Calculate the pay for the following day of a weekly time card given a wage of $14/hr.
Name
Week of: Morning
Afternoon
In
Out
In
Out
Monday 08:00 12:15
13:00 | 17:15 round to the nearest hundred

Answer:

∠A = 48.2

Step-by-step explanation:

Cos^-1(6/9

Use the Cosine with the negative one because you are finding an angle. Then, because Cosine is adjacent/hypotenuse, 6 would be your adjacent side length to theta and 9 would be the hypotenuse as it is across from the right angle.

the length of the missing side is approximately 4.73 units. Rounded to two decimal places, the answer is 4.73.

To solve this problem, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle.

Let's label the missing side as b, and use sin(A) = opposite/hypotenuse to find the length of the side opposite angle A:

sin(A) = opposite/hypotenuse

sin(33.7°) = b/8

b = 8 × sin(33.7°)

b ≈ 4.726

Therefore, the length of the missing side is approximately 4.73 units. Rounded to two decimal places, the answer is 4.73.

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A 95.2% confidence interval in z-score is a range of values that is likely to contain the true population mean with a probability of 95.2%. The z-score for a 95.2% confidence interval is approximately ±1.97.

To calculate the 95.2% confidence interval in z-score, we can use the following formula:

CI = X ± z × (σ/√n)

where X is the sample mean, σ is the population standard deviation, n is the sample size, and z is the critical value from the standard normal distribution table.

For a 95.2% confidence interval, the area in the tails of the standard normal distribution is 0.024 (0.5 - 0.952/2). Therefore, the critical z-value for a 95.2% confidence interval is approximately ±1.97.

This formula assumes that the sample follows a normal distribution, or that the sample size is large enough to satisfy the central limit theorem.

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Answer:

Step-by-step explanation:

3

x

−

4

y

+

15

=

0

.

y

=

3

4

x

−

1

2

Correct answers 5.4

Distance from point (2,3) to the line 3x+4y+9=0

=>r=

3

2

+4

2

∣3(2)+4(3)+9∣

=>r=

9+16

∣16+12+9∣

=>r=

5

27

=>r=5.4

According to the assumption the translated triangle would have vertices at (4,0), (6,2), and (5,-1).

Given,

To translate a triangle 3 units to the right and 2 units down, we would take each point of the original triangle and add 3 to the x-coordinate and subtract 2 from the y-coordinate.

For example, if the original triangle has vertices at (1,2), (3,4), and (2,1), the translated triangle would have vertices at:

(1+3, 2-2) = (4,0)

(3+3, 4-2) = (6,2)

(2+3, 1-2) = (5,-1)

So, the translated triangle would have vertices at (4,0), (6,2), and (5,-1).

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By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.

a. To construct a time series plot, you would typically use software like Excel or programming languages like Python or R.

However, I can describe the general trend based on the given data points.

Week 1: 7.35

Week 2: 7.40

Week 3: 7.55

Week 4: 7.56

Week 5: 7.60

Week 6: 7.52

Week 7: 7.52

Week 8: 7.70

Week 9: 7.62

Week 10: 7.55

Based on the data, there seems to be a slight overall upward trend in the Commodity Futures Index over the ten weeks, but there are also small fluctuations up and down throughout the period.

b. To find the best exponential smoothing coefficient (alpha) using the trial and error method, you would typically use software or programming languages to calculate the Mean Squared Error (MSE) for each alpha value.

Choose an initial value for alpha (e.g., 0.1).

Apply the exponential smoothing formula to the data series: St = α * Xt + (1 - α) * St-1, where St is the smoothed value at time t, α is the smoothing coefficient, Xt is the observed value at time t, and St-1 is the smoothed value at time t-1.

Calculate the forecast errors (Et) as the difference between the observed value and the smoothed value for each time period: Et = Xt - St-1.

Compute the Mean Squared Error (MSE) by taking the average of the squared forecast errors.

Repeat steps 2-4 with different values of alpha (e.g., 0.2, 0.3, 0.4, etc.) until you find the alpha value that minimizes the MSE.

By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.

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a. Hypotheses are: [tex]\rm H_{0}:p=0.20,H_{a}:p > 0.20[/tex]

b. Z-statistics will be 2.47 and p-value = 0.0068

c. you should proceed with plans to develop and market the upgrade.

d. Confidence interval for population proportion will be (0.23, 0.45)

e. The populations are different because they are frοm different regiοns sο prefrences οf peοple may different.

Probability is a mathematical concept that measures the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

a. Hypotheses are:

[tex]\rm H_{0}:p=0.20,H_{a}:p > 0.20[/tex]

b) Here we have following information:

[tex]\rm n=50, \hat{p}=\frac{17}{50}=0.34[/tex]

Standard deviation of the proportion is:

[tex]\rm \sigma=\sqrt{\frac{p\left ( 1-p \right )}{n}}=\sqrt{\frac{0.20\left ( 1-0.20 \right )}{50}}=0.0566[/tex]

Test statistics will be:

[tex]$ \rm z=\frac{\hat{p}-p}{\sigma}=\frac{0.34-0.20}{0.0566}=2.47[/tex]

Alternative hypothesis shows that the test is right tailed so p-value of the test is

[tex]\rm p-value = P(z > 2.47) = 1 - P(z \leq 2.47)=1-0.9932=0.0068[/tex]

c. Since p-value of the test is less than 0.05 so we reject the null hypothesis at 0.05 level of significance. So based on this sample, you should proceed with plans to develop and market the upgrade.

d. For 90% confidence interval [tex]\alpha[/tex] = 0.10 so critical value of z will be[tex]\rm z_{c}=z_{\alpha/2}=1.645[/tex]

Confidence interval for population proportion will be

[tex]\rm \hat{p}\pm z_{c}\sqrt{\frac{\hat{p}\left ( 1-\hat{p} \right )}{n}}=0.34\pm 1.645\sqrt{\frac{0.34\left (1-0.34 \right )}{50}}=0.34\pm 0.11 =(0.23, 0.45)[/tex]

e. The populations are different because they are frοm different regiοns sο prefrences οf peοple may different.

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Answer:

modes

Step-by-step explanation:

The only one of the three (median, mode, mean) that there can be more than one for a set is the mode, and in fact it has two modes: 5 and 9.

Answer: modes

The difference in centigrade degrees between a temperature of 60°F and a temperature of 78°F is 10°C.

To convert Fahrenheit (°F) to Celsius (°C), you can use the following formula:

°C = (°F - 32) / 1.8

Using this formula, we can first convert 60°F to Celsius:

°C = (60°F - 32) / 1.8

°C = (28°F) / 1.8

°C = 15.56°C

Similarly, we can convert 78°F to Celsius:

°C = (78°F - 32) / 1.8

°C = (46°F) / 1.8

°C = 25.56°C

The difference in Celsius degrees between these two temperatures is:

Δ°C = 25.56°C - 15.56°C

Δ°C = 10°C

Therefore, the difference in centigrade degrees between a temperature of 60°F and a temperature of 78°F is 10°C.

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To fold and tape a net into a solid shape, cut out the net, fold it along the lines, tape the edges together, and flatten the edges

The net cannot be fold and tape as it is given as a picture, however the general steps to fold and tape a net into a solid shape:

When the above stapes are followed, some of the nets that can be gotten from the given shape are triangular pyramids and triangles

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a. The shape of the distribution of the sample mean will be approximately normal, according to the Central Limit Theorem.

b. The standard error of the mean is given by:

SE = σ / sqrt(n)

where σ is the population standard deviation (8 seconds), and n is the sample size (26). Substituting the given values, we get:

SE = 8 / sqrt(26) ≈ 1.57 seconds

Rounded to 2 decimal places, the standard error of the mean is 1.57 seconds.

c. To find the percentage of sample means that will be greater than 152 seconds, we need to calculate the z-score corresponding to a sample mean of 152 seconds:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean (152 seconds), μ is the population mean (148 seconds), σ is the population standard deviation (8 seconds), and n is the sample size (26).

Substituting the given values, we get:

z = (152 - 148) / (8 / sqrt(26)) ≈ 1.98

Using Appendix B.1, we find that the area to the right of a z-score of 1.98 is 0.0242, or 2.42%. Therefore, approximately 2.42% of the sample means will be greater than 152 seconds.

d. To find the percentage of sample means that will be greater than 144 seconds, we need to calculate the z-score corresponding to a sample mean of 144 seconds:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean (144 seconds), μ is the population mean (148 seconds), σ is the population standard deviation (8 seconds), and n is the sample size (26).

Substituting the given values, we get:

z = (144 - 148) / (8 / sqrt(26)) ≈ -1.98

Using Appendix B.1, we find that the area to the right of a z-score of -1.98 is also 0.0242, or 2.42%. Therefore, approximately 2.42% of the sample means will be less than 144 seconds.

e. To find the percentage of sample means that will be greater than 144 but less than 152 seconds, we need to find the area between the z-scores corresponding to sample means of 144 and 152 seconds.

The z-score corresponding to a sample mean of 144 seconds is:

z1 = (144 - 148) / (8 / sqrt(26)) ≈ -1.98

The z-score corresponding to a sample mean of 152 seconds is:

z2 = (152 - 148) / (8 / sqrt(26)) ≈ 1.98

Using Appendix B.1, we find that the area to the right of a z-score of -1.98 is 0.0242, and the area to the right of a z-score of 1.98 is 0.0242. Therefore, the area between these two z-scores is:

0.5 - 0.0242 - 0.0242 = 0.4516

Multiplying by 100, we get that approximately 45.16% of the sample means will be greater than 144 but less than 152 seconds.

Answer:

Step-by-step explanation:

The sketch is a right-angled triangle.

We can use Pythagoras' theorem.

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

z=√(50^2+25^2)

z= 25√5

z=25*2.23

z=55.75

z≈56km

the distance between where he started the

journey=56km

tan(90-Θ)=25/50

cotΘ=1/2

tanΘ=2

Θ=[tex]tan^{-1} (2)[/tex]

bearing of his starting point from his end point[tex]tan^{-1} (2)[/tex]

a) The plumber should buy 13 sections of 6-meter lengths of PVC pipes.

b) Assuming that there are no errors, the left-over pipe after satisfying Jobs A and B should be 3 meters.

We can use mathematical operations of addition, division, multiplication, and subtraction to determine the two quantities above.

In the first place, the total quantity of PVC pipes required is 75 meters.

This sum is divided by 6 to obtain the sections required, which is approximated to the nearest whole number.

Finally, 75 meters are subtracted from the total quantity bought.

PVC pipes required by Job A = 45 meters

PVC pipes required by Job B = 30 meters

The total quantity of PVC pipes required = 75 meters

The length of each PVC pipe = 6 meters

The number of pipes to buy = 13 sections (75 ÷ 6)

13 sections of pipes = 78 (6 x 13)

Leftover PVC pipes = 3 meters (78 - 75)

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Answer:

Step-by-step explanation:

First you have to find out how many red were in the original 27 marbles.

If there were 7 Black and 4 Yellow - this is 11 so 27 - 11 = 16 RED.

Removing 3 Black - changes the the can to 24 marbles.

4 Black - 4 Yellow and 16 Red.

So probability the random pick is red after the removal of 3 Black is

16 out of 24.       [tex]\frac{16}{24}[/tex]  

This reduces to 2/3.

Choice (K)

Answer:

Step-by-step explanation:

There are 27 marbles which include 7 black marbles

                                                            4 yellow marbles

Therefore, the number of red marbles=27-(7+4) marbles

                                                              =16 red marbles

When 3 black marbles are removed from the can, the total number of marbles become 24, which includes the 4 black marbles,4 yellow marbles and 16 red marbles.

So, the probability of the picking of a random red marble from the can after removing the black marbles = 16÷24

This when reduced gives the result 2÷3.

Hence, the answer is option K which means the probability that it was red is 2/3.

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The random variable x is normally distributed with mean u=89 and standard deviation o=4. The indicated probability, ​P(x<8​7)= 0.3085.

In statistics, standard deviation is a measure of the amount of variation or dispersion in a set of data. It is a commonly used measure of the spread of a distribution and is calculated as the square root of the variance.

A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation indicates that the data points are more tightly clustered around the mean.

Standard deviation is an important concept in statistics and is used in many statistical analyses, such as hypothesis testing and confidence interval estimation. It is also used in finance and economics to measure the risk associated with investments and to assess the variability of economic data.

To solve this problem, we need to standardize the value using z-score formula.

z = (x - mu) / sigma

Here, x = 87, mu = 89, and sigma = 4.

z = (87 - 89) / 4 = -0.5

We can look up the corresponding area under the standard normal distribution curve using a table or calculator. The area to the left of z = -0.5 is 0.3085.

Therefore, P(x < 87) = P(z < -0.5) = 0.3085.

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Answer:

20 kilograms.

Step-by-step explanation:

First, we need to find the cost per kilogram.

We can do this by dividing 374 by 34.

That gives us 11. So, our cost per kilogram is $11.

Next, to find how many kilograms we can get with $220 dollars, we need to divide 220 by 11.

That leaves us with a final answer of 20 kilograms.

There are 9 parts to the trail, and each part is the same length.  and hence each part of the trail is 0.2 kilometers long.

The following equation may be used to determine a rectangle's perimeter:

Perimeter is equal to 2 x (length + breadth).

where "length" denotes the rectangle's length and "width" denotes the rectangle's width. In order to account for both sides of the rectangle, we add the length and breadth together and multiply the result by 2, which is the distance around the outside of the rectangle.

The distance ran by Brurt is:

1.8 km ÷ 9 = 0.2 km

Hence, each part of the trail is 0.2 kilometers long.

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Answer: Sampling is very often used in our daily life. For example, while purchasing fruits from a shop, we usually examine a few to assess the quality. A doctor examines a few drops of blood as a sample and draws a conclusion about the blood constitution of the whole body.

the probability of exactly 4 successes in 7 trials of a binomial experiment in which the probability of success is 55% is 0.29 = 29%

A binomial experiment is an experiment that has the following four properties:

The experiment consists of n repeated trials

Each trial has only two possible outcomes

The probability of success, denoted p, is the same for each trial

Each trial is independent

the probability of exactly 4 successes in 7 trials of a binomial experiment in which the probability of success is 55%

We know that the binomial distribution

[tex]^nc_rp^rq^{n-r[/tex]

Where n is the number of trails

p is the probability of success in each trial

q =1-p which is the probability of failure

Here r=4

n=7

p= 55 % =0.55

then by the formula

[tex]p(4)=^7c_4(o.55)^4(0.45)^3\\\\p(4)=35*0.091*0.091\\p(4)=0.29[/tex]

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Answer:

C) Stock Market.

Brainly wants me to put at least 20 characters and since I only provided the answer choice and was too lazy to do an explanation I am putting this here.

Answer:

C) Stock Market.

Step-by-step explanation:

The term stock market refers to several exchanges in which shares of publicly held companies are bought and sold.

Teri can drive approximately 27.47 miles per gallon of gas.

To find how many miles per gallon Teri can drive, we need to divide the total distance she can travel on a full tank of gas by the amount of gas she needs to fill the tank. This gives us the average number of miles she can travel on one gallon of gas.

Teri's car can hold 17.4 gallons of gas.

Teri can drive 478.5 miles on a full tank of gas.

Mathematically, we can represent this as:

Miles per gallon = Total distance traveled ÷ Amount of gas used

Plugging in the values we have:

Miles per gallon = [tex]\frac{478.4 miles}{17.4 gallons}[/tex]

Performing the division:

Miles per gallon = 27.47126437

Rounding to two decimal places, we get:

Miles per gallon ≈ 27.47

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The values of all angles [tex] \angle \: GAL, \angle \: LAO, \angle CAO, \: angle \: KAC[/tex] are 71°, 90°, 90°, 90° respectively.

Given angle GAK is 19°.

It is clear that angle KAL is equal to 90°.

So,

[tex] \angle KAL = {90}^{o} [/tex]

Now,

[tex] \angle \: KAL = \angle KAG+ \angle \: GAL[/tex]

So,

[tex] \angle \: GAL = {90}^{o} - {19}^{o} \\ = {71}^{o} [/tex]

Now,

[tex] \angle \: KAO = 180°[/tex]

Now

[tex] \angle \: KAL + \angle \: LAO = {180}^{o} \\ \angle \: LAO = {180}^{o} - {90}^{o} \\ \angle \: LAO = {90}^{o} [/tex]

Again,

[tex] \angle \: CAO = {180}^{o} - \angle \: LAO \\ \angle \: CAO = {180}^{o} - {90}^{o} = {90}^{o} [/tex]

Similarly,

[tex] \angle \: KAC = \angle \: KAO - \angle \: CAO \\ \angle \: KAC = {180}^{o} - {90}^{o} \\ = {90}^{o} [/tex]

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After descending [tex]21.5[/tex] feet, the scientist's new position in the submarine would be  [tex]30.8[/tex] feet below sea level.

The scientist was initially in a submarine, located 52.3 feet below sea level. This means that her position was  [tex]52.3[/tex]feet lower than the surface of the sea.

Over the next ten minutes, the scientist descended further into the ocean by [tex]21.5[/tex] feet. This means that she traveled down an additional [tex]21.5[/tex]feet from her initial position.

To calculate her new position below sea level, we can subtract the distance she traveled downward (21.5 feet) from her initial position (52.3 feet below sea level).

[tex]52.3 feet - 21.5 feet = 30.8 feet[/tex]

Therefore, after descending 21.5 feet, the scientist's new position in the submarine would be 30.8 feet below sea level.

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Prahar would need approximately [tex]2.0625[/tex] cups of sugar to make apple pies that serve [tex]22[/tex]  people. We can round this up to [tex]2 1/8[/tex] cups of sugar for practical purposes.

To determine the amount of one of the ingredients needed to make an apple pie that serves [tex]22[/tex] people, we can set up a proportion comparing the number of servings:

Number of servings of the original recipe: [tex]8[/tex]

Number of servings needed: [tex]22[/tex]

Let's choose sugar as the ingredient to calculate:

Original amount of sugar for 8 servings:  [tex]3/4[/tex] cup

Unknown amount of sugar for  [tex]22[/tex] servings: x

We can set up the proportion as follows:

[tex]8/22 = 3/4x[/tex]

To solve for x, we can cross-multiply:

[tex]8x = 22 \times 3/4[/tex]

[tex]8x = 16.5[/tex]

[tex]x = 16.5/8[/tex]

[tex]x = 2.0625[/tex]

Therefore, Prahar would need approximately [tex]2.0625[/tex] cups of sugar to make apple pies that serve [tex]22[/tex]  people. We can round this up to [tex]2 1/8[/tex] cups of sugar for practical purposes.

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The equation of a circle given by the graph is (x-3)²+(y+2)²=29,

hence option b is correct.

A circle is a closed curve that extends outward from a set point known as the center, with each point on the curve being equally spaced from the center. A circle with an (h, k) center and a radius of r has the equation:

(x-h)² + (y-k)²= r²

This is the equation's standard form. Hence, we can quickly determine the equation of a circle if we know its radius and center coordinates.

We know that the equation of the circle is -

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where (h,k) is the center of the circle and r is the radius,

given that,

center of the circle is = (3,-2)

and the radius is=√29

Hence the equation of a circle is

[tex](x-3)^2+(y+2)^2=(\sqrt29)^2\\\\(x-3)^2+(y+2)^2=29[/tex]

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Equation: [tex]y=k\sqrt{z}[/tex].

Value of the constant of proportionality (k): 6.

So if y varies directly as the square root of z, there's a constant coefficient (k) multiplying the square root of z. Why? Because taking the square root of z will reduce it's value, then a number (k) must be multiplying it to make it match the value of y.

So:

[tex]y=k\sqrt{z}[/tex]

[tex]y = 36;\\ \\z = 36;\\ \\(36)=k\sqrt{(36)}\\ \\[/tex]

[tex]\frac{36}{\sqrt{(36)}} =\frac{k\sqrt{(36)}\\ \\}{\sqrt{(36)}} \\ \\\frac{36}{6} =k\\ \\k=6[/tex]

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[tex]\blue{\huge {\mathrm{SOLVING \; EQUATIONS}}}[/tex]

[tex]{===========================================}[/tex]

[tex]{\underline{\huge \mathbb{Q} {\large \mathrm {UESTION : }}}}[/tex]

[tex]{===========================================}[/tex]

[tex] {\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}} [/tex]

[tex]\qquad\qquad\begin{aligned}\bold{y = k\sqrt{z}}\\\\\bold{\:k = 6\:\:\:}\end{aligned}[/tex]

*Please read and understand my solution. Don't just rely on my direct answer*

[tex]{===========================================}[/tex]

[tex] {\underline{\huge \mathbb{S} {\large \mathrm {OLUTION : }}}} [/tex]

From the given information, we are given that:

Since y varies directly as the square root of z, we can write this as an equation:

where:

Substitute the given values into the equation and solve for k:

[tex]\qquad\qquad\begin{aligned}\sf \red{y} &=\sf k\sqrt{\blue{z}}\\\sf \red{36} &=\sf k\sqrt{\blue{36}}\\\sf \dfrac{\red{36}}{\sqrt{\blue{36}}} &=\sf \dfrac{k \cancel{\sqrt{\blue{36}}}}{ \cancel{\sqrt{\blue{36}}}}\\\sf \dfrac{\red{36}}{\blue{6}}&=\sf k\\\sf 6& =\sf k\\\sf \bold{\:k}& = \bold{6}\:\end{aligned}[/tex]

[tex]{===========================================}[/tex]

Equations are mathematical statements that show that two quantities are equal. They typically contain an equal sign (=) and one or more variables. Equations are used to express relationships between quantities and to solve problems in various fields of science, engineering, and mathematics.

Example equations include:

To solve an equation, we want to determine the value of the variable(s) that make the equation true.

There are different techniques to solve equations, but some common steps include:

1. Simplify both sides of the equation by combining like terms and using the order of operations if necessary.

2. Isolate the variable on one side of the equation by undoing any operations that were performed on it.

3. Check the solution by plugging it back into the original equation to see if it makes the equation true.

[tex]{===========================================}[/tex]

[tex]- \large\sf\copyright \: \large\tt{AriesLaveau}\large\qquad\qquad\qquad\tt 04/01/2023[/tex]

The probability of people who run red lights is

=0.465

Probability is the measure of the likelihood or chance that an event will occur.

Probability = Possible outcome of an event ÷ Total outcome

Total number of people asked if they had run a red light = number of people that responded 'yes' + number of people that responded 'no'

Total outcome = 348+400

Total outcome = 748

Possible outcome = number of people that responded yes = 394

The probability that if a person is chosen at random, they have run a red light in the last year will be

[tex]=\frac{348}{748}\\\\=0.465[/tex]

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Answer:

A) The histogram is bimodal.

Answer:

To create a frequency distribution, we first need to determine the range of the data. The smallest measurement is 3.4 inches and the largest is 5.4 inches, so the range is 5.4 - 3.4 = 2 inches. To create 6 classes with a width of 0.4 inches, we divide the range by 0.4 and round up to the nearest integer:

Number of classes = (range / class width) rounded up = 2 / 0.4 = 5

So we will use 5 classes with a width of 0.4 inches each. The classes and their corresponding frequency counts are:

Class 1: 3.4 - 3.8 | Frequency: 3

Class 2: 3.9 - 4.3 | Frequency: 8

Class 3: 4.4 - 4.8 | Frequency: 6

Class 4: 4.9 - 5.3 | Frequency: 7

Class 5: 5.4 - 5.8 | Frequency: 1

To create a histogram, we can plot the frequency counts on the y-axis and the class intervals on the x-axis. The shape of the histogram can give us information about the distribution of the data. In this case, the histogram is bimodal, meaning there are two peaks in the data. This suggests that the data may be composed of two separate subpopulations.

See the image linked below.

To graph system of inequalities y ≥ 5x + 2 and y > -3x - 2, draw lines for y = 5x + 2 and y = -3x - 2, then shade areas above each line. The intersection of shaded areas shows solutions to the system of inequalities.

In order to graph the system of inequalities y ≥ 5x + 2 and y > -3x - 2 you would start by graphing each inequality as if it was an equality. For the first inequality, you would graph the line y = 5x + 2 and shade the area above the line because it's y 'greater than or equal to'. For the second inequality, you draw the line y = -3x - 2 and also shade the area above because it's y 'greater than'. The intersection of the shaded areas represents the solution to the system of inequalities. Therefore, the graph would have two shaded lines, with the common shaded area above both lines representing the solutions to the system of inequalities.

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To prove: Area of triangle MAP = 1/2 area of trapezium MNOP

Given: MN//OP and A is the midpoint of NO

Proof:

Let h be the height of trapezium MNOP.

The area of trapezium MNOP is given by:

Area of trapezium MNOP = (1/2) × (MN + OP) × h

Since MN//OP, we have MN = OP.

Therefore, Area of trapezium MNOP = (1/2) × 2MN × h = MN × h

Now, consider triangle MAP.

The base of triangle MAP is MA, which is half of NO.

Therefore, the base of triangle MAP is (1/2) × NO.

The height of triangle MAP is h, which is the same as the height of trapezium MNOP.

Therefore, the area of triangle MAP is given by:

Area of triangle MAP = (1/2) × (base) × (height) = (1/2) × (1/2NO) × h

Substituting NO = 2OA, we get:

Area of triangle MAP = (1/2) × (1/2 × 2OA) × h = (1/2) × OA × h

Since A is the midpoint of NO, we have OA = 1/2NO.

Therefore, Area of triangle MAP = (1/2) × (1/2NO) × h = (1/2) × (base of trapezium MNOP) × h

Hence, Area of triangle MAP = 1/2 Area of trapezium MNOP.

Therefore, the given statement is proved.

Answer: Given that MNOP is a trapezium with MN parallel to OP and A is the midpoint of NO.

To prove that the area of triangle MAP is half the area of trapezium MNOP, we need to use the following theorem:

Theorem: If a line segment joins the midpoint of one side of a triangle to a vertex opposite to that side, then the segment divides the triangle into two equal areas.

Proof:

Since A is the midpoint of NO, we can draw a line segment AP from vertex P to point A on NO as shown in the figure below:

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      M _______N

       |\      |

       |  \    |

       |    \  |

       |      \|

      O--------P

            A

We can see that triangle MAP is formed by the line segment AP and side MP of the trapezium.

Also, we can see that triangle MAN is congruent to triangle NOP because they are both right triangles with corresponding sides parallel. Therefore, they have the same area.

Since MNOP is a trapezium, the area of trapezium is given by:

Area of trapezium MNOP = (1/2)(MN+OP) × height

Since MN is parallel to OP, the height of the trapezium is the same as the height of triangle MAN and NOP.

Therefore, the area of trapezium MNOP can be written as:

Area of trapezium MNOP = (1/2)(MN+OP) × height

= (1/2)(MA+AP+OP) × height

= (1/2)(MA+MP) × height (Since AP = NO/2 = height)

So, we have proved that:

Area of trapezium MNOP = (1/2)(MA+MP) × height

Using the theorem, we know that AP divides triangle MAP into two equal areas. Therefore,

Area of triangle MAP = (1/2) × Area of triangle MAP

Substituting the above in the expression for the area of trapezium, we get:

Area of trapezium MNOP = Area of triangle MAP + (1/2) × Area of triangle MAP

= (3/2) × Area of triangle MAP

Therefore, we have:

Area of triangle MAP = (1/2) × Area of trapezium MNOP

Thus, we have proved that the area of triangle MAP is half the area of trapezium MNOP.

Step-by-step explanation: