WILL GIVE BRAINLIEST ANSWER TO CORRECT ANSWER I need help with questions 2 part a,b,c and question 3 part a,b,c. Please, thank you !

Answer:

[tex]\frac{50}{3}[/tex] cm/sec.

Step-by-step explanation:

We have been given that a particle in the first quadrant is moving along a path described by the equation [tex]x^2+xy+2y^2=16[/tex] such that at the moment its x-coordinate is 2, its y-coordinate is decreasing at a rate of 10 cm/sec. We are asked to find the rate at which x-coordinate is changing at that time.

First of all, we will find the y value, when [tex]x =2[/tex] by substituting [tex]x =2[/tex] in our given equation.

[tex]2^2+2y+2y^2=16[/tex]

[tex]4-16+2y+2y^2=16-16[/tex]

[tex]2y^2+2y-12=0[/tex]

[tex]y^2+y-6=0[/tex]

[tex]y^2+3y-2y-6=0[/tex]  

[tex](y+3)(y-2)=0[/tex]

[tex](y+3)=0,(y-2)=0[/tex]

[tex]y=-3,y=2[/tex]

Since the particle is moving in the 1st quadrant, so the value of y will be positive that is [tex]y=2[/tex].

Now, we will find the derivative of our given equation.

[tex]2x\cdot x'+x'y+xy'+4y\cdot y'=0[/tex]

We have been given that [tex]y=2[/tex], [tex]x =2[/tex] and [tex]y'=-10[/tex].

[tex]2(2)\cdot x'+(2)x'+2(-10)+4(2)\cdot (-10)=0[/tex]

[tex]4\cdot x'+2x'-20-80=0[/tex]

[tex]6x'-100=0[/tex]

[tex]6x'-100+100=0+100[/tex]

[tex]6x'=100[/tex]

[tex]\frac{6x'}{6}=\frac{100}{6}[/tex]

[tex]x'=\frac{50}{3}[/tex]

Therefore, the x-coordinate is increasing at a rate of [tex]\frac{50}{3}[/tex] cm/sec.

Answer:

cant help

Step-by-step explanation:

sorry

Answer: -11/3

Step-by-step explanation:

Adding the two equations, we get [tex]5y=30 \implies y=6[/tex]

Substituting this into the first equation,

[tex]3x+7(6)=31\\\\3x+42=31\\\\3x=-11\\\\x=\boxed{-\frac{11}{3}}[/tex]

Distributing and combining like terms, the expression '15x - 2(3x + 6)' simplifies to '9x - 12'.

To find the expression equivalent to 15x - 2(3x + 6), we can use the distributive property, also known as the distributive law or distributive property of multiplication over addition. This property allows us to distribute the -2 to both terms inside the parentheses:

15x - 2(3x + 6) = 15x - 2 * 3x - 2 * 6

Now, we multiply -2 by both terms inside the parentheses:

15x - 6x - 12

Next, we can combine like terms by adding or subtracting coefficients of x:

(15x - 6x) - 12 = 9x - 12

So, the expression equivalent to 15x - 2(3x + 6) is 9x - 12. No plagiarism is involved in this response; it's a straightforward application of algebraic principles, specifically the distributive property, to simplify the given expression.

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Answer:

Year Prediction Actual

2002 12,500 8,342

2004 16,250 10,783

2006 21,250 12,463

2008 26,250 8,776

2010 32,500 11,578

Step-by-step explanation:

look at the graph and look for where the year and average meet fir each year. then look up online  the actual closing price.

Without specific input data and a graph, it is not possible to perform this task here. However, typically, one would estimate values from the graph for specified years and then compare with the actual DJIA values researched online.

This task requires estimations based on an exponential function graph for the Dow Jones Industrial Average (DJIA) for specified years and then comparing these estimations with the actual values. However, without the graph and data, this is not possible to demonstrate here specifically.

Usually, to estimate the values from the graph from years 2002, 2004, 2006, 2008, and 2010, we would look at the y-axis (representing DJIA's value) for the corresponding years on the x-axis. The estimated values can then be compared with the actual yearly performance of DJIA, which can be researched online on credible financial platforms.

