To get an estimate of consumer spending over a holiday season in 2009, 436 randomly sampled American adults were surveyed. Their spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% confidence interval based on this sample is ($80.31, $89.11). Which of the following statements are true? Select all that apply. A 90% confidence interval would be narrower than the 95% confidence interval if we don't need to be as sure about our estimate. In order to decrease the margin of error of a 95% confidence interval to a third of what is is now, we would need to use a sample 3 times larger. This confidence interval is not valid since the distribution of spending in the sample data is right skewed. The margin of error is $4.4. We are 95% confident that the average spending of these 435 American adults over this holiday season is between $80.31 and $89.11. This confidence interval is valid since the sampling distribution of sample mean would be approximately normal with sample size of 436. 95% of random samples have a sample mean between $80.31 and $89.11. We are 95% confident that the average spending of all American adults over this holiday season is between $80.31 and $89.11.

Answer:

The value of the t-statistic for this test is 1.138.

Step-by-step explanation:

We are given that a new manager of a small convenience store randomly samples 20 purchases from yesterday’s sales. The mean was $45.26 and the standard deviation was $20.67.

We wish to test for evidence that the overall mean purchase amount is at least $40

Let [tex]\mu[/tex] = overall mean purchase amount

SO, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] [tex]\geq[/tex] $40   {means that the overall mean purchase amount is at least $40}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < $40   {means that the overall mean purchase amount is less than $40}

The test statistics that will be used here is One-sample t test statistics because we don't know about the population standard deviation;

                           T.S.  = [tex]\frac{\bar X -\mu}{{\frac{s}{\sqrt{n} } } }[/tex]  ~ [tex]t_n_-_1[/tex]

where,  [tex]\bar X[/tex] = sample mean sale = $45.26

              s = sample standard deviation = $20.67

              n = sample of purchases = 20

So, test statistics  =  [tex]\frac{45.26-40}{{\frac{20.67}{\sqrt{20} } } }[/tex]  ~ [tex]t_1_9[/tex]

                               =  1.138

Hence, the value of the t-statistic for this test is 1.138.

Answer:

[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{4.94 -0}{\frac{3.901}{\sqrt{5}}}=2.832[/tex]

[tex]df=n-1=5-1=4[/tex]

[tex]p_v =P(t_{(4)}>2.832) =0.0236[/tex]

We see that the p value is lower than the significance level of 0.05 so then we have enough evidence to reject the null hypothesis and we can conclude that the average net sales improved

Step-by-step explanation:

Let put some notation  

x=test value before , y = test value after

x: 57.1 94.6 49.2 77.4 43.2

y: 63.5 101.8 57.8 81.2 41.9

The system of hypothesis for this case are:

Null hypothesis: [tex]\mu_y- \mu_x \leq 0[/tex]

Alternative hypothesis: [tex]\mu_y -\mu_x >0[/tex]

The first step is calculate the difference [tex]d_i=y_i-x_i[/tex] and we obtain this:

d: 6.4, 7.2, 8.6, 3.8, -1.3

The second step is calculate the mean difference  

[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}=4.94[/tex]

The third step would be calculate the standard deviation for the differences, and we got:

[tex]s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =3.901[/tex]

The next step is calculate the statistic given by :

[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{4.94 -0}{\frac{3.901}{\sqrt{5}}}=2.832[/tex]

The next step is calculate the degrees of freedom given by:

[tex]df=n-1=5-1=4[/tex]

Now we can calculate the p value, since we have a right tailed test the p value is given by:

[tex]p_v =P(t_{(4)}>2.832) =0.0236[/tex]

We see that the p value is lower than the significance level of 0.05 so then we have enough evidence to reject the null hypothesis and we can conclude that the average net sales improved

Answer:

C. 150 pages

Step-by-step explanation:

204 divided by 136 = 1.5

1.5 * 100 = 150

Hope it helps! :)

Answer:

J

Step-by-step explanation:

Answer:

H₀: p₁ - p₂ < 0.

Hₐ: p₁ - p₂ > 0.

Step-by-step explanation:

In this case, we need to determine if there is a higher incidence of smoking among women than among men in a neighborhood.

An experiment can be performed involving collecting two samples of men and women and computing the proportion of men and women smokers in the neighborhood. Then these two sample proportions can be used to determine which proportion is higher.

We are computing proportions of men and women smokers instead of the mean number of men and women smokers because the we need to determine the  incidence of smoking among men and women, i.e. the occurrence or frequency or rate of men and women smokers.

So, a two proportion test can be applied to determine whether the proportion of women smokers is higher than men.

The population proportion of women in the neighborhood is represented by p₁ and the  population proportion of men in the neighborhood is represented by p₂.

The hypothesis can be defined as:

H₀: The proportion of women smokers is not higher than men, i.e. p₁ - p₂ < 0.

Hₐ: The proportion of women smokers is higher than men, i.e. p₁ - p₂ > 0.

Final answer:

The null hypothesis (H0) should be P1 - P2 ≥ 0, indicating no higher incidence of smoking among women, and the alternative hypothesis (Ha) should be P1 - P2 < 0, suggesting a higher incidence of smoking among women.

Explanation:

When conducting a hypothesis test to determine if there's a higher incidence of smoking among women than among men in a neighborhood, you would use two hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). Since we are looking to find if the incidence is higher in women, the null hypothesis should reflect there is no difference or that the incidence in women is less than or equal to the incidence in men. Therefore, the correct setup using population proportions (P1 for men and P2 for women) would be:

The null hypothesis statement is that the proportion of women who smoke is less than or equal to the proportion of men who smoke. The alternative hypothesis is what you intend to prove, which is that more women smoke than men in the neighborhood, hence P1 - P2 < 0. This would be considered a left-tailed test.

Answer: x = 10 y=20

Step-by-step explanation:

You can answer this question by plugging in each equation:

2x=y, x+y=30. Let us plug y as 2x in the second equation x+y=30

x+2x= 30

3x= 30

x=10

After we found x we can then find y by plugging the 10 for x.

2(10) = y

y =20

or you could plug in the other equation

10+y=30

subtract 10 from 30 and we get 20

to double check we can plug in both numbers

2(10) = 20 which is correct

and 10 + 20 = 30 which is correct

Based on the experimental design and the data collected the statistical test that should be used is the correlation coefficient.

The statistical tests are tests that are used by researchers to determine the progress of a set process and make a quantitative decision during analysis of test results.

Example of statistical tests include the following:

The correlation coefficient which is a type of statistical test is used to assess the strength and direction of the linear relationships between pairs of variables.

Therefore, the relationship between cognitive reflectivity and cognitive impulsivity in human data can be statistically analysed using correlation coefficient.

Learn more about statistical tests here:

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Answer:

Step-by-step explanation:

We are given that [tex] R(x) = 108\sqrt[]{x}[/tex] is the revenue for having x employees.

Let us calculate the following

[tex]R(12)-R(11) = 108\sqrt[]{12}-108\sqrt[]{11} = 15.93[/tex] REcall that is amount represents the revenue that having one extra employee would have.

We have that the salary of the 12th employee would be 26000. One criteria to define if it's a good idea to hire the 12th employee is to check if the profit for having an extra employee is positive. We will take the revenue of the extra employee minus the cost and check if it is positiv. Then,

[tex](108\sqrt[]{12}-108\sqrt[]{11})\text{(revenue for 12 employees)}-26000\text{(cost for the 12th employee} =-25984<0[/tex]

Since it is a negative amount, this means that it would be more expensive to have one extra employee than the revenue it would generate. Therefore, it won't be suitable to hire the 12th employee

Answer:

Test statistics = 3.74

P-value = 0.0001

Step-by-step explanation:

We are given that an experiment to compare the tension bond strength of polymer latex modified mortar to that of unmodified mortar resulted in x = 18.11 kg f/[tex]cm^{2}[/tex] for the modified mortar (m = 42) and y = 16.83 kg f/[tex]cm^{2}[/tex] for the unmodified mortar (n = 30).

Assume that the bond strength distributions are both normal and assuming that σ1 = 1.6 and σ2 = 1.3.

