Express the null hypothesis and the alternative hypothesis in symbolic form. The owner of a football team claims that the average attendance at games is over 63,500 , and he is therefore justified in moving the team to a city with a larger stadium.

Answer:

a)  the time speed for  distance from the television camera to the rocket is changing at rate of 360 ft/sec

b) camera’s angle of elevation changing at the rate of 0.096 radians/ sec

Step-by-step explanation:

details and step by step explanation is in the images below

image I is for Solution of a)

image 2 is for Solution of b)

Answer: (1124.5619, 1315.4381)

Step-by-step explanation:

The confidence interval for population mean[tex](\mu)[/tex] when populatin standard deviation is unknown :-

[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]

, where [tex]\overline{x}[/tex] = Sample mean

[tex]s[/tex] =Sample standard deviation

t* = Critical t-value.

Given : n= 300

Degree of freedom : df = n-1 = 299

[tex]\overline{x}=1220[/tex]

[tex]s=840[/tex]

Confidence interval = 95%

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

Using t-distribution table ,

The critical value for 95% Confidence interval for significance level 0.05 and df = 299 : [tex]t^*=t_{\alpha/2,\ df}=t_{0.025,\ 299}=1.9679[/tex]

Then, a 95% confidence interval estimate of the average debt of all cardholders will be :-

[tex]1220\pm (1.9679)\dfrac{840}{\sqrt{300}}[/tex]

[tex]=1220\pm (1.9679)\dfrac{840}{17.3205080757}[/tex]

[tex]=1220\pm (1.9679)(48.4974226119)[/tex]

[tex]\approx1220\pm 95.4381=(1220-95.438,\ 1220+95.438)\\\\=(1124.5619,\ 1315.4381)[/tex]

Hence, a 95% confidence interval estimate of the average debt of all cardholders is (1124.5619, 1315.4381) .

Answer:

a) Approximate the change in R is 0.5 ohm.

b) The actual change in R is 0.04 ohm.

Step-by-step explanation:

Given : The total resistance R of two resistors connected in parallel circuit is given by [tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]

To find :

a) Approximate the change in R ?

b) Compute the actual change.

Solution :

a) Approximate the change in R

[tex]R_1=12\ ohm[/tex] and [tex]R_2=10\ ohm[/tex]

[tex]R_1[/tex] is decreased from 12 ohms to 11 ohms.

i.e. [tex]\triangle R_1=21-11=1\ ohm[/tex]

[tex]R_2[/tex] is increased from 10 ohms to 11 ohms.

i.e. [tex]\triangle R_2=11-10=1\ ohm[/tex]

The change in R is given by,

[tex]\frac{1}{\triangle R}=\frac{1}{\triangle R_1}+\frac{1}{\triangle R_2}[/tex]

[tex]\frac{1}{\triangle R}=\frac{\triangle R_2+\triangle R_1}{(\triangle R_1)(\triangle R_2)}[/tex]

[tex]\triangle R=\frac{(\triangle R_1)(\triangle R_2)}{\triangle R_2+\triangle R_1}[/tex]

[tex]\triangle R=\frac{(1)(1)}{1+1}[/tex]

[tex]\triangle R=\frac{1}{2}[/tex]

[tex]\triangle R=0.5\ ohm[/tex]

b) The actual change in Resistance

[tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]

[tex]\frac{1}{R}=\frac{1}{12}+\frac{1}{10}[/tex]

[tex]R=\frac{10\times 12}{10+12}[/tex]

[tex]R=\frac{120}{22}[/tex]

[tex]R=5.46\ ohm[/tex]

When resistances are charged, [tex]R_1=R_2=11[/tex]

[tex]\frac{1}{R'}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]

[tex]\frac{1}{R'}=\frac{1}{11}+\frac{1}{11}[/tex]

[tex]R'=\frac{11}{2}[/tex]

[tex]R'=5.5\ ohm[/tex]

Change in resistance is given by,

[tex]C=R'-R[/tex]

[tex]C=5.5-5.46[/tex]

[tex]C=0.04\ ohm[/tex]

Final answer:

The actual change in total resistance R of a parallel circuit as R_1 is decreased from 12 ohms to 11 ohms and R_2 is increased from 10 ohms to 11 ohms is calculated using the formula 1/R = 1/R_1 + 1/R_2.

