A survey reveals that each customer spends an average of 35 minutes with a standard deviation of 10 minutes in a department store. Assuming the distribution is normal, what is the probability a customer spends less than 30 minutes in the department store?

Answer:

The question is incomplete, below is the complete question,"There are two traffic lights on Darlene's route from home to work. Let E denote the event that Darlene must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.

a) What is the probability that the individual needn't stop at either light?

b) What is the probability that the individual must stop at exactly one of the two lights? c) What is the probability that the individual must stop just at the first light?"

Answer:

A. 0.63

B. 0.24

C. 0.07

Step-by-step explanation:

Data given,

P(E) = 0.2, P(F) = 0.3, and P(E ∩ F) = 0.13.

From the question, we can conclude that the event are dependent, hence

a. P(needn't stop at either light) = 1 - P(Need to stop at either light)

P(EUF)' =1-P(EUF)

P(EUF)' =1- (P(E)+P(F) -P(E ∩ F))

P(EUF)' =1-(0.2+0.3-0.13)

P(EUF)' =1-0.37

P(EUF)' =0.63

b. P(must stop at exactly one of the two lights) = P(must stop at either light) - P(must stop at both lights)

P(must stop at exactly one of the two lights)  = P(E u F) - P(En F)

but P(E u F)=0.37,

P(En F)=0.13,

P(must stop at exactly one of the two lights) = 0.37 - 0.13 = 0.24

c. P(must stop at just the first light) = P(must stop at either light) - P(must stop at the second light)

P(must stop at just the first light) = P(E u F)-P(F)

P(must stop at just the first light) = 0.37 - 0.3 = 0.07

The question deals with the topic of Probability in Mathematics. It presents the probabilities of two events, denoted as E and F, which are stopping at the first and second traffic lights, respectively. The question also provides the concurrent occurrence of both events.

The mathematics topic this question deals with is Probability. In the scenario given, E represents the event that Darlene must stop at the first traffic light and F represents the event that she needs to stop at the second traffic light. The probabilities of these events are given as P(E)=0.2 and P(F)=0.3, respectively. Additionally, we're given that the probability of both events happening (denoted P(E ∩ F)) is 0.13.

In order to analyze the situation, we can leverage the rule of joint probability, which states that the probability of two independent events both happening is the product of their individual probabilities. However, in this case the events E and F are not independent (since the probability of the intersection P(E ∩ F) is not equal to the product of probabilities P(E)*P(F)) so we know that the occurrence of E does influence the occurrence of F, and vice versa.

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Answer:

Number of student = 25

Step-by-step explanation:

Let x be the number of student and y be the others

A new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else

[tex]x+y=55\\y=55-x[/tex]

[tex]5x+10y=425[/tex]

replace y with 55-x

[tex]5x+10y=425\\5x+10(55-x)=425\\5x+550-10x=425\\-5x+550= 425[/tex]

Subtract 550 from both sides

[tex]-5x+550= 425\\-5x= -125\\x=25[/tex]

[tex]y=55-x\\y=55-25\\y=30\\[/tex]

Number of student = 25

Answer: 25 students used the service this month.

Step-by-step explanation:

Let x represent the number of students that used the streaming service this month.

Let y represent the number of people apart from students that used the streaming service this month.

This month, the service had 55 users. It means that

x + y = 55

The new streaming service charges 5 dollars per month for students, and 10 dollars per month for everyone else. They collected a total of 425 dollars. It means that

5x + 10y = 425 - - - - - - -1

Substituting x = 55 - y into equation 1, it becomes

5(55 - y) + 10y = 425

275 - 5y + 10y = 425

- 5y + 10y = 425 - 275

5y = 150

y = 150/5 = 30

x = 55 - y = 55 - 30

x = 25

Answer:

(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more = 0.0316.

(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 0.9684 .

Step-by-step explanation:

We are given that 20% of all packages from Amazon are delivered by UPS, from which 12% of the packages that are delivered by UPS weighs 2 lbs or more and 8% of the packages that are not delivered by UPS weighs less than 2 lbs.

Firstly Let A = Package from Amazon is delivered by UPS.

                B = Packages that are delivered by UPS weighs 2 lbs or more.

So, P(A) = 0.2  and P(A') = {Probability that package is not delivered by UPS}

                                P(A') = 1 - 0.2 = 0.8

P(B/A) = 0.12 {means Probability that package weight 2 lbs or more given it

                      is delivered by UPS}

P(B'/A') = 0.08 [means Probability that package weight less than 2 lbs given

                         it is not delivered by UPS}

Since, P(B/A) = [tex]\frac{P(A\bigcap B)}{P(A)}[/tex]   ,    [tex]P(A\bigcap B)[/tex] = P(B/A) * P(A) = 0.12 * 0.2 = 0.024 .

Also P(B) { Probability that package weight 2 lbs or more} is given by;

So, P(B) = [tex]P(B\bigcap A) + P(B\bigcap A')[/tex] = P(B/A) * P(A) + P(B/A') * P(A')

             = 0.12 * 0.2 + 0.92 * 0.8 { Here P(B/A') = 1 - P(B'/A') = 1 - 0.08 = 0.92}

             = 0.76

(a) Probability that a package is delivered by UPS if it weighs 2 lbs or more  is given by P(A/B);

  P(A/B) =   [tex]\frac{P(A\bigcap B)}{P(B)}[/tex] =  [tex]\frac{0.024}{0.76}[/tex] = 0.0316

(b) Probability that a package is not delivered by UPS if it weighs 2 lbs or more = 1 - P(A/B) = 1 - 0.0316 = 0.9684 .            

Using the Last Diminisher method, in the first round, Anne gets her fair share because no one diminishes. In the second round, Doris is the only one who diminishes, thus gets her fair share. In the third round, despite Cindy and Ellen both diminishing, Ellen gets her fair share because she is later in turn order.

The Last Diminisher method is a fair division protocol used when a divisible good, like a cake in this example, needs to be divided amongst several players. This method removes discrepancies by having each player in turn reduce the piece until they don't want to diminish it further, and then giving that piece to the last to diminish.

In this case, Anne, Betty, Cindy, Doris, and Ellen are dividing the cake and playing in that order. In the first round, no one diminishes, so Anne gets her fair share of the cake. In the second round, Doris is the only one who diminishes, so she gets her fair share at the end of this round. In the third round, the last to diminish are Cindy and Ellen, but since Ellen is later in order, Ellen is the one who gets her fair share at the end of the round.

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Approximately 68.26% of the measurements fall between 46 and 74 in this distribution.

To find the proportion of measurements between 46 and 74 in a normal distribution with a mean (μ) of 60 and a standard deviation (σ) of 14, we can use the standard normal distribution (z-score) and the cumulative distribution function (CDF).

First, we need to convert the interval endpoints to z-scores using the formula:

z = (x - μ) / σ

Where x is the value in the interval, μ is the mean, and σ is the standard deviation.

For x = 46:

z₁ = (46 - 60) / 14

z₁ = -1

For x = 74:

z₂ = (74 - 60) / 14

z₂ = 1

Using the Excel functions:

=NORM.S.DIST(-1) and =NORM.S.DIST(1)

The probabilities are 0.1587 and 0.8413 respectively.

Now, we want the proportion of measurements between z₁ and z₂, which is:

Proportion = 0.8413 - 0.1587

                  ≈ 0.6826

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Final answer:

Using the Empirical Rule for a normal distribution, approximately 68% of the measurements would fall between 46 and 74, as this range lies within one standard deviation above and below the mean of 60 in a distribution with a standard deviation of 14.

Explanation:

To find the proportion of measurements between 46 and 74 in a distribution with a mean of 60 and a standard deviation of 14, we can use the Empirical Rule, assuming the distribution is normal (bell-shaped). This rule states that approximately 68% of the data lies within one standard deviation of the mean, 95% within two, and more than 99% within three.

In this case, 46 is one standard deviation below the mean (60 - 14), and 74 is one standard deviation above the mean (60 + 14). So, we would expect approximately 68% of the measurements to lie between 46 and 74.

This is because the data is likely to be distributed symmetrically around the mean in a normal distribution, and the range given includes measurements falling within one standard deviation from the mean.

Answer:

[tex]z=\frac{\bar d -0}{\frac{\sigma_d}{\sqrt{n}}}=\frac{-0.625 -0}{\frac{6.818}{\sqrt{8}}}=-0.259[/tex]

[tex]p_v =2*P(z<-0.259) =0.796[/tex]

So the p value is higher than the significance level given [tex]\alpha=0.01[/tex], then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is equal to 0. So we can conclude that we don't have significant differences between the two populations.

Step-by-step explanation:

Previous concepts  

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

Solution to the problem

Let's put some notation :

x=values popoulation 2 , y = values population 1

x: 64 69 83 60 66 84 60 81

y: 58 76 77 70 62 76 67 76

The system of hypothesis for this case are:

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

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

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

d: -6,7,-6,10,-4,-8, 7, -5

The second step is calculate the mean difference  

[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}= \frac{-5}{8}=-0.625[/tex]

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

[tex]\sigma_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n} =6.818[/tex]

The 4 step is calculate the statistic given by :

[tex]z=\frac{\bar d -0}{\frac{\sigma_d}{\sqrt{n}}}=\frac{-0.625 -0}{\frac{6.818}{\sqrt{8}}}=-0.259[/tex]

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

[tex]p_v =2*P(z<-0.259) =0.796[/tex]

So the p value is higher than the significance level given [tex]\alpha=0.01[/tex], then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is equal to 0. So we can conclude that we don't have significant differences between the two populations.

Answer: Vector b is not a linear combination

Step-by-step explanation:

First of all we put the vectors in terms of different variables, such as:

a1(1,-2,0)=(a,-2a,0);

a2(0,1,3)=(0,b,3b);

a3(6,-6,18)=(6c,-6c,18c);

To know that a vector is a linear combination we need to express it like a sum of other different vectors.

(2,-2,6)=(a,-2a,0)+(0,b,3b)+(6c,-6c,18c)

(2,-2,6)=(a+0+6c,-2a+b-6c,0+3b+18c)

We express this sum like a system of equations.

a+6c=2

-2a+b-6c=-2

3b+18c=6

We solve this system of equations and we can note that the system don't have a solution, so the vector b is not a linear combination of a1, a2, and a3.

Upon forming a system of linear equations and solving, a solution would imply that vector b is indeed a linear combination of vectors a1, a2, and a3. The observed placement of pivots in the corresponding echelon matrix backs this conclusion.

In this question, you are asked to determine if vector b is a linear combination of vectors a1, a2, and a3. A vector is a linear combination of others if it can be written as a weighed sum of those vectors. To solve this problem, we need to form a system of linear equations based on the vectors and solve this system. If all of the coefficients can be expressed as real numbers, it means that the vector b is a linear combination of a1, a2, and a3.

In this case, our system of equations looks like this:

x∗a1 + y∗a2 + z∗a3 = b

In matrix form it can be written as:

|1 0 6|x| = |2|, |-2 1 -6|y| = |-2|, |0 3 18|z| = |6|.

Solve this system through methods like Gauss-Jordan elimination or row reduction. The pivots in the corresponding echelon matrix should be in the first entry in the first column, the second entry in the second column, and the third entry in the third column.

This suggests that vector b can indeed be a linear combination of vectors a1, a2, and a3.

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Final answer:

The hypothesis test to determine the mean annual premium in Pennsylvania compared to the national mean annual premium is H0: μ ≥ 1,503 and Ha: μ < 1,503. The point estimate of the difference between the mean annual premiums is -$78.

Explanation:

(a) To determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium, we need to develop a hypothesis test. The null hypothesis (H0) states that the mean annual premium in Pennsylvania is greater than or equal to the national mean annual premium. The alternative hypothesis (Ha) states that the mean annual premium in Pennsylvania is less than the national mean annual premium. Therefore, the correct answer is:

H0: μ ≥ 1,503
Ha: μ < 1,503

(b) The point estimate in dollars of the difference between the mean annual premium in Pennsylvania and the national mean is calculated by subtracting the national mean annual premium ($1,503) from the mean annual premium in Pennsylvania ($1,425). Therefore, the point estimate is $1,425 - $1,503 = -$78.

