In a cooking contest, the mean score for taste was 20, with a standard deviation of 2.8. One of the contestants received a score of 14. Convert this score to a z score and tell if it is “usual” or “unusual.”

Samples of size 49 are selected from a population with mean 40 and standard deviation 7.5. the standard error of the sampling distribution of sample means is B. 1.07

The standard error of the sampling distribution of sample means can be calculated using the formula:

Standard Error (SE) = (Standard Deviation of the Population) / sqrt(Sample Size)

In this case, the population mean (μ) is 40, the standard deviation (σ) is 7.5, and the sample size (n) is 49.

To find the standard error, plug the values into the formula:

SE = 7.5 / sqrt(49)

SE = 7.5 / 7

SE = 1.07

Therefore, the standard error of the sampling distribution of sample means is 1.07 (Option B).

The Question was Incomplete, Find the full content below :

(normal approximation) samples of size 49 are selected from a population with mean 40 and standard deviation 7.5. the standard error of the sampling distribution of sample means is

A. 0.30

B. 1.07

C. 7.50

D. 0.82

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the correct answers are:=4, 11, 8, 12, 1, 6, 14 and 14, 10, 6, 9, 11, 8, 1 for the given data

How to solve the data

The box plot represents the distribution of a dataset using quartiles, median, and outliers. In order to create a box plot, we need to have a dataset with numerical values. Based on the options given, the data sets that can be used to create the box plot are:

4, 11, 8, 12, 1, 6, 14: This data set has all the required numerical values to create a box plot. It contains 7 values which are within the range of values shown on the x-axis of the box plot.

14, 10, 6, 9, 11, 8, 1: This data set also contains all the required numerical values to create a box plot. It has 7 values which are within the range of values shown on the x-axis of the box plot.

The remaining data sets listed are either non-numerical or do not contain values within the range shown on the x-axis. Therefore, they cannot be used to create the box plot.

In summary, the correct answers are:

4, 11, 8, 12, 1, 6, 14

14, 10, 6, 9, 11, 8, 1

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The error mason did is he has subtracted 5 from the right side of the inequality.

An inequality is a statement that one value is greater than, less than, or equal to another value. Inequalities are often represented using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "!=" (not equal to).

The given inequality is x-5<11.

Solution of the given equation is x<6.

To solve the equation the addition property of inequality is used and 5 is added to both sides of the equation.

a) The error mason did is he has subtracted 5 from the right side of the inequality. Instead of adding the number 5.

So, the error was to subtract 5 from right side.

b) The given inequality is x-5<11.

Add 5 on both the sides

x - 5 + 5 <  11 +5.

After adding we get,

x < 16.

Now plot the point 16 on the number line. Since the inequality conatins < sign, number to the left of 16 are the solution.

The correct solution on the number line is as shown in the figure.

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226 integers are present between 100 and 999, inclusive, and have the property that some permutation of its digits is a multiple of 11 between 100 and 999. Hence, option A is the correct option.

The problem statement is to find the number of multiples of 11 between 100 and 999 inclusive, where some multiples may have digits repeated twice and some may not.

To solve this problem, we can first count the number of multiples of 11 between 100 and 999 inclusive, which is 81. Some of these multiples may have digits repeated twice, and each of these can be arranged in 3 permutations. Other multiples of 11 have no repeated digits, and each of these can be arranged in 6 permutations. However, we must account for the fact that switching the hundreds and units digits of these multiples also yields a multiple of 11, so we must divide by 2, giving us 3 permutations for each of these multiples.

Thus, we have a total of 81 × 3 = 243 permutations. However, we have overcounted because some multiples of 11 have 0 as a digit. Since 0 cannot be the digit of the hundreds place, we must subtract a permutation for each of these multiples. There are 9 such multiples (110, 220, 330, ..., 990), yielding 9 extra permutations. Additionally, there are 8 multiples (209, 308, 407, ..., 902) that also have 0 as a digit, yielding 8 more permutations.

Therefore, we must subtract these 17 extra permutations from the total of 243, giving us 226 permutations in total. Hence, option A is the correct option.

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The probability of the two events, A and B, is given as follows:

C. P(A and B) = 0.15.

Conditional probability is the probability of one event happening, considering the result of a previous event. The formula is:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which:

The parameters for this problem are given as follows:

P(A|B) = 0.25, P(B) = 0.6.

Hence the probability of the two events is given as follows:

P(A and B) = P(A|B) x P(B)

P(A and B) = 0.6 x 0.25

P(A and B) = 0.15.

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The possible coordinates of the point a is (-4, 11.06) and (-4, -9.06).

Given that point, b has coordinates (5,1) and the distance between point a and point b is 15 units, we can use the distance formula to find the possible coordinates of point a.

The distance formula is given by:

distance = [tex]\sqrt({x_{2}-x_{1 })^{2} } +\sqrt({y_{2}-y_{1 })^{2}[/tex]

If we replace with the point b's coordinates, we obtain:

15 = [tex]\sqrt({5}-(-4)})^{2} } +\sqrt({1-y_{1 })^{2}[/tex]

Simplifying the equation, we get:

225 = [tex](9 + y_{1} ^{2} - 2y_{1} )[/tex]

Rearranging and solving for y1, we get:

y₁² - 2y₁ - 216 = 0

Using the quadratic formula, we get:

y₁ = 11.06

y₁ = -9.06

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Families with one or two females make up the same percentage of households as those with three or more girls. These two categories are identical to one another.

Assuming that the probability of having a boy or a girl is equal and independent for each child, there are four possible outcomes for the number of girls in a family: 0, 1, 2, or 3.

To determine the difference in the proportion of families with 1 or 2 girls compared to those with 0 or 3 girls, we need to calculate the probability of each scenario.

The probability of having 0 or 3 girls is the same because there are two ways to achieve this outcome: either all children are boys, or there is exactly one boy and two girls in the family. The probability of each of these outcomes is:

P(0 girls or 3 girls) = P(all boys) + P(1 boy, 2 girls)

[tex]= (1/2)^3 + 3(1/2)^3[/tex]

[tex]= 1/2[/tex]

The probability of having 1 or 2 girls is the complement of the probability of having 0 or 3 girls, which is:

P(1 girl or 2 girls) [tex]= 1 - P[/tex] (0 girls or 3 girls)

[tex]= 1 - 1/2[/tex]

[tex]= 1/2[/tex]

Therefore, the proportion of families with 1 or 2 girls is the same as the proportion of families with 0 or 3 girls. There is no difference between these two groups.

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Answer: The bedroom is 8 gridlines by 10 gridlines.

Step-by-step explanation:

The area of the bedroom is 8 x 10 gridlines, or 80 square gridlines.

Multiply this by 4 to get the number of square feet needed to cover the entire floor of Bedroom 1:

80 x 4 = 320 square feet

The true answer for the statement 'When conducting a simple main effects analysis is given by you use the MS within value from the original two-factor ANOVA.

Simple main effects analysis involves,

Examining the effect of one independent variable at each level of the other independent variable.

It allows us to determine the significance of the effect of one independent variable at each level of the other independent variable.

To compute the simple main effects,

We use the residual variability from the original two-factor ANOVA, which is the MS within value.

This is because the variability between the levels of the other independent variable has already been accounted for by the original two-factor ANOVA.

It is required to look only at the variability within each level of the other independent variable.

To determine the significance of the effect of the independent variable of interest.

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The y-intercept of the function is -3. The function of f(-x) is not equal to f(x) hence, it is not symmetric.

According to the Intermediate Value Theorem, there must be at least one value c in the interval (a, b) such that f(c) = k if f(x) is a continuous function on the closed interval [a, b] and if k is any number between f(a) and f(b).

