An airport official wants to prove that the proportion of delayed flights for Airline A (denoted as p1) is less than the proportion of delayed flights for Airline B (denoted as p2). Random samples for both airlines after a storm showed that 51 out of 200 flights for Airline A were delayed, while 60 out of 200 of Airline B's flights were delayed. The test statistic for this problem is -1.00. The p-value for the test statistic for this problem is:

A. p = 0.3413 B. p = 0.0668 C. p = 0.1587 D. p = 0.0228

Answer:

  b) $17,400

  d) $33,517.20

Step-by-step explanation:

a) $28,482.19 . . . . future value of all deposits

__

b) The initial deposit was $3000, and there were 144 deposits of $100 each, for a total of ...

  $3000 +144×100 = $17,400 . . . . total deposited

__

c) $558.62

__

d) 60 monthly withdrawals were made in the amount $558.62, for a total of ...

  60×$558.62 = $33,517.20 . . . . total withdrawn

_____

Additional information about (a) and (c)

(a) The future value of the initial deposit is the deposit multiplied by the interest multiplier over the period.

  A = P(1 +r/n)^(nt) = 3000(1 +.066/12)^(12·12) = 3000·1.0055^144 ≈ 6609.065

The future value of $100 deposits each month is the sum of the series of 144 terms with common ratio 1.0055 and initial value 100.

  A = 100(1.055^144 -1)/0.0055 ≈ 21,873.123

So, the total future value is ...

  $6609.065 +21873.123 ≈ $28482.188 ≈ $28,482.19

__

(c) The withdrawal amount can be found using the same formula used for loan payments:

  A = P(r/n)/(1 -(1 +r/n)^(-nt)) = $28482.19(.0055)/(1 -1.0055^-60) ≈ $558.62

The total amount deposited in the account was $17,400 including an initial investment of $3,000 and subsequent monthly payments of $100 for 12 years. The total amount withdrawn was equal to the final balance after the last deposit.

Let's tackle each question one by one:

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Answer:

Option b.

Step-by-step explanation:

Statement b would be true in this case.

Let's gather data from the question:

student council = seniors + juniors

Now, some few things to note:

1. Senior students are in their 12th grade. This is the senior year in high school.  

2. The sophomore is the 10th year in school. These are not senior year students.

Isolating the students, the sophomore + junior students are likely to be the majority here.

Some junior and sophomore students voted for the proposal so it means that the combined number will be: all senior students + some juniors + some sophomores.

Therefore, the majority of the freshmen and a majority of the sophomores voted for the proposal.

Answer:

128

Step-by-step explanation:

Answer:

a) 0.2

b) 0.28

c) 0.12

d) 0.7

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

We have the following table:

                  At Least One Tattoo No Tattoo

Age 18-29 126                              324

Age 30-50 54                               396

So in total, there are

126 + 324 + 54 + 396 = 900 people

(A) P(tattoo)

This is the probability that a randomly selected person has a tattoo.

Desired outcomes:

126 + 54 = 180

180 people have at least one tattoo

Total outcomes:

There are 900 people.

P(tattoo) = 180/900 = 0.2

(B) P(tattoo | age 18-29)

This the probability that a person aged 18-29 has a tattoo

Desired outcomes:

126 people aged 18-29 have tattoos

Total outcomes:

126 + 324 = 450 people aged 18-29

P(tattoo | age 18-29) = 126/450 = 0.28

(C) P(tattoo | age 30-50)

This the probability that a person aged 30-50 has a tattoo

Desired outcomes:

54 people aged 18-29 have tattoos

Total outcomes:

54 + 396 = 450 people aged 30-50

P(tattoo | age 18-29) = 54/450 = 0.12

(D) P(age 18-29 | tattoo)

The probability that a tattoed person is 18-29.

Desired outcomes:

126 tattoed people are 18-29

Total outcomes

126 + 54 = 180 tattoed people

P(age 18-29 | tattoo) = 126/180 = 0.7

Answer:

Option D

Step-by-step explanation:

The current GDP is a true reflective of the actual GDP per person.

The average GDP per person is given as follows:

average GDP = [tex]\frac{Current GDP}{total population}[/tex]

For example, take Germany:

Amount in millions ( current GDP) = 3,649,493

Total population = 82 110 000

GDP per person = [tex]\frac{3649493}{82110000}[/tex]

                            = 0.044

The list in the descending order will be:

U.S

Japan

Germany

The correct ordering of real GDP per person is Japan, Germany, United States.

The correct ordering of real GDP per person from highest to lowest based on the given information is Japan, Germany, United States (option B).

GDP per person is calculated by dividing the GDP (Constant USS) by the population. In this case, for 2008, the GDP per person for Japan is 5,166,281 / 127.70 = 40,442.37, for Germany is 2,091,573 / 82.11 = 25,467.29, and for the United States is 11,513,872 / 304.06 = 37,868.49.

Therefore, option A) Japan has the highest real GDP per person, followed by the United States, and then Germany.

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(a) The probability that a hand will contain exactly two wild cards is 0.076.

(b) The probability that a hand will contain two wild cards, two red cards, and three blue cards is 0.0007.

Let's solve these problems using the concept of combinations in probability.

