The formula Upper A equals 23.1 e Superscript 0.0152 tA=23.1e0.0152t models the population of a US​ state, A, in​ millions, t years after 2000.

a. What was the population of the state in​ 2000?
b. When will the population of the state reach 28.328.3 ​million?

Answer:

a) [tex]P(5.8<X<8.2)=P(\frac{5.8-7}{1.2}<Z<\frac{8.2-7}{1.2})=P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.841-0.159=0.683[/tex]

b) [tex]P(5.8<\bar X <8.2) = P(Z<3)-P(Z<-3)=0.999-0.0014=0.9973[/tex]

c) For part a we are just finding the probability that an individual baby would have a weight between 5.8 and 8.2. So we can't compare the result of part a with the result for part b.

For part b we are finding the probability that the mean of 9 babies (from random sampling) would be between 5.8 and 8.2, so on this case we have a distribution with a different deviation depending on the sample size. And for this reason we have different values

Step-by-step explanation:

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".  The letter [tex]\phi(b)[/tex] is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words: [tex]\phi(b)=P(z<b)[/tex]

Let X the random variable that represent the weights of​ full-term newborn babies of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(7,1.2)[/tex]

a. What is the probability that one newborn baby will have a weight within 1.2 pounds of the meanlong dashthat ​is, between 5.8 and 8.2 ​pounds, or within one standard deviation of the​ mean?

We are interested on this probability

[tex]P(5.8<X<8.2)[/tex]

And 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(5.8<X<8.2)=P(\frac{5.8-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{8.2-\mu}{\sigma})[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(5.8<X<8.2)=P(\frac{5.8-7}{1.2}<Z<\frac{8.2-7}{1.2})=P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.841-0.159=0.683[/tex]

b. What is the probability that the average of nine ​babies' weights will be within 1.2 pounds of the​ mean; will be between 5.8 and 8.2 ​pounds?

And 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(7,\frac{1.2}{\sqrt{9}})[/tex]

The z score on this case is given by this formula:

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

And if we replace the values that we have we got:

[tex]z_1=\frac{5.8-7}{\frac{1.2}{\sqrt{9}}}=-3[/tex]

[tex]z_2=\frac{8.2-7}{\frac{1.2}{\sqrt{9}}}=3[/tex]

For this case we can use a table or excel to find the probability required:

[tex]P(5.8<\bar X <8.2) = P(Z<3)-P(Z<-3)=0.999-0.0014=0.9973[/tex]

c. Explain the difference between​ (a) and​ (b).

For part a we are just finding the probability that an individual baby would have a weight between 5.8 and 8.2. So we can't compare the result of part a with the result for part b.

For part b we are finding the probability that the mean of 9 babies (from random sampling) would be between 5.8 and 8.2, so on this case we have a distribution with a different deviation depending on the sample size. And for this reason we have different values

Answer:

after 15%, discount = $5,015

after 10%, discount = $5,310

after 4%, discount=  $5,664

Step-by-step explanation:

1.discount of  15 % of the price listed

so 15 % of $5,900 will be=    15/100      x 5900  =    885 $

 so after discounting the price =$5,900 - $885    = $5,015

2. discount of  10 % of the price listed

so 10 % of $5,900 will be=    10/100      x 5900  =    590 $

 so after discounting the price =$5,900 - $590    = $5,310

3. discount of  4 % of the price listed

so 4 % of $5,900 will be=    4/100      x 5900  =    236 $

 so after discounting the price =$5,900 - $236  = $5,664

Answer:

[tex]\large\boxed{x=\pm\dfrac{3\pi}{20}+\dfrac{2n\pi}{5}\ \vee\ x=\pm\dfrac{\pi}{20}+\dfrac{2n\pi}{5}}[/tex]

Step-by-step explanation:

[tex][tex]\cos^2(5x)=\sin^2(5x)\qquad\text{substitute}\ t=5x\\\\\cos^2t=\sin^2t\qquad\text{use}\ \sin^2\theta+\cos^2\theta=1\to\sin^2\theta=1-\cos^2\theta\\\\\cos^2t=1-\cos^2t\qquad\text{add}\ \cos^2t\ \text{to both sides}\\\\2\cos^2t=1\qquad\text{divide both sides by 2}\\\\\cos^2t=\dfrac{1}{2}\Rightarrow \cos t=\pm\sqrt{\dfrac{1}{2}}\\\\\cos t=\pm\dfrac{\sqrt2}{2}\\\\\cos t=-\dfrac{\sqrt2}{2}\Rightarrow t=\pm\dfrac{3\pi}{4}+2n\pi\\\\\cos t=\dfrac{\sqrt2}{2}\Rightarrow t=\pm\dfrac{\pi}{4}+2n\pi[/tex]

[tex]t=5x\\\\5x=\pm\dfrac{3\pi}{4}+2n\pi\ \vee\ 5x=\pm\dfrac{\pi}{4}+2n\pi\qquad\text{divide both sides by 5}\\\\x=\pm\dfrac{3\pi}{20}+\dfrac{2n\pi}{5}\ \vee\ x=\pm\dfrac{\pi}{20}+\dfrac{2n\pi}{5}[/tex]

The trigonometric equation 2cos^2(x) + 2 = 4 has solutions x = 0 and x = π in the interval [0, 2π). To include all possible solutions, we express them as x = 2nπ and x = π + 2nπ with n as any integer.

To solve the trigonometric equation 2cos^2(x) + 2 = 4, we start by simplifying the equation:

This equation has two solutions in the interval [0, 2π): x = 0 and x = π (which are 0 and 3.1416 when rounded to four decimal places). To express the infinite set of solutions that occur at every period of cos(x), we add 2nπ to each solution, where n is any integer, giving the final solutions as x = 2nπ and x = π + 2nπ.

the complete Question is given below:

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0, 2π)

and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."

2cos^2 (x) + 2 = 4

Enter your answer in radians, as an exact answer when possible. Multiple answers should be separated by commas.

Answer:

a) (y-16)/y = -7*e∧(-0.016946*t)

b) y = 6.34

c) t = 129.66 years in 2055

Step-by-step explanation:

a) dy/dt = ky*(16-y)

Solving the differential equation we have

dy / (y*(y-16)) = -k dt

∫ dy / (y*(y-16)) = ∫ -k dt

(-1/16)*Ln (y) + (1/16)*Ln (y-16) = -k*t + C

(1/16) Ln ((y-16)/y) = -k*t + C

Ln ((y-16)/y) = -16*k*t + C  

(y-16)/y = C*e∧(-16*k*t)

If  t = 0  and y = 2

(2-16)/2 = C*e∧(0)  

C = -7   then we have

(y-16)/y = -7*e∧(-16*k*t)

In 1975 we have t = 1975 - 1925= 50 years   and  y = 4

(4-16)/4 = -7*e∧(-16*k*50)

k= - Ln (3/7) / 800 = 0.001059

Finally, the differential equation will be

(y-16)/y = -7*e∧(-16*0.001059*t)

(y-16)/y = -7*e∧(-0.016946*t)

b)  In 2015 we have t = 2015 – 1925 = 90 years

(y-16)/y = -7*e∧(-0.016946*90)

Solving the equation we get

y = 6.34

c) If  y = 9

(9-16)/9 = -7*e∧(-0.016946*t)

t = 129.66 years  in 2055

The solution for (a)[tex]a) (y-16)/y = -7*e^{(-0.016946*t)}[/tex]b) y = 6.34 and (c) the value of t is  129.66 years  in 2055

We have given that,

[tex]a) dy/dt = ky*(16-y)[/tex]

By using variable separable form we have,

A variable separable differential equation is any differential equation in which variables can be separated

Therefore by solving the differential equation we have

[tex]dy / (y*(y-16)) = -k dt[/tex]

integrating both side with respect to t

[tex]\int dy / (y*(y-16)) = \int -k dt[/tex]

Solve the integration of the above

[tex](-1/16)*ln (y) + (1/16)*ln (y-16) = -k*t + C[/tex]

[tex](1/16) ln ((y-16)/y) = -k*t + C[/tex]

[tex]ln ((y-16)/y) = -16*k*t + C[/tex]

[tex](y-16)/y = C*e^{(-16*k*t)}[/tex]

If  t = 0  and y = 2

[tex](2-16)/2 = C*e^{0}[/tex]

C = -7   then we have

[tex](y-16)/y = -7*e^{(-16*k*t)}[/tex]

In 1975 we have t = 1975 - 1925= 50 years   and  y = 4

[tex](4-16)/4 = -7*e^{(-16*k*50)[/tex]

[tex]k= - Ln (3/7) / 800 = 0.001059[/tex]

Finally, the differential equation will be

[tex](y-16)/y = -7*e^{(-16*0.001059*t)}[/tex]

[tex](y-16)/y = -7*e^{(-0.016946*t)}[/tex]

b)  In 2015 we have t = 2015 – 1925 = 90 years

[tex](y-16)/y = -7*e^{(-0.016946*90)}[/tex]

Solving the equation we get

y = 6.34

c) If  y = 9

[tex](9-16)/9 = -7*e^{(-0.016946*t)}[/tex]

Therefore we get the value of t is  129.66 years  in 2055

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Answer:

Δy = 1

dy = 1

Step-by-step explanation:

Data provided in the question:

dx = Δx

y = x

x = 1,

Δx = 1

Now,

we know,

Δy = f( x + Δx ) - f(x)

also, we have

y = f(x) = x

thus,

f( x + Δx ) =  x + Δx

Therefore,

Δy = ( x + Δx ) - x

on substituting the respective values, we get

Δy = ( 1 + 1 ) - 1

or

Δy = 1

and,

dy = f'(x) = [tex]\frac{d(x)}{dx}[/tex]

or

dy = 1

Step-by-step explanation:

(a, b) → (a, -b)

This is a reflection across the x-axis.