Answer: the bill before 6% sales tax is $92.85

Step-by-step explanation:

Let x represent the bill before the 6% sales tax.

It means that you paid 6% tax on x and the amount of tax paid would be

6/100 × x = 0.06 × x = 0.06x

Total amount that you paid for the meal including the 6% tax would be

x + 0.06x = 1.06x

If you spent $98.42 on the meal after the 6% tax, it means that

1.06x = 98.42

Dividing the left hand side and the right hand side of the equation by 1.06, it becomes

1.06x/1.06 = 98.42/1.06

x = $92.85

Answer:

(44/4)*30 = $330

Step-by-step explanation:

divide by four and multiply by 30

Answer:

No, the standard deviation for number of pages in a romance novel is six only.

Step-by-step explanation:

First we state our Null Hypothesis, [tex]H_o[/tex] : [tex]\sigma[/tex] = 6

             and Alternate Hypothesis, [tex]H_1[/tex] : [tex]\sigma[/tex] > 6

We have taken these hypothesis because we have to check whether our population standard deviation is higher than what editor hypothesized of 6 pages in a romance novel.

Now given sample standard deviation, s = 9 and sample size, n = 25

To test this we use Test Statistics = [tex]\frac{(n-1)s^{2} }{\sigma^{2} }[/tex] follows chi-square with (n-1) degree of freedom [[tex]\chi ^{2}_n__-1[/tex]]

       Test Statistics = [tex]\frac{(25-1)9^{2} }{6^{2} }[/tex] follows [tex]\chi ^{2}_2_4[/tex]  = 54

and since the level of significance is not stated in question so we assume it to be 5%.

Now Using chi-square table we observe at 5% level of significance the [tex]\chi ^{2}_2_4[/tex] will give value of 36.42 which means if our test statistics will fall below 36.42 we will reject null hypothesis.

Since our Test statistics is more than the critical value i.e.(54>36.42) so we have sufficient evidence to accept null hypothesis and conclude that our population standard deviation is not more than 6 pages which the editor hypothesized.

Answer:

Correct answer is option C i.e 28 minutes

Step-by-step explanation:

Number of students arriving at adviser's office per hour = x = 28

Number of students get approved = [tex]\frac{1}{2min}[/tex]  = 30/hour

    ∴ y = 30

Number of students on average on waiting =Lq

Lq = [tex]\frac{x^{2} }{y(y-x)}[/tex]

=  [tex]\frac{28^{2} }{30(30-28)}[/tex]

= 13.07

Average time student has to spend in

Waiting = Wq = [tex]\frac{x}{y(y-x)}[/tex]

= [tex]\frac{28}{30(30-28)}[/tex]

= 0.466 hours

= 28 minutes

So, the probabilities are: (a) 37/64, (b) 3/64, (c) 27/64, (d) 57/64

In each spin, there are four possible outcomes (nun, gimel, hei, shin), and each outcome is equally likely.

(a) Probability of getting at least one nun:

The probability of getting no nuns in a single spin is 3/4. So, the probability of getting no nuns in three spins is [tex](3/4)^3[/tex]. Therefore, the probability of getting at least one nun is 1 - [tex](3/4)^3[/tex].

Probability of getting at least one nun:

1 - [tex](3/4)^3[/tex] = [tex]1-\frac{27}{64}=\frac{37}{64}[/tex] = 0.58

(b) Probability of getting exactly 2 nuns:

The probability of getting a nun in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex].

= [tex]3 \times \frac{1}{16} \times \frac{3}{4}=\frac{3}{64}[/tex] = 0.05

(c) Probability of getting exactly 1 hei:

The probability of getting a hei in a single spin is 1/4. So, the probability of getting exactly 1 hei in three spins is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]

= [tex]3 \times \frac{1}{4} \times \frac{9}{16}=\frac{27}{64}[/tex] = 0.42

(d) Probability of getting at most 2 gimels:

The probability of getting 0 gimels is [tex](3/4)^3[/tex]. The probability of getting 1 gimel is [tex]\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2[/tex]. The probability of getting 2 gimels is [tex]\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex].

Add these probabilities to get the total probability.

