Fifty people in the civilian labor force are randomly selected and the sample average age iscomputed to be 36.4.(a) Find a 90% confidence interval for the mean age, ?, of all people in the civilian laborforce. Assume that the population standard deviation for the ages of civilian labor force is12.1 years. Interpret the confidence interval.(b) It is being claimed that the mean age of the population of civilian labor force is 40. Whatdo you conclude based on the confidence interval?

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:

Step-by-step explanation:

Answer:

0.0406, 0.8284,0.7887

Step-by-step explanation:

Given that Mattel Corporation produces a remote-controlled car that requires three AA batteries

X is N(34, 5.5)

Hence sample size of 25 would follow a t distribution with df = 24

This is because sample size <30

t distribution with df 24 would be bell shaped symmetrical about the mean and unimodal.

Std error of sample mean = std dev /sqrt n=[tex]\frac{5.5}{5} \\=1.1[/tex]

Prob (X>36) = [tex]P(t>\frac{36-34}{1.1} ) = P(t>1.82)\\= 0.04063[/tex]

i.e nearly 4.1% of the sample would have a mean useful life of more than 36 hours

X>33.5 implies [tex]t>-0.45[/tex]

=0.82837

=0.8284 proportion will have a mean useful life greater than 33.5 hours

Proportion between 33.5 and 36 hours

= [tex]0.3284+0.4593=0.7887[/tex]

The shape of the distribution of the sample mean is approximately normal due to the Central Limit Theorem. The standard error of the distribution of the sample mean can be calculated as 1.1 hours. Proportions of the sample mean falling above and between specific time intervals can be determined using z-scores and the Z-table.

The shape of the distribution of the sample mean, in this case, is approximately normal. This is because the distribution of the battery lives closely follows the normal probability distribution. When taking a sample of 25 batteries, the Central Limit Theorem states that the distribution of the sample mean approaches a normal distribution regardless of the shape of the original distribution.

The standard error of the distribution of the sample mean can be calculated using the formula: standard deviation / square root of sample size. In this case, the standard deviation is 5.5 hours and the sample size is 25. Therefore, the standard error is 5.5 / √25 = 1.1 hours.

To determine the proportion of the samples that will have a mean useful life of more than 36 hours, we need to calculate the z-score first. The formula for z-score is: (sample mean - population mean) / standard error. Plugging in the given values, we get (36 - 34) / 1.1 = 1.82. By looking up the z-score in the Z-table, we find the corresponding proportion is approximately 0.0344.

Similarly, to find the proportion of the sample that will have a mean useful life greater than 33.5 hours, we calculate the z-score: (33.5 - 34) / 1.1 = -0.45. By looking up the z-score in the Z-table, we find the corresponding proportion is approximately 0.3264.

To determine the proportion of the sample that will have a mean useful life between 33.5 and 36 hours, we subtract the proportion of the sample that will have a mean useful life greater than 36 hours from the proportion of the sample that will have a mean useful life greater than 33.5 hours. Therefore, 0.3264 - 0.0344 = 0.2920.

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Answer:

X[bar]= 115

Step-by-step explanation:

Hello!

Every Confidence interval to estimate the population mean are constructed following the structure:

"Estimator" ± margin of error"

Wich means that the intervals are centered around the sample mean. To know the value of the sample mean you have to make the following calculation:

[tex]X[bar]= \frac{Upper bond + Low bond}{2}[/tex]

[tex]X[bar]= \frac{115.6+114.4}{2}[/tex] = 115

Since both intervals were calculated with the information of the same sample, you can choose either to calculate the sample mean.

I hope it helps!

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:

Test Statistic is 2.34

Step-by-step explanation:

File Attached. kindly view it.

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:

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: 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:

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.

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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

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:

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:

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:

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:

  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:

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]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:

(a) 6.2843 billion

(b) 6.8716 billion

Step-by-step explanation:

Since there is a constant growth rate of 1.5% per year, and the population in 2002 was 6.1 billion people. The general equation for the total world population, in billions, after 2002 is:

[tex]P(t) = 6.1*(1+0.015)^t[/tex]

With t being the time, in years, after 2002.

a) What was the population in 2004?

[tex]t=2004-2002=2\\P(2) = 6.1*(1.015)^2\\P(2) = 6.2843 \ billion[/tex]

b) What was the population in 2010?

[tex]t=2010-2002=8\\P(8) = 6.1*(1.015)^8\\P(8) = 6.8716 \ billion[/tex]

Answer: 1/6

Step-by-step explanation:

Final answer:

The research question that can be answered using a chi-square goodness-of-fit test is option c: Is the distribution of political party affiliation for all registered voters in 2012 the same as stated for 2008?