(a, b) → (a, b+5)

This is a translation 5 units up.

(a, b) → (b, -a)

This is a rotation of 270° about the origin.

Answer:

If the 62 year old man makes a lump-sum withdrawal from the plan or tax structure, his investments would start incurring ordinary income taxes without attracting any other form of penalties. However, it has to be noted that prior to withdrawal of the lump-sum, his investments would grow without incurring income taxes.

Answer:

1) n=114

2) n=59

3) On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.

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".  

The population proportion have the following distribution

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 80% of confidence, our significance level would be given by [tex]\alpha=1-0.80=0.2[/tex] and [tex]\alpha/2 =0.1[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.28, z_{1-\alpha/2}=1.28[/tex]

Part 1

The margin of error for the proportion interval is given by this formula:  

[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]    (a)  

And on this case we have that [tex]ME =\pm 0.06[/tex] or 6% points, and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

Since we don't have a prior estimate of [tex]\het p[/tex] we can use 0.5 as a good estimate, replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.5(1-0.5)}{(\frac{0.06}{1.28})^2}=113.77[/tex]  

And rounded up we have that n=114

Part 2

On this case we have a prior estimate for the population proportion and is [tex]\hat p =0.85[/tex] so replacing the values into equation (b) we got:

[tex]n=\frac{0.85(1-0.85)}{(\frac{0.06}{1.28})^2}=58.027[/tex]

And rounded up we have that n=59

Part 3

On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.

Answer:

We can conclude that the drying time in hours for type A is significantly higher than the drying time for type B. And the margin above it's between 4.90 and 17.5 hours at 2% of significance.

Step-by-step explanation:

Previous concepts  

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".  

[tex]\bar X_1 =76.3[/tex] represent the sample mean 1

[tex]\bar X_2 =65.1[/tex] represent the sample mean 2

n1=11 represent the sample 1 size  

n2=9 represent the sample 2 size  

[tex]s_1 =4.5[/tex] sample standard deviation for sample 1

[tex]s_2 =5.1[/tex] sample standard deviation for sample 2

[tex]\mu_1 -\mu_2[/tex] parameter of interest.

Confidence interval

The confidence interval for the difference of means is given by the following formula:  

[tex](\bar X_1 -\bar X_2) \pm t_{\alpha/2}\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}[/tex] (1)  

The point of estimate for [tex]\mu_1 -\mu_2[/tex] is just given by:

[tex]\bar X_1 -\bar X_2 =76.3-65.1=11.2[/tex]

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:  

[tex]df=n_1 +n_2 -1=11+9-2=18[/tex]  

Since the Confidence is 0.98 or 98%, the value of [tex]\alpha=0.02[/tex] and [tex]\alpha/2 =0.01[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.01,18)".And we see that [tex]t_{\alpha/2}=\pm 2.55[/tex]  

The standard error is given by the following formula:

[tex]SE=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}[/tex]

And replacing we have:

[tex]SE=\sqrt{\frac{4.5^2}{11}+\frac{5.1^2}{9}}=2.175[/tex]

Confidence interval

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

[tex]11.2-2.55\sqrt{\frac{4.5^2}{11}+\frac{5.1^2}{9}}=5.65[/tex]  

[tex]11.2+2.55\sqrt{\frac{4.5^2}{11}+\frac{5.1^2}{9}}=16.75[/tex]  

So on this case the 98% confidence interval would be given by [tex]5.65 \leq \mu_1 -\mu_2 \leq 16.75[/tex]  

But let's assume that the confidence interval given is true 4.90 hrs < μ1 - μ2 < 17.50 hrs

What does the confidence interval suggest about the population means?

We can conclude that the drying time in hours for type A is significantly higher than the drying time for type B. And the margin above it's between 4.90 and 17.5 hours at 2% of significance.

Answer:

The percentage of regional phone companies whose operating expenses were closer to the mean than the operating expenses of Company A were to the mean is 73%.

Step-by-step explanation:

The Company A's operating expenses were $27.00. This is $2.93 less than the regional mean.

[tex]\Delta E=29.93-27.00=2.93[/tex]

The companies whose operating expenses are closer to the mean are the ones that have expenses $2.93 below or above the mean.