Explanation:

The question is asking for the change in total resistance R of a parallel circuit when the resistances R_1 and R_2 are changed from 12 ohms to 11 ohms and 10 ohms to 11 ohms, respectively. To calculate the actual change, we can use the formula:

1/R = 1/R_1 + 1/R_2

Before the change, the total resistance is:

1/R_initial = 1/12 + 1/10

After the change, it becomes:

1/R_final = 1/11 + 1/11

By calculating both 1/R_initial and 1/R_final, we determine R_initial and R_final separately and find the difference between them to get the actual change in resistance.

Answer:

Step-by-step explanation:

For a hyperbola of the form ...

  [tex]\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1[/tex]

The vertices are located at (h±a, k), and the foci are located at (h±c, k), where ...

  [tex]c=\sqrt{a^2+b^2}[/tex]

Here, we have (h, k) = (-4, -3), a=3, b=4, and c=√(9+16) = 5.

So, the points of interest are ...

Answer:

previous was correct

Step-by-step explanation:

Answer:

  9.5 ft wide by 24 ft long

Step-by-step explanation:

A graphing calculator shows there are two solutions to the system of equations ...

  2x + y = 43 . . . . . . fence length when x= width

  xy = 228 . . . . . . .  area

The solutions are ...

  (width, length) = (9.5, 24)   or   (12, 19)

Since we want the length as long as possible, the choice of dimensions is ...

  length = 24 feet; width = 9.5 feet.

Answer:

The p value obtained was a low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of of driver fatalities related to alcohol is less from 0.77 or 77%.  

Step-by-step explanation:

1) Data given and notation n  

n=53 represent the random sample taken

X=35 represent the automobile driver fatalities in a certain county involved with an intoxicated driver

[tex]\hat p=\frac{35}{53}=0.660[/tex] estimated proportion of automobile driver fatalities in a certain county involved with an intoxicated driver

[tex]p_o=0.77[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

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

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the population proportion of driver fatalities related to alcohol is less than 77% or 0.77 in Kit Carson:  

Null hypothesis:[tex]p\geq 0.77[/tex]  

Alternative hypothesis:[tex]p < 0.77[/tex]  

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

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

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

3) Calculate the statistic  

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

[tex]z=\frac{0.660 -0.77}{\sqrt{\frac{0.77(1-0.77)}{53}}}=-1.903[/tex]

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This methos is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is an unilateral lower test the p value would be:  

[tex]p_v =P(z<-1.903)=0.0285[/tex]  

So the p value obtained was a low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of of driver fatalities related to alcohol is less from 0.77 or 77%.  

Answer:

398.411

Step-by-step explanation:

Explanation has been given in the following attachments.

Final answer:

(a) Using the differential equation[tex]\( dy/dt = ky(1 - y/K) \)[/tex]  with[tex]\( y(0) = 2 \times 10^7 \)[/tex] kg, the biomass after one year is approximately [tex]\( 3.11 \times 10^7 \) kg.[/tex]

(b) Solving the equation for [tex]\( y(t) = 4 \times 10^7 \) kg yields \( t \approx 3.41 \) years.[/tex]

Explanation:

(a) To find the biomass a year later, we can use the given initial condition and the provided differential equation. Substituting[tex]\( y(0) = 2 \times 10^7 \) kg, \( k = 0.75 \), and \( K = 9 \times 10^7 \)[/tex] kg into the equation, we get:

[tex]\[ \frac{{dy}}{{dt}} = 0.75y \left(1 - \frac{{y}}{{9 \times 10^7}}\right) \][/tex]

Now, we can solve this first-order ordinary differential equation. One method is the separation of variables, then integration. Integrating both sides gives:

[tex]\[ \int \frac{{dy}}{{y(1 - \frac{{y}}{{9 \times 10^7}})}} = \int 0.75 dt \][/tex]

After solving the integrals and applying the initial condition, we find [tex]\( y(1) \approx 3.11 \times 10^7 \) kg.[/tex]

(b) To determine how long it takes for the biomass to reach [tex]\( 4 \times 10^7 \) kg,[/tex] we can use the same differential equation and solve for [tex]\( t \)[/tex] when [tex]\( y(t) = 4 \times 10^7 \) kg.[/tex]This involves solving a separable differential equation and then finding the time [tex]\( t \)[/tex] that satisfies this condition. By solving this equation, we find[tex]\( t \approx 3.41 \) years.[/tex]

Answer:

The competition can be played in 7 different ways

Step-by-step explanation:

First game : X

Second game : Y

Third game : Y

After that either X or Y can win

The competition ends with Y winning as he/she won 3 games in a row

more games are required to decide the winner

more games are required to decide the winner

The competition ends with X winning as he/she won 3 games in a row

The competition ends after the seventh game. Whoever wins seventh game wins the competition as he/she would've won 4 games in total by then.

more games are required to decide the winner

The competition ends with Y winning as he/she won 4 games in total

The competition ends after the seventh game. Whoever wins seventh game wins the competition as he/she would've won 4 games in total by then.

∴The competition can be played in 7 different ways

The problem requires combinatorial analysis to find all potential sequences of wins leading to player X or Y's victory. There are different scenarios of victory based on the number of games player X and Y wins and the sequence of their wins.

This problem showcases combinatorial analysis where we consider the different sequences of wins that leads either team to victory. In the given scenario X has already won the first game and Y has won the second and third games.

From this point, some possible winning combinations could be:

These are representative of the different sequences of wins that could occur. To calculate the total number of possible sequences, you would consider the different ways the remaining games can unfold until one player satisfies the win conditions.

https://brainly.com/question/34849457

#SPJ3

Answer:

Step-by-step explanation:

Sara's base pay is $80

(a) The 9t represents the additional amount of money that she earns if she gives instruction for t hours. 9 stands for $9 per hour

b)The term in the expression is t which stands for the number of hours that she instructs. The coefficient is 9 which stands for the hourly rate

c) The expression for 7 hours would be

We will substitute t into 80 + 9t. It becomes

80 + 9×7

d) in all, she will earn

80 + 9×7 = $143

Answer:

a) The 90% confidence interval would be given by (33.594;39.206)  

b) Since the 90% confidence interval not contains the value 40 we can say that this value at this confidence level is not the true population mean, because it's outside of the limits for the interval calculated.

Step-by-step explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Part a

[tex]\bar X=36.4[/tex] represent the sample mean  

[tex]\mu[/tex] population mean (variable of interest)  

[tex]\sigma=12.1[/tex] represent the population standard deviation  

n=50 represent the sample size  

90% confidence interval  

The confidence interval for the mean is given by the following formula:  

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)  

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]  

Now we have everything in order to replace into formula (1):  

[tex]36.4-1.64\frac{12.1}{\sqrt{50}}=33.594[/tex]  

[tex]36.4+1.64\frac{12.1}{\sqrt{50}}=39.206[/tex]  

So on this case the 90% confidence interval would be given by (33.594;39.206)  

Part b

Since the 90% confidence interval not contains the value 40 we can say that this value at this confidence level is not the true population mean, because it's outside of the limits for the interval calculated.

Answer:

[tex]\mu = 56.8[/tex]

[tex]\sigma = 12.1[/tex]

A)what is the probability that the sample mean will be More than 58 pounds

P(x>58)

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{58-56.8}{12.1}[/tex]

[tex]Z=0.09917[/tex]

refer the z table

P(x<58)=0.5359

P(X>58)=1-P(x<58)=1-0.5359=0.4641

Hence the probability that the sample mean will be More than 58 pounds is 0.4641

B)what is the probability that the sample mean will be More than 57 pounds

P(x>57)

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{57-56.8}{12.1}[/tex]

[tex]Z=0.0165[/tex]

refer the z table

P(x<57)=0.5040

P(X>57)=1-P(x<57)=1-0.5040=0.496

Hence the probability that the sample mean will be More than 57 pounds is 0.496

C)what is the probability that the sample mean will be Between 55 and 57 pound

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{57-56.8}{12.1}[/tex]

[tex]Z=0.0165[/tex]

refer the z table

P(x<57)=0.5040

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{55-56.8}{12.1}[/tex]

[tex]Z=-0.1487[/tex]

refer the z table

P(x<55)=0.4443

P(55<x<57)=P9x<57)-P(x<55) =0.5040-0.4443=0.0597

Hence the probability that the sample mean will be Between 55 and 57 pounds is 0.0597

D)what is the probability that the sample mean will be Less than 53 pounds

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{53-56.8}{12.1}[/tex]

[tex]Z=−0.314[/tex]

refer the z table

P(x<53)=0.3783

The probability that the sample mean will be Less than 53 pounds is 0.3783

E)what is the probability that the sample mean will be Less than 48 pounds

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

Substitute the values :

[tex]Z=\frac{48-56.8}{12.1}[/tex]

[tex]Z=−0.727[/tex]

refer the z table

P(x<48)=0.2358

The probability that the sample mean will be Less than 48 pounds is 0.2358

Answer:

Test Statistic is 2.34

Step-by-step explanation:

File Attached. kindly view it.

Answer:

  C.  6

Step-by-step explanation:

You can try the answers to see which satisfies the Pythagorean theorem:

  A.  4² +7² = 16+49 ≠ 117

  B.  5² +8² = 25 +64 ≠ 117

  C.  6² +9² = 36 +81 = 117 . . . . this (C) is the correct choice

and, for completeness, ...

  D.  9² +12² = 81 +144 ≠ 117

_____

Working out

You can also actually work the problem. The Pythagorean theorem tells you ...

  x² + (x+3)² = (√117)²

  2x² +6x +9 = 117

  2x² +6x = 108 . . . . subtract 9

  x² +3x = 54 . . . . . . divide by 2

  x(x +3) = 54 . . . . . . factor

Now, you can compare to your memorized times tables, where you find 6×9 = 54, so you know that x=6.

__

In case times tables are a challenge, you can continue to complete the square:

  x² +3x +1.5² = 54 +1.5²

  (x +1.5)² = 56.25 = 7.5² . . . . write as squares

  x = 7.5 -1.5 = 6 . . . . . . . . . . . square root, subtract 1.5

The value of x is 6.

_____

The other solution to the quadratic equation is x=-9, but negative segment lengths make no sense. We acknowledge, then ignore, that extraneous solution.

Answer:

Step-by-step explanation:

Confidence Interval can be calculated using P±ME where

And margin of error (ME) can be calculated using the formula

ME=[tex]\frac{z*s}{\sqrt{N} }[/tex] where

Margin of error, ME=[tex]\frac{1.645*15}{\sqrt{49} }[/tex] = 3.525

Then 90% confidence interval is $122±3.525

To interpret this, there is 90% probability that true population average price of text books is in the range $122±$3.525

Answer:

Step-by-step explanation:

Answer: she is 8.06 miles from her starting point

Step-by-step explanation:

The diagram in the attached photo describes Juliet's movement from her starting point to her current position.

A triangle ABC is formed

AB = distance that she walked towards west

BC = distance that she walked towards south

AC= x = distance that she is from her starting point

The diagram is a right angle triangle. So we can find the distance that she is from her starting point using Pythagoras theorem.

Hypotenuse^2 = opposite^2 + adjacent^2

Hypotenuse = x

Opposite = 7

Adjacent = 4

x^2 = 7^2 + 4^2

x^2 = 49 + 16 = 65

x = √65 = 8.06 miles

Answer:

The Domain is "All Real numbers", and the Range is "All real number less than or equal to 4" which coincides with option C in your problem.

Step-by-step explanation:

Recall that the Domain of a function is the set of all x-values for which there is a y-value obtained by the rule defined by the function.

In this case, whatever real number you enter for "x" in your functional expression, you will find another real value "y". That means that the actual Domain of your function is the full Real number line (all Real numbers).

Recall as well that the Range of a function is the set of y-values that are being "called" (or generated) as you use all the values for x in the Domain. In our case, it is great that they give the graph of the function (see attached image), so you can visualize that the function's graph is that of a "parabola" with branches opening downwards. You can see as well that the parabola presents a maximum value when x = -1, that means that any other value of x you use cannot give you as result a y-value that goes above that maximum.

If you evaluate that maximum value of the vertical coordinate by replacing "x" with "-1" in the actual function, you get:

[tex]f(x)=-(x+3)*(x-1)\\f(-1)=-(-1+3)*(-1-1)\\f(-1)=-(2)*(-2)\\f(-1)=4[/tex]

That means that the maximum y-value one can get from this function is "4". That is, the actual Range of the function can be any number that is smaller or equal to 4.

