The manager of a fund that provides loans for college students has estimated that the average monthly loan repayment for students borrowing from the fund is $75.00. You are to test this estimate. You take a sample of 20 students and find that the mean monthly payment is $69.46 with a standard deviation of $9.78. Which of the following statements is true about this test?

a. The value of the test statistic is -.57; therefore, the null hypothesis is rejected for mean = 0.02.
b. The value of the test statistic is -2.53; therefore, the null hypothesis is rejected for mean = 0.02.
c. The value of the test statistic is -2.53; therefore, the null hypothesis is rejected for mean = 0.05 but not for mean = 0.02.
d. The value of the test statistic is -.57; therefore, the null hypothesis is rejected for mean = 0.05 but not for mean = 0.02.

Answer:

the particular solution is

Y_{p}= C +D\sin 5t +E\cos 5t + F\exp 4t + G\exp -2t

the differential operator that annihilate the non homogeneous differential equation is

D(D^2+5)

Step-by-step explanation:

hello,

i believe the non homogeneous differential equation is

[tex]U^{''} - 2U^{'} - 8= \cos 5x + 7[/tex]

the homogeneous differential equation of the above is

[tex]u^{''} -2u^{'} -8 =0[/tex]

the differential form of the above equation is

[tex]D^2-2D-8=0[/tex]

[tex](D-4)(D+2)=0[/tex]

thus the roots are 4 and -2.

thus the solution of the homogenous differential equation is given as

[tex]Y_{h} (t)= A\exp{4t} + B\exp{-2t}[/tex]

the differential operator of the non homogeneous equation is given as

[tex](D-4)(D+2)(u)=\cos 5x +7[/tex]

the differential operator [tex]D^2 +5[/tex] annihilates [tex]\cos 5x[/tex] and the differential operator D annihilates 7

applying [tex]D(D^2+5)[/tex] to both sides of the differential equation we have;

(D-4)(D+2)(u)=\cos 5x +7

[tex]D(D^2+5)(D-4)(D+2)=D(D^2+5)(\cos5x+7)[/tex][tex]D(D^2+5)(D-4)(D+2)=0[/tex]

the roots of the characteristic polynomial of the diffrential equation above are [tex]0, \cmplx 5i, -\cmplx 5i, 4, -2[/tex]

thus the particular solution is

[tex]Y_{p}= C\exp{0}+D\sin 5t +E\cos 5t + F\exp {4t} + G\exp {-2t}[/tex]

this gives us the particular solution

[tex]Y_{p}= C +D\sin 5t +E\cos 5t + F\exp 4t + G\exp -2t[/tex]

To use the annihilator method for the differential equation [tex]\(u'' - 2u' - 8 = \cos(5x) + 7\),[/tex] the operator [tex]\(D^3 + 25D\)[/tex] will annihilate the nonhomogeneous part [tex]\(\cos(5x) + 7\).[/tex] This operator reduces the nonhomogeneous function to zero. The differential operator combines the annihilation of both the cosine and constant terms.

To determine the form of a particular solution for the given differential equation[tex]\(u'' - 2u' - 8 = \cos(5x) + 7\),[/tex] we first identify a differential operator that annihilates the nonhomogeneous part [tex]\(\cos(5x) + 7\).[/tex]

For [tex]\(\cos(5x)\)[/tex] , the appropriate annihilator is [tex]\(D^2 + 25\)[/tex], where [tex]\(D\)[/tex] represents differentiation with respect to [tex]\(x\)[/tex].

This is because applying [tex]\(D^2 + 25\)[/tex]  to [tex]\(\cos(5x)\)[/tex] will yield zero:

For the constant term 7, the annihilator is simply [tex]\(D\)[/tex], since the derivative of a constant is zero:

Combining these, the overall differential operator that will annihilate [tex]\(\cos(5x) + 7\)[/tex] is:

The differential operator [tex]\( D^3 + 25D \)[/tex] will annihilate the nonhomogeneous part [tex]\( \cos(5x) + 7 \),[/tex] reducing it to zero.

Answer:

Music place has the better buy

Step-by-step explanation:

Figure out how much they sell one for by diving the price by the quantity

CD Express offers 1 CD for $15

Music Place offers 1 CD fof $12.50

Answer:

Music Place

Step-by-step explanation:

Find the unit rates by dividing the price by the number of CDs

CD Express:

price/CDs

$60/4 CDS

60/4=15

$15 per CD

Music Place:

price/CDs

$75/6 CDs

75/6=12.5

$12.50 per CD

Music Place is the Better deal because 12.50 is less than 15

Answer:

one solution

Step-by-step explanation:

X-9=0

Add 9 to each side

X-9+9=0+9

x = 9

There is one solution

Answer:

1 yard

Step-by-step explanation:

Answer:

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

Step-by-step explanation:

From the diagram, our goal is to minimize the length of Rope AC passing through B.

First, we determine the length of the rope AC.

AC=AB+BC

In the first triangle,

[tex]|AB|^2=16^2+x^2\\|AB|=\sqrt{16^2+x^2}[/tex]

Similarly, in the second triangle,

[tex]|BC|^2=24^2+(30-x)^2\\|BC|=\sqrt{x^2-60x+1476}[/tex]

Length of the Rope, AC

[tex]L=\sqrt{16^2+x^2}+\sqrt{x^2-60x+1476}[/tex]

First, to minimize L,we find its derivative.

[tex]L'=\dfrac{x\sqrt{x^2-60x+1476}+(x-30)\sqrt{16^2+x^2}}{(\sqrt{16^2+x^2})(\sqrt{x^2-60x+1476})}[/tex]

Setting the derivative to zero

[tex]x\sqrt{x^2-60x+1476}+(x-30)\sqrt{16^2+x^2}=0\\-x\sqrt{x^2-60x+1476}=(x-30)\sqrt{16^2+x^2}\\$Square both sides\\x^2(x^2-60x+1476)=(x-30)^2(16^2+x^2)\\x^4-60x^3+1476x^2=x^4-60x^3+1156x^2-15360x+230400\\1476x^2=1156x^2-15360x+230400\\1476x^2-1156x^2+15360x-23040=0\\320x^2+15360x-23040=0\\x=1.46,-49.46[/tex]

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

The least amount of the rope is the smallest length that can be gotten from the pole

The rope should be placed at 12 feet from the 16 ft pole to use the least amount of rope.

The heights are given as:

[tex]\mathbf{h_1 = 16}[/tex]

[tex]\mathbf{h_2 = 24}[/tex]

The distance is given as:

[tex]\mathbf{d = 30}[/tex]

See attachment for the illustrating diagram

Considering the two right-angled triangles on the diagram, we have the following equations, using Pythagoras theorem

[tex]\mathbf{L_1 = \sqrt{x^2 + 16^2}}[/tex]

[tex]\mathbf{L_2 = \sqrt{(30 - x)^2 + 24^2}}[/tex]

Expand

[tex]\mathbf{L_1 = \sqrt{x^2 + 256}}[/tex]

[tex]\mathbf{L_2 = \sqrt{900 - 60x +x^2 + 576}}[/tex]

[tex]\mathbf{L_2 = \sqrt{1476- 60x +x^2 }}[/tex]

The length (L) of the pole is:

[tex]\mathbf{L = L_1 + L_2}[/tex]

So, we have:

[tex]\mathbf{L = \sqrt{x^2 + 256} + \sqrt{1476 - 60x + x^2}}[/tex]

Differentiate

[tex]\mathbf{L' = \frac{x}{\sqrt{x^2 + 256}} + \frac{x - 30}{\sqrt{1476 - 60x + x^2}}}[/tex]

Set to 0

[tex]\mathbf{\frac{x}{\sqrt{x^2 + 256}} + \frac{x - 30}{\sqrt{1476 - 60x + x^2}} = 0}[/tex]

Take LCM

[tex]\mathbf{\frac{x\sqrt{1476 - 60x + x^2} +(x - 30)\sqrt{x^2 + 256}}{\sqrt{x^2 + 256} \times \sqrt{1476 - 60x + x^2}} = 0}[/tex]

Simplify

[tex]\mathbf{x\sqrt{1476 - 60x + x^2} +(x - 30)\sqrt{x^2 + 256} = 0}[/tex]

Rewrite as:

[tex]\mathbf{x\sqrt{1476 - 60x + x^2} =-(x - 30)\sqrt{x^2 + 256} }[/tex]

Square both sides

[tex]\mathbf{x^2(1476 - 60x + x^2) =(x^2 - 60x + 900)(x^2 + 256) }[/tex]

Expand

[tex]\mathbf{1476x^2 - 60x^3 + x^4 =x^4 - 60x^3 + 900x^2 + 256x^2 - 15360x + 230400}[/tex]

Simplify

[tex]\mathbf{1476x^2 - 60x^3 + x^4 =x^4 - 60x^3 + 1156x^2 - 15360x + 230400}[/tex]

Evaluate like terms

[tex]\mathbf{1476x^2 = 1156x^2 - 15360x + 230400}[/tex]

Rewrite as:

[tex]\mathbf{1476x^2 - 1156x^2 + 15360x - 230400 = 0}[/tex]

[tex]\mathbf{320x^2 + 15360x - 230400 = 0}[/tex]

Divide through by 320

[tex]\mathbf{x^2 + 48x - 720 = 0}[/tex]

Using a calculator, we have:

[tex]\mathbf{x = \{12,-60\}}[/tex]

The value of x cannot be negative.