[tex]\left(\frac{3}{4}\right)^3+\left(\begin{array}{l}\frac{3}{1} \\\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2+\left(\begin{array}{l}\frac{3}{2} \\\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)[/tex]

[tex]=\frac{27}{64}+\frac{27}{64}+\frac{3}{64}=\frac{57}{64}[/tex] = 0.9

​

Calculating probabilities of specific outcomes when spinning a dreidel multiple times.

Dreidel Probability Calculations:

Answer:

12x + 2

Step-by-step explanation:

Let the number be represented by x.

Then five times the number = 5*x

Seven times the number = 7*x

Sum of 5 times the number minus -2 = [tex]\[5*x - (-2)\][/tex] = [tex]\[5x +2\][/tex]

Adding seven times the number to this expression yields, [tex]\[5x+2+7x\][/tex]

[tex]\[= (5+7)x+2\][/tex]

[tex]\[= 12x+2\][/tex]

So the simplified expression corresponds to 12x + 2.

Answer:

a) And if we replace we got: [tex]\bar X= 78[/tex]

[tex] s = 15.391[/tex]

b) [tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]    

[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]    

So on this case the 99% confidence interval would be given by (62.182;93.818)    

Step-by-step explanation:

Dataset given: 109 67 58 76 65 80 96 86 71 72

Part a

For this case we can calculate the sample mean with the following formula:

[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

And if we replace we got: [tex]\bar X= 78[/tex]

And the deviation is given by:

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 15.391[/tex]

Part b

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

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

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=10-1=9[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,9)".And we see that [tex]t_{\alpha/2}=3.25[/tex]

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

[tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]    

[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]    

So on this case the 99% confidence interval would be given by (62.182;93.818)    

Answer:

N = 6300 ways

She can choose the materials 6300 ways

Step-by-step explanation:

In this case order of selection is not important, so we use combination.

For solids,

She needs 2 out of 9 available solids = 9C2

For striped prints

She needs 4 out of 7 available = 7C4

For floral prints

She needs 4 out of 5 available = 5C4

The total number of ways she can choose the materials is;

N = 9C2 × 7C4 × 5C4

N = 9/(7!2!) × 7/(4!3!) × 5/(4!1!)

N = 6300 ways

To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is the product of the number of choices for each type of fabric.

To find the number of different ways Courtney can choose the materials for her quilt, we can use the concept of combinations. The total number of ways she can choose the materials is given by the product of the number of choices for each type of fabric. So, the answer is:

Total number of ways = number of ways to choose solids * number of ways to choose floral prints * number of ways to choose striped fabrics

Given that she needs 2 solids, 4 floral prints, and 4 striped fabrics, we can calculate:

Substituting these values into the formula:

Total number of ways = 36 * 5 * 35 = 6300

So, there are 6300 different ways Courtney can choose the materials for her quilt.

The answer are (a) 169 (b) 341 (c) 125 (d) 87

A Venn diagram is an illustration that uses circles to show the commonalities and differences between things or groups of things.

Given that, A department store surveyed 428 shoppers, and the following information was obtained: 216 shoppers made a purchase, and 294 were satisfied with the service they received. If 47 of those who made a purchase were not satisfied with the service,

Refer to the Venn diagram attached.

The total number of shoppers surveyed is, N = 428.

Number of shoppers who made a purchase, n (P) = 216

Number of shoppers who were satisfied with the service they received,

n (S) = 294

Number of shoppers who made a purchase but were not satisfied with the service, n(S' ∩ P)  = 47

(a) The number of shoppers who made a purchase and were satisfied with the service = n(S ∩ P)

n(S ∩ P) = n(P)-n(S'∩P)

= 216 - 47 = 169

(b) The numbers of shoppers who made a purchase or were satisfied with the service = n (P ∪ S)

n (P ∪ S) = n(P)+n(S)-n(S∩P)

= 216+294-169

= 341              

(c) The numbers of shoppers who were satisfied with the service but did not make a purchase = n(S∩P')

= n(S)-n(S∩P)

= 241-169

= 125

(d) The number of shoppers who were not satisfied and did not make a purchase  = n(S'∩P')

= N-n (S ∪ P)

= 428-341

= 87

Hence, the answer are (a) 169 (b) 341 (c) 125 (d) 87

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a. 169 shoppers made a purchase and were satisfied with the service.

b. 341 shoppers made a purchase or were satisfied with the service.

c. 125 shoppers were satisfied with the service but did not make a purchase.

d. 381 shoppers were not satisfied and did not make a purchase.