Bottom line: The Domain is "All Real numbers", and the Range is "All real number less than or equal to 4".

Answer:

its C

Step-by-step explanation:

edge 2020

Answer:

[tex]\triangle VWZ = \triangle YWZ[/tex]. Justification is given below.

Step-by-step explanation:

Given:

[tex]\angle WVZ = \angle WYZ[/tex]

WZ is the perpendicular bisector of VY.

∴[tex]\angle WZV = \angle WZY = 90\°\\and\\VZ = YZ[/tex]

In [tex]\triangle VWZ and \triangle YWZ[/tex]

[tex]\angle WVZ = \angle WYZ[/tex] Given

[tex]VZ = YZ[/tex]

[tex]\angle WZV = \angle WZY = 90\°[/tex]

∴ [tex]\triangle VWZ \cong \triangle YWZ[/tex] by ASA test

[tex]\triangle VWZ = \triangle YWZ[/tex]

Here the correspondence of the vertices of a triangle should be match hence the option is first one that is.

[tex]\triangle VWZ = \triangle YWZ[/tex]

Answer:

Step-by-step explanation:

VWZ=YWZ

Answer:

a) [tex]P(X <48500)=0.0304[/tex]

b) [tex]P(\bar X>50200)=1-0.994=0.0062[/tex]  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent the mean life span of a brand name tire, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu=50000,\sigma=800)[/tex]  

Part a

We want this probability:

[tex]P(X<48500)[/tex]

The best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X <48500)=P(Z<\frac{48500-50000}{800})=P(Z<-1.875)=0.0304[/tex]

Part b

Let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:

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

On this case  [tex]\bar X \sim N(50000,\frac{800}{\sqrt{100}})[/tex]

We want this probability:

[tex]P(\bar X>50200)=1-P(\bar X<50200)[/tex]

The best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

If we apply this formula to our probability we got this:

[tex]P(\bar X >50200)=1-P(Z<\frac{50200-50000}{\frac{800}{\sqrt{100}}})=1-P(Z<2.5)[/tex]

[tex]P(\bar X>50200)=1-0.994=0.0062[/tex]  

Final answer:

To determine the probability that a tire's lifespan is less than a certain value, you calculate its Z-score and look up the probability in a standard normal distribution table. For sample means, use the standard error to calculate Z-scores. In hypothesis testing, compare the p-value to α to decide on the claim.

Explanation:

Calculating Probabilities for Normally Distributed Tires' Lifespan

To calculate the probability that a single tire has a lifespan of less than 48,500 miles, we can use the Z-score formula. The formula for the Z-score is (X - μ) / σ, where X is the value we are looking at (48,500 miles), μ is the mean (50,000 miles), and σ is the population standard deviation (800 miles). Calculating the Z-score gives us (48,500 - 50,000) / 800 = -1.875. We can then look up this Z-score on a standard normal distribution table or use a calculator to find the corresponding probability.

For part b, looking at the probability for the mean of 100 tires being more than 50,200 miles, we would use the standard error of the mean, σ/√ n, where n is the sample size (100 in this case). We calculate the Z-score with this new standard deviation and find the probability corresponding to that Z-score using the same method.

It is essential to remember that for larger samples, the standard deviation of the mean decreases, and the calculations have to reflect this change.

Hypothesis Testing for Tires' Lifespan

Using a hypothesis test, we can determine if there is enough evidence to support or reject a claim about the population mean. In a hypothesis test, we compare the p-value to the level of significance (α = 0.05). If the p-value is lower than α, we reject the null hypothesis. In the example given with an alpha of 0.05 and a p-value of 0.0103, we would reject the null hypothesis, concluding that the average lifespan of the tires is likely less than the claimed 50,000 miles.

Answer:

Step-by-step explanation:

PQ=24

PS=19

PR=42

TQ=10

QR=19

SR=24

PT=21

SQ=20

m∠QRS=180-m∠PQR=180-106=74°

m∠PQS=m∠QSR=49°

m∠RPS=m∠PRQ=m∠QRS-m∠PRS=74-35=39°

m∠PSQ=m∠RQS=m∠PQR-m∠PQS=106-49=57°

m∠PQR=106°

m∠ QSR=49°

M∠PRS=35°

Answer: Our required probability is [tex]\dfrac{1}{7}[/tex]

Step-by-step explanation:

Since we have given that

P(Junior ) = [tex]\dfrac{1}{2}[/tex]

P(Senior) = [tex]\dfrac{1}{2}[/tex]

Let the given event be 'C' taking calculus.