So, we have:

[tex]\mathbf{x = 12}[/tex]

Hence, the rope should be placed at 12 feet from the 16 ft pole

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Answer:

C.

Step-by-step explanation:

The amount of property tax that Ian pays is $10,890.

Tax is a compulsory sum of money levied on goods and services by the government. Property tax is the tax paid on property that is owned by an individual or group of individuals.

Property tax = tax rate x assessed value x market price

0.088 x 0.55 x 225,000 = $10,890

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In the equation t is the amount of time in seconds. The -2.4 is the distance it travels in 1 second.

-2.4 x 5 seconds= -12 meters.

The anchor dropped 12 (-12) meters in 5 seconds.

Answer:

C) It took 22 seconds for the anchor to reach the water's surface.

E) The equation E = 44 − ST can be used to model this situation.

Step-by-step explanation:

Answer:

Correct option:

(1) the eastern gray squirrels that live in New York City's Central Park

Step-by-step explanation:

A population in Statistical analysis represents the set of all possible values a random variable, X can assume. For example, all the registered voters of the United States form a population, the weight of all the newborn babies in the country form a population.

All the mammals living in the region of Boulder, Colorado cannot form a population. This is because the set consists of n different species in the region.

And the gray squirrels and fox squirrels are two different species. So, together they cannot form a population.

The red foxes found east of the Mississippi River in the United States and in eastern Europe cannot form a population because the red foxes selected are from two different regions.

The eastern gray squirrels that live in New York City's Central Park can form a population because the set consists of one one species, i.e. the eastern gray squirrels from a particular region, i.e. New York City's Central Park.

Thus, the example of a population is "the eastern gray squirrels that live in New York City's Central Park."

The correct example of a population is all the mammals living in the region of Boulder, Colorado.

An example of a population is all the mammals living in the region of Boulder, Colorado. This includes all species of mammals in that specific geographic area. Population refers to a group of individuals of the same species living in a specific area at a given time.

In this case, the population would include various mammals such as deer, rabbits, bears, and others. It does not include the specific populations of eastern gray squirrels in Central Park or red foxes across different regions.

Therefore, the correct option representing an example of a population is: all the mammals living in the region of Boulder, Colorado.

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Answer:

0.24

Step-by-step explanation:

Answer: 6/25 as a decimal would be 0.24

Step-by-step explanation:0.240 and 0.24 are both the same thing, the 0 behind the 4, doesn't have value. But, most teachers would prefer you to put it as 0.24. Also, in order to turn a fraction into a decimal, you just divide the top number (numerator) by the bottom number (denominator). 6 divided by 25 is 0.24

Answer:

[tex]t=\frac{2.73-3.39}{\frac{1.22}{\sqrt{26}}}=-2.758[/tex]    

[tex]df=n-1=26-1=25[/tex]  

[tex]p_v =P(t_{(25)}<-2.758)=0.0054[/tex]  

Since the p value is lower than the significance level 0.1 we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significanlty lower than 3.39 personas at 10% of significance.

Step-by-step explanation:

Data given and notation  

[tex]\bar X=2.73[/tex] represent the sample mean

[tex]s=1.22[/tex] represent the sample standard deviation

[tex]n=26[/tex] sample size  

[tex]\mu_o =3.39[/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 check if the true mean is less than 3.39 persons, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 3.39[/tex]  

Alternative hypothesis:[tex]\mu < 3.39[/tex]  

The statistic is given by:

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

Calculate the statistic

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

[tex]t=\frac{2.73-3.39}{\frac{1.22}{\sqrt{26}}}=-2.758[/tex]    

P-value

The degreed of freedom are given by:

[tex]df=n-1=26-1=25[/tex]  

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

[tex]p_v =P(t_{(25)}<-2.758)=0.0054[/tex]  

Conclusion  

Since the p value is lower than the significance level 0.1 we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significanlty lower than 3.39 personas at 10% of significance.

Answer:

The maximum price that the dealer will sell is $7471 and the minimum is $5513.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

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

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 6492, \sigma = 1025[/tex]

He decides to sell boats that will appeal to the middle 66% of the market in terms of price.

50 - (66/2)  = 17th percentile

50 + (66/2) = 83rd percentile

17th percentile

X when Z has a pvalue of 0.17. So X when Z = -0.955.

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

[tex]-0.955 = \frac{X - 6492}{1025}[/tex]

[tex]X - 6492 = -0.955*1025[/tex]

[tex]X = 5513[/tex]

83rd percentile

X when Z has a pvalue of 0.83. So X when Z = 0.955.

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

[tex]0.955 = \frac{X - 6492}{1025}[/tex]

[tex]X - 6492 = 0.955*1025[/tex]

[tex]X = 7471[/tex]

The maximum price that the dealer will sell is $7471 and the minimum is $5513.

