1 b_3 ,a_1 b_2 -a_2 b_1)[/tex]

The last one is the math definition but we have a geometric interpretation as well.

We define the angle between two vectors a and b [tex]\theta[/tex] and we assume that [tex] 0\leq \theta \leq \pi[/tex] and we have the following equation:

[tex] |axb| = |a| |b| sin(\theta)[/tex]

And then we conclude that the cross product is orthogonal to both of the original vectors.

Some properties

Let a and b vectors

If two vectors a and b are parallel that implies [tex] |axb| =0[/tex]

If [tex] axb \neq 0[/tex] then [tex]axb[/tex] is orthogonal to both a and b.

Let u,v,w vectors and c a scalar we have:

[tex] uxv =-v xu[/tex]

[tex] ux (v+w) = uxv + uxw[/tex] (Distributive property)

[tex] (cu)xv = ux(cv) =c (uxv)[/tex]

[tex] u. (vxw) = (uxv).w[/tex]

Other application of the cross product are related to find the area of a parallelogram for two dimensions where:

[tex] A = |axb|[/tex]

And when we want to find the volume of a parallelepiped in 3 dimensions:

[tex] V= |a. (bxc)|[/tex]

Answer:

option C. (x + 4) and (x − 8)

Step-by-step explanation:

A factor is one of the linear expressions of a single-variable of the polynomial.

Given: y = (1/4)(x + 4)(x - 8)

When y = 0

∴ (1/4)(x + 4)(x - 8) = 0 ⇒ multiply both sides by 4

∴ (x + 4)(x - 8) = 0

So, the factors of the function are (x+4) and (x-8)

The answer is option C. (x + 4) and (x − 8)

The ODE has characteristic equation

[tex]r^2+6r+9=(r+3)^2=0[/tex]

with roots [tex]r=-3[/tex], and hence the characteristic solution

[tex]y_c=C_1e^{-3t}+C_2te^{-3t}[/tex]

For the particular solution, assume an ansatz of [tex]y_p=ae^{-t}[/tex], with derivatives

[tex]\dfrac{\mathrm dy_p}{\mathrm dt}=-ae^{-t}[/tex]

[tex]\dfrac{\mathrm d^2y_p}{\mathrm dt^2}=ae^{-t}[/tex]

Substituting these into the ODE gives

[tex]ae^{-t}-6ae^{-t}+9ae^{-t}=4ae^{-t}=4e^{-t}\implies a=1[/tex]

so that the particular solution is

[tex]\boxed{y(t)=C_1e^{-3t}+C_2te^{-3t}+e^{-t}}[/tex]

Answer:

Enthalpy of Vaporization of substance X = 489.15KJ/mol

Step-by-step explanation:

The Concept of Clausius Clapeyron equation is applied. This equation allows us to calculate the vapor pressure of a liquid over a some range of temperatures. The Clausius Clapeyron equation make use of the assumption that the heat of vaporization does not change as the temperature changes.

The question has data attached to it, I have added the other details and a step by step derivation and application of the Clausius Clapeyron equation was done.

Answer:

λ = 1.3252 x 10⁻⁴

Step-by-step explanation:

Since we are already given the half-life, the decay expression can be simplified as:

[tex]N(t) = N_0*e^{-\lambda t}\\\frac{N(half-life)}{N_0}=0.5[/tex]

For a half-life of t =5230 years:

[tex]0.5 = e^{(-\lambda t)} \\ln(0.5) = -\lambda * 5230\\\lambda = 1.3252*10^{-4}[/tex]

The decay-rate parameter λ for C-14 is 1.3252 x 10⁻⁴

Answer:

find derivative of function

sub in x value of point to find gradient of tangent

put gradient into y=(gradient)x+c

Sub in point and solve for c

you have found the equation of the tangent.

Answer: Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line

Step-by-step explanation:

$4.95

Step-by-step explanation:

Initial price of gas was $5.00 a month ago

Today price of a gas is $5.50 per gallon

Increase in price= $5.50-$5.00=$0.50

%increase= 0.50/5.00 *100 =10%

Current price= $5.50

decrease current price by 10% is by multiplying the current price by 90%

90/100 * 5.50 = $4.95

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Answer:

Part A: the a quadratic function that reflects the company's revenue.

R = (800+40x)(80-2x) = 64,000 + 1,600 x - 80 x²

Part B: The price should the company charge to maximize its revenue = $60

Step-by-step explanation:

company that sells 800 phones each week when it charges $80 per phone. It sells 40 more phones per week for each $2 decrease in price

Part A: Find the a quadratic function that reflects the company's revenue.