P(C|J) = 10% = 0.10

P(C|S) = 60% = 0.60

We need to find the probability that the student is a junior.

So, our required probability is given by

[tex]P(J|C)=\dfrac{P(J).P(C|J)}{P(S).P(C|S)+P(J).P(C|J)}\\\\P(J|C)=\dfrac{0.5\times 0.1}{0.5\times 0.1+0.5\times 0.6}\\\\P(J|C)=\dfrac{0.05}{0.05+0.3}\\\\P(J|C)=\dfrac{0.05}{0.35}\\\\P(J|C)=\dfrac{5}{35}\\\\P(J|C)=\dfrac{1}{7}[/tex]

Hence, our required probability is [tex]\dfrac{1}{7}[/tex]

Answer:

Step-by-step explanation:

Hello!

You need to test the hypothesis that the slope of the regression is cero.

I've run in the statistic software the given data for Y and X and estimated the regression line:

Yi= 7.82 -1.60Xi

Where

a= 7.82

b=-1.60

Sb= 3.38

The hypothesis is:

H₀: β = 0

H₁: β ≠ 0

α: 0.05

This is a two-tailed test, the null hypothesis states that the slope of the regression is cero, this means that if the null hypothesis is true, there is no linear regression between Y and X.

The statistic for this test is a Student-t

t=  b - β   ~t[tex]_{n-2}[/tex]

      Sb

The critical values are:

Left: [tex]t_{n-2; \alpha /2} = t_{2; 0.025} = -4.303[/tex]

Right: [tex]t_{n-2; \alpha /2} = t_{2; 0.975} = 4.303[/tex]

t= -1.60 - 0 = -0.47

      3.38

the p-value is also two-tailed, you can calculate it by hand:

P(t ≤ -0.47) + (1 - P(t ≤ 0.47) = 0.3423 + (1 - 0.6603) =0.6820

With the level of significance of 5%, the decision is to not reject the null hypothesis. This means that the slope of the regression is equal to cero, i.e. there is no linear regression between the two variables.

I hope this helps!

The hypothesis test, utilizing a t-statistic of 3.002 and a p-value of 0.00765, rejects the null hypothesis, demonstrating a statistically significant linear relationship. The slope coefficient (β₁ = 1.637) suggests a moderately strong relationship.

Hypothesis Test for a Linear Relationship

In this hypothesis test, we are trying to determine whether there is a statistically significant linear relationship between the independent and dependent variables. The null hypothesis (H₀) states that there is no linear relationship (β₁ = 0), while the alternative hypothesis (H₁) states that there is a linear relationship (β₁ ≠ 0).

The test statistic used in this case is the t-statistic, which is calculated as the ratio of the estimated slope (β₁) to its standard error. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming that the null hypothesis is true.

In this particular example, the t-statistic is 3.002 and the p-value is 0.00765. The level of significance, which is the threshold for rejecting the null hypothesis, is typically set at 0.05.

Since the p-value (0.00765) is less than the level of significance (0.05), we reject the null hypothesis. This means that we have statistically significant evidence to conclude that there is a linear relationship between the independent and dependent variables. In other words, the slope of the regression line is not equal to zero, indicating that there is a change in the dependent variable as the independent variable changes.

Interpretation of Results

The rejection of the null hypothesis in this case indicates that there is a statistically significant linear relationship between the independent and dependent variables. This means that we can be confident that the observed relationship is not due to chance. The magnitude of the t-statistic (3.002) suggests that the relationship is moderately strong.

The interpretation of the slope coefficient (β₁) is that for every one-unit increase in the independent variable, the dependent variable is expected to increase by 1.637 units, on average. The standard error (0.5453) indicates the variability of the slope estimate.

In conclusion, the hypothesis test provides strong evidence to support the existence of a statistically significant linear relationship between the independent and dependent variables. The magnitude of the slope coefficient indicates that the relationship is moderately strong.

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The conclusion is that there is sufficient evidence to reject the publisher's claim that over 64% of their readers own a Rolls Royce.

When conducting a hypothesis test, there are two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1). In this scenario:

Null hypothesis (H0): The proportion of readers who own a Rolls Royce is equal to or less than 64%.

Alternative hypothesis (H1): The proportion of readers who own a Rolls Royce is greater than 64%.

The significance level, denoted by α, represents the probability of rejecting the null hypothesis when it is actually true. In this case, α = 0.01, indicating that there is a 1% chance of making a Type I error (incorrectly rejecting the null hypothesis).

After conducting the hypothesis test, the advertiser decided to reject the null hypothesis. This decision suggests that the evidence from the test was significant enough to conclude that the true proportion of readers who own a Rolls Royce is greater than 64%.

In summary, based on the test results at the 0.01 level of significance, the conclusion is that the publisher's claim that over 64% of their readers own a Rolls Royce is not supported by the evidence. The data suggest that the proportion may be higher than 64%.

Let [tex]f:\Bbb R\times\Bbb R^+\to\Bbb R[/tex].

a. [tex]f[/tex] is injective if [tex]f(x_1,y_1)=f(x_2,y_2)[/tex] for any two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the two points must be identical with [tex]x_1=x_2[/tex] and [tex]y_1=y_2[/tex].

[tex]f[/tex] is not injective because we can pick two points for which [tex]f[/tex] gives the same value:

[tex]f(2,1)=\dfrac21=2[/tex]

[tex]f(4,2)=\dfrac42=2[/tex]

b. [tex]f[/tex] is surjective if there exists [tex](x,y)[/tex] for which every point in the codomain [tex]\Bbb R[/tex] is obtained by [tex]f(x,y)[/tex].

This should be somewhat obvious if you consider 3 different cases:

So [tex]f[/tex] is surjective.

c. [tex]f[/tex] is bijective if it is both injective and surjective. The conclusions above show [tex]f[/tex] is not bijective.

Answer:

2.7 PM

Step-by-step explanation:

Since, the rate of spread is proportional to the product of fraction y of people who have heard the rumour and the fraction who have not heard,

[tex]\frac{dy}{dt}=ky(1-y)[/tex]

[tex]\frac{dy}{y(1-y)}=kdt[/tex]

[tex](\frac{1}{y}+\frac{1}{1-y})dy = kdt[/tex]

Integrating both sides,

[tex]\int (\frac{1}{y}+\frac{1}{1-y})dy = \int kdt[/tex]

[tex]\ln y - \ln (1-y) = kt + C[/tex]

[tex]\ln (\frac{y}{1-y}) = kt + C[/tex]

[tex]\frac{y}{1-y}=e^{kt + C}[/tex]

[tex]y = e^{kt+C} - y e^{kt+C}[/tex]

[tex]y(1+e^{kt+C}) = e^{kt+C}[/tex]

[tex]y = \frac{e^{kt+C}}{1+e^{kt+C}}[/tex]

If t = 0, ( at 8 AM ), y = [tex]\frac{80}{2100}[/tex]

[tex]\frac{80}{2100}= \frac{e^{0+C}}{1+e^{0+C}}[/tex]

[tex]\frac{4}{105}=\frac{e^C}{1+e^C}[/tex]

[tex]4 + 4e^C = 105e^C[/tex]

[tex]4 = 101e^C[/tex]

[tex]\implies e^C=\frac{4}{101}[/tex]

Now, at noon, i.e t = 4, y = [tex]\frac{1}{2}[/tex]

[tex]\frac{1}{2}=\frac{e^{4k}.\frac{4}{101}}{1+e^{4k}.\frac{4}{101}}[/tex]

[tex]\frac{1}{2}=\frac{4e^{4k}}{101+4e^{4k}}[/tex]

[tex]101 + 4e^{4k}=8 e^{4k}[/tex]

[tex]101 = 4e^{4k}[/tex]

[tex]\frac{101}{4}=e^{4k}[/tex]

[tex](\frac{101}{4})^\frac{1}{4} = e^k[/tex]

If [tex]y = \frac{90}{100}=\frac{9}{10}[/tex]

[tex]\frac{9}{10}= \frac{(\frac{101}{4})^\frac{t}{4}\times \frac{4}{101}}{1+(\frac{101}{4})^\frac{t}{4}\times \frac{4}{101}}[/tex]

Using graphing calculator,

t ≈ 6.722,

Hence, after 6.722 hours since 8 AM, i.e. on 2.7 PM ( approx ) the 90% of the population have heard the rumour.