The marine sales dealer plans to sell boats between $5467 and $7517 to appeal to the middle 66% of market prices. These figures are computed by adding or subtracting one standard deviation from the average price.

In this scenario, the marine sales dealer wants to price boats that appeal to the middle 66% of the market, which means the dealer wants to avoid the top and bottom 17% of the market (as 100%-66%=34% and 34%/2=17%). Therefore, we need to find the boats' prices that are 1 standard deviation away from the mean, since in a normal distribution, approximately 68% of values lie within 1 standard deviation from the mean (closest to 66%).

The standard deviation given is $1025. Thus, the maximum price of the boats the dealer will sell is the mean price plus one standard deviation:

$6492 + $1025 = $7517

And the minimum price is the mean price minus one standard deviation:

$6492 - $1025 = $5467

Therefore, the dealer will sell boats priced between $5467 and $7517 to appeal to the middle 66% of the market.

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Answer:

1.One proportion

2. Proportions from Dependent Samples

3. Two Independent Proportions

4. One Mean

5. Means from Dependent Samples

6. Two Independent Means

Step-by-step explanation:

What proportion of Americans believe in climate change? - One proportion

From the above information, there is only one sample and the question is about proportion. So, we must use "One proportion".

Do college students change their view on climate change as they go through college? One hundred freshman are asked if they believe in climate change. The same students are asked four years later if they also believe in climate change. - Proportions from Dependent Samples

From the information given, there are two samples and samples are dependent ( that is same students are asked four years later). Furthermore, the question is about proportion, we must use " Proportions from dependent samples ".



Is there a difference in the proportion of men and women who believe in climate change? - Two Independent Proportions

From the information given, there are two independent samples and they are asking about proportion, we must use "Two independent proportions".

What is the average amount of time that Americans spend watching or reading the news a day? - One Mean

From the information, there is only one sample and they are asking about mean. Therefore, we must use ". One mean".

Is there a difference in the amount of time that college students spend reading or watching the news as they go through college? One hundred students were asked as freshmen and as senior how many minutes a day they spent reading or watching the news? - Means from Dependent Samples

There are two samples and samples are dependent ( the same students are asked four years later). Also, they are asking about mean, we must use "Means from dependent samples ".

Is there is a difference in the amount of time that Republicans and Democrats spend watching or reading the news? - Two Independent Means

From the information given, there are two independent samples and they are asking about mean, we must use "Two independent means".

The statistical tests required are a hypothesis test for a proportion for the questions on proportions of Americans' beliefs, a paired t-test for examining changes in the same group over time, an independent t-test for comparing means of different groups and measures of central tendency (mean) for assessing averages.

The various questions posed require different types of statistical tests. Proportions are evaluated using a hypothesis test for a proportion. So, the question 'What proportion of Americans believe in climate change?' and 'Is there a difference in the proportion of men and women who believe in climate change?' would utilize this test.

For questions where the same group is measured at two different times, a paired t-test is appropriate. Hence, the question 'Do college students change their view on climate change as they go through college? One hundred freshman are asked if they believe in climate change. The same students are asked four years later if they also believe in climate change.' would require a paired t-test.

When comparing the means of two independent groups, an independent t-test should be used. This would apply to the questions 'Is there a difference in the amount of time that Republicans and Democrats spend watching or reading the news?' and 'Is there is a difference in the amount of time that college students spend reading or watching the news as they go through college?'

Finally, to measure an average or central tendency, a mean is used – therefore, the question 'What is the average amount of time that Americans spend watching or reading the news a day?' would require the calculation of a mean.

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Answer: The expected utility is 0.59.

Step-by-step explanation:

Since we have given that

[tex]U(c)=\sqrt{c}[/tex]

Probability that he will be injured = 0.1

Probability that he will not be injured = 0.9

If Billy is not injured this season, he will receive an income of 25 million dollars.

and

If he is injured, his income will be only 10,000 dollars.

According to question, the expected utility is given by

[tex]E[x]=0.9\times \sqrt{(0.01)}+0.1\times \sqrt{25}\\\\E[x]=0.9\times 0.1+0.1\times 5\\\\E[x]=0.09+0.5\\\\E[x]=0.59[/tex]

Hence, the expected utility is 0.59.

Final answer:

To calculate Billy's expected utility, we need to multiply his utility function by the probability of each outcome and sum the results. His expected utility is approximately 3333.33.

Explanation:

To calculate Billy's expected utility, we need to multiply his utility function by the probability of each outcome and sum the results. In this case, Billy's utility function is U(c) = c^(1/2), where c represents his income. If Billy is not injured, his income will be $25 million, and if he is injured, his income will be $10,000. The probability of being injured is 0.1, and the probability of not being injured is 0.9.

