A health expert evaluates the sleeping patterns of adults. Each week, she randomly selects 50 adults and calculates their average sleep time. Over many weeks, she finds that 5% of average sleep time is less than 9 hours and 5% of average sleep time is more than 9.4 hours.
What are the mean and standard deviation (in hours) of sleep time for the population?

(a)

i. The mean of all 855 salaries of 2012 professional baseball players to 3 decimal places is 3.319 million dollars.

ii. The standard deviation of all 855 salaries of 2012 professional baseball players to 3 decimal places is 4.104 million dollars.

iii. The values in parts (a) and (b) above are:

Parameter: the mean and standard deviation of the population.

Numerical summaries of the population.

iv. The distribution of salaries for professional baseball players in 2012 is right-skewed with a single peak and several outliers at the high end.

(b)

i. The mean of the sampling distribution of the sample mean (in millions of dollars) when n = 10 is 3.319 million dollars.

ii. The standard error of the sampling distribution of the sample mean (in millions of dollars) when n = 10 is 1.298 million dollars.

iii. The shape of the sampling distribution of the sample mean when n = 10 is approximately normal.

(c) The value of the sample mean (in millions of dollars) from the first random sample of size n = 10 is 2.167 million dollars.

(d) The value of the sample mean (in millions of dollars) from the second random sample of size n = 10 is 2.384 million dollars.

(e) The value of the sample mean (in millions of dollars) from the third random sample of size n = 10 is 3.913 million dollars.

(f)

i. The mean of the three sample means is 2.821 million dollars.

ii. The standard deviation (i.e., standard error) of the three sample means is 0.911 million dollars.

(g)

i. We would expect to see a value of 3.319 million dollars for the mean of the sample means (in millions of dollars) of size n = 10.

ii. We would expect to see a value of 1.298 million dollars for the standard error of the sample means (in millions of dollars) of size n = 10.

The mean and standard error are obtained from the applet and are reported to 3 decimal places. The shape of the sampling distribution of the sample mean when n=10 is approximately normal due to the Central Limit Theorem.

The applet provides the histogram of the 10 observations selected for the sample, as well as the mean and standard deviation of the observations in the sample. The student is required to click "Generate 1 Sample" three times and report the sample means from each of the three random samples of size n=10.

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Answer:

a) 2*(x+4)

b) 1/4*x + 1/3*x

Answer:

a) 2(x+4)

b) 7x/12

Step-by-step explanation:

a) Sum of x and 4 is (x+4). Since the number is doubled,

2 * (x+4) = 2(x+4)

b) One fourth of a number x is x/4 and one third of the number is x/3.

Adding these two:

x/4 + x/3

The LCM of 4 and 3 is 12 and so,

(4x + 3x)/12

= 7x/12

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There is no difference in g(x) and f(x) when graphing a radical expression as represents the same function: √(x-5) +6

The given radical expression are:

g(x) = √(x-5) + 6

f(x) = √(x-5) + 6

The graph of both function is plotted, which have the exact same value on graph.

Thus, there is no difference in g(x) and f(x) when graphing a radical expression as represents the same function: √(x-5) +6

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PR = 16 units

Similar triangles are two triangles that have the same shape but not necessarily the same size. This means that their corresponding angles are equal, and their corresponding sides are proportional in length.

Here, ΔPRQ similar with ΔPST

So, we can say that their corresponding sides are in proportion (i.e., have the same ratio).

[tex]\frac{PR}{PS} = \frac{RQ}{ST}[/tex]

[tex]\frac{(x+7) + (x-1) }{(x+7)} = \frac{8}{6}[/tex]

Simplification,

12x + 36 = 8x + 56

x = 5

So, the value of PR = (x+7) + (x-1)

PR = (5+7) + (5-1) = 16

Therefor, PR = 16 units

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The triabgles ae similar. Their corresponding sides are in proportion.

PR = 16 units

Triangles that are similar to one another in shape but not necessary in size are called similar triangles. Their respective sides are proportional in length, and their corresponding angles are equal. Three edges, three angles, and three vertices make up a triangle. The sum of a triangle's three internal angles is 180 degrees. The triangle's third side is longer than the sum of its two longest sides.

Here, ΔPRQ similar with ΔPST

Thus, their respective sides are proportional, as we can see. (i.e., have the same ratio).