Let the number of weeks = x, and the revenue R(x)

So, the number of sold phones = 800 + 40x

And the cost of the one phone = 80 - 2x

∴ R = (800+40x)(80-2x)

∴ R = 64,000 + 1,600 x - 80 x²

Part B: What price should the company charge to maximize its revenue?

The equation of the revenue represent a parabola

R = 64,000 + 1,600 x - 80 x²

The maximum point of the parabola will be at the vertex

see the attached figure

As shown, the maximum will be at the point (10, 72000)

Which mean, after 10 weeks

The number of sold phones = 800 + 40*10 = 1,200 phones

The price of the phone = 80 - 2 * 10 = 80 - 20 = $60

So, the price should the company charge to maximize its revenue = $60

In this exercise we have to use the knowledge of quadratic function to calculate the value of the company in this way we can say uqe;

A) [tex]R= 64,000 + 1,600 X - 80 X^2[/tex]

B)[tex]V= \$60[/tex]

A) First, we find the a quadratic function that reflects the company's revenue:

[tex]R = (800+40X)(80-2X)\\ R = 64,000 + 1,600X - 80 X^2[/tex]

B)The equation of the revenue represent a parabola:

[tex]R = 64,000 + 1,600X - 80 X^2[/tex]

As shown, the maximum will be at the point 10, which mean, after 10 weeks the number of sold phones

[tex]S = 800 + 40*10 = 1,200 \\P=1,200/200=60[/tex]

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Answer:

21

Step-by-step explanation:

7! / (5! x 2!) = 42/2 = 21

other explanation:

TTHHHHH THTHHHH THHTHHH THHHTHH THHHHTH THHHHHT ... 6

HTTHHHH HTHTHHH ........................................................................................... 5

HHTTHHH HHTHTHH ............................................................................................4

HHHTTHH HHHTHTH .............................................................................................3

HHHHTTH HHHHTHT .............................................................................................2

HHHHHTT ...................................................................................................................1

6+5+4+3+2+1 = 21

The number of different ways should be 21.

Since there is  7-letter permutations can be formed from 5 identical H's and two identical T's

So,

[tex]= 7! \div (5! \times 2!) \\\\= 42 \div 2[/tex]

= 21

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Answer:

There are 252 possible outcomes.

Step-by-step explanation:

For each economic course, the student can select nine mathematic courses.

For each mathematic couse, the student can select four computer courses.

There are 7 economic courses.

So in all, there are 9*4*7 = 252 possible outcomes, that is, the number of different ways which the student can select his courses.

The confidence interval for the mean fluoride ion concentration in toothpaste at a 95% confidence level is [tex]$0.14 \pm 0.06\%$[/tex].

To calculate the confidence interval for the mean fluoride ion concentration in toothpaste, we use the formula:

[tex]\[ \text{Confidence interval} = \text{Mean} \pm \left( \text{Critical value} \times \frac{\text{Standard deviation}}{\sqrt{\text{Sample size}}} \right) \][/tex]

Given:

- Mean (sample mean) = 0.14%

- Standard deviation = 0.05%

- Sample size (replicate measurements) = 5

- Confidence level = 95%

We need to find the critical value corresponding to a 95% confidence level. Since the sample size is small (n < 30), we use a t-distribution and degrees of freedom [tex]\(df = n - 1 = 5 - 1 = 4\)[/tex].

From the t-distribution table or a statistical calculator, the critical value for a 95% confidence level with 4 degrees of freedom is approximately 2.776.

Now, we can calculate the confidence interval:

[tex]\[ \text{Confidence interval} = 0.14 \pm \left( 2.776 \times \frac{0.05}{\sqrt{5}} \right) \][/tex]

[tex]\[ \text{Confidence interval} = 0.14 \pm \left( 2.776 \times \frac{0.05}{\sqrt{5}} \right) \]\[ \text{Confidence interval} = 0.14 \pm 0.06 \][/tex]

So, the confidence interval is [tex]$0.14 \pm 0.06\%$[/tex].

Therefore, the correct option is [tex]$0.14( \pm 0.06) \%$[/tex].

Complete Question:

A measurement of fluoride ion in tooth paste from 5 replicate measurements delivers a mean of 0.14 % and a standard deviation of 0.05 %. What is the confidence interval at 95 % for which we assume that it contains the true value?