The fraction of companies that are closer to the mean is equal to the proability of having expenses between those two limits:

[tex]z_1=(M-\mu)/\sigma=-2.93/2.65=-1.105\\\\z_2=+1.105[/tex]

[tex]P(|z|\leq1.105)=P(z\leq 1.105)-P(z<-1.105)=0.86542-0.13458=0.73[/tex]

To find the percentage of companies with operating expenses closer to the mean than Company A, calculate the Z score for Company A. Then find the percentile, and double it to account for both sides of the normal distribution.

To find out the percentage of regional phone companies that had operating expenses closer to the mean than the ones of Company A, we need to calculate the Z-score for Company A. The

Z-score

is a measure of how many standard deviations an element is from the mean. In this particular case, you can calculate the Z score using the formula:

Z = (X - μ) / σ

, where X is the value from the dataset (in this case, Company A's operating expenses), μ is the mean of the dataset, and σ is the standard deviation of the dataset.

So using the data from the question:

This will give you a percentage that can be understood as the percentage of regional phone companies which had operating expenses closer to the mean than Company A's were to the mean.

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For this case we must simplify the following expression:

[tex](\frac {3} {4} + \frac {4} {5} i) - (\frac {1} {2} - \frac {3} {10} i) =[/tex]

By law of multiplication signs we have to:

[tex]- * + = -\\- * - = +\\\frac {3} {4} + \frac {4} {5} i- \frac {1} {2} + \frac {3} {10} i =[/tex]

We add similar terms:

[tex]\frac {3} {4} - \frac {1} {2} + \frac {4} {5} i + \frac {3} {10} i =\\\frac {3 * 2-4 * 1} {4 * 2} + \frac {4 * 10 + 5 * 3} {10 * 5} i =\\\frac {2} {8} + \frac {55} {50} i =[/tex]

We simplify:

[tex]\frac{1}{4}+\frac{11}{10}i[/tex]

Answer:

[tex]\frac{1}{4}+\frac{11}{10}i[/tex]

Answer:

-any value in an interval or collection of intervals

Step-by-step explanation:

Discrete random variables are only integers in the interval.

Continuous random variables are all the values(integers, fractional, negative, positive) in an interval, or a collection of intervals.

The correct answer is:

-any value in an interval or collection of intervals

A continuous random variable can assume any value in an interval or collection of intervals. Unlike discrete random variables, which can only take integer values, continuous variables can take a range of values, like measurements such as height.

A continuous random variable is a type of variable in statistics that can assume any value within an interval or collection of intervals. This is different from a discrete random variable which can only take on a finite or countable number of values (often integer values).

For example, if we measure the height of a person, it could be any value within the feasible range, say between 0 meter and 3 meters. This is a continuous variable because there are infinite possibilities between 0 and 3, including measurements like 1.73 meters or 2.45 meters etc. It's not limited to integer values (like 1 m, 2 m, etc.), nor only fractional values, nor just positive integers within an interval.

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Answer:

a. 10,000 Hours

Step-by-step explanation:

MTBF (Mean Time Between Failures) can be calculated using the formula:

[tex]MTBF={(L-F)}*{n}[/tex] where

[tex]{MTBF=(200-100)}*{100}=10000[/tex]

Final answer:

The MTBF for a set of devices tested by the manufacturer, with failures evenly distributed from 100 to 200 hours, is 15,000 hours, which is calculated using the sum of an arithmetic series formula.

Explanation:

The student is asking about the mean time between failures (MTBF) for a set of devices. MTBF is a basic measure of reliability for repairable items and represents the average time expected between failures. To calculate MTBF, you divide the total operational time by the number of failures. In this case, the device manufacturer tests 100 devices, with the first failing after 100 hours and the last after 200 hours, which suggests a linear distribution of failures over time.

Assuming a linear and even distribution of failures from 100 to 200 hours for the 100 devices, you would calculate the total operational time before each device failed and then divide by the number of devices. The calculation would be the sum of an arithmetic series:

MTBF = (100 hours + 101 hours + ... + 200 hours) / 100 devices

We can use the formula for the sum of an arithmetic series:

S = n/2 * (a1 + an)

Where:

n is the number of terms in the series (here, 100 devices),

a1 is the first term in the series (100 hours),

an is the last term in the series (200 hours),

Now apply the formula:

MTBF = 100/2 * (100 + 200)

MTBF = 50 * 300

MTBF = 15,000 hours

So, the answer is b. 15,000 hours.

Answer:

B is greater than A.

Step-by-step explanation:

We have been given that 40% of A is equal to $300 and 30% of B is equal to $400. We are asked to compare both quantities.

Let us find A and B using our given information.

[tex]\frac{40}{100}*A=300[/tex]

[tex]0.40A=300[/tex]

[tex]\frac{0.40A}{0.40}=\frac{300}{0.40}[/tex]

[tex]A=750[/tex]

Similarly, we will find B.

[tex]\frac{30}{100}*B=400[/tex]

[tex]0.30*B=400[/tex]

[tex]\frac{0.30B}{0.30}=\frac{400}{0.30}[/tex]

[tex]B=1333.3333[/tex]

We know that smaller percent of big number is greater than bigger percent of a small number.

Therefore, B is greater than A.

Answer:

Step-by-step explanation:

40% Of A = 300

30% of B = 400

40a / 100 = 300, 40a = 3  divide both sides by 3 ,

then a = 13.33 approximately 13

30b = 400 = 400, 30b = 4 divide both sides by 30, then b =7.5

comparing both results where a = 13 , b = 7.5

a is greater than b with difference of 5.5

To form a larger 4-inch cube, Clare will need 512 wooden cubes with an edge length of 1/2 inch. We find this by dividing the volume of the large cube (64 cubic inches) by the volume of the small one (1/8 cubic inch).

Clare is building a larger cube with an edge length of 4 inches, consisting of smaller wooden cubes each with an edge length of 1/2 inch. To solve this problem, we need to find out how many smaller cubes make up the volume of the larger cube. The volume of a cube is found by multiplying the length of an edge by itself three times, or cube the edge length.

So first, we calculate the volume of the large cube which is 4in * 4in * 4in = 64 cubic inches.

Next, we calculate the volume of a small cube which is (1/2in) * (1/2in) * (1/2in) = 1/8 cubic inch.

Finally, we divide the volume of the large cube by the volume of the small cube to find out how many small cubes are needed. Thus, 64 cubic inches / 1/8 cubic inch = 512. So, Clare will need 512 small wooden cubes to construct her larger cube.

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To find the number of little cubes needed, divide the volume of the larger cube by the volume of each little cube.

To find the number of little cubes Clare needs to build a larger cube with an edge length of 4 inches, we need to

determine the volume of the larger cube and divide it by the volume of each little cube.

The volume of the larger cube is calculated by multiplying the length of one side by itself three times (4 x 4 x 4 = 64 cubic inches).

The volume of each little cube is calculated by multiplying the length of one side by itself three times (1/2 x 1/2 x 1/2 = 1/8 cubic inches).

To find the number of little cubes needed, we divide the volume of the larger cube by the volume of each little cube (64 / (1/8) = 512 little cubes).

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Answer

b. Type II error

Step-by-step explanation:

The hypothesis was formulated as  

the solution can be seen in the attached document

In probability, if two events are mutually exclusive (cannot occur at the same time), the probability of either event occurring is calculated by adding the probabilities of each event separately.

The concept being referred to in your question is related to probability within the study of mathematics. When two events are mutually exclusive, it means they cannot occur at the same time. For example, when rolling a die, the events of throwing a '6' and a '3' are mutually exclusive; you cannot throw both on a single toss.

In reference to your question, the probability that one event or the other occurs, given that they are mutually exclusive, is calculated by adding the probabilities of each individual event. So if the probability of Event A is P(A) and the probability of Event B is P(B), then the probability that either A or B will occur (P(A U B)) equals P(A) + P(B).

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Answer:

Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2

Step-by-step explanation:

Set up hypotheses as

[tex]H_0: p1 = p2\\H_a: p1>p2[/tex]

(Right tailed t test)

Alpha = 0.05

Sample               I                II                Total

X                        117             141              248

N                        249           312            561

p                        0.4700     0.4519        0.4420

p difference = 0.0181    

Std dev = [tex]\sqrt{p(1-p)(\frac{1}{n_1} +\frac{1}{n_2} )}[/tex]

=0.0427

z statistic = 0.424

p value = 0.33724

Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2

Final answer:

To test whether p1 is greater than p2 using hypothesis testing, one needs to state the null and alternative hypotheses, calculate the test statistic and p-value, and then decide whether to reject or accept the null hypothesis based on the level of significance and the p-value. Calculating the test statistic and p-value requires specific formulas and is not provided here.

Explanation:

Conducting a Hypothesis Test for Two Population Proportions

A student would like assistance in conducting a hypothesis test for two population proportions. The steps to perform such a test include:

State the null and alternative hypotheses: The null hypothesis (H0) is that there is no difference between the two population proportions (p1 - p2 = 0), while the alternative hypothesis (Ha) posits that there is a difference (p1 > p2).

The random variable (P') represents the difference between the two sample proportions.

Calculate the test statistic: Using the provided sample sizes and successful outcomes, the test statistic is calculated using a formula based on the Z-distribution.

Calculate the p-value: The p-value is determined from the test statistic, indicating the probability of observing such a result if the null hypothesis were true.

At the 5 percent level of significance, compare the p-value to the alpha value of 0.05 to make a decision. If the p-value is less than alpha, reject the null hypothesis.

