Suppose the selling price of homes is skewed right with a mean of 350,000 and a standard deviation of 160000 If we record the selling price of 40 randomly selected US homes what will be the shape of the distribution of sample means what will be the mean of this distribution what will be the standard deviation of this distribution

I don't know how to do that. Only one angle is given and no relationships between the triangles or any lines are given.

Answer:

The answer is 50.

Step-by-step explanation:

Answer:

-5,-4,-3,-2,-1,0,1,2,3,4,5

Step-by-step explanation:

The sample space for the two balanced dice, a red die, and a green die is given below in the pair (G,R)

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

For each pair, Y=G-R is presented below:

0 -1 -2 -3 -4 -5

1 (2,2) (2,3) (2,4) (2,5) (2,6)

2 (3,2) (3,3) (3,4) (3,5) (3,6)

3 (4,2) (4,3) (4,4) (4,5) (4,6)

4 (5,2) (5,3) (5,4) (5,5) (5,6)

5 (6,2) (6,3) (6,4) (6,5) (6,6)

The first column and row is representative of the values which will be obtained throughout the table.

Therefore, the possible values of the random variable Y are:

-5,-4,-3,-2,-1,0,1,2,3,4,5

The possible values of Y are its sample space, and the values are 0, -1, -2, -3, -4, -5, 5, 4, 3, 2 and 1

The sample space of the green die is:

[tex]G= \{1,2,3,4,5,6\}[/tex]

The sample space of the red die is:

[tex]R= \{1,2,3,4,5,6\}[/tex]

When the numbers on both dice are combined, we have the following possible outcomes

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) , (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) , (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) , (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) , (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) , (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Subtract the second outcomes from the first, to get Y

(0) (-1) (-2) (-3) (-4) (-5) , (-1) (0) (-1) (-2) (-3) (-4) , (2) (1) (0) (-1) (-2) (-3) , (3) (2) (1) (0) (-1) (-2) , (4) (3) (2) (1) (0) (-1) , (5) (4) (3) (2) (1) (0)

List out the unique numbers:

(0) (-1) (-2) (-3) (-4) (-5) (5) (4) (3) (2) (1)

Hence, the possible values of Y are 0, -1, -2, -3, -4, -5, 5, 4, 3, 2 and 1

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Answer:

7 inches hope this helps

Step-by-step explanation:

Answer:

7 inches

Step-by-step explanation:

12 inches is a foot

Answer:

The maximal margin of error associated with a 90% confidence interval for the true population mean watermelon weight is of 3.46 ounces.

Step-by-step explanation:

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1-0.9}{2} = 0.05[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].

So it is z with a pvalue of [tex]1-0.05 = 0.95[/tex], so [tex]z = 1.645[/tex]

Now, find the margin of error M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

In this problem:

[tex]\sigma = 13.3, n = 40[/tex]

So

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

[tex]M = 1.645*\frac{13.3}{\sqrt{40}}[/tex]

[tex]M = 3.46[/tex]

The maximal margin of error associated with a 90% confidence interval for the true population mean watermelon weight is of 3.46 ounces.

Final answer:

The maximal margin of error for a 90% confidence interval for the mean watermelon weight is calculated using the z-score for the confidence level, the known population standard deviation, and the sample size, resulting in 3.463 ounces.

Explanation:

To find the maximal margin of error associated with a 90% confidence interval for the true population mean watermelon weight when the mean weight of the watermelons is 66 ounces, the population standard deviation is 13.3 ounces, and the sample size is 40, we use the formula for the margin of error E = z * (σ / sqrt(n)), where z is the z-score corresponding to the confidence level, σ is the population standard deviation, and n is the sample size.

For a 90% confidence interval, the z-score is approximately 1.645. Plugging in the values, we get:

E = 1.645 * (13.3 / sqrt(40))
= 1.645 * (2.105)
= 3.463 ounces.

The maximal margin of error is therefore 3.463 ounces, which means the interval is 66 ± 3.463 ounces for the true population mean weight of watermelons with 90% confidence.

18/10

Step-by-step explanation:

Answer:

18/10

Step-by-step explanation:

Just did the question

Answer:

2 (2x - 3) x (2x + 3)

Answer

Step-by-step explanation:

A=2πrh+2πr2=2·π·3·6+2·π·32≈169.646

Answer:

x = 1 or x = 3/2

Step-by-step explanation:

Solve for x:

(4 x^2 - 10 x + 6)/(4 x + 2) = 0

Multiply both sides by 4 x + 2:

4 x^2 - 10 x + 6 = 0

The left hand side factors into a product with three terms:

2 (x - 1) (2 x - 3) = 0

Divide both sides by 2:

(x - 1) (2 x - 3) = 0

Split into two equations:

x - 1 = 0 or 2 x - 3 = 0

Add 1 to both sides:

x = 1 or 2 x - 3 = 0

Add 3 to both sides:

x = 1 or 2 x = 3

Divide both sides by 2:

Answer: x = 1 or x = 3/2

Answer:

Soda x 2

Apple Juice x 1

Fruit Punch x 1

Lemonade x 1

and

Soda x 1

Apple Juice x 1

Fruit Punch x 1

Lemonade x 2

Step-by-step explanation:

Amounts:

Soda 2L

Apple Juice 2.5L

Fruit Punch 1.5L

Lemonade 2L

S + A + FP + L = 8L

So we need 2 L more. The only way of doing this is by either having 2 lots of soda or 2 lots of lemonade.

The combination of beverages she can purchased;

Soda = 2 container

Apple juice = 1 container

Fruit punch= 1 container

Lemonade = 1 container

The quantity of Beverages present  at the store ;

Soda = 2000ml = 2L

Apple Juice = 2.5L

Fruit Punch = 1.5L

Lemonade =  2L

We have to buy  exactly 10 liters but it should be at least one container of each beverage.

Soda + Apple juice + Fruit Punch + Lemonade = 8L

So we need 2 L more.

The only way of doing this is by either having 2 container of soda or 2 container of lemonade.

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Answer:

y = -3x+8

Step-by-step explanation:

We can use the slope intercept form of an equation

y = mx+b  where m is the slope and b is the y intercept

y = -3x+b

Substitute the point into the equation

-1 = -3(3)+b

-1 = -9+b

Add 9 to each side

-1+9 = -9+9+b

8 =b

y = -3x+8

Answer:

Type II error

Step-by-step explanation:

Type 1 error occurs when:

We reject a True Null Hypothesis

Type 2 error occurs when:

We fail to reject a wrong Null Hypothesis.

The given hypothesis are:

[tex]H_{o}: \mu=20\\\\ H_{a}:\mu>20[/tex]

Level of significance = α = 0.05

The calculated z test statistic = z = 1.47

In order to make a decision we first need to convert z = 1.47 to its equivalent p-value. From the z-table the p value for z score being greater than 1.47 comes out to be:

p-value = 0.0708

Since, p-value is greater than the level of significance, we fail to reject the Null Hypothesis.

It is given that the true value of μ is 25. If the true value of μ is 25, then the Null hypothesis was false. But from the test we performed, we failed to reject the Null Hypothesis.

Since, we failed to reject a False Null Hypothesis, the conclusion resulted in a Type II error.

Answer:

a) H0: Null Hypothesis: P1 = P2

HA: Alternative Hypothesis:P1> P2

The P-value is less than the significance level.

Therefore, Reject the null hypothesis

In conclusion, there is enough evidence to support the claim that vinyl gloves have a greater virus leak rate than latex gloves.

Step-by-step explanation:

(a) From the information given, we can identify the null hypothesis and alternative hypothesis.

H0: Null Hypothesis: P1 = P2

HA: Alternative Hypothesis:P1> P2

(b) n1 which is size of sample 1 = 279

p1 which is proportion of sample 1= 0.60

n2 which is size ofsample 2 = 279

p2 which is proportion of sample 2 = 0.14

P=n1p1+n2p2 / n1+n2

=279 × 0.60+279 × 0.14/279+279

=0.37

Q = 1- P = 0.63

SE= √PQ(1/n1+1/n2)

= √ 0.37 × 0.63(1/279+1/279)=0.0409

So,

Test statistic is:

Z = (p1 - p2) /SE

= (0.60 - 0.14)/0.0409

= 11.3049

(c) The able of Area Under Standard Normal Curve gives the following area =

0.5 approximately.

So,

P-value = 0.5 - 0.5 nearly =0 nearly

(d)From the information derived, the P-value is less than the significance level.

Therefore, Reject the null hypothesis

In conclusion, there is enough evidence to support the claim that vinyl gloves have a greater virus leak rate than latex gloves.

Answer:

0.0445937

Step-by-step explanation:

-Given that the sample statistic has a mean of 406 grams, standard deviation of sq root(225) and the null statistic is 411 grams.

-Assuming normal distribution, the test statistic is calculated as:

[tex]z=\frac{Sample \ statistic-Null \ statistic}{\sigma/\sqrt{n}}\\\\=\frac{406-411}{\sqrt{225/26}}\\\\=-1.6997[/tex]

-we then find the p-value of the test statistic from the z-tables:

P-value=0.0445937

The value of x that maximizes the volume V is x = 2 inches.

This means that by cutting 2-inch squares from each corner of the 24-inch square cardboard, you'll create an open box with the largest possible volume.

We have,

To solve this problem, we need to express the volume of the open box in terms of the length x of the side of the square cut from each corner, and then find the value of x that maximizes this volume.

Let's denote:

Side length of the original square cardboard = 24 inches

Side length of the cut square from each corner = x inches

The dimensions of the resulting box would be:

Length = (24 - 2x) inches (since we're removing x from both sides)

Width = (24 - 2x) inches (same as length)

Height = x inches

The volume V of the box can be calculated by multiplying these dimensions:

V = Length * Width * Height

V = (24 - 2x) * (24 - 2x) * x

Now, we'll simplify this expression for V:

V = x * (24 - 2x)²

To find the value of x that maximizes V, we need to find the critical points of the function and then analyze the behaviour around those points.

Take the derivative of V with respect to x:

dV/dx = 24x - 12x²

Set the derivative equal to zero and solve for x to find the critical points:

24x - 12x² = 0

12x(2 - x) = 0

This gives us two critical points: x = 0 and x = 2.

Now we need to determine which critical point corresponds to a maximum value of V.

To do this, we can analyze the second derivative of V with respect to x:

d²V/dx² = 24 - 24x

Evaluate the second derivative at each critical point:

For x = 0: d²V/dx² = 24 (positive value)

For x = 2: d²V/dx² = 24 - 24(2) = -24 (negative value)

Since the second derivative is negative at x = 2, it indicates that this critical point corresponds to a maximum.

Therefore,

The value of x that maximizes the volume V is x = 2 inches.

This means that by cutting 2-inch squares from each corner of the 24-inch square cardboard, you'll create an open box with the largest possible volume.

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To find the value of x that maximizes the volume of the box, we need to express the volume as a function of x. Then, we can take the derivative of the volume function with respect to x, set it equal to zero, and solve for x. Finally, we substitute the value of x back into the volume function to find the maximum volume.

To find the value of x that maximizes the volume of the box, we need to express the volume as a function of x. Let's start by finding the dimensions of the box after cutting out the squares from each corner. Since the original square has sides of 24 inches, each side of the base of the box will be 24 - 2x inches. The height of the box will be x inches.

The volume of a box is given by the formula V = length x width x height. In this case, the length and width of the base of the box are the same, so we can simplify the formula to V = (24 - 2x)^2 * x.

To find the value of x that maximizes the volume, we can take the derivative of the volume function with respect to x, set it equal to zero, and solve for x. Once we find the value of x, we can substitute it back into the volume function to find the maximum volume.

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Answer:

1. B

2. D

3. E

4. A

5. C

6. F  

Step-by-step explanation:  

1. The projection onto the x-axis is given by T(x, y) = (x, o)  =(1 0 0 0) B  

2. Counter-clockwise rotation by π/2 radians C

= (0 - 1 1 0) D  

3. Clockwise rotation by π/2 radians

= (0 1 - 1 0) E  

4. Reflection about the y-axis

= (-1 0 0 1) A  

5. Reflection about the x-axis  = (1 0 0 - 1) C  

6. Reflection about the line y=x

(0 1 1 0) F  

For every line in a plane, there is a linear transformation that reflects the vector about that line. The easiest way to answer a question like this is to figure out where the standard basic vector is, e1 and e2. Write the answers at the column of the matrix. Letting As be the matrix corresponding to the linear transformation s. It is easier to see that e1 gets carried to e2 and e2 gets carried to - e1

As= (0 - 1 1 0)

The answer identifies and correlates different types of linear transformations in ℝ2 to ℝ2 space with their corresponding 2×2 matrices, considering operations such as projection onto axis, clockwise and counter-clockwise rotations, and reflections about axes or a line.

The question is about matching linear transformations with their associated 2×2 matrices.

The projection onto the x-axis given by T(x,y)=(x,0) would be represented by a matrix that eliminates the y-component, so its matrix is (1000). Counter-clockwise rotation by π/2 radians corresponds to the matrix (01−10), as it reverses the entries and changes the sign of the y-component. Clockwise rotation by π/2 radians corresponds to the matrix (0−110), as it also switches the entries, but with a positive sign for the y-component. Reflection about the y-axis inverts the sign of the x-component, corresponding to the matrix (−1001). Reflection about the x-axis influences the sign of the y-component, thus corresponding to the matrix (010-1). Lastly, reflection about the line y=x equates to exchange the roles of x and y and hence represented by the matrix (0110).

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Answer:

Step-by-step explanation:

Just for some extra closure lol the answer is indeed 23. :)

Answer:

33.12 rad/s

Step-by-step explanation:

We are given that

Diameter=d=23.3 in

Radius,=[tex]r=\frac{d}{2}=\frac{23.3}{2}=11.65 in=11.65\times 0.0254= 0.29591 m[/tex]

1 in=0.0254 m

We have to find the angular speed of drum would have to spin.

Force=[tex]mg[/tex]

Centripetal force=[tex]m\omega^2 r[/tex]

[tex]mg=m\omega^2 r[/tex]

[tex]\omega^2=\frac{g}{r}[/tex]

[tex]\omega=\sqrt{\frac{g}{r}}[/tex]

Where [tex]g=9.8m/s^2[/tex]

[tex]\omega=\sqrt{\frac{9.8}{0.29591}[/tex]

[tex]\omega=33.12 rad/s[/tex]

Answer:

A. 24024

B. 364

Step-by-step explanation:

(a) There are 14*13*12*11 = 24024 ways to select the officers...

(b) Since the officers can also serve on the committee, the sampling

    is with replacement, so then there are (14 choose 3) ways to

    select the committee members... 364 ways to be exact

Answer:

(Y-X), X, Y, (X+Y), (X+Y)+Y, ...

Step-by-step explanation:

FIBONACHI SEQUENCE IS A SPECIAL MATHEMATICAL SEQUENCE IN W/C YOU HAVE TO ADD THE LAST AND THE NEXT TERM TO GET THE FOLLOWING TERM, IF SO.. TO GET THE LAST TERM, JUST REDUCE THE 3RD TERM TO YOUR 2ND TERM

TO GET THE 4RTH AMD 5TH TERM, JUST ADD THE FLLOWING CONSECITIVE TERM AS SHOWN IN THE ANSWER

Complete Question:

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with ? = 0.001 millimeters. A random sample of 15 rings has a mean diameter of \bar{X}= 74.106. Construct a 99% two-sided confidence interval on the true mean piston diameter and a 95% lower confidence bound on the true mean piston diameter.

(Round your answers to 3 decimal places.)

(Calculate the 99% two-sided confidence interval on the true mean piston diameter.

Answer:

99% true sided confidence Interval on the true mean Piston diameter = (74.105, 74.107)

Step-by-step explanation:

Check the attached file for the complete solution

Answer:

50 all you have to do is subtract the least and the biggest number

Step-by-step explanation:

Answer:

1/3

Step-by-step explanation:

For the 1st fraction, since 4 × 3 = 12,

3 /4  =  3 × 3/  4 × 3 =  9/ 12

Likewise, for the 2nd fraction, since 12 × 1 = 12,

5 /12  =  5 × 1 /12 × 1  =  5 /12

Subtract the two fractions: 9 /12  -  5 /12  =  9 - 5 /12  =  4 /12

So next you simplify the answer how many 4 go in 4 and how many go in 12 the simplified answer is the answer is  1/3

Hope this helps

Answer:

P (Type I Error) = P (Revokes the license | Restaurant is not in gross violation)

Step-by-step explanation:

A type I error occurs when we reject a true null hypothesis (H₀). It is the probability of rejecting the null hypothesis when the null hypothesis is true.

The type I error is also known as the level of significance. It is denoted by α .

P (Type I Error) = P (Rejecting H₀ | H₀ is true) = α.

In this case, the food inspector uses a hypothesis testing framework to evaluate whether regulations are not being met.

He decides the the restaurant's license to serve food will be revoked if the restaurant is in gross violation.

So the hypothesis is defined as:

H₀: The restaurant is not in gross violation.

Hₐ: The restaurant is in gross violation.

The type I error will be committed by the food inspector if he concludes that the the restaurant is in gross violation and revokes their license, when in fact the restaurant is not in gross violation.

α = P (Revokes the license | Restaurant is not in gross violation)

A Type 1 Error in this context would be when the food safety inspector incorrectly identifies the restaurant as violating sanitation regulations when in reality, it was following all necessary practices. It’s the wrongful rejection of the null hypothesis.

In statistical hypothesis testing, a Type 1 Error occurs when a true null hypothesis is rejected. In the context of the food safety inspector's investigation, the null hypothesis would likely be that the restaurant is abiding by all required sanitation practices. Therefore, a Type 1 Error would be if the food safety inspector concludes that the restaurant is in serious violation of sanitation regulations and revokes its license, but in actuality, the restaurant was not violating any regulations, i.e., the inspector incorrectly identified the restaurant as unclean and unhealthy.

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Answer:

a) [tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]

And we can find this probability using the normal standard table or excel and we got:

[tex]P(z<-1.387)=0.0827[/tex]

b) [tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]

And we can find this probability using the complement rule and the normal standard table or excel and we got:

[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the time for the step 1 and Y the time for the step 2, we define the random variable R= X+Y for the total time and the distribution for R assuming independence between X and Y is:

[tex]R \sim N(40+60 = 100,\sqrt{2^2 +3^2}= 3.606 s)[/tex]  

Where [tex]\mu=65.5[/tex] and [tex]\sigma=2.6[/tex]

We are interested on this probability

[tex]P(R<95)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{R-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]

And we can find this probability using the normal standard table or excel and we got:

[tex]P(z<-1.387)=0.0827[/tex]

Part b

[tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]

And we can find this probability using the complement rule and the normal standard table or excel and we got:

[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]

Given Information:  

Mean = μ = 40 + 60 = 100 minutes  

Standard deviation = σ = 2² + 3² = 13 minutes  

Required Information:  

a. P(X < 95) = ?

b. P(X > 110) = ?

Answer:  

a. P(X < 95) = 0.0823

b. P(X > 110) = 0 .0028

Explanation:  

a)

Let random variable X represents the time in minutes of wheel throwing and firing.

The probability that a piece of pottery will be finished within 95 minutes means,

P(X < 95) = P(Z < (x - μ)/√σ)  

P(X < 95) = P(Z < (95 - 100)/√13)  

P(X < 95) = P(Z < -1.39)  

The z-score corresponding to -1.39 is 0.0823

P(X < 95) = 0.0823

Therefore, there is 8.23%  probability that a piece of pottery will be finished within 95 minutes.

b)

P(X > 110) = 1 - P(X < 110)  

P(X > 110) = 1 - P(X < (x - μ)/√σ)

P(X > 110) = 1 - P(X < (110 - 100)/√13)

P(X > 110) = 1 - P(X < 2.77)

The z-score corresponding to 2.77 is 0.9972

P(X > 110) = 1 - 0.9972

P(X > 110) = 0 .0028

Therefore, there is 0.28%  probability that a piece of pottery will take longer than 110 minutes.

Step-by-step explanation:

[tex] {5}^{17} \times {5}^{2} [/tex]

Now adding powers

[tex] {5}^{17 + 2} [/tex]

[tex] {5}^{19} [/tex]

Hope it will help :)

Answer:

5x1^19

Step-by-step explanation:

Basically, the only way to have the 5 be to the power of 1 in this equation is to put it in scientific notation, or in other words, multiply 5 by 1 to the power of 19.

Answer:

Q1 or the 25th percentile is 204.5

Step-by-step explanation:

Combine all data values in TI-84, excel, or other software and make a box and whisker plot. The Q1 is the first vertical line on the plot

Answer:

25 20

Step-by-step explanation:

D. Width = 25 inches, height = 20 inches

Answer:64 pi units

What is the ratio of the arc length to the circumference?Answer: 1/4

What is the measure of central angle XOY?Answer: 90 degrees

Answer:

1. 64 pi units

2. 1/4

3. 90°

Step-by-step explanation: edge

Answer:

At 0.05 level of significance, the abrasive wear of material 1 exceeds that of material 2 by more than 2 units

Step-by-step explanation:

We hypothesize that mean difference between abrasive wear of material 1 and material 2 is greater than 2.

So we write the null hypothesis [tex]H_0 : \mu_1 - \mu_2 >2[/tex],

and the alternative hypothesis [tex]H_1: \mu_1 - \mu_2 \leq 2[/tex].

We will find the T-score as well as the p-value. If the p-value is less than the level of significance, we will reject the null hypothesis, i.e. we will conclude that the abrasive wear of material 1 is less than that of material 2. Otherwise, we will accept the null hypothesis.

Since the variance is unknown and assumed to be equal, we will use the pooled variance

[tex]s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2} = 20.05[/tex],

where [tex]n_1 = 12, n_2 =10, s_1 =4, s_2 = 5[/tex].

The mean of material 1 and material 2 are [tex]\mu_1 =85, \mu_2=81[/tex] respectively and mean difference [tex]d[/tex] is equal to 4. The hypothesize difference [tex]d_0[/tex] is equal to 2.

To find the T-score, we use the following formula

[tex]T = \frac{d - d_0}{\sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2} }}[/tex]

Substituting all the values into the T-score formula gives us [tex]T = 1.04[/tex], and the respective p-value is equal to 0.31. This means we have enough statistical evidence not to reject the null hypothesis, and at 5% significance level, the abrasive wear of material 1 exceeds that of material 2 by more than 2 units.

Step-by-step explanation:

To find the sum of first 20 terms,

a = 6, r = 3

By formula, [tex]S_{n} = \frac{a(r^{n}-1) }{r-1}[/tex]

substitute the values in the above formula, the equation becomes,

[tex]S_{20} = \frac{6(3^{20}-1) }{3-1}[/tex]

[tex]S_{20} = \frac{6(3^{20}-1) }{2}[/tex]

[tex]S_{20}[/tex] = [tex]{3(3^{20}-1) }[/tex]