Answer: 90% confidence interval would be (0.448, 0.552).

Step-by-step explanation:

Since we have given that

x = 125

n = 250

So, [tex]\hat{p}=\dfrac{x}{n}=\dfrac{125}{250}=0.5[/tex]

We need to find the 90% confidence interval.

so, z = 1.64

So, interval would be

[tex]\hat{p}\pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.5\pm 1.64\sqrt{\dfrac{0.5\times 0.5}{250}}\\\\=0.5\pm 0.0519\\\\=(0.5-0.0519,0.5+0.0519)\\\\=(0.448,0.552)[/tex]

Hence, 90% confidence interval would be (0.448, 0.552)

Final answer:

The 90% confidence interval is (0.462, 0.538). To calculate a confidence interval, use the formula p' ± z * √(p'q'/n). In this case, with x = 125, n = 250, and 90% confidence, the interval is (0.462, 0.538).

Explanation:

The 90% confidence interval is (0.462, 0.538).

To construct a confidence interval for a population proportion, you can use the formula for confidence interval: p' ± z * √(p'q'/n), where p' = x/n, q' = 1 - p', and z corresponds to the confidence level.

In this case, given x = 125, n = 250, and 90% confidence level, the confidence interval is (0.462, 0.538).

Answer: 1/6

Step-by-step explanation:

Answer: [tex]H_a: \mu >75[/tex]

Step-by-step explanation:

Alternative hypothesis [tex](H_a)[/tex]: It is a statement which always indicates that there is a significant difference between the groups being tested.

It always contains by < , > or ≠ sign .

Given claim  : Is the average speed on that stretch of highway significantly more than 75mph?

Here objective is whether  the average speed on that stretch of highway significantly more than 75mph.

Let [tex]\mu[/tex] be the population mean speed on that stretch of highway significantly more than 75mph.

Then, [tex]H_a: \mu >75[/tex]

Hence , the correct alternative hypothesis would be :  [tex]H_a: \mu >75[/tex]

The correct alternative hypothesis for this question is that the average speed on the highway is significantly greater than 75mph.

The correct alternative hypothesis for this question would be that the average speed on that stretch of highway is significantly greater than 75mph.

To determine if the average speed is significantly more than 75mph, a statistical test can be conducted. This test would compare the speeds of the 40 recorded vehicles to the 75mph speed limit and calculate the average speed. If the average speed is significantly higher than 75mph, it would support the alternative hypothesis.

For example, if the average speed of the 40 recorded vehicles is found to be 80mph with a small p-value, it would indicate that the average speed on that stretch of highway is significantly greater than 75mph.

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Answer

The answer and procedures of the exercise are attached in a microsoft word document.  

Explanation  

Please consider the data provided by the exercise. If you have any question please write me back. All the exercises are solved in a single sheet with the formulas indications.  

Answer:

a)  p(x) = (-1/30)*x + 30

b) xe = 240

c) The consumers' surplus at that demand is $ 960.00

Step-by-step explanation:

Given info

p₁ = 20

x₁ = 300

p₂ = 18

x₂ = 360

a) We can apply the following equation

p - p₁ = m*(x - x₁)

where

m is the slope, which can be obtained as follows

m = (p₂-p₁) / (x₂-x₁)

⇒ m = (18-20) / (360-300) = -2/60 = -1/30

then

p-20 = (-1/30)*(x-300)

⇒ 30*p - 600 = -x + 300

⇒ p(x) = (-1/30)*x + 30

b) If p(x) = 22

⇒  22 = (-1/30)*x + 30

⇒  xe = 240 passengers

c) If x = 0   ⇒  p(0) = (-1/30)*(0) + 30 = 30

then

we can get h as follows

h = p(0) - p(240) = 30 - 22 = 8

if b = x(equlibrium) = 240

we apply

A = b*h/2  ⇒  A = 240*8/2 = 960

The consumers' surplus at that demand is $ 960.00