The Type I error would occur if the null hypothesis is incorrectly rejected when it is actually true.

The Type II error would occur if the null hypothesis is not rejected when it is actually false.

The test statistic and p-value cannot be calculated from the information provided without the appropriate formulas or statistical tools.

Decision Making and Errors

If the alpha level is greater than the p-value, the null hypothesis should be rejected, indicating there is evidence to suggest p1 is greater than p2.

Failure to reject the null hypothesis when it is false constitutes a Type II error.

Answer:

[tex]t=\frac{27.8-28.5}{\frac{4.1}{\sqrt{100}}}=-1.707[/tex]    

[tex]p_v =P(t_{99}<-1.707)=0.0455[/tex]

If we compare the p value with a significance level for example [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we reject the null hypothesis, so there is not enough evidence to conclude that the mean for the consumption is less than 28.5 gallons at 0.1 of significance, so we can reject the claim that person consume more than 28.5 gallons.

Step-by-step explanation:

Data given and notation    

[tex]\bar X=27.8[/tex] represent the mean for the account balances of a credit company

[tex]s=4.1[/tex] represent the population standard deviation for the sample    

[tex]n=1000[/tex] sample size    

[tex]\mu_o =28.5[/tex] represent the value that we want to test  

[tex]\alpha=0.1[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)    

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

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to determine if the mean for the person consume is more than 28.5 gallons, the system of hypothesis would be:    

Null hypothesis:[tex]\mu \geq 28.5[/tex]    

Alternative hypothesis:[tex]\mu < 28.5[/tex]    

We don't know the population deviation, so for this case we can use the t test to compare the actual mean to the reference value, and the statistic is given by:    

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic    

We can replace in formula (1) the info given like this:    

[tex]t=\frac{27.8-28.5}{\frac{4.1}{\sqrt{100}}}=-1.707[/tex]    

Calculate the P-value    

First we need to calculate the degrees of freedom given by:

[tex]df=n-1=100-1=99[/tex]

Since is a one-side lower test the p value would be:    

[tex]p_v =P(t_{99}<-1.707)=0.0455[/tex]

In Excel we can use the following formula to find the p value "=T.DIST(-1.707,99)"  

Conclusion    

If we compare the p value with a significance level for example [tex]\alpha=0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we reject the null hypothesis, so there is not enough evidence to conclude that the mean for the consumption is less than 28.5 gallons at 0.1 of significance, so we can reject the claim that person consume more than 28.5 gallons.

Answer:

We need to check the conditions in order to use the normal approximation.

[tex]np=100*0.2=20 \geq 10[/tex]

[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]

If we check the conditions with the estimated proportion we got:

[tex]n\hat p=100*0.41=41 \geq 10[/tex]

[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]

So we see that we satisfy the conditions and then we can apply the approximation.

Step-by-step explanation:

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

[tex]X \sim Binom(n=100, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

We need to check the conditions in order to use the normal approximation.

[tex]np=100*0.2=20 \geq 10[/tex]

[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]

If we check the conditions with the estimated proportion we got:

[tex]n\hat p=100*0.41=41 \geq 10[/tex]

[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]

So we see that we satisfy the conditions and then we can apply the approximation.

Answer: d. random variable.

Step-by-step explanation:

A random variable is a numerical description of outcomes of an experiment, it can be used to represent the possible values of a past experiment or yet-to-be-performed experiments. It is a variable whose values depends on outcome of a random occurrence. Random variables also allows the calculation of probability of an occurrence or result in a particular experiment.

The numerical description of the outcome of an experiment is best described as a random variable. It is not referred to as a descriptive statistic, a probability function, or variance.

In statistics, a numerical description of the outcome of an experiment is referred to as a random variable. The term random variable refers to a function that assigns a real number to each outcome of an experiment conducted according to a certain probability distribution. This term is central to probability theory and statistics, in which numerical results of random variables are analyzed to understand underlying processes or to make predictions.

On the other hand, descriptive statistics summarize and organize characteristics of a data set. A probability function is a mathematical function that provides the probabilities of occurrence of different possible outcomes. Variance is a measurement of spread between numbers in a data set.

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Answer: Houston residences

Step-by-step explanation:

Here , The objective of the researcher is to determine the average number of vehicles are registered to a typical Houston residence.

Clearly , the population is this situation is "Houston residences" having vehicles.

Note : 250 randomly selected residences are defining the sample of the entire population of Houston residences which is a subset of population.

Hence, the correct answer is Houston residences.

Answer:

C. No. In warm weather, more children will go outside and play

Step-by-step explanation:

The correct answer is option C: No. In warm weather, more children will go outside and play.