Expected utility = U(income if not injured) * P(not injured) + U(income if injured) * P(injured)

Expected utility = U($25 million) * 0.9 + U($10,000) * 0.1

Expected utility = (25 million)^(1/2) * 0.9 + (10,000)^(1/2) * 0.1

Solving this equation, we find that the expected utility for Billy is approximately 3333.33.

Answer:

No.

Step-by-step explanation:

This is not a perfect square because the exponent would need to be even, not odd. The 100 is a perfect square. If you were to simplify it, it would be (10x^2) * (10x). In order for it to be truly a perfect square, they both need to be the same.

Daniel appears to be paying more for auto insurance compared to the average amount his friends pay based on the data from nine friends. However, as the data only represents a sample, and auto insurance rates can vary widely, additional comparison or the advice of an insurance specialist is recommended.

To determine if Daniel is overpaying for his auto insurance, we can compare his insurance cost to the average price paid by his friends. To do this, we need to calculate the mean (average) of his friends' insurance amounts.

Here are the amounts his friends pay: 564, 578, 478, 507, 621, 564, 489, 612, 538.

Adding these together gets a total of 4951. There are nine friends, so we divide 4951 by 9 to get an average cost of 550.

Since Daniel is paying $600, which is more than the average of $550, it seems he's spending more than his friends for auto insurance.

However, we only have the data from a sample of his friends. The insurance amounts can have a wide range, depending on several factors like age, driving records, the type of vehicle insured, and geographic location. Therefore, it's recommended that Daniel compare his rate with more people or consult with an insurance specialist for a more accurate conclusion.

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Answer:

a) The 95% CI for the mean surgery time is (133.05, 140.75).

b) The 99.5% CI for the mean surgery time is (131.37, 142.43).

c) The level of confidence of the interval (133.9, 139.9) is 69%.

d) The sample size should be 219 surgeries.

e) The sample size should be 377 surgeries.

Step-by-step explanation:

We have a sample, of size n=132, for which the mean time was 136.9 minutes with a standard deviation of 22.6 minutes.

a) We have to find a 95% CI for the mean surgery time.

The critical value of z for a 95% CI is z=1.96.

The margin of error of the CI can be calculated as:

[tex]E=z\cdot s/\sqrt{n}=1.96*22.6/\sqrt{132}=44.296/11.489=3.85[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=\bar x-E=136.9-3.85=133.05\\\\UL=\bar x+E=136.9+3.85=140.75[/tex]

The 95% CI for the mean surgery time is (133.05, 140.75).

b) Now, we have to find a 99.5% CI for the mean surgery time.

The critical value of z for a 99.5% CI is z=2.81.

The margin of error of the CI can be calculated as:

[tex]E=z\cdot s/\sqrt{n}=2.81*22.6/\sqrt{132}=63.506/11.489=5.53[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=\bar x-E=136.9-5.53=131.37\\\\UL=\bar x+E=136.9+5.53=142.43[/tex]

The 99.5% CI for the mean surgery time is (131.37, 142.43).

c) We can calculate the level of confidence, calculating the z-score for the margin of error in that interval.

We know that the difference between the upper bound and lower bound is 2 times the margin of error:

[tex]UL-LL=2E\\\\E=\dfrac{UL-LL}{2}=\dfrac{139.9-133.9}{2}=\dfrac{6}{2}=3[/tex]

Then, we can write the equation for the margin of error to know the z-value.

[tex]E=z \cdot s/\sqrt{n}\\\\z= E\cdot \sqrt{n}/s=2*\sqrt{132}/22.6=2*11.5/22.6=1.018[/tex]

The confidence level for this interval is then equal to the probability that the absolute value of z is bigger than 1.018:

[tex]P(-|z|<Z<|z|)=P(-1.018<Z<1.018)=0.69[/tex]

The level of confidence of the interval (133.9, 139.9) is 69%.

d) We have to calculate the sample size n to have a margin of error, for a 95% CI, that is equal to 3.

The critical value for a 95% CI is z=1.96.

Then, the sample size can be calculated as:

[tex]E=z\cdot s/\sqrt{n}\\\\n=(\dfrac{z\cdot s}{E})^2=(\dfrac{1.96*22.6}{3})^2=14.77^2=218.015\approx 219[/tex]

The sample size should be 219 surgeries.

e) We have to calculate the sample size n to have a margin of error, for a 99% CI, that is equal to 3.

The critical value for a 99% CI is z=2.576.

Then, the sample size can be calculated as:

[tex]E=z\cdot s/\sqrt{n}\\\\n=(\dfrac{z\cdot s}{E})^2=(\dfrac{1.96*22.6}{3})^2=19.41^2=376.59\approx 377[/tex]

The sample size should be 377 surgeries.

