25. Michael is studying population changes in two

types of birds living on an island. Compare the

populations by finding and interpreting the

average rates of change over the interval [0, 18]


Bird A


Time(months):

6

12

18


Population:

(thousands)

8.3

8.6

8.8

9.1


Bird B

y= 3.6(1.06)x

Answer:

-2/7 0r 2/-7 or the negative sign is in the middle

Step-by-step explanation:

(-3/4)(-4/7)= 3/7

3/7 x -2/3= -2/7

Given:

Given that the graph OACE.

The coordinates of the vertices OACE are O(0,0), A(2m, 2n), C(2p, 2r) and E(2t, 0)

We need to determine the midpoint of EC.

Midpoint of EC:

The midpoint of EC can be determined using the formula,

[tex]Midpoint=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})[/tex]

Substituting the coordinates E(2t,0) and C(2p, 2r), we get;

[tex]Midpoint=(\frac{2t+2p}{2},\frac{0+2r}{2})[/tex]

Simplifying, we get;

[tex]Midpoint=(\frac{2(t+p)}{2},\frac{2r}{2})[/tex]

Dividing, we get;

[tex]Midpoint=(t+p,r)[/tex]

Thus, the midpoint of EC is (t + p, r)

Hence, Option A is the correct answer.

Answer:

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 90% confidence interval for this case would be (38.01, 44.29) and is given.

The best interpretation for this case would be: We are 90% confident that the true average is between $ 38.01 and $ 44.29 .

And the best option would be:

The store manager is 90% confident that the average amount spent by all customers is between S38.01 and $44.29

Step-by-step explanation:

Assuming this complete question: Which statement gives a valid interpretation of the interval?

The store manager is 90% confident that the average amount spent by the 36 sampled customers is between S38.01 and $44.29.

There is a 90% chance that the mean amount spent by all customers is between S38.01 and $44.29.

There is a 90% chance that a randomly selected customer will spend between S38.01 and $44.29.

The store manager is 90% confident that the average amount spent by all customers is between S38.01 and $44.29

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".

Solution to the problem

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 90% confidence interval for this case would be (38.01, 44.29) and is given.

The best interpretation for this case would be: We are 90% confident that the true average is between $ 38.01 and $ 44.29 .

And the best option would be:

The store manager is 90% confident that the average amount spent by all customers is between S38.01 and $44.29

The problem is an Optimization problem in Calculus that is solved by representing the volume of the open box in terms of a single variable using the fixed cost. With the volume equation, we can use calculus to find the optimal dimensions.

The question is about maximizing the volume of an open box given a fixed cost and considering that the base of the box is 1.5 times as expensive as the sides. This problem comes under the branch of mathematics known as Optimization in Calculus. The volume V of an open box (a box without a top) is given by the product of its length, width, and height (V = lwh).

In this problem, the total cost is fixed, hence, the sum of the cost of the base and the cost of the sides is a constant. We can say that cost = C = (Base Cost) + (Sides Cost) = 1.5lw + 2.0lh + 2.0wh. We can express the width w in terms of l and h using the cost equation, and then substitute in the volume equation to write V in terms of a single variable. This enables the use of calculus to optimize the volume. It is beyond the scope of this answer to give a complete solution, but essentially, you would differentiate to obtain an equation, and solve for the optimal dimensions.

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Answer:

B C E are the answers

Step-by-step explanation:

hope it helps

Only the equations -1(x/4) + 3/4 = 12 and (-x/4) + 3/4 = 12 are equivalent to the original equation. The other equations provided do not hold the same properties of distribution and are not equivalent to the original equation.

When comparing these equations to the original, we must take into consideration the properties of distribution, a crucial component of algebra. The original equation is -1/4(x) + 3/4 = 12. Let's go through the options one by one:

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Answer:  

We are 85% confident that the proportion of adults who dreaded valentines day is within the range of 8.8% to 21.2%.

Step-by-step explanation:  

Proportion of adults who dreaded valentines day = 15% = 0.15

The margin of error with 85% confidence = 6.2% = 0.062

The confidence interval is given by

p ± margin of error

0.15 ± 0.062

Lower limit = 0.15 - 0.062

Lower limit = 0.088

Lower limit = 8.8%

Upper limit = 0.15 + 0.062

Upper limit = 0.212

Upper limit = 21.2%

So it means that we are 85% confident that the proportion of adults who dreaded valentines day is within the range of 8.8% to 21.2%.

Answer:

5/6

Step-by-step explanation:

2/3=4/6

4/6+1/6=5/6

Answer:

That would be 5/6

Step-by-step explanation:

You multiply the denominator and the numerator by 2 to get a common denominator of 6. You add the 4 to the 1 to get 5/6

Answer:

E(rP) = 4% + 5.50% x β(M1) + 10.92% x β(M2)

Step-by-step explanation:

let us recall from the following statement:

The  two independent economic factors are M1 and M2

Th risk free rate = 4%

The standard deviation of all stocks of  independent firm specific components is =49%

P = portfolios for A and B

Now,

What is the expected relationship of return-beta

The Expected return-beta relationship E(rP) =  % +  βp₁ +  βp₂

E(rA) = 4% + 1.6 * M1 + 2.4* M2 = 39%

E(rB) = 4% + 2.3 * M1 + (-0.7)* M2 = 9%

Therefore

Solving for M1 and M2 using excel solver, we have M1 = 5.50% and M2 = 10.92%

E(rP) = 4% + 5.50% x β(M1) + 10.92% x β(M2)

The question pertains to finance and investment analysis. It emphasizes the CAPM model, which combines systematic risk measured by beta and market risk premium to calculate expected returns on portfolios. It also highlights that diversification reduces firm-specific risks.

The question deals with the concept of portfolio return, beta coefficients, and firm-specific risk, which are important aspects of finance and investment analysis. The expected return on portfolios A and B, can be calculated using the CAPM model, which states that expected return equals the risk-free rate plus the portfolio's beta (which measures systematic risk) multiplied by the market risk premium (difference between the expected market return and the risk-free rate). To compute this, the beta coefficients need to be multiplied with their respective economic factors, and the results obtained are added together.

For portfolio A, the expected return would be calculated like this: Return = Risk-Free rate + β1*M1 + β2*M2 = 4 + (1.6 * M1) + (2.4 * M2).

For portfolio B, the calculation would be similar: Return = Risk-Free rate + β1*M1 + β2*M2 = 4 + (2.3 * M1) - (0.7 * M2). The negative beta on M2 indicates that the portfolio's return would decrease when M2 increases, hence it has an inverse relationship with the portfolio return. The independent firm-specific component would not affect the return as per the assumption that the portfolios are well diversified; diversification reduces, but not completely eliminates, the firm-specific risk.

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Answer:

[tex]y = 2x -9[/tex]

Step-by-step explanation:

To find the equation of a line using slope and a point, first use the slope to create the basic line using the slope and work from there.

For instance, the base equation here is

[tex]y = 2x[/tex]

This line passes through the point (0, 0).

You can then plug in a value for x. In this case, use the value of 3, as it corresponds with your question.

[tex]y = 2 * 3[/tex]

A point on the line of y = 2x would thus be (3, 6).

To make the y-value equal -3, you must then subtract from the original equation. There are 9 units between 6 and -3, so you must subtract nine units in the equation. You should get this at the end:

[tex]y = 2x -9[/tex]

The subject of this question is Science in Middle School. The student is learning about scientific models by making a model of air flow in their classroom or room in their house.

The subject of this question is Science and the grade level is Middle School. In this question, the student is learning about scientific models by making a model of how air flows through their classroom or a room in their house. The student can use this activity to gain a better understanding of how air moves in closed spaces and the factors that affect its flow.

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Answer:

The minimum sample size required is 25 so that margin of error is no more than 3 minutes.  

Step-by-step explanation:

We are given the following in the question:

Mean, μ = 42 minutes

Standard Deviation, σ = 9 minutes.

We want to build a 90% confidence interval such that margin of error is no more than 3 minutes.

Formula for margin of error:

[tex]z_{critical}\times \dfrac{\sigma}{\sqrt{n}}[/tex]

[tex]z_{critical}\text{ at}~\alpha_{0.10} = 1.64[/tex]

Putting values, we get.

[tex]z_{critical}\times \dfrac{\sigma}{\sqrt{n}}\leq 3\\\\1.64\times \dfrac{9}{\sqrt{n}}\leq 3\\\\\dfrac{1.64\times 9}{3}\leq \sqrt{n}\\\\4.92\leq \sqrt{n}\\\Rightarrow n\geq 24.2064\approx 25[/tex]

Thus, the minimum sample size required is 25 so that margin of error is no more than 3 minutes.

Final answer:

To construct a confidence interval for the true mean daily fees collected at the parking garage, the normal distribution should be used. Using a 90% confidence level, the confidence interval is calculated as $116 to $136. Based on the confidence interval, it is possible that the consultant's prediction of $130 per day could be correct.

Explanation:

(a) To construct the confidence interval, we should use the normal distribution. This is because the sample size is large enough (14 days) and the Central Limit Theorem applies, which states that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

(b) To construct a 90% confidence interval, we need to find the critical value corresponding to a 90% confidence level. The critical value for a 90% confidence level can be found using a z-table. From the z-table, we find that the critical value is approximately 1.645. The margin of error is calculated as the critical value multiplied by the standard deviation of the sample mean, which is the standard deviation divided by the square root of the sample size. In this case, the margin of error is (1.645 x 15) / √14. We can then construct the confidence interval by subtracting and adding the margin of error to the sample mean. The 90% confidence interval for the true mean daily fees collected at this parking garage is approximately $116 to $136.

(c) Based on the confidence interval, we can see that the range of $116 to $136 includes the predicted average of $130 per day by the consultant. Therefore, it is possible that the consultant could have been correct.

Answer:

42

Step-by-step explanation:

work backwards And use the opposite operation

start with 8 x 4 = 32 + 40 = 72/2 = 36 + 6 = 42

Answer:

[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

The quotient is the number which is generated when we perform division operations on two numbers.

The quotient of 2/5 and 4/5.

The quotient of 2/5 and 4/5 is determined in the following steps given below.

[tex]\rm Quotient =\dfrac{\dfrac{2}{5}}{\dfrac{4}{5}}\\\\Quotient =\dfrac{2}{5}\times \dfrac{5}{4}\\\\ Quotient =\dfrac{2}{4}\\\\Quotient =\dfrac{1}{2}[/tex]

Hence, the quotient of 2/5 and 4/5 is 1/2.

Learn more about quotient here;

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Answer:

[tex]z = \frac{21.3-21}{\frac{1.2}{\sqrt{50}}}= 1.77[/tex]

Step-by-step explanation:

Data given and notation  

[tex]\bar X=21.3[/tex] represent the sample mean

[tex]\sigma=1.2[/tex] represent the population standard deviation

[tex]n=50[/tex] sample size represent the value that we want to test

[tex]\alpha[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

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

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 21, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 21[/tex]  

Alternative hypothesis:[tex]\mu > 21[/tex]  

If we analyze the size for the sample is > 30 and we know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

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

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

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

[tex]z = \frac{21.3-21}{\frac{1.2}{\sqrt{50}}}= 1.77[/tex]

9514 1404 393

Answer:

  (x, y, z) = (20, 4, 5)

Step-by-step explanation:

The last two equations allow y and z to be expressed in terms of x, so we have ...

  3x +2(5y) -3(4z) = 40

  3x +2(x) -3(x) = 40

  x = 20 . . . . . . . . . . . . divide by the coefficient of x

  y = 20/5 = 4

  z = 20/4 = 5

The solution is (x, y, z) = (20, 4, 5).

Answer:

Using the normal probability distribution, with a capacity of 350, it is enough for all abused on 90.82% of nights.

274 shelters will be needed.

Step-by-step explanation:

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

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

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

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

In this problem, we have that:

[tex]\mu = 250, \sigma = 75[/tex]

If the city’s shelters have a capacity of 350, will that be enough places for abused women on 95% of all nights?

What is the percentile of 350?

This is the pvalue of Z when X = 350.

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

[tex]Z = \frac{250 - 150}{75}[/tex]

[tex]Z = 1.33[/tex]

[tex]Z = 1.33[/tex] has a pvalue of 0.9082.

Using the normal probability distribution, with a capacity of 350, it is enough for all abused on 90.82% of nights.

If not, what number of shelter openings will be needed?

The 95th percentile, which is X when Z = 1.645. So

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

[tex]1.645 = \frac{X - 150}{75}[/tex]

[tex]X - 150 = 1.645*75[/tex]

[tex]X = 274[/tex]

274 shelters will be needed.

The current shelter capacity of 350 will not be sufficient for 95% of nights according to the normal distribution of the number of women in shelters each night. To have enough capacity for 95% of the nights, the city's shelter would need approximately 397 bed openings.

The subject of this question relates to a discipline in statistics called normal distribution. Essentially, we have the mean number of women in shelters each night (250) and the standard deviation (75). Since the council wants the capacity to be enough for 95% of nights, we need to find the number corresponding to the 95th percentile in this normal distribution.

In a normal distribution, 95% of the data falls within 1.96 standard deviations of the mean. So, we will calculate the upper limit of the capacity using the following formula: Upper Limit = Mean + (1.96 * Standard Deviation)

By substituting the given mean and standard deviation (250 and 75 respectively), we get: Upper Limit = 250 + (1.96 * 75) = 397.

So, a shelter capacity of 350 will not be sufficient to house most abused women on 95% of all nights. The city's shelters would need 397 openings to meet this requirement.

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Answer:

y = 8

Step-by-step explanation:

2y + 3 = 19

2y = 19 - 3

2y = 16

y = 16/2

y = 8

Hopefully this help u

Answer: y = 8

Step-by-step explanation: To solve for y, we must first isolate the term containing y which in this problem is 2y.

Since 3 is being added to 2y, we subtract 3 from

both sides of the equation to isolate the 2y.

On the left, the +3 and -3 cancel

out and on the right, 19 - 3 is 16.

So we have 2y = 16.

Now we can finish things off by just dividing

both sides of the equation by 2.

On the left the 2's cancel and on

the right, 16 divided by 2 is 8.

So y = 8.

Answer:

40 in^2

Step-by-step explanation:

The area of the trapezoid is found by

A = 1/2 (b1+b2)h  where b1 and b2 are the lengths of the bases and h is the height

   = 1/2 (7+3)*8

   = 1/2 (10)*8

   = 40 in^2

Answer:

ok the answer is 40

Step-by-step explanation:

a+b/2*h

7+3/2*8

7+3=10

10/2=5

5*8=40

Answer:

its c I just id the test

Step-by-step explanation:

Answer:

58.1377674149945 inches. (Round to whatever you need)

Step-by-step explanation:

Here we turn to Pythagoras theorem.

Because there is a right angled triangle,

Let the two shorter sides (length and width) be a and b and the longer side (diagonal) be c

A² + B² = C²

38² + 44² = 3380inches² (this is not the area. It's how Pythagoras theorem works)

Then the square root of this is 58.1377674149945 inches. (Round to whatever you need)

So that is the length of the diagonal

Answer:

By the Chebyshev Theorem, 75% probability of a randomly selected state having between 21 and 53 multi-million dollars

Step-by-step explanation:

We have no information about the distribution, so we use the Chebyshev's theorem to solve this question.

Chebyshev Theorem:

75% of the data within 2 standard deviations of the mean.

89% of the data within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 37

Standard deviation = 8

What is the probability of a randomly selected state having between 21 and 53 multi-million dollars

21 = 37 - 2*8

So 21 is 2 standard deviations below the mean.

53 = 37 + 2*8

So 52 is 2 standard deviations above the mean.

By the Chebyshev Theorem, 75% probability of a randomly selected state having between 21 and 53 multi-million dollars

Final answer:

To summarize data in a significance test for two different treatments with a quantitative response variable, the mean and standard deviation for each treatment group are computed separately to look for statistically significant differences.

Explanation:

When conducting a significance test to determine if there is a difference between two treatments with a quantitative response variable, and treatments are given to different experimental units, we summarize the data by computing the mean and standard deviation of each treatment group separately. This approach involves comparing the two sets of data from the treatment groups to see if there is a statistically significant difference in their means, which could suggest an effect of the treatments. This methodology is part of inferential statistics, where researchers use the collected sample data to make inferences about the population from which the sample was drawn.

Given:

Given that the triangle ABC is similar to triangle FGH.

We need to determine the value of x.

Value of x:

Since, the triangles are similar, then their sides are proportional.

Thus, we have;

[tex]\frac{AC}{FH}=\frac{AB}{GF}=\frac{BC}{GH}[/tex]

Let us consider the proportion [tex]\frac{AB}{GF}=\frac{BC}{GH}[/tex] to determine the value of x.

Substituting AB = 9 cm, GF = 13.5 cm, BC = 15 cm and GH = x, we get;

[tex]\frac{9}{13.5}=\frac{15}{x}[/tex]

Cross multiplying, we get;

[tex]9x=15 \times 13.5[/tex]

[tex]9x=202.5[/tex]

 [tex]x=22.5 \ cm[/tex]

Thus, the value of x is 22.5 cm

Hence, Option F is the correct answer.

Answer:

22.5

Step-by-step explanation:

Answer:

[tex]19.1-3.355\frac{1.5}{\sqrt{9}}=17.42[/tex]    

[tex]19.1+3.355\frac{1.5}{\sqrt{9}}=20.78[/tex]    

And the best option would be:

C. [17.42,20.78]

Step-by-step explanation:

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=19.1[/tex] represent the sample mean

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

s=1.5 represent the sample standard deviation

n=9 represent the sample size  

Solution to the problem

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=9-1=8[/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(0.005,8)".And we see that [tex]t_{\alpha/2}=[/tex]

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

[tex]19.1-3.355\frac{1.5}{\sqrt{9}}=17.42[/tex]    

[tex]19.1+3.355\frac{1.5}{\sqrt{9}}=20.78[/tex]    

And the best option would be:

C. [17.42,20.78]

Answer:

The margin of error increases as the level of confidence increases because the larger the expected proportion of intervals that will contain the​ parameter, the larger the margin error.

Step-by-step explanation:

Margin of Error is a statistical measure of random sampling error insurvey results.

Level of confidence reflects percentage range around sample mean, that can be expected to contain population actual parameter.

High level of confidence means larger range band of confidence interval, supposed to contain the population parameter. This further implies  high expected variation between sample statistic & actual parameter i.e Margin Error increases.

The margin of error increases as the level of confidence increases because the smaller the expected proportion of intervals that will contain the​ parameter, the larger the margin of error (C).

The correct answer is C. The margin of error increases as the level of confidence increases because the smaller the expected proportion of intervals that will contain the​ parameter, the larger the margin of error. When we calculate a confidence interval, we need to balance our desire for a higher level of confidence with the need for a smaller margin of error.

For example, let's say we want to estimate the proportion of students at a school who prefer chocolate ice cream. If we want to be extremely confident in our estimate, let's say 99% confident, there will be a larger margin of error since we need to consider a smaller proportion of intervals that will contain the true proportion.

On the other hand, if we were only interested in being 90% confident, there would be a smaller margin of error since we can include a larger proportion of intervals that will contain the true proportion.

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Given:

Given that the length of the side of the square is 12 cm.

The  given figure consists of two circles with radius of 3 cm each.

We need to determine the area of the shaded region.

Area of the square:

The area of the square can be determined using the formula,

[tex]A=s^2[/tex]

Substituting s = 12, we get;

[tex]A=12^2[/tex]

[tex]A=144 \ cm^2[/tex]

Thus, the area of the square is 144 square cm.

Area of the two circles:

The area of the circle can be determined using the formula,

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

Substituting r = 3, we get;

[tex]A=(3.14)(3)^2[/tex]

[tex]A=(3.14)(9)[/tex]

[tex]A=28.26 \ cm^2[/tex]

The area of 2 circles is given by

[tex]A=2(28.26)[/tex]

[tex]A=56.52 \ cm^2[/tex]

Thus, the area of the two circles is 56.52 square cm.

Area of the shaded region:

The area of the shaded region can be determined by subtracting the area of the two circles from the area of the square.

Thus, we have;

Area = Area of square - Area of two circles.

Substituting the values, we get;

[tex]Area = 144- 56.52[/tex]

[tex]Area=87.5 \ cm^2[/tex]

Thus, the area of the shaded region is 87.5 square cm.

Answer:

38,38,380

Step-by-step explanation:

GIVEN: A club is considering changing its laws. In an initial straw vote on the issue, [tex]24[/tex] of the [tex]40[/tex] members of the club favored the change and [tex]16[/tex] did not. A committee of six is to be chosen from the [tex]40[/tex] club members to devote further study to the issue.

TO FIND: How many committees of six can be formed from the club membership.

SOLUTION:

Total number of members [tex]=40[/tex]

Total members to be chosen [tex]=6[/tex]

To select committee of [tex]6[/tex] members from [tex]40[/tex] [tex]=^{40}C_6[/tex]

                                                                       [tex]=\frac{40!}{34!6!}[/tex]

                                                                      [tex]=38,38,380[/tex]

Hence 38,38,380 different committee can be formed.

Answer:

6 at tens place is 10 times 6 at unit place.

Step-by-step explanation:

We are given that  a number

50,866

Underlined digits are 66.

We  have to find the relationship between underlined digits.

Unit place =6

Tens place=

Tens place value=[tex]6\times 10=60[/tex]

Unit pace value=6

6 at tens place is 10 times 6 at unit place.

This is required relation relation between underlined digits 66.

Answer:

The mean calculated for this case is [tex]\bar X=584[/tex]

And the 95% confidence interval is given by:

[tex]584-2.776\frac{86.776}{\sqrt{5}}=476.271[/tex]    

[tex]584+2.776\frac{86.776}{\sqrt{5}}=691.729[/tex]    

So on this case the 95% confidence interval would be given by (476.271;691.729)    

Step-by-step explanation:

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  

Solution to the problem

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 mean and the sample deviation we can use the following formulas:  

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)  

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)  

The mean calculated for this case is [tex]\bar X=584[/tex]

The sample deviation calculated [tex]s=86.776[/tex]

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=5-1=4[/tex]

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,4)".And we see that [tex]t_{\alpha/2}=2.776[/tex]

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

[tex]584-2.776\frac{86.776}{\sqrt{5}}=476.271[/tex]    

[tex]584+2.776\frac{86.776}{\sqrt{5}}=691.729[/tex]    

So on this case the 95% confidence interval would be given by (476.271;691.729)    

Final answer:

The point estimate of the population mean of SAT math scores, calculated from the sample scores (570, 620, 710, 540, and 480), is 584.

Explanation:

The subject of this question is Mathematics, specifically focusing on statistics and the SAT examination scores. To calculate the point estimate for the given simple random sample of SAT Mathematics test scores (570, 620, 710, 540, and 480), we need to find the sample mean. This can be done by adding all the scores together and dividing by the number of students in the sample, which is five in this case.

Point Estimate calculation:

Add all the scores together: 570 + 620 + 710 + 540 + 480 = 2920

Divide by the number of students: 2920 / 5 = 584

The point estimate of the population mean of SAT math scores is 584.