Correlation means that two variables are related, but it does not necessarily mean that one causes the other. In this case, there is a strong correlation between temperature and the number of skinned knees on playgrounds, but it does not tell us that warm weather causes children to trip.

Option A is incorrect because warm weather does not directly cause children to go outside and play. It may be a factor, but it is not the sole reason. Option B is incorrect because warm weather does not cause fewer children to trip and suffer skinned knees. In fact, the correlation suggests that more children are likely to be outside playing in warm weather, which could potentially increase the number of skinned knees. Option D is incorrect because warm weather does not directly cause more children to trip and suffer skinned knees.

The correlation suggests that the increase in skinned knees is likely due to more children being outside and playing, rather than the warm weather itself. To summarize, the strong correlation between temperature and the number of skinned knees on playgrounds indicates that in warm weather, more children will go outside and play. However, it does not tell us that warm weather causes children to trip.

To know more about temperature here

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This ensures that if both project 2 and project 4 are selected, project 5 must also be selected. Since x5 is a binary variable, 2x5 is equivalent to x5, ensuring that project 5 is selected when the condition is met. (option d)

Let's formulate constraints for each of the given statements:

a. Out of projects 1, 2, 4, and 6, CRT’s management wants to select exactly two projects.

Let [tex]\(x_i\)[/tex] be a binary decision variable representing whether project i is selected or not, where [tex]\(i = 1, 2, 4, 6\).[/tex]

The constraint can be formulated as:

[tex]\[x_1 + x_2 + x_4 + x_6 = 2\][/tex]

This ensures that exactly two out of the listed projects are selected.

b. Project 2 can be selected only if project 3 is selected and vice-versa.

Let [tex]\(x_2\) and \(x_3\)[/tex] be binary decision variables representing whether project 2 and project 3 are selected, respectively.

The constraints can be formulated as:

[tex]\[x_2 \leq x_3\]\[x_3 \leq x_2\][/tex]

These constraints ensure that if project 2 is selected, project 3 must also be selected, and vice versa.

c. Project 5 cannot be undertaken unless both projects 3 and 4 are also undertaken.

Let [tex]\(x_3\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 3, project 4, and project 5 are selected, respectively.

The constraint can be formulated as:

[tex]\[x_5 \leq x_3 + x_4\][/tex]

This ensures that if project 5 is selected, both project 3 and project 4 must also be selected.

d. If projects 2 and 4 are undertaken, then project 5 must also be undertaken.

Let [tex]\(x_2\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 2, project 4, and project 5 are selected, respectively.

The constraint can be formulated as:

[tex]\[x_2 + x_4 \leq 2x_5\][/tex]

The probability density function (pdf) is;

f(x)=[tex]\frac{1}{b - a}[/tex] for a ≤ x ≤ b.

while the mean is given as [tex]\frac{a + b}{2}[/tex]

And the standard deviation given as [tex]\sqrt{\frac{(b - a)^{2} }{12} }[/tex]

Answer: -None of these choices.

PDF = 1/(b-a)

Step-by-step explanation:

The probability density function f(x) of a random variable X that has a uniform distribution between a and b is given by:

PDF = 1/(b-a) for X€[a,b]

otherwise zero.

Answer:

F=y+z=4/6.25

Step-by-step explanation:

First, we have to consider that in the problem model we have only two possible states: sunny and cloudy. Now, according to the information given in the statement, we also have the behavior of the last two days. In any case, we can have four possible transitional states:

Today-Yesterday(S=Sunny, C=cloudy)

1) ST and SY (Sunny today and sunny yesterday)

2) ST and CY.

3) CT and SY.

4) CT and CY.

Now, according to the statement, the probabilities given for the four states can be expressed by the following matrix:

[tex]\left[\begin{array}{cccc}0.6&0&(1-0.6)&0\\0.5&0&(1-0.5)&0\\0&0.4&0&(1-0.4)\\0&0.2&0&(1-0.2)\end{array}\right][/tex]

Now, making w, x, y, z as the transition probabilities for the four states mentioned, we then have that:

x=0.6w+0.5x

w=1.25x (1)

x=0.4y+0.2z (2)

y=0.4w+0.5x

y= 0.4(1.25x)+0.5x=x

y=x (3)

replacing 3 in 2:

y=0.4y+0.2x

x=3y (4)

And as w+x+y+z= 1 (no more possible combinations):

w+x+y+z=1 (5)

So, replacing the expressions obtained previously in equation 5, we have finally that:

1.25x+x+x+3x=1

x=1/6.25=y

z=3x=3/6.25

So, the fraction of sunny days is given by:

F=y+z=4/6.25

Answer: [tex]x=\dfrac{25}{3}[/tex]

Step-by-step explanation:

The given equation : [tex]\log_2(3x+7)=5[/tex]