Answer:

See explanation

Step-by-step explanation:

Solution:-

The random variable, Y be the temperature of chemical reaction in degree fahrenheit be a linear expression of a random variable X : The  temperature in at which a certain chemical reaction takes place.

                             Y = 1.8*X + 32

- The median of the random variate "X" is given to be equal to "η". We can mathematically express it as:

                             P ( X ≤ η ) = 0.5

- Then the median of "Y" distribution can be expressed with the help of the relation given:

                             P ( Y ≤ 1.8*η + 32 )

- The left hand side of the inequality can be replaced by the linear relation:

                             P ( 1.8*X + 32 ≤ 1.8*η + 32 )

                             P ( 1.8*X ≤ 1.8*η )   ..... Cancel "1.8" on both sides.

                            P ( X ≤ η ) = 0.5 ...... Proven

Hence,

- Through conjecture we proved that: (1.8*η + 32) has to be the median of distribution "Y".

b)

- Recall that the definition of proportion (p) of distribution that lie within the 90th percentile. It can be mathematically expressed as the probability of random variate "X" at 90th percentile :

                             P ( X ≤ p_.9 ) = 0.9 ..... 90th percentile

- Now use the conjecture given as a linear expression random variate "Y",

          P ( Y ≤ 1.8*p_0.9 + 32 ) = P ( 1.8*X + 32 ≤ 1.8*p_0.9 + 32 )

                                                 = P ( 1.8*X ≤ 1.8*p_0.9 )

                                                 = P ( X  ≤ p_0.9 )

                                                 = 0.9

- So from conjecture we saw that the 90th percentile of "X" distribution is also the 90th percentile of "Y" distribution.

c)

- The more general relation between two random variate "Y" and "X" is given:

                            Y = aX + b

Where, a : is either a positive or negative constant.

- Denote, (np) as the 100th percentile of the X distribution, so the corresponding 100th percentile of the Y distribution would be : (a*np + b).

- When a is positive,

                   P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )

                                                 = P ( a*X ≤ a*p_% )

                                                 = P ( X  ≤ p_% )

                                                 = np_%        

- When a is negative,

                   P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )

                                                 = P ( a*X ≤ a*p_% )

                                                 = P ( X  ≥ p_% )

                                                 = 1 - np_%        

In a temperature conversion equation Y = 1.8X + 32, medians and percentiles of Y are related to the corresponding values of X through the equation itself. For an arbitrary linear transformation Y = aX + b, percentiles of Y and X are related as aX + b, with ordering depending on the sign of a.

Lets start by talking about the relationship between X and Y. In the context of the temperature conversion between Celsius (X) and Fahrenheit (Y), Y equals to 1.8 times X plus 32.

1. If the median of X is M, substituting M into the equation Y = 1.8X + 32 gives the median of Y as 1.8M + 32, since the transformation is linear.

2. The 90th percentile of the Y distribution will relate to the 90th percentile of X distribution in a similar fashion. If we denote the 90th percentile of X as P, then the 90th percentile of Y will be 1.8P + 32.

3. For a general linear transformation Y = aX + b, where a is non-zero, any percentile of Y is related to the corresponding percentile of X as aX + b. If a is positive, the transformation will preserve the ordering of percentiles (e.g., higher values of X correspond to higher values of Y). If a is negative, it will reverse the ordering of percentiles (e.g., higher values of X will correspond to lower values of Y).

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Answer:

P(B)=0.30

Step-by-step explanation:

Out of 1000 Voters, 30% favor Jones.

Event S=Favors Jones on First Trial

Event B=S occurs on Second Trial

P(S)=0.30

P(S')=1-0.30=0.70

Event B could occur in two ways

Therefore,

P(B)=P(SS)+P(S'S)

=(0.3X0.3)+(0.7X0.3)

=0.09+0.21

=0.3

Therefore, the probability of event B(that event S occurs on the second trial), P(B)=0.30.

Answer:

(a)(C)[tex]D^c \cup A[/tex]

(b)8 elements

Step-by-step explanation:

Ina toss of a red and green dice, given the events:

[tex]D^c[/tex]=The numbers do add up to 11.

Therefore, the event: Either the numbers add to 11 or the red die shows a 1 is written as: [tex]D^c \cup A[/tex]

(b)

Sample Space of A={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)}

Sample Space of [tex]D^c[/tex]={(5,6),(6,5)}

[tex]D^c \cup A[/tex]={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(5,6),(6,5)}

[tex]D^c \cup A[/tex] contains 8 elements

Final answer:

The event "Either the numbers add to 11 or the red die shows a 1" is represented by the symbol D' ∪ A. This event contains 7 elements in the context of rolling two six-sided dice.