[tex]$0.14( \pm 0.06) \%$[/tex]

[tex]$0.14( \pm 6.2) \%$[/tex]

[tex]$0.14( \pm 0.07) \%$[/tex]

[tex]$0.14( \pm 0.69) \%$[/tex]

Answer:

The growth factor b is 1.04

Step-by-step explanation:

we know that

In this problem we have a exponential function of the form

[tex]y=a(b^x)[/tex]

where

y ---> is the number of students enrolled at a college

x ----> the number of years

a is the initial value or y-intercept

b is the growth factor

b=(1+r)

r is the rate of change

we have

[tex]a=15,000\ students[/tex]

[tex]r=4\%=4/100=0.04[/tex]

[tex]b=1+0.04=1.04[/tex]

therefore

The exponential function is equal to

[tex]y=15,000(1.04^x)[/tex]

Answer:

d. No, you should examine the situation to identify lurking variables that may be influencing both variables

Step-by-step explanation:

Hello!

Finding out that there is a regression between two variables is not enough to claim that there is a causation relationship between the two of them. First you have to test if other factors are affecting the response variable, if so, you have to control them or test how much effect they have. Once you controled all other lurking variables you need to design an experiment, where only the response and explanatory variables are left uncontroled, to learn if there is a regression and its strenght.

If after the experiment, you find that there is a significally strog relationship between the variables, then you can imply causation between the two of them.

I hope it helps!

Answer:

The random variables in this case are discrete since they have a Multinomial distribution.

The probability mass function for a discrete random variable X is given by:

[tex]P(X=x_{i} )[/tex]

Where are [tex]x_{i}[/tex] are possible values of X.

The joint probability mass function of two discrete random variables X and Y is defined as

P(x,y) =P(X=x,Y=y).

It follows that, The joint probability mass function of [tex]X_{3} , X_{4}[/tex] is :

[tex]P(X_{3}, X_{4} ) = P( X_{3} = x_{3}, X_{4} = x_{4} ) =\frac{1}{8} +\frac{3}{8} =\frac{1}{2}[/tex]

In a Multinomial Distribution, variables are independent. Hence, the joint mass function of a pair (X3 , X4) is the product of their individual mass functions. Their specific joint mass function equals P(X3=x3)P(X4=x4) = (n choose x3)(1/8)^x3(3/8)^x4 for x3+x4 ≤ n and x3, x4 ≥ 0.

This problem relates to the concept of a Multinomial Distribution in probability theory. The Multinomial Distribution describes the probabilities of potential outcomes from a multinomial experiment.

In this particular case, you are given that (X1, X2, X3, X4) follows a Multinomial Distribution with parameters n (number of trials) and 4 categories, with known probabilities 1/6, 1/3, 1/8, and 3/8 respectively.

You are asked to derive the joint mass function of the pair (X3, X4). This is actually very straightforward. Due to the properties of a multinomial distribution, these two variables are independent and the joint mass function is simply the product of the individual mass functions.

So the joint mass function of X3 and X4 would be P(X3=x3, X4=x4) = P(X3=x3)P(X4=x4) = (n choose x3)(1/8)^x3(3/8)^x4 provided x3+x4 ≤ n and each of x3,x4 are nonnegative.

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Answer:

1.33 x 10⁻⁸ seconds

Step-by-step explanation:

Assuming that the speed of light is 299,792,458 m/s, and that in order to bounce of Bill's forehead, hit the mirror and return back to his eyes, light must travel 4 meters (distance to the mirror and back) the time that it takes for light to travel is:

[tex]t=\frac{4}{299,792,458} \\t=1.33*10^{-8}[/tex]

It takes 1.33 x 10⁻⁸ seconds.

Answer:

a) [tex] E(u|X=1)= E(v|X=1) + E(X|X=1) = E(v) +1 = 0 +1 =1+[/tex]

b) [tex]E(Y| X=1)= E(1|X=1) + E(X|X=1) + E(u|X=1) = E(1) + 1 + E(v) + 0 = 1+1+0=2[/tex]

c) [tex] E(u|X=2)= E(v|X=2) + E(X|X=2) = E(v) +2 = 0 +2 =2[/tex]

d) [tex]E(Y| X=2)= E(1|X=2) + E(X|X=2) + E(u|X=2) = E(2) + 2 + E(v) + 2 = 2+2+2=6[/tex]

e) [tex] E(u|X) = E(v+X |X) = E(v|X) +E(X|X) = E(v) +E(X) = 0+1=1[/tex]

f) [tex] E(Y|X) = E(1+X+u |X) = E(1|X) +E(X|X) + E(u|X) = 1+1+1=3[/tex]

g) [tex]E(u) = E(v) +E(X) = 0+1=1[/tex]

h) E(Y) = E(1+X+u) = E(1) + E(X) +E(v+X) = 1+1 + E(v) +E(X) = 1+1+0+1 = 3[/tex]

Step-by-step explanation:

For this case we know this:

[tex] Y = 1+X +u[/tex]

[tex] u = v+X[/tex]

with both Y and u random variables, we also know that:

[tex] [tex] E(v) = 0, Var(v) =1, E(X) = 1, Var(X)=2[/tex]

And we want to calculate this:

Part a

[tex] E(u|X=1)= E(v+X|X=1)[/tex]

Using properties for the conditional expected value we have this:

[tex] E(u|X=1)= E(v|X=1) + E(X|X=1) = E(v) +1 = 0 +1 =1[/tex]

Because we assume that v and X are independent

Part b

[tex]E(Y| X=1) = E(1+X+u|X=1)[/tex]

If we distribute the expected value we got:

[tex]E(Y| X=1)= E(1|X=1) + E(X|X=1) + E(u|X=1) = E(1) + 1 + E(v) + 0 = 1+1+0=2[/tex]

Part c

[tex] E(u|X=2)= E(v+X|X=2)[/tex]

Using properties for the conditional expected value we have this:

[tex] E(u|X=2)= E(v|X=2) + E(X|X=2) = E(v) +2 = 0 +2 =2[/tex]

Because we assume that v and X are independent

Part d

[tex]E(Y| X=2) = E(1+X+u|X=2)[/tex]

If we distribute the expected value we got:

[tex]E(Y| X=2)= E(1|X=2) + E(X|X=2) + E(u|X=2) = E(2) + 2 + E(v) + 2 = 2+2+2=6[/tex]

Part e

[tex] E(u|X) = E(v+X |X) = E(v|X) +E(X|X) = E(v) +E(X) = 0+1=1[/tex]

Part f

[tex] E(Y|X) = E(1+X+u |X) = E(1|X) +E(X|X) + E(u|X) = 1+1+1=3[/tex]

Part g

[tex]E(u) = E(v) +E(X) = 0+1=1[/tex]

Part h

E(Y) = E(1+X+u) = E(1) + E(X) +E(v+X) = 1+1 + E(v) +E(X) = 1+1+0+1 = 3[/tex]

Answer:

a) Mean = 2.8

b) Median = 3

c) Mode = 4

d) Mid range = 2.5

e) Option C) Only the mode makes sense since the data is nominal.  

Step-by-step explanation:

We are given the following data set in the question:

1, 4, 4, 4, 2, 1, 4, 3, 1, 4, 4, 3, 3, 1

a) Mean

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{39}{14} = 2.78 \approx 2.8[/tex]

b) Median

[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]

Sorted data:

1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4

[tex]\text{Median} = \dfrac{7^{th}+8^{th}}{2} = \dfrac{3+3}{2} = 3[/tex]

c) Mode

Mode is the observation with highest frequency. Since 4 appeared maximum time  

Mode = 4

d) Mid range

It is the average of the smallest and largest observation of data.

[tex]\text{Mid Range} = \dfrac{1+4}{2} = 2.5[/tex]

e) Measure of center

Option C) Only the mode makes sense since the data is nominal.

Answer:

a) The 95% confidence interval for the population proportion of college students who work to pay for tuition andliving expenses is (0.4239, 0.5161).

b) The 99% confidence interval for the population proportion of college students who work to pay for tuition and living expenses is (0.4094, 0.5306).

c)The margin of error increases as the confidence level increases.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 450, p = 0.47[/tex]

a) Provide a 95% confidence interval for the population proportion of college students who work to pay for tuition andliving expenses.

95% confidence interval

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.47 - 1.96\sqrt{\frac{0.47*0.53}{450}} = 0.4239[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.47 + 1.96\sqrt{\frac{0.47*0.53}{450}}{119}} = 0.5161[/tex]

The 95% confidence interval for the population proportion of college students who work to pay for tuition andliving expenses is (0.4239, 0.5161).

b. Provide a 99% confidence interval for the population proportion of college students who work to pay for tuition andliving expenses.

95% confidence interval

So [tex]\alpha = 0.01[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.47 - 2.575\sqrt{\frac{0.47*0.53}{450}} = 0.4094[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.47 + 2.575\sqrt{\frac{0.47*0.53}{450}}{119}} = 0.5306[/tex]

The 99% confidence interval for the population proportion of college students who work to pay for tuition and living expenses is (0.4094, 0.5306).

c What happens to the margin of error as the confidence is increased from 95% to 99%?

The margin of error is the subtraction of the upper end by the lower end of the interval, divided by 2. So

95% confidence interval

[tex]M = \frac{(0.5161 - 0.4239)}{2} = 0.0461[/tex]

99% confidence interval

[tex]M = \frac{(0.5306 - 0.4094)}{2} = 0.0606[/tex]

The margin of error increases as the confidence level increases.