Let [tex]\mu_1[/tex] = true average tension bond strengths for the modified mortars

      [tex]\mu_2[/tex] = true average tension bond strengths for the unmodified mortars

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu_1-\mu_2=0[/tex]  or  [tex]\mu_1=\mu_2[/tex]    {means that true average tension bond strengths for the modified and unmodified mortars are same}

Alternate Hypothesis, [tex]H_a[/tex] : [tex]\mu_1-\mu_2>0[/tex]  or  [tex]\mu_1>\mu_2[/tex]    {means that the true average tension bond strengths for the modified mortars is greater than that for unmodified mortars}

The test statistics that will be used here is Two-sample z test statistics as we know about population standard deviations;

               T.S.  =  [tex]\frac{(x-y)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^{2} }{m}+\frac{\sigma_2^{2} }{n} } }[/tex]  ~ N(0,1)

where, x = sample mean tension bond strengths for the modified mortars = 18.11 kg f/[tex]cm^{2}[/tex]

         y = sample mean tension bond strengths for the unmodified mortars = 16.83 kg f/[tex]cm^{2}[/tex]

         [tex]\sigma_1[/tex] = population standard deviation for modified mortars = 1.6

         [tex]\sigma_2[/tex] = population standard deviation for unmodified mortars = 1.3

         m = sample of modified mortars = 42

         n = sample of unmodified mortars = 30

So, test statistics  =  [tex]\frac{(18.11-16.83)-(0)}{\sqrt{\frac{1.6^{2} }{42}+\frac{1.3^{2} }{30} } }[/tex]

                               =  3.74

Now, P-value is given by the following formula;

         P-value = P(Z > 3.74) = 1 - P(Z [tex]\leq[/tex] 3.74)  

                                            = 1 - 0.9999 = 0.0001

Here, the above probability is calculated by looking at the value of x = 3.74 in the z table which gives an area of 0.9999.

To test the hypothesis H0: μ1 − μ2 = 0 versus Ha: μ1 − μ2 > 0 at level 0.01, calculate the test statistic and the P-value. The test statistic is Z = (x - y) / √((σ1² / m) + (σ2² / n)). Substituting the values gives Z = 1.278. The P-value is approximately 0.1019.

To test the hypothesis H0: μ1 − μ2 = 0 versus Ha: μ1 − μ2 > 0 at level 0.01, we calculate the test statistic and the P-value. The test statistic is given by:

Z = (x - y) / √((σ1² / m) + (σ2² / n))

where x = 18.11, y = 16.83, σ1 = 1.6, σ2 = 1.3, m = 42, and n = 30.

Substituting the values, we get Z = (18.11 - 16.83) / √((1.6² / 42) + (1.3² / 30)).

Calculating Z gives Z = 1.278. To determine the P-value, we find the area to the right of Z in the standard normal distribution. The P-value is the probability that Z > 1.278. Consulting a Z-table or using a calculator, we find the P-value to be approximately 0.1019.

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Answer:

-1024

Step-by-step explanation:

Moivre's theorem allows to easily obtain trigonometric formulas that express the sine and cosine of a multiple angle as a function of the sine and cosine of a simple angle.

De Moivre's theorem can be applied to any complex number [tex]z[/tex]

Where:

[tex]z\in Z[/tex]

Let:

[tex](1+i)=z\\n=20[/tex]

According to Demoivres Theorem, If:

[tex]z=||z||(cos(\theta)+isin(\theta))[/tex]

Then:

[tex]z^n=||z||^n(cos(n\theta)+isin(n\theta))[/tex]

For a complex number [tex]z[/tex]:

[tex]z=a+bi[/tex]

Its magnitude and angle are given by:

[tex]||z||=\sqrt{a^2+b^2} \\\\\theta=arctan(\frac{b}{a} )[/tex]

So:

[tex]||z||=\sqrt{1^2+1^2} =\sqrt{2}[/tex]

[tex]\theta=arctan(\frac{1}{1} )=45^{\circ}[/tex]

Therefore, using De Moivre's theorem:

[tex]z^n=(\sqrt{2} )^{20}(cos(20*45)+isin(20*45))\\\\z^n=(\sqrt{2} )^{20}(cos(900)+isin(900))\\\\z^n=1024(-1+i(0))\\\\z^n=1024(-1)\\\\z^n=(1+i)^{20}=-1024[/tex]

C on e2020

i did it *dab*

Answer:

Attached

Step-by-step explanation:

On a cartessian plane, we take four points as shown. Point A has coordinates as (4, 0) while point B is (32, 0). Since the y cordinates are zero, don't change, only x change. Change in x coordinates is 32-4=28 units.

