8) Answer below and show all work. (3 points) 4.5x a.) What does x equal? b.) What is the measure of ABC? c.) What is the measure of 2DBC?, D W​

The probability that a randomly selected observation is between 23 and 85 is 31/35 or approximately 0.886.

To find the probability that a randomly selected observation is between 23 and 85, we need to calculate the area under the probability density function (PDF) between 23 and 85.

Since X is a uniform random variable, the PDF is a horizontal line with height 1/(90-20) = 1/70 over the interval [20, 90].

The area under the PDF between 23 and 85 is the area of a rectangle with width 85-23 = 62 and height 1/70, which is (62)(1/70) = 31/35.

Therefore, the probability that a randomly selected observation is between 23 and 85 is 31/35 or approximately 0.886.

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Mr. Robbin's commission rate is 15%.

A commission is a service charge assessed by a broker or investment advisor for providing investment advice or handling purchases and sales of securities for a client.

We know that:

commission = rate * sales

where "commission" is the amount earned in commission, "rate" is the commission rate, and "sales" is the total amount of sales.

In this case, we have:

commission = $31.29

sales = $208.60

Substituting these values into the formula, we get:

$31.29 = rate * $208.60

Solving for the rate, we get:

rate = $31.29 / $208.60

= 0.15 or 15%

Therefore, Mr. Robbin's commission rate is 15%.

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The width of the triangle is 12 inches, and the height of the triangle is 44 inches.

Let w = the width of the triangle and h = the height.

The area of the triangle = 264 in²

Area of a triangle is given as A = (1/2) * base * height

Since the base of the triangle is the width "w," we can write the equation as:

264 = (1/2) * w * h

Therefore:

h = 4w - 4

Substitute the value of "h" in terms of "w" into the area equation:

264 = (1/2) * w * (4w - 4)

Now, we can solve for "w":

264 = 2w² - 2w

2w² - 2w - 264 = 0

Divide the entire equation by 2 to simplify:

w² - w - 132 = 0

factorize:

(w - 12)(w + 11) = 0

w = 12 or w = - 11

The width of the triangle cannot be negative, therefore:

Width "w" of the triangle = 12 inches.

Now, we can find the height "h" using the equation we derived earlier:

h = 4w - 4

h = 4(12) - 4

h = 44 in.

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There is not sufficient evidence to support the claim that the mean score of students at XYZ is significantly different from 100.

The effect size is small, which means that the difference between the sample mean, and the population mean is not very large relative to the variability within the sample.

Test may not have sufficient statistical power to detect a meaningful difference in mean scores between the population and the sample.

We want to test the hypothesis that the mean score of students at XYZ is not significantly different from 100, at a significance level of 0.01.

Let's start by calculating the sample mean and sample standard deviation:

[tex]Sample mean (\bar x) = (80+60+120+140+200+70+30+180+70+150+20+50+170+90+130)/15 = 105.33[/tex]

Sample standard deviation (s) = 52.411

The test statistic for a two-tailed t-test with n-1 degrees of freedom is:

[tex]t = (\bar x - \mu ) / (s / \sqrt n)[/tex]

where μ is the hypothesized population mean.

Plugging in the values, we get:

[tex]t = (105.33 - 100) / (52.411 / \sqrt 15) = 1.208[/tex]

The critical t-value for a two-tailed test with 14 degrees of freedom and α =[tex]0.01 is \±2.977.[/tex]

Since our calculated t-value (1.208) falls within the acceptance region (between -2.977 and 2.977), we fail to reject the null hypothesis.

To compute the effect size, we can use Cohen's d:

[tex]d = (\bar x - \mu) / s[/tex]

where μ is the population mean and s is the sample standard deviation.

Plugging in the values, we get:

[tex]d = (105.33 - 100) / 52.411 = 0.104[/tex]

To estimate the power of the hypothesis test, we need to know the effect size, sample size, and significance level.

We have already calculated the effect size (d = 0.104), and the significance level is [tex]\alpha = 0.01.[/tex]

Assuming a sample size of n = 15, we can use the power curve chart to find the power of the test.

The chart shows that for a two-tailed t-test with n = 15 and α = 0.01, the power is approximately 0.26 when the effect size is 0.104.

Based on the power curve chart, the power of the test is relatively low (0.26), which means that there is a high probability of a Type II error (failing to reject the null hypothesis when it is actually false).

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It will take approximately 8 months to pay off the loan (rounded up to the nearest month).

a. After 6 months, the student will owe $5,383.38 (to the nearest cent).

