Two bicyclists are 7/8 of the way through a mile long tunnel when a train approaches the closer end at 40 mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast did they ride?

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.

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:

[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:

b. can be calculated by subtracting the fitted values from the actual values.

Step-by-step explanation:

OLS residuals -  it stands for ordinary least square. it is used to determine the missing value in the regression analysis. OLS works on one purpose that is to minimize the difference between the observed response and predict response.

The basic difference between Residual sum of square(RSS) and OLS is that RSS is used to predict how good is model while OLS  is considered as the method which is used to construct model>

Answer:

(a) 1 in 365 or 0.2740%

(b) 0.8227%

Step-by-step explanation:

(a) For any given birthday date of the first person, there is a 1 in 365 chance that the second person shares the same birthday, therefore the probability that the first two people share a birthday is:

[tex]P = \frac{1}{365}=0.2740\%[/tex]

(b) There are four possibilities that at least two people share a birthday, first and second, first and third, second and third, all three share a birthday. Therefore, the probability that at least two people share a birthday is:

[tex]P =3* \frac{1}{365}+ (\frac{1}{365})^2\\ P=0.8227\%[/tex]

a) The probability that the first two people share a birthday is approximately 0.0027. b) The probability that at least two people share a birthday is approximately 0.9901.

let's solve each part step by step:

To calculate this probability, we can consider the scenario where the first person is born on any day of the year (365 possibilities) and the second person must share the same birthday. So, the probability that the second person shares the same birthday as the first is 1/365.

Therefore, the probability that the first two people share a birthday is [tex]\( \frac{1}{365} \).[/tex]

To find this probability, we can use the complement rule: the probability of the event happening is 1 minus the probability of the event not happening.

The probability that no two people share a birthday can be found by considering the birthday of each person and ensuring that they all have different birthdays. For the first person, any of the 365 days is possible. For the second person, there are 364 days remaining. For the third person, there are 363 days remaining. So, the probability that no two people share a birthday is:

[tex]\[ \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \][/tex]

To find the probability that at least two people share a birthday, we subtract this probability from 1:

[tex]\[ 1 - \left( \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \right) \][/tex]

[tex]\[ = 1 - \frac{365 \times 364 \times 363}{365^3} \][/tex]

[tex]\[ \approx 1 - \frac{479,664,580}{48,627,125} \][/tex]

[tex]\[ \approx 1 - 0.009882 \][/tex]

[tex]\[ \approx 0.990118 \][/tex]

So, the probability that at least two people share a birthday is approximately ( 0.990118 ).

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%)

The answer is 9 1/5 simplified.

Too lazy to make up an explanation or change it to a mixed number even tho it is easy...

Answer:

Step-by-step explanation:

5³/10 +3 9/10

Convert to improper fraction

53/10 +39/10

L. C. M =10

=92/10

=9²/10

=9¹/5

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:

S={R,A,N}

Step-by-step explanation:

There are three letters in a word RAN i.e. R, A and N. The sample space consists of all possible outcomes of an experiment. So, the sample for selecting a letter from word RAN will be

Sample Space=S={R,A,N}

As, there are three possible letters that can be selected n(S)=3

Thus, the required sample in the set notation is

S={R,A,N}.

Final answer:

The sample space for selecting a letter from the word RAN is S = {R,A,N}, which includes all the individual letters of the word as separate and distinct outcomes.

Explanation:

The sample space for the experiment of randomly selecting one of the letters in the word RAN is a set of all possible outcomes of this experiment. Using set notation, we can describe this sample space as follows:

S = {R,A,N}

Each letter represents a unique outcome in the sample space, which in this case consists of the three letters that make up the word. Since the word RAN has no repeated letters, each letter is a distinct outcome. To list this sample space in set notation we simply enclose the elements within braces and separate them with commas, without spaces.

Answer:

c₁ = 1/2 cos⁻¹ (2/π) = 0.44

c₂ = -1/2 cos⁻¹ (2/π) = -0.44

Step-by-step explanation:

the average value of f(x)=2 cos(2x) on ( − π/ 4 , π/ 4 ) is

av f(x) =∫[2*cos(2x)] dx /(∫dx) between limits of integration − π/ 4 and π/ 4

thus

av f(x) =∫[cos(2x)] dx /(∫dx) = [sin(2 * π/ 4 ) - sin(2 *(- π/ 4 )] /[ π/ 4 -  (-π/ 4)]

= 2*sin (π/2) /(π/2) = 4/π

then the average value of f(x) is 4/π . Thus the values of c such that f(c)= av f(x) are

 4/π = 2 cos(2c)  

2/π = cos(2c)

c = 1/2 cos⁻¹ (2/π) = 0.44

c= 0.44

since the cosine function is symmetrical  with respect to the y axis then also c= -0.44 satisfy the equation

thus

c₁ = 1/2 cos⁻¹ (2/π) = 0.44

c₂ = -1/2 cos⁻¹ (2/π) = -0.44

The two values are,

[tex]c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi}) or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi}) [/tex]

Given that,

[tex]f(x)=2cos(2x)[/tex]

[tex]f_{avg}=\frac{1}{\frac{\pi}{2} }\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}2cos(2x)dx\\ =\frac{8}{2\pi} sin\frac{\pi}{2} \\ =\frac{4}{\pi} [/tex]

[tex]f(c)=2cos(2c)=\frac{4}{\pi} \\ cos2c=\frac{2}{\pi} \\ c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\[/tex]

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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:

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:

Step-by-step explanation:

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:

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:

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:

-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:

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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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:

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:

a) Let X the random variable that represent the blood pressure for people of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(128,23)[/tex]  

Where [tex]\mu=128[/tex] and [tex]\sigma=23[/tex]

b) [tex]P(X\geq 135)=P(\frac{X-\mu}{\sigma}\geq \frac{135-\mu}{\sigma})=P(Z\geq \frac{135-128}{23})=P(Z\geq 0.304)[/tex]

And we can find this probability using the complement rule:

[tex]P(Z\geq 0.304)=1-P(Z<0.304)[/tex]

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

[tex]P(Z\geq 0.304)=1-P(Z<0.304)= 1-0.619=0.381 [/tex]

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".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the blood pressure for people of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(128,23)[/tex]  

Where [tex]\mu=128[/tex] and [tex]\sigma=23[/tex]

Part b

We are interested on this probability

[tex]P(X\geq 135)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

If we apply this formula to our probability we got this:

[tex]P(X\geq 135)=P(\frac{X-\mu}{\sigma}\geq \frac{135-\mu}{\sigma})=P(Z\geq \frac{135-128}{23})=P(Z\geq 0.304)[/tex]

And we can find this probability using the complement rule:

[tex]P(Z\geq 0.304)=1-P(Z<0.304)[/tex]

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

[tex]P(Z\geq 0.304)=1-P(Z<0.304)= 1-0.619=0.381 [/tex]

In this context, the random variable is the blood pressure of people in China. The probability that a person in China has a blood pressure of 135 mmHg or more is about 38.21%, calculated using the Z-score and standard Z-table.

The subject of this question is the study of normal distribution and probability in statistics, a branch of mathematics.

a.) The random variable in this context is the blood pressure of people in China.

b.) To find the probability that a person in China has a blood pressure of 135 mmHg or more, we need to convert this to a Z-score. The Z-score is calculated by subtracting the mean from the individual score and then dividing by the standard deviation. Therefore, Z = (135 - 128) / 23 = 0.30.

Using a standard Z-table, the probability corresponding to Z=0.30 is about 0.6179. However, because we want the probability of a person having a blood pressure that's 135 mmHg or more (greater than the mean), we must subtract this value from 1. Thus, the probability is about 1 - 0.6179 = 0.3821, or 38.21%.

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Answer:

Kamala had the higher Z-score, so she had the best GPA when compared to other students at his school.

Step-by-step explanation:

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

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

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

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

In this problem, we have that:

Three students, graded on different curves. I will find whoever has the higher Z-score, and this is the one which had the best GPA.

Thuy 2.7 3.2 0.8

So the student GPA is 2.7, the Average GPA at the school was 3.2 and the standard deviation was 0.8.

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

[tex]Z = \frac{2.7 - 3.2}{0.8}[/tex]

[tex]Z = -0.625[/tex]

Vichet 87 75 20

So the student GPA is 87, the Average GPA at the school was 75 and the standard deviation was 20.

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

[tex]Z = \frac{87 - 75}{20}[/tex]

[tex]Z = 0.6[/tex]

Kamala 8.6 8 0.4

So the student GPA is 8.6, the Average GPA at the school was 8 and the standard deviation was 0.4.

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

[tex]Z = \frac{8.6 - 8}{0.4}[/tex]

[tex]Z = 1.5[/tex]

Kamala had the higher Z-score, so she had the best GPA when compared to other students at his school.

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: 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

Answer:

the instantaneous rate of change of f(x) at x=(-3) is f'(x=(-3))= (-3)

Step-by-step explanation:

for f(x)=x²+3*x

the rate of change of f(x) is

f'(x)=df(x)/dx = 2x + 3

since the derivative of x² is 2x and the derivative of 3*x is 3.

Then at x=(-3)

f'(x=(-3))= 2*(-3) +3 = (-3)

then the instantaneous rate of change of f(x) at x=(-3) is f'(x=(-3))= (-3)

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

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Answer:

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

Step-by-step explanation:

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.

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