Using logarithmic property : [tex]\log_a N=M\to N=a^M[/tex]

The given equation will be equivalent to [tex](3x+7)=2^5[/tex]

[tex]\Rightarrow\ 3x+7=32[/tex]

Subtract 7 from both sides , we get

[tex]3x=25[/tex]

Divide both sides by 3 , we get

[tex]x=\dfrac{25}{3}[/tex]

Hence, the solution is [tex]x=\dfrac{25}{3}[/tex]

Answer:

177,827.941

Step-by-step explanation:

The magnitude of an earthquake is given by:

[tex]R=log(\frac{l}{l_0})[/tex]

The intensity of the Nephi earthquake is:

[tex]3.0=log(\frac{l_N}{l_0})\\10^3 =\frac{l}{l_0} \\l_N=1,000*l_0[/tex]

The intensity of the San Francisco earthquake is

[tex]8.25=log(\frac{l_S}{l_0})\\10^{8.25} =\frac{l_S}{l_0} \\l_S=177,827,941*l_0[/tex]

The intensity ratio is:

[tex]r= \frac{l_S}{l_N}=\frac{177,827,941*l_0}{1,000*l_0}\\r= 177,827.941[/tex]

The San Francisco earthquake was 177,827.941 times as intensive as the Nephi earthquake.

When comparing the intensity of the San Francisco earthquake (magnitude 8.25) and the Nephi earthquake (magnitude 3.0) on the Richter Scale, we find that the San Francisco earthquake was approximately 177,827.94 times more intensive than the Nephi earthquake.

To compare the intensity of two earthquakes using the Richter Scale, we need to apply the formula:
R = log10 I/Io, where R is the Richter magnitude, I is the intensity of the earthquake, and Io is a reference intensity.

Now, we know that the Richter scale is a base-10 logarithmic scale. This means that each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.

The earthquakes in question have magnitudes of 3.0 and 8.25. So, the intensity of the San Francisco earthquake compared to the Nephi earthquake is 10^(8.25-3) = 10^5.25.

Therefore, the San Francisco earthquake was approximately 177,827.94 times more intensive as the Nephi earthquake.

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Answer:

Step-by-step explanation:

the sign here means division

7 divided by the number in the box equals to 8

simply meaning 7 multiplied by 8 will give the number

7 x 8 = 56  

The probability that the fisherman stays for more than two hours is approximately 0.4512. The probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043. The expected number of fish that the fisherman catches is 0.6.

To solve this problem, we can use the concept of a Poisson process and the properties of the Poisson distribution.

(a) We want to find the probability that the fisherman stays for more than two hours. Since the fisherman quits at the end of the second hour if he has caught at least one fish, he will stay for more than two hours only if he hasn't caught any fish in the first two hours. Using the Poisson distribution, the probability of catching zero fish in two hours is given by P(X=0)=e^(-lambda*t)*(lambda^0)/(0!)=e^(-0.6*2)≈0.5488. Therefore, the probability that the fisherman stays for more than two hours is 1 - P(X=0) = 1 - 0.5488 ≈ 0.4512.

(b) The total time the fisherman spends fishing can be between two and five hours. This can happen in three ways: fishing for 2 hours without catching fish (P(X=0)) and then fishing for 3 more hours until catching the first fish (P(X=1)); fishing for 3 hours without catching fish (P(X=0)) and then fishing for 2 more hours until catching the first fish (P(X=1)); fishing for 4 hours without catching fish (P(X=0)) and then fishing for 1 more hour until catching the first fish (P(X=1)). Using the Poisson distribution, we can calculate the probabilities for each case: P(X=0) = e^(-0.6*2) ≈ 0.5488, P(X=1) = e^(-0.6*3)*(0.6^1)/(1!) ≈ 0.3293. Adding up these probabilities, we get P(X=0)*P(X=1) + P(X=0)*P(X=1) + P(X=0)*P(X=1) = 0.5488*0.3293 + 0.5488*0.3293 + 0.5488*0.3293 ≈ 0.5043. Therefore, the probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043.

(c) The expected number of fish that the fisherman catches can be found using the formula for the mean of a Poisson distribution, which is equal to the rate parameter lambda. In this case, lambda = 0.6, so the expected number of fish is 0.6.

(d) To find the expected total fishing time given that the fisherman has been fishing for four hours, we need to condition on the event that the fisherman hasn't caught any fish in the first four hours. Using the Poisson distribution, the probability of catching zero fish in four hours is given by P(X=0) = e^(-lambda*t)*(lambda^0)/(0!) = e^(-0.6*4) ≈ 0.3012. Therefore, the expected total fishing time given that the fisherman has been fishing for four hours is 4 + (1/P(X=0)) = 4 + (1/0.3012) ≈ 7.32 hours.

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