Explanation:

The research question that can be answered using a chi-square goodness-of-fit test in this scenario is option c: Is the distribution of political party affiliation for all registered voters in 2012 the same as stated for 2008?

In a chi-square goodness-of-fit test, we compare the observed frequencies (in this case, the distribution of political party affiliation in 2008) with the expected frequencies (the distribution we are testing, which is the distribution of political party affiliation in 2012). If the observed frequencies differ significantly from the expected frequencies, we can conclude that there is a significant difference between the two distributions.

In this case, the observed distribution is 36% Democrats, 27% Republicans, and 37% Independents in 2008. We want to test if this distribution is the same as the distribution in 2012. By performing a chi-square goodness-of-fit test, we can determine if there is a significant difference between the two distributions.

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:

The point estimate for the population standard deviation of the length of the walking canes is 0.16.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].

In this problem

The point estimate of the standard deviation is the standard deviation of the sample.

In a sample of 83 walking canes, the average length was found to be 34.9in. with a standard deviation of 1.5. By the Central Limit Theorem, we have that:

[tex]s = \frac{1.5}{\sqrt{83}} = 0.16[/tex]

The point estimate for the population standard deviation of the length of the walking canes is 0.16.

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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Answer:

67,757 errors per million opportunities

Step-by-step explanation:

Assuming that each customer can only make a single complaint (1 error opportunity per customer), the number of errors per million opportunities (EPMO) is given by:

[tex]EPMO = \frac{complaints}{guests}*1,000,000 \\EPMO = \frac{29}{428}*1,000,000 \\EPMO = 67,757[/tex]

Refington hotel should expect 67,757 complaints per million guests.

[tex]\bf \cfrac{1}{x-3}-6\implies \stackrel{\textit{using the LCD of x-3}}{\cfrac{1-(x-3)6}{x-3}}\implies \cfrac{1-6x+18}{x-3}\implies \cfrac{-6x+19}{x-3}[/tex]

for the vertical asymptote, we simply zero out the denominator and solve for "x"

x - 3 = 0

x = 3   <---- that's the only vertical asymptote

for the horizontal asymptote, well, let's notice the degrees of the numerator and denominator, in this case, the degree of the numerator is 1, and the degree of the denominator is 1, thus when that occurs, the horizontal asymptote occurs at the fraction from the leading terms' coefficients.

[tex]\bf \cfrac{-6x+19}{1x-3}\implies \stackrel{\textit{horizontal asymptote}}{\cfrac{-6}{1}\implies -6 = y}[/tex]

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:

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

Final answer:

The slightly higher proportion of Americans who indicated they are afraid to fly in a random sample is not necessarily evidence of an increased fear of flying overall. This variation could be due to sample variability, and without statistical analysis like hypothesis testing or confidence interval construction, one cannot conclude a genuine increase in fear of flying among Americans.

Explanation:

A study conducted by an organization found that the proportion of Americans who were afraid to fly in 2006 was 0.10. When a random sample of 1,400 Americans results in 154 individuals indicating that they are afraid to fly, it might initially seem like evidence that the proportion of Americans afraid to fly has increased. However, this is not necessarily indicative of an overall trend. To understand why we need to consider statistical variability and the concept of a confidence interval.

Statistical fluctuations in samples can lead to results that differ from the actual population proportion. In this case, the sample proportion of individuals afraid to fly is roughly 0.11 (154/1400), only slightly higher than the reported 0.10 from 2006. Without conducting a hypothesis test or constructing a confidence interval around the sample proportion, it's not possible to definitively say whether this difference is statistically significant or just due to random chance.

Furthermore, sample size plays a crucial role in the reliability of estimates. Although a sample size of 1,400 may seem large, the inherent randomness in sample selection could still lead to results differing from the true population proportion. In summary, a slight increase in the sample proportion of individuals afraid to fly, compared to a previous study, is not conclusive evidence of a trend without further statistical analysis to support such a claim.

The data from a sample of 1,400 Americans revealing 154 individuals who are afraid to fly does not necessarily indicate an increase in the proportion of Americans with this fear compared to 2006. Statistically significant evidence through a hypothesis test is required to make such a conclusion since sample data can fluctuate due to random chance.

A random sample of 1,400 Americans resulted in 154 indicating that they are afraid to fly, which might suggest an increase from the 2006 study that showed a 0.10 proportion of Americans had this fear. However, this is not necessarily evidence that the proportion of Americans who are afraid to fly has increased due to potential sampling error or the natural fluctuation inherent in sample data. To determine if there is a statistically significant increase, one would need to perform a hypothesis test comparing the sample proportion to the known proportion of 0.10.

Variability can occur in survey results, and a single sample might not represent the entire population accurately. Also, the difference observed might be a result of random chance. Therefore, it's essential to conduct statistical tests to infer whether the observed change is meaningful and not due to sampling variability.

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%.