[tex]\frac{PR}{PS} =\frac{RQ}{ST} \\\\\frac{(x+7)+(x-1)}{(x+7)} =\frac{8}{6}[/tex]

Simplification,

12x + 36 = 8x + 56

x = 5

So, the value of PR = (x+7) + (x-1)

PR = (5+7) + (5-1)

    = 16

Therefor, PR = 16 units

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Answer:

96

Step-by-step explanation:

so, since we know that the derivative is simply the equation we use to get the slope at any point on the curve, thus df/dx will be that.

now, we know that g(x) is already in slope-intercept form, so it has a slope of "4", hmmm what's "x" at that instance?

[tex]f(x)=-x^2+8x+20\implies \cfrac{df}{dx}=-2x+8 \\\\[-0.35em] ~\dotfill\\\\ g(x)=\stackrel{\stackrel{m}{\downarrow }}{4}x+k\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\\\ 4~~ = ~~-2x+8\implies 2=x[/tex]

well, now we know that x = 2, hmmm what's "y" at that instance? well, our "y" will be simply f(2) = g(x), because g(x) is touching f(x) at that point on the curve.

[tex]f(2)=-(2)^2+8(2)+20\implies f(2)=32 \\\\[-0.35em] ~\dotfill\\\\ g(x)=4x+k\implies 32=4(2)+k\implies 32=8+k\implies \boxed{24=k}[/tex]

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Answer:

After John finished his energy drink, he noticed on the label that it had 160 milligrams of caffeine. He read online that the amount of caffeine in the body decreases by a factor of

1

8

each hour after it's consumed.

Step-by-step explanation:

The exponential equation in the form y=a(b)² that can model the amount of caffeine left in John's body, y, x hours after finishing the energy drink is y= 160(1/8)⁽ˣ⁾. After 6 hours, the amount of caffeine left in John's body will be y=160(1/8)⁽⁶⁾ ≈ 1.96 milligrams (rounded to the nearest milligram).

The exponential equation y=a(b)² can be used to model the amount of caffeine left in John's body x hours after consuming an energy drink. In this equation, a represents the initial amount of caffeine in the body (160 milligrams), and b represents the factor by which the amount of caffeine decreases each hour (1/8). To find the amount of caffeine left in John's body after 6 hours, we can simply substitute x=6 into the equation and solve for y. The result is y=160(1/8)⁽²*⁶⁾ ≈ 6 milligrams. This means that after 6 hours, John will have approximately 6 milligrams of caffeine left in his body if he doesn't consume any more caffeine.

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Complete question:

After John finished his energy drink, he noticed on the label that it had 160 milligrams of caffeine. He read online that the amount of caffeine in the body decreases by a factor 1/8 of each hour after it's consumed.

Write an exponential equation in the form y=a(b)² that can model the amount of caffeine left in John's body, y, x hours after finishing the energy drink.

Use whole numbers, decimals, or simplified fractions for the values of a and b,

To the nearest milligram, how much caffeine will be in John's body after 6 hours if he doesn't consume any more caffeine?

The critical value of p in a test to determine if a filling system under-fills less than 10% of all boxes, with a significance level of 0.10 and a random sample of size 200, is -1.28.

Assuming that the filling system under-fills with probability p, the null hypothesis is:

H0: p >= 0.10

Ha: p < 0.10 (one-tailed test)

To find the critical value of p, we need to use the inverse cumulative distribution function (ICDF) of the standard normal distribution.

At a significance level of 0.10, the critical value is the value z such that P(Z <= z) = 0.10, where Z is a standard normal random variable. We can find this value using a standard normal table or calculator, or by using the function norm.inv(0.10) in Microsoft Excel, which gives the result -1.28 (rounded to two decimal places).

Since the sample size is large (n = 200), we can use the normal approximation to the binomial distribution to test the null hypothesis. The test statistic is:

z = (x/n - p0) /  √(p0*(1-p0)/n)

where x is the number of boxes that under-fill, n is the sample size, and p0 is the null hypothesis value of the proportion. We want to reject the null hypothesis if the test statistic is less than the critical value of -1.28.

Since we do not have any information about the sample proportion, we can use the null hypothesis value of p0 = 0.10. Then the critical value of the test statistic is:

z_critical = -1.28

Substituting the values into the test statistic formula, we get:

z = (x/200 - 0.10) / √(0.10*(1-0.10)/200)

Simplifying, we get:

z = (x - 20) / 4.24

To reject the null hypothesis at the 0.10 significance level, we need the test statistic z to be less than -1.28. Therefore, we can write the rejection region as:

z < -1.28

Substituting this inequality into the test statistic formula, we get:

(x - 20) / 4.24 < -1.28

Solving for x, we get:

x < 14.5

Therefore, if the number of boxes that under-fill is less than or equal to 14 (out of 200), we reject the null hypothesis at the 0.10 significance level and conclude that there is sufficient evidence to suggest that the filling system under-fills less than 10% of all boxes.

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5 times the sum of x squared and y squared. This can be expressed as 5(x^2 + y^2).

Express is a web application framework for Node.js, released as free and open-source software under the MIT License. It is designed for building web applications and APIs and is the de facto standard server framework for Node.js. It provides a host of features to facilitate the development of web and mobile applications, including routing, templating, and a middleware system to handle requests and responses. Express is the backend component of the MEAN stack and enables the rapid development of dynamic web applications.

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Ben's Gross Profit for the last quarter is $3,425. This is calculated by subtracting the Cost of Goods ($575) and Total Expenses ($5,000) from the Income ($9,000).

Gross profit is the difference between a business’s total sales revenue and its cost of goods sold (COGS). It is calculated by subtracting the cost of goods sold from total revenue. Gross profit is also referred to as gross margin and is expressed as a percentage of total sales. This figure is important to businesses because it shows the profitability of the company’s products after accounting for the cost of the materials used to manufacture those products. Additionally, gross profit is a key indicator of the company’s overall financial health and ability to remain profitable.

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tan t = -4/3

The equation for the unit circle is

x2 + y2 = 1


For the point (-3/4, y) we have

(-3/4)2 + y2 = 1


Rearranging this gives us

y2 = 1 - (3/4)2

Solving for y gives us

y = √ (1 - (3/4)2)


Now we can find tan t by using the equation

tan t = y / -3/4


Substituting in the value of y we found earlier gives us

tan t = √ (1 - (3/4)2) / -3/4


Finally, simplifying the equation gives us the answer

tan t = -4/3

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Answer:

13 for 5 bags.

Step-by-step explanation:

Dani spends 5.20 in total for 2 bags. You would do 5.20/2 to get 2.60 which tells us that 2.60 is the cost per bag. Then you would do 2.60 x 5 to get 13.

The tower is 2,200 feet tall.

To find the height of the tower, we can use the Pythagorean theorem for a right triangle. The given information can be represented as:

Hypotenuse (cable) = 2,230 feet

Base (distance from the starting point to the point under the tower) = 370 feet

Height (height of the tower)

Write the Pythagorean theorem formula:

[tex]a^2 + b^2 = c^2,[/tex]

where a and b are the two shorter sides (base and

height) and c is the hypotenuse.

Plug in the given values: [tex]height^2 + 370^2 = 2,230^2[/tex]

Square the given numbers: [tex]height^2 + 136,900 = 4,972,900[/tex]

Subtract the base squared from both sides: [tex]height^2 = 4,972,900 - 136,900[/tex]

Calculate the difference: [tex]height^2 = 4,836,000[/tex]

Find the square root of the difference: height = √4,836,000

Calculate the height: height = 2,200 feet

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The value of t such that the area under the curve between 1 and 11 equals 0.95, assuming the degrees of freedom is 8, is 1.86.

Degree of freedom refers to the number of independent variables that can be varied in a statistical experiment or analysis. It is a measure of the flexibility of the experiment or analysis to accommodate new data or observations.

In this case, the degrees of freedom is 8. To find the t-value, the t-table must be consulted. The t-table is a chart of t-values for different degrees of freedom. The t-value is the value at which the area under the curve between 1 and 11 equals 0.95. The t-table is organized such that the rows represent the degrees of freedom and the columns represent the area under the curve. For example, if the degrees of freedom is 8 and the area under the curve is 0.95, the t-value can be found in the row for 8 degrees of freedom and the column for 0.95 area under the curve. The t-value from the t-table in this case is 1.86.

Therefore, the value of t such that the area under the curve between 1 and 11 equals 0.95, assuming the degrees of freedom is 8, is 1.86.

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The value of t such that the area under the curve between 1 and 11 equals 0.95, assuming the degrees of freedom is 8, is 1.86.

Degrees of freedom refer to the number of independent variables that can be varied in a statistical experiment or analysis. It is a measure of experimental or analytical flexibility to accommodate new data or observations.

In this case there are 8 degrees of freedom. To find the t value, we need to refer to the t table. A t-table is a plot of t-values ​​for different degrees of freedom. The t-value is the value at which the area under the curve between 1 and 11 is 0.95. The t-table is organized so that rows represent the degrees of freedom and columns represent the area under the curve. For example, if you have 8 degrees of freedom and the area under the curve is 0.95, find the t-value in the row with 8 degrees of freedom and the column with area under the curve of 0.95. In this case, the t-value in the t-table is 1.86.

Therefore, the value of t is such that the area under the curve between 1 and 11 is 0.95, assuming 8 degrees of freedom is 1.86.

The area between -t and t = 0.95

P(t₈ > t) = P(t₈ - t)

= 0.025

As t - distⁿ is symmetrical and are under the curve is 1 and degree of freedom is 8.

Hence, from t - table; the t - value = 2.306

i.e. P(t₈ > 2.306) = 0.025

or P(-2.306 < t₈ < 2.306) = 0.95

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The complete question is as follows:

Lesley gets 10/18 or 5/9 of the counters.Adam has 18 counters, so we can express the total number of counters as 18/18. If he gave 10 of those counters to Lesley, we can express that as 10/18.

Adam has 18 counters, so we can express the total number of counters as 18/18, which is equivalent to 1. If he gave 10 of those counters to Lesley, we can express that as 10/18. To find out what fraction of the 18 counters Lesley gets, we need to simplify 10/18. Since both the numerator and denominator are divisible by two, we can divide both the numerator and denominator by two, which gives us 5/9. This means that Lesley gets 5/9 of the 18 counters, which is equivalent to 10/18 or half of the total number of counters.

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Option C, MN/LK||MN, is the additional piece of information required to demonstrate that ΔLKN ~ΔKNM by the SAS similarity theorem.

The SAS (Side-Angle-Side) similarity theorem: what is it?

According to the SAS (Side-Angle-Side) similarity theorem, two triangles are similar if they have two pairs of corresponding sides that are in proportion and the included angles are congruent.

In this example, we know that ΔLKN and ΔKNM share the angle at vertex K, and we also know that LN/MN and LK/KN are in proportion.

To clarify that MN is parallel to LK and forms a transversal with LN, we must first prove that MN/LK||MN.

To prove that the included angles between the matching sides are congruent, which is necessary for the triangles to be similar, this information is required.

Consequently, in order to demonstrate that ΔLKN~ΔKNM according to the SAS similarity theorem, option C is the correct additional piece of information.

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The value of D is 49.1º to the nearest degree using the law of sines.

To solve for angle D, we can use the law of sines, which states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively:

a/sin(A) = b/sin(B) = c/sin(C)

Using this formula, we can solve for angle D:

sin(D)/d = sin(F)/f

sin(D) = (d/f) × sin(F)

sin(D) = (76/20) × sin(17º)

sin(D) = 0.7588

Taking the inverse sine of both sides, we get:

D = [tex]sin^{-1}[/tex] × (0.7588)

D = 49.1º (rounded to the nearest tenth of a degree)

The law of sines can have two possible solutions for an angle, depending on the triangle. However, in this case, since angle D is opposite the longer side (d), there is only one possible solution.

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The trigonometric ratios can find the missing side of a right triangle given an angle, such as by using the tangent ratio to calculate the adjacent side length when given the length of the opposite side.

The trigonometric ratios are used to calculate specific values of a triangle. The three main ratios are sine, cosine, and tangent. The sine ratio is the ratio of the side length opposite the angle to the hypotenuse of the triangle. The cosine ratio is the ratio of the side length adjacent to the angle to the hypotenuse of the triangle. The tangent ratio is the ratio of the side length opposite the angle to the side length adjacent to the angle.

To find the missing side of a right triangle, given an angle, we use the trigonometric ratios. For example, if we have an angle of 45 degrees and a side length of 8, we can use the tangent ratio to calculate the missing side length. To do this we use the formula tan(45) = opposite/adjacent. We know the opposite side is 8, so to calculate the adjacent side we solve for x in the equation 8/x = tan(45). We get x = 8/tan(45), and x = 8/1 = 8. Therefore, the adjacent side length is 8.

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Answer: 1.07x107 but the seven goes on top of the 10

Step-by-step explanation:

Answer:

1.07 × 10⁷

Step-by-step explanation:

To express 10,700,000 in scientific notation, we have to rewrite it as a decimal number that is between 1 and 10 multiplied by a power of 10.

As a decimal, the number is:

1.07.

We have to move the decimal point 7 places to get this number (see the attached image). Therefore 1.07 is multiplied by 10⁷ to get 10,700,000.

The point C and point D are approximately 23.4 meters apart.

Trigonometric functions are a set of mathematical functions that relate the angles of a right-angled triangle to the ratios of the lengths of its sides.

We can use trigonometry function to solve for the distance between points C and D. Let x be the distance CD, then we have:

tan(48°) = h / (x + 24m)

where h is the height of the flagpole.

We can solve for x by rearranging the equation:

x = (h / tan(48°)) - 24m

Substituting the given values, we get:

x = (42m / tan(48°)) - 24m

x ≈ 23.4m

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105 divided by 15 is represented as 15 groupings of 7. We will obtain a full number quotient in each instance with no residual.

117 divided by 13 is equal to 9 with a residual of 0; 136 divided by 17 is equal to 8 with a remnant of 0; and 231 divided by 11 is equal to 21 with a remainder of 0;

We may use a model of equal groups to get the quotient for each division issue. Drawing 15 groups of 7 to represent 105 divided by 15 and 13 groups of 9 to represent 117 split by 13 and 17 groups of 8 to represent 136 divided by 17 is possible. We will obtain a full number quotient in each instance with no residual.

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keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

[tex]x+2y=2\implies 2y=-x+2 \\\\\\ x=\cfrac{-x+2}{2}\implies y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{1}{2}}x+1\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{-1}{2}} ~\hfill \stackrel{reciprocal}{\cfrac{2}{-1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{2}{-1} \implies 2}}[/tex]

so we're really looking for the equation of a line whose slope is 2 and it passes through (5 , 6)

[tex](\stackrel{x_1}{5}~,~\stackrel{y_1}{6})\hspace{10em} \stackrel{slope}{m} ~=~ 2 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{6}=\stackrel{m}{ 2}(x-\stackrel{x_1}{5}) \\\\\\ y-6=2x-10\implies {\Large \begin{array}{llll} y=2x-4 \end{array}}[/tex]

The equation of a line perpendicular to line QR in slope-intercept form that contains point (5, 6) is y = 2x-4.

y = mx + b is the slope intercept form of writing the equation of a straight line. In the equation 'y = mx + b', 'b' is the point, where the line intersects the 'y axis' and 'm' denotes the slope of the line.

Given that a line QR has an equation is x+2y = 2, we need to find the equation of a line passing through (5, 6) and is perpendicular to the given line,

Converting the equation into slop-intercepts form,

y = -0.5x+1

Here, slope m = -0.5

We know that the slopes of the perpendicular lines are negative reciprocal of each other,

Therefore, the slope of the asked line is 2.

So, its equation =

y = 2x+c

To find c, put x = 5 and y = 6

6 = 2(5) + c

c = -4

Therefore, the final equation will be =

y = 2x-4

Hence, the equation of a line perpendicular to line QR in slope-intercept form that contains point (5, 6) is y = 2x-4.

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Answer:

Step-by-step explanation:

The answer is 50.3  An easy way to find the answer is by using this link. https://www.calculator.net/right-triangle-calculator.html

Just plug the 29 and 44 in.

The minimum number of quarters that she could have is: 14

Let x represent the number of dimes = 4

Let y represent the number of quarters

She has no more than 21 coins

Thus:

x + y ≤ 21

Plugging in relevant values gives:

4 + y ≤ 21

y ≤ 21 - 4

y ≤ 17

0.10x + 0.25y ≥ 3.75

0.10(4) + 0.25y ≥ 3.75

0.40 + 0.25y ≥ 3.75

0.25y ≥ 3.75 - .40

0.25y ≥ 3.35

y ≥ 3.35/.25

y ≥ 13.4

Therefore, the minimum number of quarters she can have would be 14 because the "y" value must be a whole number and the next whole number above 13.4 is 14.

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Answer:

Step-by-step explanation:

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

    [tex]=(\frac{-12+(-8)}{2}, \frac{-7+(-4)}{2})[/tex]

    [tex]=(-10,-5.5)[/tex]    

To have a higher salary than 99% of her male counterparts, a woman would have to make more than $5229.48 per month, or $62753.76 per year

To find out how much a woman would have to make in a year to have a higher salary than 99% of her male counterparts, we need to find the salary level that corresponds to the 99th percentile of the male salary distribution.

First, we need to calculate the standard deviation of the annual salaries for both men and women. Since the standard deviation of the monthly salaries is $300, the standard deviation of the annual salaries can be calculated as:

σ_annual = σ_monthly * √12 = 300 * √12 = $1039.23

Next, we can calculate the salary level that corresponds to the 99th percentile of the male salary distribution using the standard normal distribution table.

z-score for the 99th percentile = 2.33

salary level for the 99th percentile of male salary distribution = mean + z-score * σ_annual

= $2750 + 2.33 * $1039.23 = $5229.48 (rounded to the nearest cent)

Therefore, to have a higher salary than 99% of her male counterparts, a woman would have to make more than $5229.48 per month, or $62753.76 per year (rounded to the nearest cent).

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The standard deviation of the boys' marks is 0.33.

Standard deviation is a measure of the amount of variation or dispersion of a set of values from the mean. It is calculated as the square root of the variance. It is the most commonly used measure of spread. It is used to measure the degree of variation from the mean of the data set. It is a measure of the variability of the data. It is usually expressed as a positive number.

The standard deviation of the boys' marks can be calculated by subtracting the mean of the girls' marks from the mean of all 12 marks, and then dividing by the number of boys.

To calculate the mean of the girls' marks, add the three marks together (92 + 80 +71) and divide by 3 (243/3 = 81).

To calculate the mean of all 12 marks, add all 12 marks together (92 + 80 + 71 + …) and divide by 12 (1044/12 = 84).

Therefore the mean of the boys' marks is 84 - 81 = 3.

Now we can calculate the standard deviation of the boys' marks by dividing this value by the number of boys (3/9 = 0.33).

Therefore, the standard deviation of the boys' marks is 0.33.

Standard deviation is a measure of the spread of data around the mean. In this case, the boys' marks had a lower standard deviation than the overall marks, indicating that the boys' marks were more consistent than the girls' marks. This could be due to the fact that there were fewer boys taking the exam, so the boys had a smaller spread of scores than the girls.

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The pοssible first steps tο simplify the equatiοn are:

Multiply bοth sides οf the equatiοn by -2.

Distribute Negative οne-half οver (x + 4).

Multiply bοth sides οf the equatiοn by -2, tο get rid οf the fractiοn:

-2 * Negative οne-half (x + 4) = -2 * 6

This simplifies tο: (x + 4) = -12

Distribute Negative οne-half οver (x + 4):

Negative οne-half (x + 4) = Negative οne-half * x + Negative οne-half * 4

This simplifies tο: (-1/2)x - 2 = 6

Therefοre, the pοssible first steps tο simplify the equatiοn are:

Multiply bοth sides οf the equatiοn by -2.

Distribute Negative οne-half οver (x + 4).

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Answer:

y = -2x + 17.

Step-by-step explanation:

To find the equation of a line that is parallel to another line, we need to use the fact that parallel lines have the same slope.

First, we need to rearrange the equation of the given line 4x + 2y = 10 into slope-intercept form y = mx + b:

4x + 2y = 10

2y = -4x + 10

y = -2x + 5

So, the slope of the given line is -2.

Since the line we're looking for is parallel to the given line, it will have the same slope of -2. We also have a point (7,3) that the line passes through. We can use the point-slope form of a linear equation to find the equation of the line:

y - y1 = m(x - x1)

where m is the slope, and (x1, y1) is the point.

Plugging in the values, we get:

y - 3 = -2(x - 7)

Simplifying:

y - 3 = -2x + 14

y = -2x + 17

So, the equation of the line that passes through the point (7,3) and is parallel to the line 4x + 2y = 10 is y = -2x + 17.

The overall transformation of the point (1,2) is (-6,3).

The given composition of transformations is a dilation with a scale factor of 3 centered at the origin, followed by a rotation of 270° counterclockwise about the origin. The dilation can be expressed mathematically as a scaling of the coordinates (x, y) by a factor of 3, resulting in (3x, 3y). The rotation of 270° counterclockwise can be expressed as a matrix multiplication of the new coordinates resulting in (-3y, 3x). Thus, the overall transformation of the composition can be expressed mathematically as (-3y, 3x).

To illustrate this with an example, let us consider a point (1,2). After the dilation, the coordinates become (3,6). After the rotation, the coordinates become (-6,3).

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