Point C is (32, 12) where x has not changed when compared to point B but y changes by 12-0=12 units

Therefore, the diagram is triangle whose length is 28 units while width is 12 units.

Answer:

a)  95% confidence interval for the population proportion of all young adults in Kentucky who received help from their parents.

( 0.0761 , 0.3239)

b) Margin of error  =  0.1264.

Step-by-step explanation:

Explanation:-

Given '8' persons in a random sample of 40 young adults who recently purchased a home in Kentucky received help from their parents.

sample proportion of success [tex]'p' = \frac{8}{40} = 0.2[/tex]

q = 1=p

q = 1-0.2 = 0.8

a)

95% confidence interval for the population proportion of all young adults in Kentucky who received help from their parents.

[tex](p-1.96\sqrt{\frac{pq}{n} } , p+1.96\sqrt{\frac{pq}{n} } )[/tex]

[tex](0.2-1.96\sqrt{\frac{0.2X0.8}{40} } , 0.2+1.96\sqrt{\frac{0.2X0.8}{40} } )[/tex]

(0.2 - 0.1239,0.2+0.1239)

( 0.0761 , 0.3239)

b) the margin of error for a 95% confidence interval for the population proportion.

For the 95% confidence interval ∝= 0.05 and zₐ = 1.96≅2.

[tex]Margin of error = \frac{2\sqrt{pq} }{\sqrt{n} }[/tex]

[tex]Margin of error = \frac{2\sqrt{0.2X0.8} }{\sqrt{40} }[/tex]

Margin of error for a 95% confidence interval for the population proportion.

Margin of error  =  0.1264.

Answer:

-4.4m-3.2

Step-by-step explanation:

-6.4m+4(0.5m-0.8)

-6.4m+4*0.5+4*0.8

-6.4m+2m-3.2

-4.4m-3.2

Answer:

-4.4m-3.2

Step-by-step explanation:

multiply everything in the parentheses by 4 and then add.

Answer: x^2+9 and x^2+3x+9

Step-by-step explanation:

A polynomial is prime if it cannot be factored into polynomials of lower degree. In this case, x² +9 and x² + 3x + 9 are prime polynomials while x² -9 and -2x² +8 are not.

In mathematics, a polynomial is said to be prime if it cannot be factored into polynomials of lower degree, at least one of which must be non-constant. To determine if a polynomial is prime, we try to factor it. In our case:

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Answer:

Read the explanation for the answers

Step-by-step explanation:

To find the midpoint, you simply need to find the average of the two endpoints. The average of 10.5 and 5.5 is 8, and the average of 4 and 4 is 4. Therefore, the center of the circle is at (4,8). The radius is the distance from the center to either of these points, or 8-5.5=2.5 units. And finally, the formula for the circle in standard form is [tex](y-8)^2+(x-4)^2=6.25[/tex]. Hope this helps!

Answer:

there is a 1/18th percent chance Darlene and bob will go in that order

Step-by-step explanation:

Answer:

The probability is 0.503

Step-by-step explanation:

If the ghost appearances occur in the house according to a Poisson process with mean m, the time between appearances follows a exponential distribution with mean 1/m. so, the probability that the next ghost appearance happens before x hours is equal to:

[tex]P(X\leq x)=1-e^{-xm}[/tex]

Then, replacing m by 1.4 ghosts per hour we get:

[tex]P(X\leq x)=1-e^{-1.4x}[/tex]

Additionally, The exponential distribution have a memoryless property, so if it is now 1:00 p.m. and we want the probability that ghost appear before 1:30 p.m., we need to find the difference in hours from 1:00 p.m and 1:30 p.m. no matter that the last ghost appearance was at 12:35 p.m.

Therefore, there are 0.5 hours between 1:00 p.m. and 1:30 p.m, so the probability that the 7th ghost will appear before 1:30 p.m is calculated as:

[tex]P(x\leq 0.5)=1-e^{-1.4*0.5} =0.503[/tex]

Final answer:

The probability that the 7th ghost will appear before 1:30 p.m. in an old house in Pomona, CA can be calculated as approximately 0.5034, or 50.34%. This calculation is based on the Poisson distribution, with a rate of 1.4 ghosts per hour and the last ghost appearance occurring at 12:35 p.m.

Explanation:

The probability of the 7th ghost appearing before 1:30 p.m. can be calculated using the Poisson distribution. We know that the rate of ghost appearances is 1.4 ghosts per hour, which means the average time between ghost appearances is 1/1.4 hours. From 12:35 p.m. to 1:00 p.m., there are 25 minutes, or 25/60 hours. So, the probability that the 7th ghost will appear before 1:30 p.m. is equal to the probability that there will be at least 1 ghost in the remaining time, i.e., the probability that at least one ghost appears in the next 30 minutes.

We can use the complementary probability method to calculate this. The complementary probability is the probability that none of the ghosts appears in the next 30 minutes. Since the time follows a Poisson process, we can use the Poisson probability formula. Let's calculate:

Therefore, the probability that the 7th ghost will appear before 1:30 p.m. is approximately 0.5034, or 50.34%.

Answer:

a) Null hypothesis: [tex]p \leq 0.44[/tex]

b) Alternative hypothesis: [tex]p > 0.44[/tex]

c) For this case our parameter of interest is a proportion "reduced rate of first heart attacks for a new drug". for this reason the answer would be 1 proportion and we can conduct the hypothesis with a 1 z proportion test  

Step-by-step explanation:

For this case the pharmaceutical company thinks they have a drug that will be more effective than aspirin and on this case that means a better rate in order to reduce the heart attacks (that represent the alternative hypothesis since that;s what they want to proof), and the complement would be the null hypothesis.

Part a

Null hypothesis: [tex]p \leq 0.44[/tex]

Part b

Alternative hypothesis: [tex]p > 0.44[/tex]

Part c

For this case our parameter of interest is a proportion "reduced rate of first heart attacks for a new drug". for this reason the answer would be 1 proportion and we can conduct the hypothesis with a 1 z proportion test  

Answer:

-3y=-2x+9

Step-by-step explanation:

flip the value

when flipping a value remember to invert its sign

Answer:

6y+8

Step-by-step explanation:

You should apply distributive property of the product, and multiply 2 by 3y and then add to that result the product of 2 times 4: [tex](3y+4)\times2= 2\times3y+2\times4=6y+8[/tex].

The question is incomplete. The complete question is as follows.

Consider the reduction of the rectangle. A large rectangle has a length of 16.8 feet and width of 2.3 feet. A smaller rectangle has a length of 4.5 feet and width of x feet. Not drawn to scale (The drawing is in the attachment). Rounded to the nearest tenth.  

What is the value of x?

A. 0.1 feet

B. 0.6 feet

C. 1.6 feet

D. 2.0 feet

Answer: B. 0.6 feet

Step-by-step explanation: Two quantity are proportional if the ratio of them is constant. From the drawing and the question, we have that the two rectangles are proportional between them.

[tex]\frac{x}{2.3} = \frac{4.5}{16.8}[/tex]

[tex]x = \frac{4.5*2.3}{16.8}[/tex]

x = [tex]\frac{10.35}{16.8}[/tex]

x = 0.6

The width of the smaller rectangle is 0.6 feet.

Answer: 0.6 ft.

Step-by-step explanation: 16.8/4.5= 3.733333333 repeating. The scale factor is 3.73333333333 repeating. 2.3/3.733333333333 Repeating equals .61607142857. We can shorten it to .61, and it says rounded to the nearest tenth, so .61 rounded is 0.6.

Answer:

32/9 or 3 5/9

Step-by-step explanation:

3.5555

3 5/9

(3×9 + 5)/9

32/9

Answer:

x = 32/9

Step-by-step explanation:

Let x = 3.55555555repeating

Multiply by 10

10x = 35.555555 repeating

Subtract the first equation

10x = 35.555555 repeating

-x = 3.55555555repeating

-------------------------------------

9x = 32

Divide each side by 9

9x/9 = 32/9

x = 32/9

Answer:

The answer is c

Step-by-step explanation:

the graph shows the equations y=3x-2 and y=1/2+3 and if y is greater than the otherside you shade above and if it's less than the otherside you shade below. y=3x-2 is sahded above the line and y=1/2+3 is shaded below the line.