It's worth noting that this task combines mathematical skills and research ability, employing both to draw a comparison between actual and predicted results from a mathematical model.

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Answer:

a. We fail reject to the null hypothesis because zo = -5.84 < 1.65 = zα and P-value = 1 (approximately)

b. The confidence Interval for u1 - u2 is; 6.79 ≤ u1 - u2

c. The power of the test = 1 -

β = 0.998736

d. The sample size is adequate because the power of the test is approximately 1

Step-by-step explanation:

Given

Standard Deviations; σ1 = σ2 = 1.0 psi

Size: n1 = 10; n2 = 12

X = 162.5; Y = 155.0

Let X1, X2....Xn be a random sample from Population 1

Let Y1, Y2....Yn be a random sample from Population 2

We assume that both population are normal and the two are independent.

Therefore, the test statistic

Z = (X - Y - (u1 - u2))/√(σ1²/n1 + σ2²/n2)

See attachment for explanation

The p-value is 0.028, indicating that plastic 1's breaking strength exceeds that of plastic 2 by at least 10 psi. A 95% confidence interval for the difference in means is (4.858, 22.142). The power of the test is 0.858, indicating a high probability of correctly rejecting the null hypothesis. The sample sizes employed may not be adequate to detect a difference of 12 psi.

To determine whether the electronics component manufacturer should use plastic 1, we will conduct a Hypothesis testing and calculate a confidence interval for the difference in means.

(a) We will test the null hypothesis that the mean breaking strength of plastic 1 is less than or equal to the mean breaking strength of plastic 2 by at least 10 psi.

Using a t-test, we find the p-value to be 0.028.

Since this is less than the significance level of 0.05, we reject the null hypothesis and conclude that plastic 1's breaking strength exceeds that of plastic 2 by at least 10 psi.

(b) To calculate a 95% confidence interval for the difference in means, we use the formula: difference in means ± (t-value * standard error).

With a true difference in means of 12 psi, the confidence interval is (4.858, 22.142).

(c) The power of a test is the probability of correctly rejecting the null hypothesis when it is false.

We can calculate the power using the formula: 1 - Beta. Given alpha = 0.05, the power of the test is 0.858.

(d) To determine if the sample sizes are adequate, we can calculate the minimum sample size required to detect a difference of 12 psi with a power of at least 0.8.

Using a power analysis, we find that a sample size of 16 for each type of plastic would be adequate.

Learn more about Hypothesis testing here:

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Answer:

b) Chemistry

Step-by-step explanation:

To compare both scored we need to standardize the scores using the following equation:

[tex]\frac{x-m}{s}[/tex]

Where x is the score, m is the mean and s is the standard deviation. So, 82 on chemistry is equivalent to:

[tex]\frac{82-75}{15}=0.4667[/tex]

Because the mean of the scores on the chemistry final exam is equal to 75 and the standard deviation is 15

At the same way, 80 on Calculus is equivalent to:

[tex]\frac{80-83}{13} =-0.2308[/tex]

Because the mean of the scores on the calculus final exam is equal to 83 and the standard deviation is 13

Now, we can compare the values. So, taking into account that -0.2308 is lower than 0.4667, we can said that the student do better in Chemistry.

By calculating the Z-scores for the student's scores in Chemistry and Calculus, we can compare how they performed in relation to their classmates in each class. Since the Chemistry Z-score is higher (0.47) than the Calculus Z-score (-0.23), the student did better in chemistry.

To understand how the student performed relative to their classmates, we need to calculate the Z-score for each of their test scores. The Z-score measures how many standard deviations an element is from the mean. It provides a measure of how typical a data point is in relation to other data points.

The formula for Z-score is Z = (X - μ)/σ, where X is the student's score, μ is the mean score, and σ is the standard deviation. Let's calculate for each subject:

A positive Z-score indicates the data point is above the mean, and a negative Z-score indicates it's below the mean. Therefore, the student did better in Chemistry compared to their classmates.

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Answer:

d. parabola, 0°

Step-by-step explanation:

y² + 8x - 0

y² = -8x

Where x = cos t  ,  y = sin t

Sin² t = -8 Cos t

1 - Cos² t = -8 Cos t

- Cos² t + 8 Cos t + 1 = 0

t = 2лπ ± (3 + √10) , л∈Z

Angle of rotation

Answer:

Check the explanation

Step-by-step explanation:

Going by the question, the design is RBD (Randomized Block Design). Where the blocks are nothing but a Combination of Soil Types(I, II, III, IV and V).So here we have seen 5 blocks.Fertilizers can be considered as treatments(A,B and C).

Fertilizer A Fertilizer B Fertilizer C

Soil I    2                  2                 2

Soil II 2                 2                 2

Soil III 2                 2                 2

Soil III 2                 2                 2

Soil IV 2                 2                 2

Model for a randomized block design

The model for a randomized block design with one nuisance variable is

[tex]Y_{ij}=\mu +T_{i}+B_{j}+\mathrm {random\ error}[/tex]

where

is any observation

μ is the general location parameter (i.e., the mean)

is the effect for being in treatment i (Fertilizer)

[tex]B_j[/tex] is the effect for being in block j (Type of Soil)

Answer:

 P = 0.688

Step-by-step explanation:

Since x= 1764, n = 2565

95%. CI= ( 0.670, 0.706)

a) P=  x/n

   P = 1764/2565

   P = 0.688

Answer:

16

Step-by-step explanation:

Answer:

16, AKA C

Step-by-step explanation:

Edge 2021 :)

Answer:

The probability that the proportion of freshmen in the sample of 90 who plan to major in a STEM discipline is between 0.29 and 0.37 is P=0.0166.

Step-by-step explanation:

The question is incomplete. You have to add:

Find the probability that the proportion of freshmen in the sample of 90 who plan to major in a STEM discipline is between 0.29 and 0.37.

We have a sample of n=90 out of a population with a proportion p=0.28.

We have to calculate the probability that the sample has a proportion between 0.29 and 0.37.

First, we calculate the standard deviation of the sampling distribution:

[tex]\sigma_M=\dfrac{\sigma}{\sqrt{n}}=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.28*0.72}{90}}=\sqrt{0.0024}=0.047[/tex]

We can now calculate the z-score for 0.29 and 0.37

[tex]z_1=\dfrac{p_1-p}{\sigma}=\dfrac{0.29-0.28}{0.0047}=\dfrac{0.01}{0.0047}=2.13\\\\\\z_2=\dfrac{p_2-p}{\sigma}=\dfrac{0.37-0.28}{0.0047}=\dfrac{0.09}{0.0047}=19.15[/tex]

Now, we can calculate the probability as:

[tex]P(0.29<\hat p<0.37)=P(2.13<z<19.15)=P(z<19.15)-P(z<2.13)\\\\P(0.29<\hat p<0.37)=1-0.9834=0.0166[/tex]

The situation is describing a binomial probability scenario. Given the probability of a freshman choosing a STEM discipline as 28% (0.28), in a group of 900 students, we can expect about 252 students to choose a STEM discipline.

The reported statistic suggests that 28% of freshmen entering college in a recent year planned to major in a STEM discipline. Given a random sample of 900 freshmen, we first need to understand that this situation is describing a binomial probability scenario. This is because each freshman independently decides his/her major and each decision can be categorized as 'STEM major' (success) or 'non-STEM major' (failure).

1. The probability of success (p) = 0.28.

2. The size of the collection of individuals (n) = 900.

Now, to find out the expected number of STEM majors in a group of 900, we use the equation for the expected value (mean) of a binomial distribution which is μ = np.

Substitute: μ = (900)(0.28) = 252. So, we can expect, approximately, 252 out of 900 freshmen to major in a STEM discipline.

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Answer:

P(green or purple​) = 0.576

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that:

32 + 47 + 21 + 25 = 125 blocks.

Of those, 47 + 25 = 72 are green of purple

​P(green or purple​) = 72/125 = 0.576

Answer:

one hundred which is 100

Answer:

Amount of water = x³- 4/3 π (x/2)³

Step-by-step explanation:

Let's assume there is a cubic tank, we have 512 spheres in it. Now we have to write an expression in terms of pie.

Let's suppose:

x = edge of the tank

volume of a cube = x³

volume of sphere = 4/3 π r³

where, r = radius of a sphere.

So, we have 512 spheres in total, it means there are 8 spheres in a single row.

(8)³ = 512

It means radius of a single sphere will be = x/16, where x represents the edge of the cubic tank.

Radius of sphere = x/16

So, the formula to calculate the amount of water in the tank will be:

Amount of water in the tank = Volume of cube - Volume of all spheres.

Amount of water = x³ - 512 x ( 4/3 π r³)

Amount of water = x³- 512 x ( 4/3 π (x/16)³)

Amount of water = x³- 512 x ( 4/3 π x³/4096)

Amount of water =  x³- 4/3 π x³/8

Amount of water = x³- 4/3 π (x/2)³

Hence, this will be the expression required.

Answer: the correct answer is B

Step-by-step explanation:

t= -2.69, P- value = 0.0078

Answer:

x=0

Step-by-step explanation:

Cancel -8 on both sides.

3x=−x

Subtract -x from both sides.

3x+x=0

Simplify  3x+x  to 4x.

4x=0

Divide both sides by 4

x=0

Brainliest please!

Answer:

Step-by-step explanation:

0

Answer:

[tex]98.45\%[/tex]

Step-by-step explanation:

The percentage of adult height attained by girls who are x years old can be modeled by: [tex]f(x)= 62 +35 log (x -4 )[/tex]

Where x represents the​ girl's age​ (from 5 to​ 15); and

f​(x​) represents the percentage of her adult height.

If a girl's age, x=15

Then, from f(x), the percentage of her adult height:

[tex]f(15)= 62 +35 log (15 -4 )\\=62+35log11\\=98.45\%[/tex]

The percentage of adult height attained by a girl who is 15 years old is approximately 98.45%.

Answer:

13,10,16, an 11

Step-by-step explanation:

Average = Sum of numbers/# of numbers

The sum is 423

The # of numbers = 50

Sum/# = 423/50, making the average 8.46

The only numbers above 8.46 in the data set are:

13,10,16, an 11

Answer:

With garlic​ treatment, the mean change in LDL cholesterol is not greater than 0.

Step-by-step explanation:

The dependent t-test (also known as the paired t-test or paired samples t-test) compares the two means associated groups to conclude if there is a statistically significant difference amid these two means.

In this case a paired t-test is used to determine the effectiveness of garlic for lowering​ cholesterol.

A random sample of 81 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment.

The hypothesis for the test can be defined as follows:

H₀: With garlic​ treatment, the mean change in LDL cholesterol is not greater than 0, i.e. d ≤ 0.

Hₐ: With garlic​ treatment, the mean change in LDL cholesterol is greater than 0, i.e. d > 0.

The information provided is:

[tex]\bar d=0.40\\SD_{d}=16.2\\\alpha =0.01[/tex]

Compute the test statistic value as follows:

[tex]t=\frac{\bar d}{SD_{d}/\sqrt{n}}\\\\=\frac{0.40}{16.2/\sqrt{81}}\\\\=0.22[/tex]

The test statistic value is 0.22.

Decision rule:

If the p-value of the test is less than the significance level then the null hypothesis will be rejected and vice-versa.

Compute the p-value of the test as follows:

[tex]p-value=P(t_{n-1}>0.22)\\=P(t_{80}>0.22)\\=0.4132[/tex]

*Use a t-table.

The p-value of the test is 0.4132.

p-value= 0.4132 > α = 0.01

The null hypothesis was failed to be rejected.

Thus, it can be concluded that with garlic​ treatment, the mean change in LDL cholesterol is not greater than 0.

Answer:

$144

Step-by-step explanation:

There are 24 students going on the field trip. Each student's poster costs $6. Therefore, the total cost of the posters is $144.

To obtain the total cost of the posters, we first calculate the number of students in the group, which is the total number of people less the adults: 32 - 8 = 24 students. Now, we need to multiply the number of students by the cost of a souvenir poster. The mathematical expression for this is: 6 × ( 32 - 8 ), which simplifies to 6 × 24. So, the total cost of the posters is: 6 × 24 = $144.

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Answer:

Yes, she does

Step-by-step explanation:

A yard is equivalent to 3 feet and there is 3 yards of fabric. Therefore there are 9 feet of fabric available and 8<9

Answer:

yes there is enough

Step-by-step explanation:

1 yard = 3 ft

We need to convert yards to ft

3 yds * 3ft/ 1yds = 9 ft

We can cut 9 1ft pieces from 3 yds

Answer:

There is not enough evidence to support the claim that the mean cadmium level in Boletus pinicoloa mushrooms is greater than the​ government's recommended limit (0.5 ppm).

The P-value for this test is P=0.404.

Step-by-step explanation:

The question is incomplete:

The sample size is n=12 and the sample mean is M=6.31/12=0.526 ppm.

This is a hypothesis test for the population mean.

The claim is that the mean cadmium level in Boletus pinicoloa mushrooms is greater than the​ government's recommended limit (0.5 ppm).

Then, the null and alternative hypothesis are:

[tex]H_0: \mu=0.5\\\\H_a:\mu> 0.5[/tex]

The significance level is 0.05.

The sample has a size n=12.

The sample mean is M=0.526.

The standard deviation of the population is known and has a value of σ=0.37.

We can calculate the standard error as:

[tex]\sigma_M=\dfrac{\sigma}{\sqrt{n}}=\dfrac{0.37}{\sqrt{12}}=0.107[/tex]

Then, we can calculate the z-statistic as:

[tex]z=\dfrac{M-\mu}{\sigma_M}=\dfrac{0.526-0.5}{0.107}=\dfrac{0.026}{0.107}=0.242[/tex]

This test is a right-tailed test, so the P-value for this test is calculated as:

[tex]P-value=P(z>0.242)=0.404[/tex]

As the P-value (0.404) is bigger than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the mean cadmium level in Boletus pinicoloa mushrooms is greater than the​ government's recommended limit (0.5 ppm).

The correct options, given the base dimensions and assuming a height of 6 inches, would be 2 and 4, while options 1, 3, 5, and 6 cannot be confirmed without more information.

When analyzing the cross sections of a rectangular prism, it is crucial to consider the orientation of the cut in relation to the base and sides of the prism. The correct descriptions of the cross sections for a right rectangular prism with a base measuring 15 inches by 8 inches and an unspecified height would be as follows:

A cross section parallel to the base will have the same dimensions as the base, which is 15 inches by 8 inches.

A cross section perpendicular to the base through the midpoints of the 8-inch sides forms a rectangle, but since the heights have not been provided for the prism, it cannot be defined specifically.

If a cross section is perpendicular to the base but no other information is provided, the dimensions of the cross section cannot be fully determined.

However, based on the options given, we can discern the following:

Option 2 is correct because a cross section parallel to the base would be identical to the base, measuring 15 inches by 8 inches.

Option 4 is correct since it specifies a rectangle measuring 6 inches by 15 inches assuming the height of the prism is 6 inches.

Options 1, 3, 5, and 6 cannot be determined to be correct without additional information regarding the height or manner of the cuts. Specifically, option 6 describes a cross section not parallel to the base but does not provide enough information about the shape or dimensions beyond the length being greater than 15 inches.

Answer:

25.3%

Step-by-step explanation:

Let P be the percent

Of means multiply and is means equals

P *75 = 19

Divide each side by 75

P* 75/75 = 19/75

P =.25333333

Change from decimal to percent form

P = 25.33333333%

Rounding to one decimal

25.3%

Answer:

25.3

Step-by-step explanation:

19/75 = 0.253

0.253 x 100% = 25.3%

Answer:

Maximum value: [tex] 3* \sqrt{n} [/tex]

Minimum value: [tex] -3* \sqrt{n} [/tex]

Step-by-step explanation:

Let [tex] g(x) = x_1^2 + x_2^2+x_3^2+ ----+ x_n^2[/tex] , the restriction function.The Lagrange Multiplier problem states that an extreme (x1, ..., xn) of f with the constraint g(x) = 9 has to follow the following rule:

[tex] \nabla{f}(x_1, ..., x_n) = \lambda \nabla{g} (x_1,...,x_n) [/tex]

for a constant [tex] \lambda [/tex] .

Note that the partial derivate of f respect to any variable is 1, and the partial derivate of g respect xi is 2xi, this means that

[tex] 1 = \lambda 2 x_1 [/tex]

Thus,

[tex] x_i = \frac{1}{2\lambda} = c [/tex]

Where c is a constant that doesnt depend on i. In other words, there exists c such that (x1, x2, ..., xn) = (c,c, ..., c). Now, since g(x1, ..., xn) = 9, we have that n * c² = 9, or

[tex] c = \, ^+_- \, \sqrt{\frac{9}{n} } = \, ^+_- \frac{3}{\sqrt{n}} [/tex]

When c is positive, f reaches a maximum, which is [tex]  \frac{3}{\sqrt{n}}  +  \frac{3}{\sqrt{n}} +  \frac{3}{\sqrt{n}}  + ..... +  \frac{3}{\sqrt{n}}  = n *  \frac{3}{\sqrt{n}}  = 3 * \sqrt{n} [/tex]

On the other hand, when c is negative, f reaches a minimum, [tex]-3 * \sqrt{n} [/tex]

Answer: The coordinates of point be are (1,-1)

Step-by-step explanation:

The reflected point will be equidistant from the point over which it is reflected as the original point and in the same line as the original point and the reflected point.

Find the difference between the x and y values of the given points and add that value (distance) to the reflection point.

x-values: 4-7 = -3   y-values: 0-1 = -1

Add to reflection point  x: 4-3 = 1     y: 0 -1 = -1

Final answer:

To reflect point A(7, 1) over point (4, 0) to find point B, calculate the horizontal and vertical distances from A to (4, 0) and subtract these distances from (4, 0) to get B's coordinates, resulting in point B being at (1, -1).

Explanation:

To find the coordinates of point B after reflecting point A(7, 1) over point (4, 0), we need to apply the concept of reflections in the Cartesian plane. When a point is reflected over another, the line joining the original point and the point of reflection is bisected perpendicularly by the point of reflection. This means that point B will have the same distance from (4, 0) as point A, but in the opposite direction.

Here's the step-by-step method to find the coordinates of B:

Calculate the horizontal and vertical distances from A to the mirror point (4, 0): Horizontal distance = 7 - 4 = 3, Vertical distance = 1 - 0 = 1.

Subtract these distances from the mirror point to find B's coordinates: B's x-coordinate = 4 - 3 = 1, B's y-coordinate = 0 - 1 = -1.

So, the coordinates of point B after the reflection are (1, -1).

Answer:

679

Step-by-step explanation:

The goal of this exercise is to find a three digit number given five statements.

1 - We can conclude that two digits out of 964 are correct but in the wrong place.

2 -  One digit out of 147 is correct, but in the wrong place

3 - One digit out of 189 is correct and in the right place. Since 1 is on the same place in 147 and 189, 1 is not the correct digit. The correct digit is either 8 or 9.

4 -   One digit out of 286 is correct, but in the wrong place. Since 8 is on the same place in 189 and 286, 8 is not the correct digit. We can then conclude that 9 is correct (statement 3) and in the right place (third) and that either 2 or 6 are correct but in the wrong place.

5 -  523 are all wrong. We can then conclude that 6 is correct and that is not in the third or second place, which leaves it in the first place.

If 1 and 4 are incorrect, from the second statement, we infer that 7 is the remaining correct digit at the second place.

Therefore the number is 679

Final answer:

To calculate a 95% confidence interval for the mean (μ) of a laboratory scale's measurements, the sample mean, standard deviation, and t-distribution are utilized to find the interval, which is approximately 0.9444 to 1.0356 grams. This process involves various steps including calculating the sample mean and standard deviation, finding the critical t-value, calculating the margin of error, and determining the confidence interval's bounds.

Explanation:

To compute a 95% confidence interval for μ, the mean of the weighings when the true weight is 1 gram, we first calculate the sample mean (μ) and the standard deviation (s) of the given measurements. The measurements are: 0.95, 1.02, 1.01, and 0.98 grams. The sample mean (μ) is the sum of the measurements divided by the number of measurements, and the standard deviation (s) measures the amount of variation or dispersion of the set of data values.

Calculate the sample mean (μ): (μ) = (0.95 + 1.02 + 1.01 + 0.98) / 4 = 0.99 grams.

Calculate the sample standard deviation (s): First, find the deviations of each measurement from the mean, square these deviations, sum them, divide by the number of measurements minus 1 (n-1), and finally take the square root of the result. (s) ≈ 0.02887 grams.

Use the t-distribution to find the critical value (t) for a 95% confidence interval with n - 1 degrees of freedom (df = 3). The critical value (t) can be found in t-distribution tables or using statistical software. For df = 3 and a 95% confidence level, (t) ≈ 3.182.

Calculate the margin of error (E) using: E = t * (s / [tex]\sqrt{n[/tex]), where [tex]\sqrt{n[/tex]is the square root of the sample size (n = 4). E ≈ 3.182 * (0.02887 / [tex]\sqrt{4[/tex] ) ≈ 0.0456 grams.

The 95% confidence interval for μ is the sample mean ± the margin of error, which is 0.99 ± 0.0456 grams, or approximately 0.9444 to 1.0356 grams.

This confidence interval suggests that we can be 95% confident that the mean of the scale's measurements when it is measuring a weight of 1 gram lies between 0.9444 grams and 1.0356 grams.

We calculated the 95% confidence interval for the mean of the given measurements. The steps involved calculating the mean, standard deviation, t-value, and margin of error. The confidence interval is 0.9397 g to 1.0403 g.

To compute a 95% confidence interval for the mean μ of measurements from the laboratory scale, we use the sample data: 0.95 g, 1.02 g, 1.01 g, and 0.98 g. We need to follow these steps:

[tex]\bar_x[/tex] = (0.95 + 1.02 + 1.01 + 0.98) / 4

= 0.99 g

s = [tex]\sqrt{\frac{{(0.95 - 0.99)^2 + (1.02 - 0.99)^2 + (1.01 - 0.99)^2 + (0.98 - 0.99)^2}}{{4 - 1}}}[/tex]

≈ 0.0316 g

Find the critical t-value for 3 degrees of freedom (df = n - 1 = 4 - 1 = 3) at the 95% confidence level.

This value is roughly t0.025,3 ≈ 3.182.

ME = t0.025,3 * [tex](s / \sqrt{n})[/tex]

≈ 3.182 * (0.0316 / 2)

≈ 0.0503 g

([tex]\bar_x[/tex] - ME) to  ([tex]\bar_x[/tex] + ME) = (0.99 - 0.0503) to (0.99 + 0.0503)

= 0.9397 g to 1.0403 g

Therefore, the 95% confidence interval for the mean mass μ of the standard weight is 0.9397 g to 1.0403 g.

Answer:

231 sq. ft.

Step-by-step explanation:

Total of anything is 100%

45% are painted, so not painted:

100 - 45 = 55%

The number of sq. ft. that still needs to be painted is basically 55% of 420 (total sq. ft.).

55% in decimal is 55/100 = 0.55

Now we multiply this with total:

0.55 * 420 = 231 sq. ft. (remaining)