To calculate this, we can use the formula:

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

where:
A = the remaining balance after 6 months
P = the initial loan amount ($8,000)
r = annual percentage rate (3% or 0.03)
n = number of times compounded in a year (12 since it is compounded monthly)
t = time in years (6 months is 0.5 years)
PMT = the monthly payment ($1,000)

Plugging in these values, we get:

A = 8,000(1 + 0.03/12)^(12*0.5) - 1,000[((1 + 0.03/12)^(12*0.5) - 1) / (0.03/12)]

A = $5,383.38 (rounded to the nearest cent)

b. To find out how many months it will take to pay off the loan, we need to keep making the monthly payments until the remaining balance is $0.

Using the same formula as above, we can solve for t:

8,000(1 + 0.03/12)^(12t) - 1,000[((1 + 0.03/12)^(12t) - 1) / (0.03/12)] = 0

Simplifying this equation, we get:

t = log(1 + (1,000/8,000)(0.03/12)) / (12 log(1 + 0.03/12))

t = 7.46 months

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The critical value of t is approximately 1.895.

The critical value of t is approximately 2.626.

What is Confidence Interval?

In statistics, a confidence interval is a range of values calculated from a sample of data that is likely to contain the true value of an unknown population parameter with a certain level of confidence, usually expressed as a percentage. It is a measure of the precision and reliability of an estimate.

a) Using a t-distribution calculator or a t-table with 7 degrees of freedom and a 90% confidence level, the critical value of t is approximately 1.895.

b) Using a t-distribution calculator or a t-table with 103 degrees of freedom and a 99% confidence level, the critical value of t is approximately 2.626.

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The confidence interval (0.51, 0.68), determine, the value of E is 0.085, which is option (b).

The value of E, also known as the margin of error, can be determined using the formula:

E = (upper bound of CI - lower bound of CI) / 2

In this case, the upper bound of the confidence interval is 0.68 and the lower bound is 0.51, so:

E = (0.68 - 0.51) / 2 = 0.085

The margin of error is a measure of the precision of an estimate, and it indicates the amount by which the estimate may differ from the true population value. The smaller the margin of error, the more precise the estimate. The margin of error is affected by the sample size and the level of confidence. A larger sample size and a higher level of confidence will result in a smaller margin of error.

It's important to note that the margin of error only provides a range of plausible values for the true population parameter, and it doesn't guarantee that the true value falls within that range. Therefore, it's always important to interpret the results of a study or survey with caution and consider other factors that may impact the accuracy of the estimate.

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Answer:

A, up I think

Step-by-step explanation:

The explorer is currently on a white circle with an X inside, and they must go through all the white circles exactly once. This means that their next move must be to either the white circle to the left or to the right of the X. However, the white circle to the right of the X is already connected to the black circle, which means that the explorer cannot use that path. Therefore, their next move must be to the white circle to the left of the X.Looking at the maze, the only way to get to the white circle to the left of the X is by going up. Therefore, the correct answer is (A) arrow pointing up.

Answer: 30 black empty squares

Step-by-step explanation: Let’s solve this problem step by step:

There are 64 squares on a chessboard and 16 of them are covered by chess pieces, so there are 64 - 16 = 48 empty squares.

The ratio of white empty squares to black empty squares is 3:5. This means that for every 3 white empty squares, there are 5 black empty squares.

The total ratio of white to black empty squares is 3 + 5 = 8.

To find out how many of the 48 empty squares are black, we can divide the total number of empty squares by the total ratio and then multiply by the ratio for black empty squares: (48 / 8) * 5 = 30.

So, there are 30 black empty squares on the chessboard.

The current would be 1.5 feet per second fast at a depth of two feet

Drawing the line of best fit, we have the following points

(0, 1.8) and (10, 0.5)

The equation is represented as

y = mx + c

Where

c = y when x = 0

So, we have

y = mx + 1.8

Using the points, we have

10m + 1.8 = 0.5

So, we have

m = -0.13

So, the equation is

y = -0.13x + 1.8

At two feet deep, we have

y = -0.13 * 2 + 1.8

Evaluate

y = 1.54

Approximate

y = 1.5

Hence, the velocity is 1.5 feet per second

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(a) By giving each registered voter in the city a special number, and then choosing 200 numbers at random from the list, a straightforward random sample of registered voters in the City of Madison might have been obtained.

(b) Based on the available data, the true percentage of registered voters in the City of Madison who cast ballots in non-presidential elections has an 80% confidence interval of (0.533, 0.687).

(c) The interval means that approximately 80% of the calculated 80% confidence intervals for the proportion of voters who participate in non-presidential elections would contain the actual proportion of voters who participate in non-presidential elections if we were to take numerous random samples of 200 registered voters from the City of Madison.

(d) We cannot conclude that fewer than half of registered voters in the City of Madison vote in non-presidential elections, as the lower bound of the confidence interval is 0.533 which is greater than 0.5.

(a) A simple random sample of registered voters in the City of Madison could have been gathered by assigning each registered voter in the city a unique number, and then using a random number generator to select 200 numbers from the list.

(b) To construct an 80% confidence interval for the true proportion of registered voters in the City of Madison who vote in non-presidential elections, we can use the following formula:

[tex]CI = \hat{p} +/- z*(\sqrt{(\hat{p}(1-\hat{p})/n)} )[/tex]

where [tex]\hat{p}[/tex]  is the sample proportion (122/200 = 0.61),

z is the critical value from the standard normal distribution corresponding to an 80% confidence level (z = 1.28), and n is the sample size (200).

Plugging in the values, we get:

[tex]CI = 0.61 +/- 1.28* \sqrt{((0.61*(1-0.61)/200))}[/tex]

CI = 0.61 ± 0.077

Therefore, the 80% confidence interval for the true proportion of registered voters in the City of Madison who vote in non-presidential elections is (0.533, 0.687).

(c) We can interpret this interval as follows: If we were to take many random samples of 200 registered voters from the City of Madison and calculate a 80% confidence interval for the proportion of voters who vote in non-presidential elections, about 80% of these intervals would contain the true proportion of voters who vote in non-presidential elections.

(d) Since the lower bound of the confidence interval is 0.533, which is greater than 0.5, we cannot conclude that fewer than half of registered voters in the City of Madison vote in non-presidential elections.

However, we can say with 80% confidence that the true proportion of voters who vote in non-presidential elections is somewhere between 0.533 and 0.687.

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Integral ≈ 904

Let's evaluate the integral of the given function:

∫[-5(-x³ - 10x² - 32x - 36)]dx from -2 to 2 First, we can distribute the -5 to each term inside the parentheses: ∫(5x³ + 50x² + 160x + 180)dx

Now, let's find the antiderivative of the function: Antiderivative: (5/4)x^4 + (50/3)x³ + 80x² + 180x + C

Now, we'll evaluate the antiderivative at the limits of integration, 2 and -2:

F(2) = (5/4)(2^4) + (50/3)(2^3) + 80(2^2) + 180(2) F(-2) = (5/4)(-2^4) + (50/3)(-2^3) + 80(-2^2) + 180(-2)

Now, subtract F(-2) from F(2) to get the integral value: Integral = F(2) - F(-2) Perform the arithmetic operations to get the final answer: Integral ≈ 904

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Answer:

(x,y) become (-x,-y), so all you have to do is take those coordinates and make them negative unless they already are negative. For example: E (-1,3) would become E (1,-3) since two negatives cancel each other out. For the rest do this: F (1,4) becomes F (-1,-4). Do the same strategy for the rest of the points, then graph your answer.

Step-by-step explanation:

Answer:

the equation of the line that passes through (3, 14) and is parallel to the line that passes through (10, 2) and (25, 15) is y = (13/15)x + 10.

Step-by-step explanation:

Parallel line equation.

Piyush Soni

Write an equation for the line that passes through (3,14) and is parallel to the line that passes through (10,2) and (25,15)

To find the equation of the line that passes through (3, 14) and is parallel to the line that passes through (10, 2) and (25, 15), we first need to find the slope of the line passing through (10, 2) and (25, 15), which we will call m1.

The slope of the line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

So, for the line passing through (10, 2) and (25, 15), we have:

m1 = (15 - 2) / (25 - 10) = 13/15

Since we want a line parallel to this one, the slope of our new line will be the same. Let's call this slope m2.

m2 = 13/15

Now, we can use the point-slope form of the equation of a line to find the equation of the line passing through (3, 14) with slope m2:

y - y1 = m2(x - x1)

where x1 = 3 and y1 = 14

Plugging in the values, we get:

y - 14 = (13/15)(x - 3)

Simplifying, we get:

y = (13/15)x + 50/5

or

y = (13/15)x + 10

Therefore, the equation of the line that passes through (3, 14) and is parallel to the line that passes through (10, 2) and (25, 15) is y = (13/15)x + 10.

The equation of the line that passes through (3,14) and is parallel to the line that passes through (10,2) and (25,15) is y = (13/15)x  171/15.

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

Using the points (10,2) and (25,15), we have:

slope = (15 - 2) / (25 - 10) = 13 / 15

Since the line we want is parallel to this line, it will have the same slope. So, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where m is the slope, and (x1, y1) is the given point (3,14).

Substituting m = 13/15 and (x1,y1) = (3,14).

y - 14 = (13/15)(x - 3)

Expanding and rearranging.

15y - 210 = 13x - 39

15y = 13x - 39 + 210

15y = 13x + 171

y = (13/15)x  171/15

Thus,

The equation of the line that passes through (3,14) and is parallel to the line that passes through (10,2) and (25,15) is y = (13/15)x  171/15.

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The vertex of the parabola y = -x² + 6x + 1 is (3, 10).

To find the vertex of the parabola y = -x² + 6x + 1, we first need to find the derivative of the function. The derivative of a function is another function that describes the rate of change of the original function at each point. In this case, we will find the derivative of y with respect to x, denoted by f'(x).

To find the derivative of y, we can use the power rule of differentiation, which states that the derivative of xⁿ with respect to x is nxⁿ⁻¹. Using this rule, we can find the derivative of y = -x² + 6x + 1 as follows:

f'(x) = -2x + 6

Now that we have the derivative, we can find the x-coordinate of the vertex by setting f'(x) = 0 and solving for x:

-2x + 6 = 0

2x = 6

x = 3

Therefore, the x-coordinate of the vertex is x = 3. To find the y-coordinate, we can substitute x = 3 into the original equation of the parabola:

y = -x² + 6x + 1

y = -(3)² + 6(3) + 1

y = -9 + 18 + 1

y = 10

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The value of function are,

⇒ f' (x) = ( sin x - x cos x) - 5 sec² x

And, ⇒ y' = 45x² - 20x + 3

Given that;

Function is,

⇒ f(x) = x sin x - 5 tan x

And, y = (3x - 2) (5x² + 1)

Now, We can simplify as;

⇒ f(x) = x sin x - 5 tan x

Differentiate as;

⇒ f' (x) = ( sin x - x cos x) - 5 sec² x

And, For y = (3x - 2) (5x² + 1)

Differentiate as;

⇒ y' = (3x - 2) (10x) + (5x² + 1) (3)

⇒ y' = 30x² - 20x + 15x² + 3

⇒ y' = 45x² - 20x + 3

Thus, The value of function are,

⇒ f' (x) = ( sin x - x cos x) - 5 sec² x

And, ⇒ y' = 45x² - 20x + 3

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The American founding father and litterateur Thomas Paine referred to as King George III" the Royal Brute of high-quality Britain."

Paine was a main figure within the American Revolution and was a fierce advocate of american independence from British rule. He wrote a series of influential pamphlets, which includes" Common feel" and" The Rights of man," which helped to excite assist for the progressive reason.

In his thoughts, Paine often blamed the British monarchy and its leaders, including King George III, whom he saw as a dictator and an oppressor. His slicing phrases helped to rally reinforcement for the purpose of America self-reliance and performed a significant element in shaping the path of history.

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In light of the geometry puzzle, the tiles that correspond to the right boxes are mentioned below:

m∠COB = 114°

The measure of arc CE = 124°

m∠EOB = 122°

m∠DFA = 58°

A branch of mathematics known as geometry is concerned with the study of objects' forms, angles, dimensions, and sizes. A very significant area of mathematics that is covered in many levels of instruction is geometry. It enables us to relate to things and forms more efficiently, as well as their angles and measurements.

The calculation is attached in the file.

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Answer:

-x³ + 2x + 7

2x^5+x^4-x³+6x²+3x-3

Step-by-step explanation:

For #10:

(f-g)(x) = f(x) - g(x)

x³-2x+3 - (2x³ + 4x - 4)

-x³ + 2x + 7

For #11:

(f·g)(x) = f(x) * g(x)

(x³+3)(2x²+x-1) = 2x^5+x^4-x³+6x²+3x-3

The problem involves classifying 414 elephants into three groups: alpha, beta, and gamma. The equations and solutions for the three unknowns are x + y + 2 = 414 and x = 1/5(y + 2). The system is solved using matrices. It is explained why the group of 414 elephants must have 63 alpha, regardless of the distribution of the other two types.  the required number of the other two types is 250 gamma and 101 beta.

From the given information, we can set up two equations as follows

x + y + z = 414 (where x, y, and z represent the number of elephants in the alpha, gamma, and beta groups, respectively)

And, x = 1/5(y + 2)

To solve the system in terms of a parameter t, we can represent it in matrix form as follows

[tex]\left[\begin{array}{ccc}1&1&1\\1/5&-1&0\\\end{array}\right][/tex] [tex]\left[\begin{array}{cc}x\\y\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}414\\0\\\end{array}\right][/tex]

Using matrix operations, we can solve for x and y in terms of z, which gives us

x = 83 - z/5

y = 331/5 + z/5

The total number of elephants in the alpha group is 69, regardless of the distribution of the other two types because we know that x + y + z = 414 and x = 1/5(y + 2). Substituting the second equation into the first, we get

(1/5)y + (1/5)2 + y + z = 414

Simplifying the equation, we get

6y + 5z = 2060

Since the total number of elephants is 414, we know that z = 414 - x - y. Substituting this into the equation above, we get

6y + 5(414 - x - y) = 2060

Simplifying this equation, we get

x + y = 69

Therefore, the number of alpha elephants is always 69, regardless of the distribution of the other two types.

If there are only 101 beta elephants available, we can set up a new equation as follows

x + y + z = 414

z = 101

Substituting z = 101 into the first equation, we get

x + y = 313

Using the equation x = 1/5(y + 2) from part a, we can solve for x and y, which gives us

x = 63

y = 250

Therefore, there are 63 alpha, 250 gamma, and 101 beta elephants.

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Graduate student conducts survey about study habits, and scatter plot represents data point of one student and orange line represents linear relationship between 2 variables.

In this scenario, the graduate student is conducting a survey on undergraduate study habits by collecting data on students from different years in college. The data collected is a sample of four students, which may not represent the entire population of undergraduate students on campus.

The graduate student collects data by asking the students how many hours of course work they do for each class in a typical week. This data is then used to create a scatter plot, which shows the relationship between the year in college and hours of course work per class.

In the scatter plot, each blue circle represents one student's data point, and the orange line represents the linear relationship between the two variables. The equation for the orange line is Y = –2X + 9, where Y represents the hours of course work per class and X represents the year in college.

It is important to note that the accuracy of the survey results depends on the representativeness of the sample collected. A larger sample size and a more diverse sample may provide more accurate results in survey.

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The probability of X being less than 70 is approximately 0.0006.

The standard normal distribution.

Transform X into a standard normal variable Z:

[tex]Z = (X - \mu) / \sigma[/tex]

Substituting the given values, we have:

[tex]Z = (70 - 83) / 4 = -3.25[/tex]

Using a standard normal table or calculator, we can find

The probability:

[tex]P(X < 70) = P(Z < -3.25) = 0.0006[/tex]

The probability of X being less than 70 is approximately 0.0006.

the usual distribution of normals.

Change X into the typical normal variable Z:

Z = (X - \mu) / \sigma

If we substitute the values provided, we get:

Z = (70 - 83) / 4 = -3.25

We may determine the probability using a calculator or a normal table to find:

Z = (70 - 83) / 4 = -3.25

X has a about 0.0006 likelihood of being less than 70.

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Answer:

a. 61%

b. $1315.77

Step-by-step explanation:

a. find percent = part/whole = 732/1200 = 0.61 = 61%

b. 61% of $2157 = 2157 x 0.61 = $1315.77

The number of oats that can be used to make 150 cookies is 800 g of oats.

A proportion in mathematics is a claim that two ratios are equal. The relationship between two quantities is called a ratio, and it is typically stated as a fraction.

Geometry, physics, and finance are just a few of the mathematical and science fields where proportions are applied. They are particularly helpful for forming predictions based on patterns or relationships seen, and for comparing two quantities that are in different units or scales.

We can see that;

80 g of oats can be used to produce 15 cookies

x g of oats can produce 150 cookies

x = 80 * 150/15

= 800g

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The minimum value of f(x, y) = 43x² + 11y² subject to the constraint x² + y² = 324 is 3564.

To find the minimum value, we use the method of Lagrange multipliers. Define a function L(x, y, λ) = 43x² + 11y² - λ(x² + y² - 324). Take partial derivatives with respect to x, y, and λ and set them to zero:

1. ∂L/∂x = 86x - 2λx = 0
2. ∂L/∂y = 22y - 2λy = 0
3. ∂L/∂λ = x² + y² - 324 = 0

From equations (1) and (2), we get x = y = 0 or λ = 43 for x and λ = 11 for y. Substituting λ = 43 into equation (3) gives x² + y² = 324. Solving for x and y, we get x = 18 and y = 6. Substituting these values into f(x, y), we obtain f(18, 6) = 3564, which is the minimum value.

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The probability that Brandon ordered a pepperoni pizza is 1/6.

To find the probability that Brandon ordered a pepperoni pizza, we need to first determine the fraction of pizzas sold that were pepperoni.

We know that 1/6 of the pizzas sold were cheese, which means that 5/6 of the pizzas sold were not cheese. So, if we let x be the total number of pizzas sold in the week, then (5/6)x is the number of pizzas sold that were not cheese.

Of those non-cheese pizzas, 1/5 were pepperoni. So the total number of pepperoni pizzas sold would be (1/5)(5/6)x = (1/6)x.

Therefore, the probability that Brandon ordered a pepperoni pizza is (1/6)x / x = 1/6.

So the answer is: The probability that Brandon ordered a pepperoni pizza is 1/6.

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The result is sometimes statistically significant. The correct option is b. sometimes.

Statistical significance is determined by comparing the observed value (in this case, FDATA) to a predetermined threshold, typically referred to as the alpha level or significance level. If the observed value exceeds the alpha level, then the result is considered statistically significant, meaning that the observed value is unlikely to have occurred by chance alone.

In this case, the given value of FDATA is 0.9. However, without knowing the context of the statistical analysis being conducted, it is not possible to determine whether this value is statistically significant or not. The determination of statistical significance depends on various factors, such as the sample size, the research question, the type of statistical test being used, and the desired level of confidence.

Therefore, without additional information about the specific context and analysis being performed, it is not possible to definitively state whether a value of FDATA = 0.9 is statistically significant or not. The result could be statistically significant in some situations (when compared to an appropriate alpha level), and not statistically significant in other situations.

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(a) A normal quantile plot of the data is illustrated below.

(b) There are some slight deviations from the line, particularly for the data points at the extremes of the distribution (68 and 82 inches).

In this case, we have the heights of 7 NBA players: 68, 69, 78, 82, 75, 73, and 80 inches. To construct the normal quantile plot, we first calculate the mean and standard deviation of the data:

Mean = (68 + 69 + 78 + 82 + 75 + 73 + 80) / 7 = 75.43

Standard deviation = 5.98

Next, we calculate the z-scores for each data point:

z1 = (68 - 75.43) / 5.98 = -1.24

z2 = (69 - 75.43) / 5.98 = -1.07

z3 = (78 - 75.43) / 5.98 = 0.43

z4 = (82 - 75.43) / 5.98 = 1.09

z5 = (75 - 75.43) / 5.98 = -0.07

z6 = (73 - 75.43) / 5.98 = -0.41

z7 = (80 - 75.43) / 5.98 = 0.75

We can then plot the z-scores against the corresponding quantiles of a standard normal distribution. The z-score of -1.24 corresponds to the 10th percentile of the standard normal distribution, while the z-score of 1.09 corresponds to the 86th percentile.

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Check the picture below.

The chance that the average of 49 boxes weighs less than 27.5 ounces is smaller.

We can solve this problem using the central limit theorem. Since the weights of individual boxes have a normal distribution with mean 28 ounces and standard deviation 2 ounces, the distribution of sample means of 49 boxes will also be normal with mean 28 ounces and standard deviation (2/√49) ounces, which simplifies to 0.2857 ounces.

(a) To find the chance that one box has a weight less than 27.5 ounces, we can standardize the weight using the formula z = (x - μ) / σ, where x is the weight we are interested in, μ is the mean weight (28 ounces), and σ is the standard deviation (2 ounces).

So,

z = (27.5 - 28) / 2 = -0.25

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -0.25 is approximately 0.4013.

(b) To find the chance that 49 boxes have an average weight less than 27.5 ounces, we can standardize the sample mean using the formula z = (x - μ) / (σ / √n), where x is the sample mean weight (27.5 ounces), μ is the mean weight (28 ounces), σ is the standard deviation (2 ounces), and n is the sample size (49 boxes).

So,

z = (27.5 - 28) / (2 / √49) = -1.75

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -1.75 is approximately 0.0401.

Comparing the two probabilities, we can see that the chance that 49 boxes have an average weight less than 27.5 ounces (0.0401) is smaller than the chance that one box has a weight less than 27.5 ounces (0.4013).

Therefore, the answer is (a) The chance that the average of 49 boxes weighs less than 27.5 ounces is smaller.

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