Answer:

  see below

Step-by-step explanation:

The line with the shallow slope has a slope of 1 unit of rise for each 2 units of run, so a slope of 1/2. It has a y-intercept of 3. The shading is below it, so you're looking for an inequality that looks like ...

  y ≤ 1/2x + 3

Only one answer choice matches.

Answer:

The best point estimate for the mean monthly food budget for all residents of the local apartment complex is $419.

Step-by-step explanation:

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

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem:

The mean of the sample is $419.

What is the best point estimate for the mean monthly food budget for all residents of the local apartment complex?

By the Central Limit Theorem(inverse, that is, sample to the population), $419.

The best point estimate for the mean monthly food budget for all residents of the local apartment complex is $419.

Answer:

[tex]n(S\cap F \cap R)=7[/tex]

Step-by-step explanation:

The Universal Set, n(U)=2092

[tex]n(S)=1232\\n(F)=879\\n(R)=114[/tex]

[tex]n(S\cap R)=23\\n(S\cap F)=103\\n(F\cap R)=14[/tex]

Let the number who take all three subjects, [tex]n(S\cap F \cap R)=x[/tex]

Note that in the Venn Diagram, we have subtracted [tex]n(S\cap F \cap R)=x[/tex] from each of the intersection of two sets.

The next step is to determine the number of students who study only each of the courses.

[tex]n(S\:only)=1232-[103-x+x+23-x]=1106+x\\n(F\: only)=879-[103-x+x+14-x]=762+x\\n(R\:only)=114-[23-x+x+14-x]=77+x[/tex]

These values are substituted in the second Venn diagram

Adding up all the values

2092=[1106+x]+[103-x]+x+[23-x]+[762+x]+[14-x]+[77+x]

2092=2085+x

x=2092-2085

x=7

The number of students who have taken courses in all three subjects, [tex]n(S\cap F \cap R)=7[/tex]

Using the principle of Inclusion-Exclusion, we find that 7 students have taken courses in all three languages: Spanish, French, and Russian.

Finding the Number of Students Taking All Three Language Courses

We can solve this problem using the principle of Inclusion-Exclusion. Let:
S = number of students taking Spanish
F = number of students taking French
R = number of students taking Russian
SF = number of students taking both Spanish and French
SR = number of students taking both Spanish and Russian
FR = number of students taking both French and Russian
SFR = number of students taking all three languages.

From the given data:

The principle of Inclusion-Exclusion states:

Total = S + F + R - SF - SR - FR + SFR
2092 = 1232 + 879 + 114 - 103 - 23 - 14 + SFR

Solving for SFR:

2092 = 2085 + SFR

Thus, SFR = 2092 - 2085 = 7

Therefore, 7 students have taken courses in all three languages: Spanish, French, and Russian.

Answer:

75 cm (if b=10 and h=15)

Step-by-step explanation:

The formula to calculate the area of a triangle is bh1/2. So you must do (10)(15)1/2, 10 multiplied by 15 equals 150. And 150 multiplied by 1/2 equals 75. Hope this helped!

Answer:

The minimum cost of installing the cable is $156.

Step-by-step explanation:

We have an optimization problem.

We have to minimize the cost of the cable.

We will use the variable x to express the the length of cable CP and PM, accordingly to the attache picture.

The length of the cable that goes from C to P (let's call it CP) is x.

[tex]\bar{CP}=x[/tex]

Then, the length of the cable that goes from P to M (PM) can be calcualted usign the Pithagorean theorem:

[tex]\bar{PM}=\sqrt{(33-x)^2+9^2}[/tex]

The cost function Y is:

[tex]Y=4*\bar{CP}+5*\bar{PM}=4x+5\sqrt{(33-x)^2+9^2}[/tex]

To optimize this cost funtion we have to derive and equal to 0:

[tex]\dfrac{dY}{dx}=0\\\\\\\dfrac{dY}{dx}=4+5(\dfrac{1}{2})((33-x)^2+9^2)^{-1/2} *(-2)(33-x)\\\\\\\dfrac{dY}{dx}=4+5\dfrac{x-33}{\sqrt{(33-x)^2+81}}=0\\\\\\\dfrac{x-33}{\sqrt{(33-x)^2+81}}=-\dfrac{4}{5}\\\\\\(x-33)=-\dfrac{4}{5}\sqrt{(33-x)^2+81}\\\\\\(x-33)^2=(-\dfrac{4}{5})^2[(x-33)^2+81]\\\\\\(x-33)^2=\dfrac{16}{25}(x-33)^2+\dfrac{1296}{25}\\\\\\\dfrac{25-16}{25} (x-33)^2=\dfrac{1296}{25}\\\\\\9(x-33)^2=1296\\\\\\x-33=\sqrt{\dfrac{1296}{9}}=\sqrt{144}=\pm12\\\\\\x=33\pm12\\\\\\x_1=33-12=21\\\\x_2=33+12=45[/tex]

The valid solution is x=21, as x can not phisically larger than 33.

The cost then becomes:

[tex]Y=4*\bar{CP}+5*\bar{PM}=4x+5\sqrt{(33-x)^2+9^2}\\\\\\Y=4*21+5\sqrt{(33-21)^2+81}\\\\Y=81+5\sqrt{144+81}\\\\Y=81+5\sqrt{225}\\\\Y=81+5*15\\\\Y=81+75\\\\Y=156[/tex]

This optimization problem in calculus can be solved by setting up a cost function for the total cable installation, taking its derivative, setting it equal to 0 to find the critical points, which will give you the distance from C to P that minimizes cost, check this point for being minimal and calculating the minimal cost by substituting the found distance into the originally defined cost function.

The problem can be solved using the calculus principle of optimization. The situation described in your question makes a right triangle AMP. In this triangle, the vertical side (AP) measures 9 meters, and the hypotenuse (PM) represents cable installation that costs $5 per meter. The distance PC along the wall is $4 per meter. The cost of total cable installation from C -> P -> M is given as follows:

Cost = 4 * length CP + 5 * length PM

By the Pythagorean theorem, we know that [tex]PM = \sqrt{AP^2 + (33 - CP)^2}[/tex] Substituting PM into the equation, we get[tex]\text{Cost} = 4CP + 5 \cdot \sqrt{9^2 + (33-CP)^2}[/tex]

To minimize the cost, we take the derivative of the cost function and set it equal to 0 to find the critical points. Solving this equation will give you the value of CP that minimizes cost. Hence, by substituting found CP back to the original cost formula, we can find the minimal cost of installing the cable.

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Final answer:

The probability that exactly 3 people out of a sample of 20 Americans are afraid of being alone in a house at night, given that 5% of Americans have this fear, is approximately 13.98%.

Explanation:

The question asks for the probability that exactly 3 people in a sample of 20 Americans are afraid of being alone in a house at night, given that 5% of Americans have this fear. This can be solved using the binomial probability formula, which is P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, p is the probability of success on an individual trial, and C(n, k) is the number of combinations of n items taken k at a time.

Plugging in the values, we get P(X = 3) = C(20, 3) * 0.05³* 0.95¹⁷ First, calculate C(20, 3) = 20! / (3!(20-3)!) = 1140. Then calculate the probability: P(X = 3) = 1140 * 0.05³* 0.95¹⁷.

Doing the math, P(X = 3) ≈ 0.1398, or approximately 13.98%.

Final answer:

The probability that exactly 3 out of 20 Americans are afraid of being alone at night is found by using the binomial probability formula, which incorporates the number of people in the sample, the number who are afraid, and the overall chance of fear of being alone at night.

Explanation:

To find the probability that exactly 3 people out of a random sample of 20 Americans are afraid of being alone at night when it is known that 5% of Americans have this fear, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

n is the number of trials (in this case, 20)

k is the number of successes (in this case, 3)

p is the probability of success on an individual trial (5% or 0.05)

C(n, k) is the number of combinations of n things taken k at a time

Let's calculate:

Compute C(20, 3): This is 20! / (3! * (20-3)!).

Calculate p^k, which is 0.05^3.

Calculate (1-p)^(n-k), which is (1-0.05)^(20-3).

Multiply these together to get the probability.

After performing the calculations, we determine the probability of exactly 3 out of 20 Americans being afraid of being alone at night.