Remember, [tex]C(n, k)[/tex] denotes the number of ways to choose k items from a set of n items, and is calculated as

[tex]C(n,k)=\frac{n!}{k!(n-k)!}[/tex]

where "!" denotes factorial. For example, [tex]5! = 5 \times 4 \times 3 \times 2 \times 1[/tex]

(a) The probability that a hand will contain exactly two wild cards:

The total number of ways to choose 7 cards out of 108 is [tex]C(108, 7)[/tex].

The number of ways to choose 2 wild cards out of 8 is [tex]C(8, 2)[/tex].

The number of ways to choose the remaining 5 cards out of the 100 non-wild cards is [tex]C(100, 5)[/tex].

So, the probability is [tex]\frac{C(8, 2) \times C(100, 5)}{C(108, 7)} \approx 0.076[/tex]

(b) The probability that a hand will contain two wild cards, two red cards, and three blue cards:

The number of ways to choose 2 wild cards out of 8 is [tex]C(8, 2)[/tex].

The number of ways to choose 2 red cards out of 25 is [tex]C(25, 2)[/tex].

The number of ways to choose 3 blue cards out of 25 is [tex]C(25, 3)[/tex].

So, the probability is [tex]\frac{C(8, 2) \times C(25, 2) \times C(25, 3)}{C(108, 7)} \approx 0.0007[/tex]

Answer:

Option C) 23.6°

Step-by-step explanation:

we know that  

In this problem the triangle ABC is a right triangle

see the attached figure to better understand the problem

[tex]sin(A)=\frac{BC}{AB}[/tex] ----> by SOH (opposite side divided by the hypotenuse)

substitute the given values

[tex]sin(A)=\frac{6}{15}[/tex]

using a calculator

[tex]A=sin^{-1}(\frac{6}{15})=23.6^o[/tex]

Answer:

0.008 is the probability that a computer will take more than 42 seconds to boot up.                                  

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 30 seconds

Standard Deviation, σ = 5 second

We are given that the distribution of time taken for a computer to boot up is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

a) P(computer will take more than 42 seconds to boot up)

P(x > 42)

[tex]P( x > 42) = P( z > \displaystyle\frac{42 - 30}{5}) = P(z > 2.4)[/tex]

[tex]= 1 - P(z \leq 2.4)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(x > 42) = 1 - 0.992 = 0.008[/tex]

0.008 is the probability that a computer will take more than 42 seconds to boot up.

Answer:

the mean is 82.75

Step-by-step explanation:

Amount                   Frequency             Mid Point           fx

0-19.99                             16                    9.995            159.92  

20.00-39.99                    13                   29.995          389.935

40.00-59.99                     21                    49.995        1049.895

60.00-79.99                      19                   69.995         1329.905

80.00-99.99                       12                89.995           1079.94

100.00-119.99                     10                109.995          1099.95

120.00-139.99                     7                 129.995              909.965

140.00-159.99                      8                  149.995            1199.96

160.00-179.99                       7                  169.995          1189.965

180,00-199.99                      1                    189.995         189.995

200.00-219.99                    3                    209.995        629. 985

220.00-239.99                    2                   229.995         459.99

240.00-259.99                    1                   249.995         240.995

∑                                         120                                           9930.4

Mean = ∑fx/∑x

Mean = 9930.4/120=82.7533 = 82.75

Answer: The mean amount spent on Valentine's day is;

$82.83

Step-by-step explanation: To find the mean amount we first arrange the numbers in a frequency table, then solve.

STEP1:

AMOUNT FREQUENCY

0-19.99 16

20.00-39.99 13

40.00-59.99 21

60.00-79.99 19

80.00-99.99 12

100.00-119.99 10

120.00-139.99 7

140.00-159.99 8

160.00-179.99 7

180,00-199.99 1

200.00-219.99 3

220.00-239.99 2

240.00-259.99 1

STEP 2: Find the center of each amount, to do this we have to find the average value of the amounts.

For the first amount is;

(0+19.99)/2 = 9.995

For the second amount is;

(20+39.99)/2 =29.995

Solving this for all the amount. Therefore the table comes

AMOUNT FREQUENCY

9.995 16

29.995 13

49.995 21

69.995 19

89.995 12

109.995 10

129.995 7

149.995 8

169.995 7

189.995 1

209.995 3

229.995 2

249.995 1

STEP 3: multiple each amount in step 2 with the frequency.

For the first amount;

9.995×16 = 159.92

For the second amount;

29.995×13= 389.935

For the third amount;

49.995×21= 1049.895

For the fourth amount;

69.99×19= 1329.81

For the fifth amount;

89.995×12=1079.94

For the six amount;

109.995×10= 1099.95

For the sixth amount;

129.995×7= 909.965

For the seventh amount;

149.995×8= 1199.96

For the eight amount;

169.995×7= 1189.965

For the ninth amount;

189.995×1= 189.995

For the tenth amount;

209.995×3= 629.985

For the eleventh amount;

229.995×2=459.99

For the twelveth amount;

249.995×1= 249.995

STEP 4: Sum up all the answers from the multiplication in step 3

Therefore;

159.92+389.935+1049.895+1329.81+1079.94+1099.95+909.965+1199.96+1189.965+189.995+629.985+459.99+629.985+459.99+249.995 = 9939.995

STEP 5: divide the sum of the value seen in step 4 with the total number of frequency to get the mean value.

The total number of frequency is 120

Therefore;

9939.305÷120=82.827541

Take the value to two decimal place, it becomes;

$82.83 this is the mean value of money spent on Valentine's day.

Answer:

The right answer is option 3.

lim x → −3 f(x) = [infinity] means the values of f(x) can be made arbitrarily large by taking x sufficiently close to (but not equal to) −3.

Step-by-step explanation:

The limit of a function is a fundamental concept concerning the behavior of that function near a particular input.

A function f assigns an output f(x) to every input x. We say the function has a limit L at an input a: this means f(x) gets closer and closer to L as x moves closer and closer to a. More specifically, when f is applied to any input sufficiently close to a, the output value is forced arbitrarily close to L.

That is,

lim x → a f(x) = L

Hope this helps!

The limit lim x → −3 f(x) = [infinity] means that as x values get closer to -3 (without becoming -3), the value of the function f(x) goes towards infinity i.e., it grows without bound. This is akin to certain function behaviors near a value at which an asymptote is present. However, the second part about f(x) values getting close to 0 seems contradiction to the first statement.

The statement lim x → −3 f(x) = [infinity] is related to a concept in Calculus known as a limit. When we say that the limit of f(x) as x approaches -3 is infinity, we mean that as we make x values closer and closer to -3 (without letting x actually be -3), the value of the function f(x) becomes larger and larger without bound, i.e., approaches infinity.

This is similar to some function behaviors near an asymptote. For example, the function y = 1/x has a vertical asymptote at x = 0, where y approaches infinity as x approaches zero from either direction. Here, as x gets arbitrarily close to 0, the value of y = 1/x gets arbitrarily large, or 'approaches infinity'.

On the other hand, when the question states, 'The values of f(x) can be made arbitrarily close to 0 by taking x sufficiently close to (but not equal to) -3', it signifies the tendency of the function values to get closer and closer to 0 as x gets closer to -3. This indicates a certain limit behavior, but it seems to be contradictory with the first part where the limit was stated to be infinity. It is important to scrutinize the function's properties and behavior around x = -3 carefully.

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Answer:

Bit rate = 440 kBits/sec

Step-by-step explanation:

Band width = W= 22 kHz

Number of levels = L = 1024 levels

Bit per sample:

 [tex]m=log_2 L\\\\m =log_2(1024)\\\\m=10 bits/sample[/tex]

Ideal pulses are sent at the Nyquist rate then bit rate = 2 x W x m

[tex]bit\,\,rate= 2\times 22\times 10^3\times 10\\\\bit\,\,rate= 440\times 10^3 bits\,sec^{-1}[/tex]

bit rate = 440 kBits/sec

Answer:

-1.28 ft/s

Step-by-step explanation:

We are given that

The height of tip of  fisherman's rod from the water surface=y=8 ft

[tex]\frac{dz}{dt}=-1ft/sec[/tex]

We have to find the rate at which the fish is approaching the base of the dock when x=10 ft

[tex]z=\sqrt{x^2+y^2}[/tex]

By Pythagoras theorem

[tex]Hypotenuse=\sqrt{base^2+(perpendicular\;side)^2}[/tex]

Substitute x=10 and y=8

[tex]z=\sqrt{(10)^2+8^2}=\sqrt{164}=2\sqrt{41}ft[/tex]

[tex]x^2+y^2=z^2[/tex]

Differentiate w.r.t t

[tex]2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt}[/tex]

[tex]x\frac{dx}{dt}+y\frac{dy}{dt}=z\frac{dz}{dt}[/tex]

Substitute the values

[tex]10\frac{dx}{dt}+8(0)=2\sqrt{41}\times (-1)[/tex]

[tex]\frac{dy}{dt}=0[/tex]

Because he never moves the rod.

[tex]\frac{dx}{dt}=\frac{-2\sqrt{41}}{10}=-1.28 ft/s[/tex]

Hence, the fish is approaching the base of the dock at the rate  of 1.28 ft/s

Answer:

Friend A

[tex]\hat p_A= \frac{100}{400}=0.25[/tex]

[tex]z=\frac{0.25 -0.333}{\sqrt{\frac{0.333(1-0.333)}{400}}}\approx -3.47[/tex]  

Friend B

[tex]\hat p_B= \frac{20}{120}=0.167[/tex]

[tex]z=\frac{0.167 -0.333}{\sqrt{\frac{0.333(1-0.333)}{120}}}\approx -3.80[/tex]  

Friend C

[tex]\hat p_C= \frac{65}{300}=0.217[/tex]

[tex]z=\frac{0.217-0.333}{\sqrt{\frac{0.333(1-0.333)}{300}}}\approx -4.17[/tex]  

So then the best solution for this case would be:

-3.47 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)

Step-by-step explanation:

Data given and notation

n represent the random sample taken

X represent the number of scissors selected for each friend

[tex]\hat p=\frac{X}{n}[/tex] estimated proportion of  scissors selected for each friend

[tex]p_o=\frac{1}{3}=0.333[/tex] is the value that we want to test

[tex]\alpha[/tex] represent the significance level

z would represent the statistic (variable of interest)

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

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the proportion that the friend will pick scissors is less than 1/3 or 0.333, the system of hypothesis would be:  

Null hypothesis:[tex]p\geq 0.333[/tex]  

Alternative hypothesis:[tex]p < 0.333[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

Friend A

[tex]\hat p_A= \frac{100}{400}=0.25[/tex]

[tex]z=\frac{0.25 -0.333}{\sqrt{\frac{0.333(1-0.333)}{400}}}\approx -3.47[/tex]  

Friend B

[tex]\hat p_B= \frac{20}{120}=0.167[/tex]

[tex]z=\frac{0.167 -0.333}{\sqrt{\frac{0.333(1-0.333)}{120}}}\approx -3.80[/tex]  

Friend C

[tex]\hat p_C= \frac{65}{300}=0.217[/tex]

[tex]z=\frac{0.217-0.333}{\sqrt{\frac{0.333(1-0.333)}{300}}}\approx -4.17[/tex]  

So then the best solution for this case would be:

-3.47 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)

Step-by-step explanation:

From the above illlustration,

Let x, be the number of footballs produced in the morning shift,

y, the number of baseball in the morning shift,

z, the number of football in the evening shift,

t, the number of baseball in the evening shift.

Minimizing the objective function,

min {20(x+y) + 25(z + t)}

Therefore, since the number of labor hours is for both shifts(morning and evening shifts), we add the following constraints:

0.75x + 2y ≤ 5000

0.75z + 2t ≤ 2000

Remember, the amount of leather available in the shifts is also limited. The following constraints are got:

7x + 15y ≤ 15000

7z + 15t ≤1 4000

Also, adding the constraints for the use of inner plastic lining, we have:

0.5x + 2y ≤ 2000

0.5z + 2t ≤ 1500

Modelling their demands through the following constraints:

x + z ≥ 1500

y + t ≥ 1200

Also, we are producing whole number of baseballs or footballs but we only, so

x, y, z, t ∈Z.

Finally,

min20(x + y) + 25(z + t)

0.75x + 2y ≤ 5000

0.75z + 2t ≤ 2000

7x + 15y ≤ 15000

7z + 15t ≤ 14000

0.5x + 2y ≤ 2000

0.5z + 2t ≤ 1500

x + z ≥ 1500

y + t ≥ 1200

y, x, t, z ∈ Z.

Answer:

249 per 100,000

Step-by-step explanation:

The projected incidence rate of heart disease per 100,000 in the United States in 2020 is determined by the number of estimated new cases of heart disease multiplied by 100,000 and divided by the estimated population of the United States in 2020:

[tex]R=\frac{850,000*100,000}{341,672,244}=248.78[/tex]

Rounding up to the next whole unit, the projected incidence is 249 per 100,000.

Final answer:

To calculate the incidence rate of heart disease per 100,000 in the U.S. in 2020, divide the number of new cases (850,000) by the U.S. population (341,672,244), then multiply by 100,000, resulting in an incidence rate of approximately 248.7.

Explanation:

To calculate the projected incidence rate of heart disease per 100,000 in the United States in 2020, follow these steps:

First, determine the total number of new heart disease cases. Researchers estimated 850,000 new cases of heart disease will be recorded.

Second, determine the population of the United States at the midpoint of 2020, which is 341,672,244.

Finally, calculate the incidence rate using the formula: \((\frac{Number\ of\ new\ cases}{Population}) \times 100,000=Incidence\ Rate\ per\ 100,000\). Therefore, \((\frac{850,000}{341,672,244}) \times 100,000\) results in an incidence rate of approximately 248.7 new cases per 100,000 people.

Understanding this incidence rate can help in assessing the burden of heart disease on the U.S. population and guiding health policy decisions.

Answer:

A. 1. P(A∪B)=0.85

2. P(A∩C)=0.045

3. P(A/B)=0.3

4. P(B∪C)=0.70

B. Event B and Event C are dependent

Step-by-step explanation:

A. As events are mutually exclusive, so,

P(A∪B)=P(A)+P(B)

P(A∩B)=P(A)*P(B)

1. P(A∪B)=?

P(A∪B)=P(A)+P(B)=0.3+0.55=0.85

P(A∪B)=0.85

2. P(A∩C)

P(A∩C)=P(A)*P(C)=0.30*0.15=0.045

P(A∩C)=0.045

3. P(A/B)

P(A/B)=P(A∩B)/P(B)

P(A∩B)=P(A)*P(B)=0.30*0.55=0.165

P(A/B)=P(A∩B)/P(B)=0.165/0.55=0.3

P(A/B)=0.3

4.  P(B∪C)

P(B∪C)=P(B)+P(C)=0.55+0.15=0.70

P(B∪C)=0.70

B.

The event B and C are mutually exclusive and events B and event C are dependent i.e. P(B and C)≠P(B)P(C)

The events are mutually exclusive i.e. P(B and C)=0

whereas  P(B)*P(C)=0.55*0.15=0.0825

Mutually exclusive events are independent only if either one of two or both events has zero probability of occurring.

Thus, event B and C are dependent

A) 1:P(A∪B)=0.85

2: P(A∩C)=0.045

3: P(A/B)=0.3

4: P(B∪C)=0.70

B) Events B and C are dependent events.

Since all three events are mutually exclusive:

So, P(A∪B)=P(A)+P(B)

P(A∪B) = 0.30+0.55

P(A∪B) = 0.85

P(A∩B)=P(A)P(B)

P(A∩B) = 0.30*0.55 = 0.165

P(A/B)=P(A∩B)/P(B)

P(A/B) = 0.165/0.55 = 0.3

Similarly, P(A∩C) =0.045

P(BUC) = 0.70

Events B and C are dependent events because they will be independent only if there is zero possibility of their occurrence.

Therefore, A) 1:P(A∪B)=0.85

2: P(A∩C)=0.045

3: P(A/B)=0.3

4: P(B∪C)=0.70

B) Events B and C are dependent events.

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Answer:

[tex]E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49[/tex]  

[tex]Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899[/tex]  

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{0.8899}=0.943[/tex]  

Step-by-step explanation:

For this case we have the following data given:

X      1    2    3    4    5

F     74   10  11    3    2

The total number of values are 100, so then we can find the empirical probability dividing the frequency by 100 and we got the followin distribution:

X          1          2        3         4          5

P(X)     0.74   0.10   0.11    0.03    0.02

Previous concepts

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".  

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).  

And the standard deviation of a random variable X is just the square root of the variance.  

Solution to the problem

In order to calculate the expected value we can use the following formula:  

[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]  

And if we use the values obtained we got:  

[tex]E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49[/tex]  

In order to find the standard deviation we need to find first the second moment, given by :  

[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]  

And using the formula we got:  

[tex]E(X^2)=1^2 *0.74 +2^2 *0.1 +3^2 *0.11 +4^2 0.03 +5^2 *0.02=3.11[/tex]  

Then we can find the variance with the following formula:  

[tex]Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899[/tex]  

And then the standard deviation would be given by:  

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{0.8899}=0.943[/tex]  

Answer:

The instantaneous rate of change of h(t) = 2t² + 2 at the point t = −1 is -4.

Step-by-step explanation:

If two distinct points [tex]P(x_1,y_1)[/tex] and [tex]Q(x_2,y_2)[/tex] lie on the curve [tex]y=f(x)[/tex], the slope of the secant line connecting the two points is

[tex]m_{sec}=\frac{y_2-y_1}{x_2-x_1}=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]

If we let the point [tex]x_2[/tex] approach [tex]x_1[/tex], then Q will approach P along the graph f(x). The slope of the secant line through points P and Q will gradually approach the slope of the tangent line through P as

[tex]m_{tan}= \lim_{x_2 \to x_1}\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]

And this is the instantaneous rate of change of the function f(x) at the point [tex]x_1[/tex].

From the information given, we know that the point P [tex](-1,2(-1)^2+2)=(-1,4)[/tex] lies on the curve [tex]h(t) = 2t^2 + 2[/tex].

If Q is the point [tex](t, 2t^2 + 2)[/tex] we can find the slope of the secant line PQ for the following values of t. Because we choose values that are getting closer and closer to −1.

[tex]\begin{array}{c}-0.9&-0.99&-0.999&-0.9999\\-1.1&-1.01&-1.001&-1.0001\\\end{array}\right[/tex]

Let the point P be [tex](x_2=-1, y_2=4)[/tex] and the point Q be [tex](x_1=t, y_1=2t^2+2)[/tex]. So,

[tex]m=\frac{4-(2t^2+2)}{-1-t}\\\\m=-\frac{2\left(t+1\right)\left(t-1\right)}{-1-t}\\\\m=2\left(t-1\right)[/tex]

Next, substitute the value of x in the formula of the slope

[tex]m=2(-0.9-1)=-3.8[/tex]

Do this for the other values of x.

Below, there is a table that shows the values of the slope.

From the table, as t approaches -1 from the left side (-0.9 to -0.9999), the slopes are approaching to -4 and as t approaches -1 from the right side (-1.1 to -1.0001), the slopes are approaching to -4. The value of the slope at P(-1,4) is then m = -4.

Final answer:

To estimate the instantaneous rate of change of h(t) at t = -1, we calculate the slopes of secant lines near that point and observe the pattern to approximate the slope of the tangent line, which represents the instantaneous velocity.

Explanation:

To estimate the instantaneous rate of change of the function h(t) = 2t² + 2 at t = −1, we need to calculate the slope of the tangent line at that point. We can approximate this slope using secant lines connecting points increasingly closer to t = −1.

Let's select two points close to t = −1, say t1 = −1.1 and t2 = −0.9, and compute the slope of the secant line:

Slope of secant line = (h(t2) − h(t1)) / (t2 − t1) = (3.62 − 4.42) / (-0.9 + 1.1) = −0.8 / 0.2 = −4.

To get a more accurate approximation, we'd choose points even closer to t = −1 and observe the pattern. We can assume that the slope of the tangent line, which represents the instantaneous velocity, is approximately equal to the slopes of the secant lines as they converge to a single value.

Answer:

0.749

Step-by-step explanation:

The probability that at least one of the three taken balls is blue is the inverse of the probability that none of the three taken balls is blue, aka all 3 of the taken balls are red. The probability of this to happen is

In the first pick: 13/20 chance of this happens

In the 2nd pick: 12/19 chance of this happens

In the 3rd pick: 11/18 chance of this happens

So the probability of picking up all 3 red balls is

[tex]\frac{13*12*11}{20*19*18} = \frac{1716}{6840} = 0.251[/tex]

So the probability of picking up at least 1 blue ball is

1 - 0.251 = 0.749

Answer:

a) Null hypothesis: [tex] p \leq 0.113[/tex]

Alternative hypothesis: [tex] p >0.113[/tex]

b) [tex]p_v =P(z>1.07)=0.1423[/tex]  

c) So the p value obtained was a high low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 5% of significance the proportion of workers belonged to unions  is not significantly higher than 0.113.  

Step-by-step explanation:

Part a

For this case we want to check the following system of hypothesis:

Null hypothesis: [tex] p \leq 0.113[/tex]

Alternative hypothesis: [tex] p >0.113[/tex]

Part b

Data given and notation

n=400 represent the random sample taken

X=52 represent the workers belonged to unions

[tex]\hat p=\frac{52}{400}=0.13[/tex] estimated proportion of workers belonged to unions

[tex]p_o=0.113[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

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

Concepts and formulas to use  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.13 -0.113}{\sqrt{\frac{0.113(1-0.113)}{400}}}=1.07[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

[tex]p_v =P(z>1.07)=0.1423[/tex]  

Part c

So the p value obtained was a high low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 5% of significance the proportion of workers belonged to unions  is not significantly higher than 0.113.  

Answer:

- 21 x + 24 y + 15 z =120

Step-by-step explanation:

Given that

Po(3,-2,5), Qo (-3,-1,-5), and Ro (0,-4,4) ,These are the point in the space.

We know that equation of a plane is given as

[tex]\begin{vmatrix}x-x_1 & y-y_1 &z-z_1 \\ x_2-x_1 & y_2-y_1 &z_2-z_1 \\ x_3-x_1 &y_3-y_1 & z_3-z_1\end{vmatrix}=0\\[/tex]

[tex]\begin{vmatrix}x-0 & y+4 &z-4 \\ 3-0 & -2+4 &5-4 \\ -3-0 &-1+4 & -5-4\end{vmatrix}=0.[/tex]

[tex]\begin{vmatrix}x & y+4 &z-4 \\ 3 & 2 &1 \\ -3 &3 & -9\end{vmatrix}=0.[/tex]

Now by solving above determinate we get

x( -18 -3 ) -(y+4 ) ( -27 +3 ) + ( z- 4) (9+6) = 0

-21 x +24 y -24 x 4 + 15 z - 24 = 0

- 21 x + 24 y + 15 z -120 = 0

- 21 x + 24 y + 15 z =120

Therefore the equation of the plane will be

- 21 x + 24 y + 15 z =120

The description corresponds to an observational study as the analyst is merely observing and analyzing the existing data (earnings per share) to make an investment decision, there's no control or manipulation of the variables involved.

The given description corresponds to an observational study. This is because the stock analyst is merely observing and analyzing the existing earnings per share of the stocks from a group of twenty and then making an investment decision based on this data. There is no manipulation or control of variables, which are defining characteristics of an experiment.

In an experiment, the researchers would have actively influenced the earnings per share (the variable) in some way to gauge the effect of that influence. However, in this case, the analyst is simply observing the earnings per share as they are to select a stock for investment.

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Answer:

a. r = 5t

b. [tex]A = \pi r^2[/tex]

c. [tex]A = \pi (5t)^2[/tex]

d. [tex]A = 25\pi t^2[/tex]

Step-by-step explanation:

a. Since the radius is increasing at a constant rate of 5 cm per second.

r = 5t

where r is the radius at time t (seconds)

b. Area of circle [tex]A = \pi r^2[/tex]

c. We can substitute r = 5t into the area formula to have

[tex]A = \pi r^2 = \pi (5t)^2[/tex]

d. In expand form

[tex]A = \pi (5t)^2 = 25\pi t^2[/tex]

a.  The expression is R = 5t

b.  The area of the circle in terms of R is A = πR²

c.  The area of the circle in terms of t is A = π(5t)²

d.  The area of the circle in terms of t in the expanded form is A = 25π×t²

It is a system of an equation in which the highest power of the variable is always 1. A one-dimension figure that has no width. It is a combination of infinite points side by side.

It is a locus of a point drawn at an equidistant from the center. The distance from the center to the circumference is called the radius of the circle.

Given

R = 5t where R is the radius, and t be the time.

Thus, the answer will be

a.  The expression will be

R = 5t

b.  The area of the circle in terms of R will be

Area = πR²

c.  The area of the circle in terms of t will be

Area = πR²

Area = π(5t)²

d.  The area of the circle in terms of t in the expanded form will be

Area = πR²

Area = π(5t)²

Area = 25π×t²

More about the linear system link is below.

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Answer:

'3'. x = 3 + 0.3333333333y Simplifying x = 3 + 0.3333333333y

Step-by-step explanation:

Answer:

a) Equation is

[tex]y = 0.1135x+1.590[/tex]

b) Slope = 0.1135 represents the change in y for a unit change in x

i.e. When nitrate content is increasedby 1, absorbence is increased by 0.1135

Step-by-step explanation:

Nitrates Absorbence

x y

50 7

50 7.6

100 12.7

200 24

400 47

800 93

1200 138

1600 183

2000 231

2000 226

SUMMARY OUTPUT        

Regression Statistics        

Multiple R 0.999911043        

R Square 0.999822094        

Adjusted R Square 0.999799856        

Standard Error 1.2890282        

Observations 10        

Coefficients

Intercept 1.589782721

x 0.113500259

we get regression line as

y = 0.1135x+1.590

a) Equation is

[tex]y = 0.1135x+1.590[/tex]

b) Slope = 0.1135 represents the change in y for a unit change in x

i.e. When nitrate content is increasedby 1, absorbence is increased by 0.1135

The difference between the volumes in cubic inches is option A) 72pi

Step-by-step explanation:

Volume = 4/3 π(6)³

⇒ 4/3(216)π

⇒ 4[tex]\times[/tex]72π

⇒ 288π cubic inches

Volume = π(6)²(6)

⇒ 6³π

⇒ 216π cubic inches

Difference between the volumes = 288π - 216π = 72π

The difference in volume between the two solids is 226.08in^3

Data;

The volume of a sphere is given as

[tex]v = \frac{4}{3} \pi r^3\\[/tex]

Let's substitute the values and find the volume

[tex]v = \frac{4}{3}*3.14*6^3\\v = 904.32in^3[/tex]

The formula of volume of a cylinder is given as

[tex]v = \pi r^2 h\\[/tex]

Let's substitute the values into the equation and solve

[tex]v = 3.14 * 6^2 * 6\\v = 678.24in^3[/tex]

The difference in volume between the two solids is

[tex]volume of sphere - volume of cylinder = 904.32 - 678.24 = 226.08in^3[/tex]

The difference in volume between the two solids is 226.08in^3

Learn more on volume of sphere and cylinder here;

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Answer:

There is a 25.52% probability of observating 4 our fewer succesful recommendations.

Step-by-step explanation:

For each recommendation, there are only two possible outcomes. Either it was a success, or it was a failure. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

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

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

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

And p is the probability of X happening.

In this problem we have that:

[tex]p = 0.553, n = 10[/tex]

If the claim is correct and the performance of recommendations is independent, what is the probability that you would have observed 4 or fewer successful:

This is

[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)[/tex]

In which

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

[tex]P(X = 0) = C_{10,0}.(0.553)^{0}.(0.447)^{10} = 0.0003[/tex]

[tex]P(X = 1) = C_{10,1}.(0.553)^{1}.(0.447)^{9} = 0.0039[/tex]

[tex]P(X = 2) = C_{10,2}.(0.553)^{2}.(0.447)^{8} = 0.0219[/tex]

[tex]P(X = 3) = C_{10,3}.(0.553)^{3}.(0.447)^{7} = 0.0724[/tex]

[tex]P(X = 4) = C_{10,4}.(0.553)^{4}.(0.447)^{6} = 0.1567[/tex]

[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.0003 + 0.0039 + 0.0219 + 0.0724 + 0.1567 = 0.2552[/tex]

There is a 25.52% probability of observating 4 our fewer succesful recommendations.

Answer:

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

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

b) [tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]

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

[tex]df=n-1=42-1=41[/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(t_{(41)}>4.905) =7.6x10^{-6}[/tex]

So the p value is lower than any significance level given, so then we can conclude that we can to reject the null hypothesis that the difference between the two mean is equal to 0.

c) The confidence interval is given by:

[tex] \bar d \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]

For this case we have a confidence of 1-0.05 = 0.95 so we need 0.05 of the area of the t distribution with 41 df on the tails. So we need 0.025 of the area on each tail, and the critical value would be:

[tex] t_{crit}= 2.02[/tex]

And if we find the interval we got:

[tex] 850- 2.02* \frac{1123}{\sqrt{42}}=499.96[/tex]

[tex] 850+ 2.02* \frac{1123}{\sqrt{42}}=1200.03[/tex]

We are confident 95% that the difference between the two means is between 499.96 and 1200.03

So we have enough evidence to conclude that one mean is higher than the other one, without conduct another hypothesis test because the confidence interval for the difference of means not contain the value of 0. And for this case the groceries would have a higher mean

Step-by-step explanation:

Previous concepts

A paired t-test is used to compare two population means where you have two samples in  which observations in one sample can be paired with observations in the other sample. For example  if we have Before-and-after observations (This problem) we can use it.  

Part a

Let put some notation  

x=test value for 1 , y = test value for 2

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:

The second step is calculate the mean difference  

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

This value is given [tex] \bar d = 850[/tex]

The third step would be calculate the standard deviation for the differences.

This value is given also [tex] s_d = 1123[/tex]

Part b

The 4 step is calculate the statistic given by :

[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]

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

[tex]df=n-1=42-1=41[/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(t_{(41)}>4.905) =7.6x10^{-6}[/tex]

So the p value is lower than any significance level given, so then we can conclude that we can to reject the null hypothesis that the difference between the two mean is equal to 0.

Part c

The confidence interval is given by:

[tex] \bar d \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]

For this case we have a confidence of 1-0.05 = 0.95 so we need 0.05 of the area of the t distribution with 41 df on the tails. So we need 0.025 of the area on each tail, and the critical value would be:

[tex] t_{crit}= 2.02[/tex]

And if we find the interval we got:

[tex] 850- 2.02* \frac{1123}{\sqrt{42}}=499.96[/tex]

[tex] 850+ 2.02* \frac{1123}{\sqrt{42}}=1200.03[/tex]

We are confident 95% that the difference between the two means is between 499.96 and 1200.03

So we have enough evidence to conclude that one mean is higher than the other one, without conduct another hypothesis test because the confidence interval for the difference of means not contain the value of 0. And for this case the groceries would have a higher mean

Answer:

H_o : u_d = 0 , H_1 : u_d ≠ 0

H_o rejected , p < 0.01

[ 499.969 < d < 1200.301 ] , d = 850

Step-by-step explanation:

Given:

- Difference in mean d = 850

- Standard deviation s = 1123

- The sample size n = 42

- Significance level a = 0.05

Solution:

- Set up and Hypothesis for the difference in means test as follows:

            H_o : Difference in mean u_d= 0

            H_1 : Difference in mean u_d ≠ 0

- The t test statistics for hypothesis of matched samples is calculated by the following formula:

           t = d / s*sqrt(n)

Hence,

           t = 850 / 1123*sqrt(42)

           t = 4.9053

  Thus, the test statistics t = 4.9053.

- The p-value is the probability of obtaining the value of the test statistics or a value greater.

 Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:

            p < 2*0.05 ----> 0.01

 Thus,  p < 0.05  ....... Hence, H_o is rejected

- Set up and Hypothesis for the difference in means test as follows:

            H_o : Difference in mean u_d =< 0

            H_1 : Difference in mean u_d > 0

- The t test statistics for hypothesis of matched samples is calculated by te following formula:

           t = d / s*sqrt(n)

Hence,

           t = 850 / 1123*sqrt(42)

           t = 4.9053

  Thus, the test statistics t = 4.9053.

 Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:

            p < 0.005

 Thus,  p < 0.05  ....... Hence, H_o is rejected

Hence, the point estimate is d = $850

- The interval estimate of the difference between two population means is calculated by the following formula:

             d +/- t_a/2*s / sqrt(n)

Where CI = 1 - a = 0.95 , a = 0.05 , a/2 = 0.025

Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:

              t_a/2 = t_0.025 = 2.020

Therefore,

              d - t_a/2*s / sqrt(n)

              850 - 2.020*1123 / sqrt(20)

             = 499.969

And,

              d + t_a/2*s / sqrt(n)

              850 + 2.020*1123 / sqrt(20)

             = 1200.031

- The 95% CI of the difference between two population means is:

              [ 499.969 < d < 1200.301 ]

Answer:

the bridge has a height y₀ = 6.94 m

Step-by-step explanation:

The position y of the loose bolt is given by (0,y) where

y = y₀ - 1/2*g*t²

where

y₀ = initial position of the bolt (height of the bridge) , g= gravity , t=time

and the position x of the car is given by (x,0) where

x= x₀  + v*t

where

x₀= initial position of the car

v= car's velocity

then in order for the bolt to hit the windshield they should be at  x=0 and y=0 at the same time , then

0= x₀  + v*t

t= -x₀/v

replacing in the equation for y

0 = y₀ - 1/2*g*t²

0 = y₀ - 1/2*g*(-x₀/v)²

0 = y₀ - 1/2*g*x₀²/v²

y₀ =  1/2*g*x₀²/v²

replacing values

y₀ =  1/2*g*x₀²/v² = 1/2* 9.8m/s² * (-25 m)²/(21 m/s)² = 6.94 m

then the bridge has a height y₀ =6.94 m

We have assumed that

- The bolt has no horizontal velocity ( only vertical velocity) , starts from rest and neglected air friction

- Neglecting the height of the car , position of the windshield and size of the loose bolt

Answer: the third option is the correct answer.

Step-by-step explanation:

Looking at the line plot,

There are 3 bags of oranges that weigh 3 7/8 pounds each. Converting 3 7/8 to improper fraction, it becomes 31/8 pounds. Therefore, the weight of the three bags would be

3 × 31/8 = 93/8 pounds

There are 2 bags of oranges that weigh 4 pounds each. Therefore, the weight of the four bags would be

2 × 4 = 8 pounds

There are 3 bags of oranges that weigh 4 1/8 pounds each. Converting 4 1/8 to improper fraction, it becomes 33/8 pounds. Therefore, the weight of the three bags would be

3 × 33/8 = 99/8 pounds

There are 2 bags of oranges that weigh 4 2/8 pounds each. Converting 4 2/8 to improper fraction, it becomes 34/8 pounds. Therefore, the weight of the three bags would be

2 × 34/8 = 68/8

Therefore, the total number of oranges would be

93/8 + 33/8 + 4 + 102/8 = (93 + 64 + 99 + 68)/8 = 324/8 = 40 1/2 pounds