The Intermediate Value Theorem in calculus is frequently employed to demonstrate the reality of a function's roots or zeros. The Intermediate Value Theorem may be used to determine that there is at least one value c in the interval (a, b) such that f(c) = 0 if we know that a function is continuous on the interval [a, b] and that f(a) and f(b) have opposite signs.

The given function is: ​f(x) = x³ + 3x²- x - 3.

Using synthetic division we have:

-1 | 1 3 -1 -3

| -1 -2 3

|___________

1 2 -3 0

The polynomial can be factored as:

f(x) = (x + 1)(x² + 2x - 3)

The quadratic equation can be factored as:

x² + 2x - 3

x² + 3x - x - 3

(x + 3)(x - 1)

Now, set x = 0:

f(0) = 0³ + 3(0)² - 0 - 3 = -3

Therefore, the y-intercept is (0, -3).

For symmetry we have:

Substituting -x for x:

f(-x) = (-x)³ + 3(-x)² - (-x) - 3x = -x³ + 3x² + x - 3x

This is not equal to f(x), so the function is not symmetric about the y-axis.

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The height divides the shape into 2 parts : a trapezoid and a right-angled triangle.

A(trapezoid) = [(a+b)h]/2, where a;b;h are the length, width and height respectively.

-> A(trapezoid) = [(15.3+17.4) x 6]/2 = 98.1 (yd2)

A(triangle) = lh/2, where l is the base.

-> A(triangle) = 2.1 x 6 : 2 = 6.3 (yd2)

So, the area of the figure is 98.1 + 6.3 = 104.4 (yd2)

Answer:

18.84 miles

Step-by-step explanation:

Circumference = 2πr

= 2 × 3.14 × 3

= 18.84 miles

The exact circumference of the circle with radius 3 miles is 6π or 18.84 miles (approx).

The radius of a circle is 3 miles. What is the circumference?The formula to calculate the circumference of a circle is given as:

Circumference = 2πr, where r is the radius of the circle and π is a constant value, approximately equal to 3.14. Substituting the given value of r in the formula, we have:

Circumference = 2π(3)

Circumference = 6π

Therefore, the exact circumference of the circle is 6π miles. To simplify this answer in its simplest form, we can use the value of π as 3.14 (approximately).Circumference = 6π = 6(3.14) = 18.84Therefore, the exact circumference of the circle is 18.84 miles (approx).

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Answer:

False

Step-by-step explanation:

15.026

1 = tens

5 = ones

0 = tenths

2 = hundredths

6 = thousandth

This number is read as fifteen and twenty-six thousandths.

So, the answer is False.

(i) The graph is attached below. (ii) the minimum value of the function is h = 30, when x = 5. (iii) the horizontal distance traveled by the rider between the two splashes is 2 × 9 m = 18 m.

Acceleration is a physical quantity that measures the rate at which the velocity of an object changes over time. It is a vector quantity, meaning that it has both magnitude and direction.

Acceleration is defined as the change in velocity of an object divided by the time interval over which the change occurred. It is typically measured in meters per second squared (m/s²) in the metric system, or in feet per second squared (ft/s²) in the imperial system.

If the velocity of an object is increasing, its acceleration is said to be positive. If the velocity is decreasing, the acceleration is said to be negative or deceleration. If the velocity is constant, there is no acceleration.

(i) Using the given scale, we can plot the graph of h = 1.2x² - 12x + 30 for 0 ≤ x ≤ 10 as follows:

Parabolic graph is attached below.

(ii) The constant term 30 in the equation represents the minimum height of the pirate ship ride above ground level. This is because the equation is in the form of a quadratic function in standard form, y = ax² + bx + c, where the vertex of the parabola is at the point (-b/2a, c - b²/4a). In this case, the vertex is at (5, 0), which means that the minimum value of the function is h = 30, when x = 5.

(iii) We are given that a rider is 19.2 m above ground level, which means that h = 19.2. We can substitute this value into the equation and solve for x:

19.2 = 1.2x² - 12x + 30

1.2x² - 12x + 10.8 = 0

x² - 10x + 9 = 0

(x - 1)(x - 9) = 0

Therefore, the rider is at a height of 19.2 m when he is either 1 m or 9 m away from the starting point. Since the rider swings back and forth, the distance between the two splashes is twice the distance from the starting point to either splash. Therefore, the horizontal distance traveled by the rider between the two splashes is:

2 × 9 m = 18 m

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The equation of the given circle is expressed as: x² + y² - 24x + 14y + 112 = 0

The standard form of expression for the equation of a circle is expressed as:

(x - h)² + (y - k)² = r²

where:

(h, k) are coordinates of the center of the circle

r is radius

We are given the parameters:

Coordinates of center = (12, -7)

Diameter = 18

Radius = 18/2 = 9

Thus, equation of circle is:

(x - 12)² + (y - (-7))² = 9²

x² - 24x + 144 + y² + 14y + 49 = 81

x² + y² - 24x + 14y + 112 = 0

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The required measures of the acute angles in the given right triangle with side lengths 9, 12, and 15 is equal to 36.87 degrees and 53.13 degrees approximately.

Right triangle with side lengths of 9, 12 and 15.

Check whether Pythagorean theorem holds true,

which states that the sum of the squares of the two shorter sides equals the square of the hypotenuse.

9^2 + 12^2 = 15^2

Simplifying this equation, we get,

⇒81 + 144 = 225

⇒225 = 225

This implies,

Pythagorean theorem holds for this triangle,

And by the converse of the Pythagorean theorem,

Triangle is a right triangle with the right angle opposite the side with length 15.

The acute angles of a right triangle are complementary.

Which means that their sum is 90 degrees.

Calculate the measures of the acute angles in this triangle,

Use trigonometric functions.

Use the sine function to find one of the acute angles,

sin(θ) = opposite/hypotenuse

         = 9/15

         = 0.6

Taking the inverse sine function of both sides, we get,

θ = sin^(-1)(0.6)

⇒ θ ≈ 36.87 degrees.

Sum of the acute angles is 90 degrees, the other acute angle is equals to,

90 - 36.87 = 53.13 degrees.

Therefore, the measures of the acute angles of the triangle are approximately 36.87 degrees and 53.13 degrees.

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Step-by-step explanation:

3a² -6a -4a +8

By Grouping method,

= 3a(a-2)-4(a-2)

= (3a-4)(a-2)

Answer:

(3a − 4)(a − 2)

Step-by-step explanation:

i got it on my quiz

Group and factor out the greatest common factor (GCF), then combine.(3a−4)(a−2)

The measure of angle a for the given rectangular piece of paper folded along ∠a ≅ ∠z is 46.5°.

A straight angle has an measure of 180° and looks like a straight line, so it is a mathematical way of representing a straight line. The angle at which the arms extend in opposite directions from the vertex and join to form 180°. Whenever two rays come together they form an angle, and the angle made by two rays in opposite directions is called a right angle.

Given,

∠a ≅ ∠z

∠HMG = 87°

Since all three angles form a straight angle:

∠a + ∠z + ∠HMG = 180°

As, ∠a ≅ ∠z

∠a + ∠z = 2∠a

Now,

2∠a + ∠HMG = 180°

2∠a + 87° = 180°

2∠a = 180° - 87°

2∠a = 93°

m∠a = 46.5°

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The rectangular sheet of paper with the given folds along a and z has an angle of 46.5°.

A straight angle is a mathematical representation of a straight line since it has a measure of 180° and seems to be a straight line. the arc formed by the arms joining to form a 180° angle as they extend in opposing directions from the vertex. A right angle is the designation for the angle created by two rays travelling in the opposing directions. Angles are formed whenever two rays come together.

In geometry, there are primarily six different types of angles. The names of all angles together with their characteristics are:

• Acute Angle: It ranges from 0° to 90°.

• Obtuse Angle: It ranges from 90° to 180°

• Right Angle: The angle that is exactly 90 degrees.

• Straight Angle: The angle that is exactly 180 degrees

• Reflex Angle: The angle that is larger than 180 degrees and less than 360 degrees

• Complete Rotation: the full 360-degree rotation

Given,

∠a ≅ ∠z

∠HMG = 87°

Since all three angles form a straight angle:

∠a + ∠z + ∠HMG = 180°

As, ∠a ≅ ∠z

∠a + ∠z = 2∠a

Now,

2∠a + ∠HMG = 180°

2∠a + 87° = 180°

2∠a = 180° - 87°

2∠a = 93°

m∠a = 46.5°

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the ratio of the number of minutes he lifted weights to the total number of minutes of his entire workout is 1:2 option B is correct.

Ratios can be written in three forms:

A to B

A:B

A/B

Ratios are also simplified by reducing to lowest terms like fractions are.

This problem's ratio is:

minutes lifted weights to total minutes workout

The number of minutes lifting weights is in the question: 25.

To find the total minutes of his workout, add the number of minutes he spent for all of the activities:

Total minutes = warm-up + lifting weights + treadmill

Total minutes = 10 + 25 + 15

Total minutes = 50

The ratio before simplifying is 25÷50.

This ratio can be reduced to lowest terms. Both sides are divisible by 25.

[tex]\frac{25}{25}=1[/tex]

[tex]\frac{50}{25} = 2[/tex]

The ratio in the lowest terms is 1/2.

It can also be written as 1 to 2 or 1:2.

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To compare the ratios 17:23 and 5:9, you can do it by either finding their decimal equivalents or by cross-multiplying.

Finding decimal equivalents:

Divide each part of the ratio by the sum of its parts.

For 17:23:

17 ÷ (17+23) = 17 ÷ 40 = 0.425

For 5:9:

5 ÷ (5+9) = 5 ÷ 14 = 0.3571

Now compare the decimals:

0.425 > 0.3571

So, the ratio 17:23 is greater than 5:9.

Cross-multiplying:

Cross-multiply the two ratios and compare the products.

For 17:23 and 5:9, multiply 17 by 9 and 23 by 5:

17 * 9 = 153

23 * 5 = 115

Now compare the products:

153 > 115

So, the ratio 17:23 is greater than 5:9.

In both methods, the result is the same: 17:23 is greater than 5:9.

The scale is 1 : 30. A rectangle is shown. The length of the rectangle is labeled 3 inches. The width of the rectangle is labeled 4 inches. Show your work to determine the area of the room in square feet.First, we need to convert the measurements of the rectangle from inches to feet, since the question asks for the area in square feet. 3 inches = 3/12 feet = 0.25 feet 4 inches = 4/12 feet = 0.33 feet Now, we can use the scale to determine the actual dimensions of the room. If 1 inch on the drawing represents 30 inches in real life, then: 1 foot on the drawing represents 30 feet in real life So, the length of the room is: 1 foot on the drawing = 30 feet in real life 3 inches on the drawing = 3/12 feet = 0.25 feet in real life 0.25 feet x 30 = 7.5 feet And the width of the room is: 1 foot on the drawing =

The p-value for testing if the proportion who are unemployed differs between the two groups is 0.

To calculate the p-value for testing if the proportion who are unemployed differs between the two groups, we can use a two-sample test of proportions

Let's define

Group 1: High School or Some College

Group 2: Bachelor's or Higher

We want to test if the proportion of unemployed individuals differs between Group 1 and Group 2.

First, we need to calculate the proportion of unemployed individuals in each group

Group 1: 41/601 = 0.06822

Group 2: 14/875 = 0.016

Next, we need to calculate the pooled proportion

Pooled proportion = (41 + 14) / (601 + 875) = 0.043

Now we can calculate the test statistic

Test statistic = (0.06822 - 0.016) / sqrt(0.043 × (1 - 0.043) × (1/601 + 1/875)) = 7.57

Using a two-tailed test with a significance level of 0.05, we can find the critical value from a normal distribution table or calculator. For a two-tailed test with a significance level of 0.05, the critical value is approximately 1.96.

Since the test statistic (7.57) is greater than the critical value (1.96), we reject the null hypothesis that the proportion of unemployed individuals is the same in both groups.

Finally, we can calculate the p-value as the probability of getting a test statistic as extreme or more extreme than the one we observed, assuming the null hypothesis is true. Since this is a two-tailed test, we need to double the area to the right of the test statistic (7.57) under the standard normal curve

p-value = 2 × P(Z > 7.57) = 0

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The given question is incomplete, the complete question is:

The following table is based off a survey of employment from 2018. What is the p- value for testing if the proportion who are unemployed differs between the two groups? Give your answer to three decimal places. Total Unemployed 41 Employed 834 875 High School or Some College Bachelor's or Higher Total 14 587 601 55 1421 1476

Answer:

y>6

Step-by-step explanation:

5y>30

y>6

therefore, y can be any number that is larger than 6 (e.g. 7, 10 etc)

9t + 4t can be simplified by combining the like terms (terms with the same variable and exponent). The coefficients of the two terms (9 and 4) are added to get the coefficient of the simplified term:

9t + 4t = (9 + 4)t = 13t

Therefore, the expression that is equivalent to 9t + 4t is 13t.

The extent to which an instrument is consistent within itself is called reliability.

Reliability refers to the consistency or stability of the measurements obtained using a specific instrument or test. In other words, it is the degree to which a tool or instrument measures the same way each time it is used under the same conditions. If an instrument is reliable, it produces consistent results across repeated trials or measurements.

Reliability is an important aspect of measurement because it determines the accuracy of the results obtained using the instrument. An unreliable instrument can produce inconsistent or inaccurate results, which can lead to erroneous conclusions or decisions.

Therefore, it is crucial to establish the reliability of an instrument before using it to make important decisions or draw conclusions based on the measurements obtained. There are several methods for measuring reliability, including test-retest reliability, inter-rater reliability, and internal consistency reliability.

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The exact area of the shaded region is = 42.25π - 84.50

What do you mean by Radius?

A radius is a line segment that has one endpoint in the circle's centre and the other terminus on the circumference of the circle. Radius = Diameter of circle.

Given that radius OA and OB = 13 cm

Area of sector AOB is 42.25π cm²

now,area of Δ AOB =1/2 ×base×height

= 1/2 × 13 ×13

=1/2 × 169

= 84.5cm²

now area of shaded region

= Area of sector AOB - area of Δ AOB

= 42.25 л cm²- 84.50 cm²

Therefore the area of shaded region is 42.25 л cm²- 84.50 cm²

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Answer:

34.1%.

Step-by-step explanation:

Given: Weekly wages at a certain factory are normally distributed with a mean of $400 and a standard deviation of $50.

This problem tells us that the distribution of weekly wages at the factory is normally shaped.

68.2% of wages fall within the standard deviation of the mean.

Since the mean is $400, this means that 68.2% of wages fall between $400 +/- $50, or between $350-$450.

Instead, we are interested in the probability of a wage being between $350-$400.

Said differently, this is between the mean ($400) and one standard deviation below the mean ($350).
The fact that the wages are normally distributed also means that the shape of the distribution is symmetrical above and below the mean.

So the probability of wages being between $350-$400 is half of 68.2%, or 34.1%.

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Blake's recorded result of a probability model in a 12 times die roll:

Part A: 0 %

Part B: 25 %

Part C:  41.67 %

To complete the probability model

Part A:

To find the experimental probability of rolling a 3, count the number of times 3 appears in the table and divide by the total number of rolls:

Number of 3's rolled: 0

Total number of rolls: 12

Experimental probability of rolling a 3: 0/12 = 0/1 = 0

So the probability of rolling a 3 is 0, or 0/1 as a fraction, 0 as a decimal number, and 0% as a percentage.

Part B:

To find the experimental probability of rolling a 6, we need to count the number of times 6 appears in the table and divide by the total number of rolls:

Number of 6's rolled: 3

Total number of rolls: 12

Experimental probability of rolling a 6: 3/12 = 1/4 = 0.25

So the probability of rolling a 6 is 1/4 as a fraction, 0.25 as a decimal number, and 25% as a percentage.

Part C:

To find the experimental probability of rolling a number less than 4, add the experimental probabilities of rolling a 1, 2, or 3:

Experimental probability of rolling a 1: 2/12 = 1/6 = 16.67%

Experimental probability of rolling a 2: 3/12 = 1/4 = 25%

Experimental probability of rolling a 3: 0/12 = 0/1 = 0

Experimental probability of rolling a number less than 4: 16.67% + 25% + 0 = 41.67%

So the probability of rolling a number less than 4 is 5/12 as a fraction, 0.4167 as a decimal number, and 41.67% as a percentage.

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Image transcribed:

16. For the three-part question that follows, provide your answer to each question in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as Part A, Part B, and Part C.

Blake rolled a die 12 times. He recorded the results in the table below.

Results

6 | 2 | 1 | 4

4 | 3 | 1 | 5

2 | 2 | 6 | 6

Then, Blake created the probability model shown below from the data in the chart. Blake was only able to complete part of the model.

Probability Model

1 | 2 | 3 | 4 | 5 | 6

P(1) 17% | P(2)-25% | ? | P(4) 17% | P(5) 8% | ?

Part A: Help Blake complete the probability model by finding the experimental probability of rolling a 3. Provide your answer as a fraction, a decimal number, and a percent.

Part B: Help Blake complete the probability model by finding the experimental probability of rolling a 6. Provide your answer as a fraction, a decimal number, and a percent.

Part C: Use your probability model to find the experimental probability of rolling a number less than 4. Provide your answer as a fraction, a decimal number, and a percent.

the time required for an investment of $5000 to grow to $[tex]7200[/tex] at an interest rate of  [tex]7.5[/tex] percent per year, compounded quarterly, is approximately  [tex]3.79[/tex] years.

o find the time required for an investment of $5000 to grow to $7200 at an interest rate of 7.5 percent per year, compounded quarterly, we can use the formula for compound interest:

[tex]A = P(1 + r/n)^(nt)[/tex]

where:

A = the final amount (in this case, $7200)

P = the principal amount (in this case, $5000)

r = the annual interest rate (in decimal form, so 7.5% = 0.075)

n = the number of times the interest is compounded per year (in this case, quarterly, so n = 4)

t = the number of years (which we need to find)

Plugging in the values, we get:

[tex]7200 = 5000(1 + 0.075/4)^(4t)[/tex]

Now we can solve for t by isolating it on one side of the equation.

Dividing both sides by 5000:

Taking the natural logarithm of both sides:

[tex]ln(7200/5000) = ln((1 + 0.075/4)^(4t))[/tex]

Using the property of logarithms that ln(a^b) = b * ln(a):

[tex]ln(7200/5000) = 4t\times ln(1 + 0.075/4)[/tex]

Dividing both sides by [tex]4 \times ln(1 + 0.075/4):[/tex]

[tex]t = ln(7200/5000) / (4 * ln(1 + 0.075/4))[/tex]

Using a calculator, we can find the value of t to be approximately 3.79 years (rounded to two decimal places).

Therefore, the time required for an investment of $5000 to grow to $7200 at an interest rate of 7.5 percent per year, compounded quarterly, is approximately 3.79 years.

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