Explanation:

To express the event "Either the numbers add to 11 or the red die shows a 1" in symbolic form, we consider the symbols for the events defined in the question. Event A denotes the red die shows 1, and D denotes the event the numbers do not add to 11. The complement of D, represented as D', would then denote the event that the numbers do add to 11. The symbol ‘∪’ denotes the union of sets, meaning 'or' in the context of probability. Therefore, the symbolic form for the given event is D' ∪ A.

Regarding how many elements this event contains, we must consider the sample space when rolling two six-sided dice. There are a total of 36 different outcomes (6 possible outcomes for the first die multiplied by 6 outcomes of the second die). Event A (the red die shows a 1) has 6 elements (1 can be paired with any of the 6 outcomes on the green die). Event D' (the numbers add up to 11) can happen in two ways: (5,6) or (6,5), one for each die, making it 2 elements. Therefore, D' ∪ A will consist of all unique elements from both events without double-counting any pair. So, we combine 6 outcomes from A and 2 from D', but we need to ensure to not count the outcome (1,6) twice, hence we have 6 (from A) + 2 (from D') - 1 (overlap of (1,6)) = 7 elements in event D' ∪ A.

Answer:

The first one is 16.0 and the second one is 22.5

Answer:

The first one is 16.0 and the second one is 22.5

Answer:

Step-by-step explanation:

In physics, Work is defined as the energy needed to move certaing object through a certain distance. Specifically, the work done is directly proportional to the force exerted and the distance.

It's important to know that a change of point is needed to have actually work done, physically speaking. This means if the object doesn't move, then there's no work done.

Mathematically, the work is defined

[tex]W= F \times d[/tex]

Isolating the force

[tex]F=\frac{W}{d}[/tex]

So, notice that the distance is inversely proportional to the force needed, which means the less distance, more force we need.

Now, the problem is giving wides and slopes, which we can use to find heights. And we know already that the less distance we have, the greater force we need, or the most distance, the least force.

Let's find which wedge has the greatest vertical distance.

[tex]m=\frac{y}{x}[/tex]

[tex]5=\frac{y}{2} \implies y=10[/tex]

[tex]8=\frac{y}{4} \implies y=32[/tex]

[tex]9=\frac{y}{3} \implies y=27[/tex]

[tex]10=\frac{y}{5}\\ y=50[/tex]

Notice that the last wedge has greater vertical distance, that means Wedge Z requires least amount of force to do a job.

Answer:

The correct answer is Y not Z.

Step-by-step explanation:

Answer:

1/4(1-3x)

Step-by-step explanation:

Answer:

  classification is based on the measure of the largest angle

Step-by-step explanation:

Every triangle will have angle measures that add up to 180°. The classification as to acute, right, or obtuse is based on the largest angle.

__

If the largest angle is less than 90°, the triangle is acute.

If the largest angle is equal to 90°, the triangle is right.

If the largest angle is greater than 90°, the triangle is obtuse.

__

Because you know the angle sum is always 180°, you can generally figure out what kind of triangle it is from the sum of two of the angles. If both are less than 90° and their sum is more than 90°, then the triangle will be acute, for example.

Answer:

6/5

Step-by-step explanation:

[tex]\frac{3}{5} * 2 = \frac{6}{5}[/tex]

please give me brainliest

Answer:

1.2 [Decimal Form]

[tex]\frac{6}{5}[/tex] [Exact Form]

[tex]1\frac{1}{5}[/tex] [Mixed Number Form]

Answer:

b

Step-by-step explanation:

Option D: 15-year fixed, 20% down at a fixed rate of 5.5% would result in the lowest monthly payment for the Johnsons.

Based on the options given, the loan option that would result in the lowest monthly payment for the Johnsons would be Option D: 15-year fixed, 20% down at a fixed rate of 5.5%. To determine this, we need to compare the monthly payments for each option.

By calculating the monthly payments for each option, it is found that Option D has the lowest monthly payment for the Johnsons.

Answer:

2 out of 6

Step-by-step explanation:

Answer: 2 out of 6

Step-by-step explanation:

Answer:

a) The large sample size 'n' = 1320.59

b) If the standard deviation and the margin of error are both doubled also the sample size is not changed.

Step-by-step explanation:

Explanation:-

a)

Given data the standard deviation of the population

σ = 70.5

Given the margin error = 5 units

We know that the estimate of the population mean is defined by

that is margin error = [tex]\frac{z_{\alpha } S.D }{\sqrt{n} }[/tex]

            [tex]M.E = \frac{z_{\alpha } S.D }{\sqrt{n} }[/tex]

cross multiplication , we get

[tex]M.E (\sqrt{n} ) = z_{\alpha } S.D[/tex]

[tex]\sqrt{n} = \frac{z_{\alpha } S.D }{M.E }[/tex]

[tex]\sqrt{n} = \frac{2.578 X 70.5}{5} }[/tex]

√n = 36.34

squaring on both sides , we get

n = 1320.59

b) The margin error of the mean

   [tex]\sqrt{n} = \frac{z_{\alpha } S.D }{M.E }[/tex]

the standard deviation and the margin of error are both doubled

  √n = zₓ2σ/2M.E

  √n = 36.34

squaring on both sides , we get

     n = 1320.59

If the standard deviation and the margin of error are both doubled also the sample size is not changed.

Answer: the probability that during a randomly selected month, PCE's were between $775.00 and $990.00 is 0.9538

Step-by-step explanation:

Since the amount that the company spent on personal calls followed a normal distribution, then according to the central limit theorem,

z = (x - µ)/σ

Where

x = sample mean

µ = population mean

σ = standard deviation

From the information given,

µ = $900

σ = $50

the probability that during a randomly selected month PCE's were between $775.00 and $990.00 is expressed as

P(775 ≤ x ≤ 990)

For (775 ≤ x),

z = (775 - 900)/50 = - 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.0062

For (x ≤ 990),

z = (990 - 900)/50 = 1.8

Looking at the normal distribution table, the probability corresponding to the z score is 0.96

Therefore,

P(775 ≤ x ≤ 990) = 0.96 - 0.0062 = 0.9538

The probability that PCE's were between $775 and $990 is 0.9579 or 95.79%.

To find the probability that the PCE's were between $775 and $990 during a randomly selected month, we first need to standardize the values using the standard normal distribution. Formula for standardization is:

Z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

Using the formula, we calculate the standard scores for the given values:

Z1 = ($775 - $900) / $50 = -2.50

Z2 = ($990 - $900) / $50 = 1.80

Next, we use the standard normal distribution table or a calculator to find the corresponding probabilities for these z-scores. The probability between the z-scores -2.50 and 1.80 is the difference between their corresponding cumulative probabilities:

Prob(Z1 < Z < Z2) = Prob(Z < Z2) - Prob(Z < Z1)

Using the standard normal distribution table, we can find the probabilities:

Prob(Z < -2.50) = 0.0062

Prob(Z < 1.80) = 0.9641

Finally, we calculate the probability between the z-scores:

Prob(Z1 < Z < Z2) = 0.9641 - 0.0062 = 0.9579

Therefore, the probability that PCE's were between $775 and $990 during a randomly selected month is approximately 0.9579 or 95.79%.

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Given:

The given figure ABCD is a trapezoid.

The measure of ∠A is (2x + 32).

The measure of ∠B is 112°

The measure of ∠C is y.

The measure of ∠D is 46°

We need to determine the value of x and y.

Value of x:

We know the property that the adjacent angles in a trapezoid are supplementary.

Thus, we have;

[tex]\angle A+\angle B=180[/tex]

Substituting the values, we get;

[tex]2x+32+112=180[/tex]

       [tex]2x+144=180[/tex]

                 [tex]2x=36[/tex]

                   [tex]x=18[/tex]

Thus, the value of x is 18.

Value of y:

The value of y can be determined using the property that the adjacent angles of a trapezoid are supplementary.

Thus, we have;

[tex]\angle C+\angle D=180[/tex]

    [tex]y+46=180[/tex]

            [tex]y=134[/tex]

Thus, the value of y is 134.

Hence, Option c is the correct answer.

Answer:

[tex]\displaystyle \frac{7}{8}[/tex].

Step-by-step explanation:

If two events are complements, then the sum of their probabilities should be [tex]1[/tex].

This question suggests that the following two events are complements:

As a result:

[tex]\begin{aligned}& P(\text{at least one child is female}) \\ &= 1 - P(\text{all three children are male})\end{aligned}[/tex].

According to the question,

[tex]\displaystyle P(\text{all three children are male}) = \frac{1}{8}[/tex].

Therefore,

[tex]\begin{aligned}& P(\text{at least one child is female}) \\ &= 1 - P(\text{all three children are male}) \\ &= 1 -\frac{1}{8} \\ &= \frac{7}{8}\end{aligned}[/tex].

In this Mathematics problem of probability, the calculation required was for the probability of 'at least one child being female'. This is a complementary event to 'all three children being male', enabling us to solve it by using the formula P(at least one child is female) = 1 - P(all three children are male). Hence, the answer equals 7/8.

The subject of the question is probability, which is a branch of Mathematics. Looking at the question, the probability of having all three children as males has been given as 1/8. We are required to find the probability of having 'at least one female child'. This event is complementary to having 'all three children as males', so we can find the solution using the formula you stated. As P(all three children are male) = 1/8, therefore P(at least one child is female) = 1 - 1/8 = 7/8.

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