Let's break down the information given:

Total shoppers surveyed = 428

Shoppers who made a purchase = 216

Shoppers satisfied with the service = 294

Shoppers who made a purchase and were not satisfied = 47

We are asked to find:

a. Shoppers who made a purchase and were satisfied with the service.

To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers who made a purchase:

216 − 47 = 169

b. Shoppers who made a purchase or were satisfied with the service.

To find this, we add the shoppers who made a purchase and the shoppers who were satisfied, but we need to be careful not to count the overlap twice (those who made a purchase and were satisfied):

216+294−169=341

c. Shoppers who were satisfied with the service but did not make a purchase.

To find this, we subtract the shoppers who made a purchase and were satisfied from the total shoppers who were satisfied:

294−169=125

d. Shoppers who were not satisfied and did not make a purchase.

To find this, we subtract the shoppers who made a purchase and were not satisfied from the total shoppers surveyed:

428−47=381

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Answer:

116,280 ways

Step-by-step explanation:

The number of ways of assigning the accounts to each of the salesperson is computed by combination

Number of ways = n combination r = n!/(n-r)!r!

n = 21, r = 7

Number of ways = 21 combination 7 = 21!/(21-7)!7! = 21!/14!7! = 116,280 ways

Let [tex]t[/tex] be the number of minutes.

The first tank starts with 1500 liters, and loses 4 liters per minute, so after [tex]t[/tex] minutes there will be

[tex]1500-4t[/tex]

liters of water.

The second tank is filled at 6 liters per second, i.e.

[tex]6\times 60=360[/tex] liters per minute.

So, there will be

[tex]300+360t[/tex]

liters of water in the second tank after [tex]t[/tex] minutes.

The two quantities will be equal when

[tex]1500-4t=300+360t \iff 1200=364t \iff t=\dfrac{1200}{364}\approx 3.3[/tex]

so, approximately, after 3.3 minutes.

Answer: it will take about 3.3 minutes for both tanks to have the same amount of water.

Step-by-step explanation:

Let x represent the number of minutes it will take both tanks to have same amount of water.

A water tank has 1,500 liters of water. It has a leak, losing 4 liters per minute. This means that in x minutes, the volume of water in the tank would be

1500 - 4x

At the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per second. Converting 6 liters per second to minutes, it becomes 60 × 6 = 360 liters per minute. This means that in x minutes, the volume of water in the tank would be

300 + 360x

For both tanks to have same amount of water, then

1500 - 4x = 300 + 360x

360x + 4x = 1500 - 300

364x = 1200

x = 1200/364 = 3.3 minutes

Answer:

Are the events A and B disjoint? Yes

Step-by-step explanation:

Disjoint events are those events that cannot occur at the same time, i.e. for events X and Y to be disjoint, [tex]P(X\cap Y)=0[/tex].

The event A is defined as the number of students whose age is at most 32.

And event B is defined as the number of students whose age is at least 37.

The events A and B are disjoint events.

The sample space for the intersection of these two events is:

Sample space of (A ∩ B) = 0

As there are no common terms in both the sample.

Hence proved, events A and B are disjoint.

Answer:

Kelly will owe $1320.92 to the fabulous fence company.

Step-by-step explanation:

There is a cost related to the number of hours and a cost per feet. So the total cost is:

[tex]T = C_{h} + C_{f}[/tex]

In which [tex]C_{h}[/tex] is the cost related to the number of hours and [tex]C_{f}[/tex] is the cost related to the number of feet.

Cost per hour

Each hour costs $15.57.

They work for 6 hours total. So

[tex]C_{h} = 15.57*6 = 93.42[/tex]

Cost per feet

Each feet costs $3.25.

Kelly needs 350 feet. So

[tex]C_{f} = 350*3.25 = 1137.5[/tex]

The total cost is:

[tex]T = C_{h} + C_{f} = 93.42 + 1137.5 = 1230.92[/tex]

Kelly will owe $1320.92 to the fabulous fence company.

Answer:

A) C(d,m) = 40 + 55d + 0.13m

B) $448

Step-by-step explanation:

Let 'd' be the number of days and 'm' the number of miles driven.

A) The cost function that describes a fixed amount of $40, added to a variable amount of $55 per day (55d) and a variable amount of 13 cents per mile (0.13m) is:

[tex]C(d,m) = 40 +55d +0.13m[/tex]

B) If  d = 5 and m =600, the total cost is:

[tex]C(5,600) = 40 +55*6 +0.13*600\\C(5,600)=\$448[/tex]

The cost is $448.

The formula for the cost, C of renting a car as a function of the number of days, d, and the number of miles driven, m is C f(d, m) = 40 + 55d + 0.13m

Given:

Application fee = $40

cost per day = $55

cost per mile = $0.13

let

Total cost = C

Number of days = d

Number of miles = m

Total cost, C = 40 + 55d + 0.13m

C f(d, m) = 40 + 55d + 0.13m

= 40 + 55(5) + 0.13(600)

= 40 + 275 + 78

= $393

Therefore, the cost of renting the car given the number of days is $393

Read more:

https://brainly.com/question/16450098

Answer:

2√6 ft

Step-by-step explanation:

Tan Ф = opposite/ adjacent

tan 60  = t / 2√2 ft

tan 60 = √3

t = (tan 60 )(2√2 ft)

t = (√3)(2√2 ft)  = 2√6 ft

Option B:

[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

Solution:

Given sound level = 120 decibel

To find the intensity of a fire alarm:

[tex]$\beta=10\log\left(\frac{I}{I_0} \right)[/tex]

where [tex]I_0=1\times10^{-12}\ \text {watts}/ \text m^2}[/tex]

Step 1: First divide the decibel level by 10.

120 ÷ 10 = 12

Step 2: Use that value in the exponent of the ratio with base 10.

[tex]10^{12}[/tex]

Step 3: Use that power of twelve to find the intensity in Watts per square meter.

[tex]$10^{12}=\left(\frac{I}{I_0} \right)[/tex]

[tex]$10^{12}=\left(\frac{I}{1\times10^{-12}\ \text {watts}/ \text m^2} \right)[/tex]

Now, do the cross multiplication,

[tex]I=10^{12}\times1\times\ 10^{-12} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1\times\ 10^{12-12} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

[tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex]

Option B is the correct answer.

Hence [tex]I=1.0\times\ 10^{0} \ \text {watts}/ \text m^2}[/tex].

Final answer:

To find the intensity of a sound at 120 dB, we use the formula SIL = 10 log(I / I0). With I0 as 10⁻¹² W/m², we find that I = 1.0 x 10⁰ W/m², corresponding to choice B.

Explanation:

To determine the intensity of a fire alarm that has a sound level of 120 decibels (dB), we use the relationship between sound intensity level and intensity in watts per meter squared (W/m²). The formula to convert decibel level to intensity is:

SIL = 10 log(I / I0)

Where SIL is the sound intensity level in decibels, I is the intensity of the sound, and I0 is the reference intensity, usually taken as 10⁻¹² W/m², the threshold of human hearing. To find the unknown intensity I, we can rearrange the formula:

I = I0 × 10(SIL/10)

For a sound level of 120 dB, the calculation would be:

I = 10⁻¹² W/m² × 10¹²⁰/¹⁰

I = 10⁻¹² W/m² × 10¹²

I = 1.0 × 10⁰ W/m²

Therefore, the correct answer is B. 1.0 x 10⁰ watts/m².

Answer:

(a) There are 70 different ways set up 4 computers out of 8.

(b) The probability that exactly three of the selected computers are desktops is 0.305.

(c) The probability that at least three of the selected computers are desktops is 0.401.

Step-by-step explanation:

Of the 9 new computers 4 are laptops and 5 are desktop.

Let X = a laptop is selected and Y = a desktop is selected.

The probability of selecting a laptop is = [tex]P(Laptop) = p_{X} = \frac{4}{9}[/tex]

The probability of selecting a desktop is = [tex]P(Desktop) = p_{Y} = \frac{5}{9}[/tex]

Then both X and Y follows Binomial distribution.

[tex]X\sim Bin(9, \frac{4}{9})\\ Y\sim Bin(9, \frac{5}{9})[/tex]

The probability function of a binomial distribution is:

[tex]P(U=k)={n\choose k}\times(p)^{k}\times (1-p)^{n-k}[/tex]

(a)

Combination is used to determine the number of ways to select k objects from n distinct objects without replacement.

It is denotes as: [tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]

In this case 4 computers are to selected of 8 to be set up. Since there cannot be replacement, i.e. we cannot set up one computer twice or thrice, use combinations to determine the number of ways to set up 4 computers of 8.

The number of ways to set up 4 computers of 8 is:

[tex]{8\choose 4}=\frac{8!}{4!(8-4)!}\\=\frac{8!}{4!\times 4!} \\=70[/tex]

Thus, there are 70 different ways set up 4 computers out of 8.

(b)

It is provided that 4 computers are randomly selected.

Compute the probability that exactly 3 of the 4 computers selected are desktops as follows:

[tex]P(Y=3)={4\choose 3}\times(\frac{5}{9})^{3}\times (1-\frac{5}{9})^{4-3}\\=4\times\frac{125}{729}\times\frac{4}{9}\\ =0.304832\\\approx0.305[/tex]

Thus, the probability that exactly three of the selected computers are desktops is 0.305.

(c)

Compute the probability that of the 4 computers selected at least 3 are desktops as follows:

[tex]P(Y\geq 3)=1-P(Y<3)\\=1-[P(Y=0)+P(Y=1)+P(Y=2)]\\=1-[({4\choose 0}\times(\frac{5}{9} )^{0}\times (1-\frac{5}{9} )^{4-0}+({4\choose 1}\times(\frac{5}{9} )^{1}\times (1-\frac{5}{9} )^{4-1}+({4\choose 2}\times(\frac{5}{9} )^{2}\times (1-\frac{5}{9} )^{4-2}]\\=1-0.59918\\=0.40082\\\approx0.401[/tex]

Thus, the probability that at least three of the selected computers are desktops is 0.401.

We can take the square root of both sides, adding a plus/minus sign of the right hand side:

[tex]\sqrt{(y+150)^2}=\pm\sqrt{50}\iff y+150 = \pm\sqrt{50}[/tex]

Then, we subtract 150 from both sides:

[tex]y=\pm\sqrt{50}-150[/tex]

So, the two solutions are

[tex]y_1 = \sqrt{50}-150,\quad y_2 = -\sqrt{50}-150[/tex]

Answer:

x

=

1

+

i

√

79

8

,

1

−

i

√

79

8

Step-by-step explanation:

Volume of the cone is 117.5 π ft³

Step-by-step explanation:

Lateral area of the cone = πrs

s is the slant height = 15 ft

From the above formula, we can find the radius as, 5 ft.

Volume of the cone = π r² h/3

s = √ (5²+ h²)

Squaring on both sides, we will get,

s² = 15² = (5² + h²)

 15² - 5² = h²

225 - 25 = 200 = h²

h = √200 = 14.1 ft

Volume = π × 5² × 14.1  / 3 = 117.5 π ft³

Answer:

a) [tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]

b) The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean

Step-by-step explanation:

Part a

The best point of estimate for the population mean is the sample mean given by:

[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

Since is an unbiased estimator [tex] E(\bar X) = \mu[/tex]

Data given: 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1

So for this case the sample mean would be:

[tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]

Part b

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[/tex] represent the sample mean for the sample  

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

s represent the sample standard deviation

n represent the sample size  

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

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

The margin of error is given by this formula:

[tex] ME=t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]    (2)

And for this case we know that ME =0.89 with a confidence of 95%

So then the limits for our confidence level are:

[tex] Lower= \bar X -ME= 1.75- 0.89=0.86[/tex]

[tex] Upperr= \bar X +ME= 1.75+0.89=2.64[/tex]

So then the best answer for this case would be:

The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean