The table shows the relationship, "Taiga reads 250 words Which equation models this relationship?
per minute."
O wm = 250
The independent variable, the number of minutes he
O w = 250m
reads, causes a change in the dependent variable, the
O m = 250w
number of words read.
O w + m = 250
Minutes
(m)
Words
(w)
250
500
750
1000

Final answer:

Oscar's balloon, which has an 8-inch diameter, will hold approximately 268.1 cubic inches of helium, calculated using the volume formula for a sphere.

Explanation:

Oscar needs to calculate the volume of a sphere-shaped balloon to determine how much helium it can hold. To find the balloon's volume, we use the formula for the volume of a sphere, which is V = ⅓πd³, where V is the volume, π is approximately 3.14159, and d is the diameter of the sphere. Since the balloon has a diameter of 8 inches, its radius r is 4 inches (which is half of the diameter).

Plugging the radius into the formula, we get: V = ⅓π(4 inches)³ = ⅓π(64 inches³) = 268.0826 inches³. Therefore, Oscar's balloon will hold approximately 268.1 cubic inches of helium to the nearest tenth, making the correct answer C. 268.1 in.³

Answer:

[tex]12087.7-1.96\frac{518}{\sqrt{10}}=11766.64[/tex]    

[tex]12087.7+1.96\frac{518}{\sqrt{10}}=12408.76[/tex]    

So on this case the 95% confidence interval would be given by (11767;12409)    

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

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".

[tex]\bar X[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

Data:  12390 12296 11916 11713 11936 11553 12000 12428 12354 12291

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   (1)

In order to calculate the mean and the sample deviation we can use the following formulas:  

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)  

The mean calculated for this case is [tex]\bar X=12087.7[/tex]

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]

Now we have everything in order to replace into formula (1):

[tex]12087.7-1.96\frac{518}{\sqrt{10}}=11766.64[/tex]    

[tex]12087.7+1.96\frac{518}{\sqrt{10}}=12408.76[/tex]    

So on this case the 95% confidence interval would be given by (11767;12409)    

To compute a 95% confidence interval for the mean of the population, use the formula (sample mean) +/- (critical value) * (standard deviation / sqrt(sample size)).

To compute a 95% confidence interval for the mean of the population, we can use the formula:

(sample mean) +/- (critical value) * (standard deviation / sqrt(sample size))

Given the sample data and the standard deviation, we can find the sample mean by taking the average of the incomes. The critical value can be found using a z-table or calculator. With a sample size of 10, the standard deviation is divided by sqrt(10). Plugging in the values, we get a 95% confidence interval of ($11627, $12460).

Answer:

C: 12.56 inches

Step-by-step explanation:

We know that the minute hand can move an equivalent of 60 minutes in any one revolution.

-10 minutes movement is equal to 1/6 the total distance and the circumference covered in that time is calculated as:

[tex]C=\pi D\\\\=\frac{1}{6}\pi \times (12\times 2)\\\\\\=12.56\ in[/tex]

Hence, over 10 minutes the minutes hand moves 12.56 inches away.

The tip of a 12 inch minute hand will move approximately 12.56 inches over the course of 10 minutes, which aligns with option C in your given choices.

The subject of this question is Mathematics, specifically geometry and involves calculating the length of an arc within a circle. The minute hand of a clock can be thought of as the radius of a circle, with a full rotation of the hand representing a complete circle. The minute hand moves 360 degrees in 60 minutes (or 6 degrees per minute), so over 10 minutes, the minute hand will move 60 degrees.

Now, the length of that portion of the circle (the arc length) is calculated using the formula: (2πr)(θ/360), where r is the radius (half of the diameter, or 12 inches in this case), and θ is the angle in degrees. When you plug in the respective values, you will find that the minute hand of the clock moves an approximate distance of 12.56 inches, which corresponds to option C in your given choices.

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Answer:

  E(3, -5)

Step-by-step explanation:

In a rectangle, the diagonals are the same length and bisect each other. That means their midpoints are the same. Then ...

  (F +C)/2 = (A +E)/2

  E = F +C -A

  E = (-5, 1) +(6, -1) -(-2, 5) = (-5+6+2, 1-1-5)

  E = (3, -5) . . . . . . . E is chosen so that the midpoint of AE is that of FC

__

To prove the figure is a rectangle, we can show the lengths of the diagonals are the same. Using the distance formula, ...

  FC = √((6-(-5))^2 +(-1-1)^2) = √(11^2 +2^2) = √125

  AE = √((3-(-2))^2 +(-5-5)^2) = √(5^2 +10^2) = √125

The diagonals are the same length and have the same midpoint, so the figure is a rectangle.

Answer: There are 4 more wrenches in the toolbox then the screwdrivers.

Step-by-step explanation: Add the 6 wrenches Sid put in the toolbox with the 8 wrenches Bella added to get 14 wrenches in total. Then, subtract the 10 screwdrivers from the 14 wrenches to get 4 wrenches.

Final answer:

Bella added 8 wrenches to the toolbox, making a total of 14 wrenches. There were initially 10 screwdrivers, so there are now 4 more wrenches than screwdrivers.

Explanation:

Calculating the Difference Between Wrenches and Screwdrivers in a Toolbox

Initially, there are 10 screwdrivers and 6 wrenches in the toolbox. Bella adds 8 more wrenches, which brings the total number of wrenches to 6 + 8, which equals 14 wrenches. The question asks how many more wrenches there are than screwdrivers. To find this, we subtract the number of screwdrivers from the number of wrenches:

14 wrenches - 10 screwdrivers = 4 more wrenches than screwdrivers in the toolbox.

The first one is KL and the second one is FG

Side GE corresponds to side EJ, which is 24 meters long and side JK corresponds to side LK, which is 60/7 meters (or approximately 8.57 meters) long.

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .

To determine which side of triangle EFG corresponds to side GE

we need to find the ratio of the lengths of corresponding sides.

The sides that share vertex E are EF and EJ, so we can write:

EF / EJ = FG / FJ = EG / EK

Substituting the given values, we get:

6 / ? = 5 / ? = 7 / 20

To solve for the missing value, we can cross-multiply and simplify:

6 × 20 = 5 × x

x = 24

To determine which side of triangle KLJ corresponds to side JK, we can use the same approach.

The sides that share vertex J are JL and JF, so we can write:

JL / JF = LK / EF = LJ / FG

Substituting the given values, we get:

24 / ? = 20 / 6 = 28 / 5

Cross-multiplying and simplifying:

24 × 5 = x × 28

x = 60 / 7

Therefore, side GE corresponds to side EJ, which is 24 meters long and side JK corresponds to side LK, which is 60/7 meters (or approximately 8.57 meters) long.

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Answer:

  $7.54

Step-by-step explanation:

Using the two-point form of the equation of a line, we can write the equation for the number tickets (t) as a function of price (p).

  t = (t2 -t1)/(p2 -p1)(p -p1) +t1

  t = (930 -1160)/(7 -5)(p -5) +1160

  t = -230/2(p -5) +1160

  t = -115p +1735 = -115(p -15 2/23)

The revenue from ticket sales will be the product of the price and the number of tickets sold:

  r = pt = p(-115)(p -15 2/23)

This is the equation of a downward-opening parabola with zeros at p=0 and p=15 2/23. The vertex of the parabola (maximum revenue) will be found at a ticket price halfway between these values. The price for maximum revenue is ...

  (0 +15 2/23)/2 = 7 25/46 ≈ 7.54

Rosalie should charge $7.54 per ticket to obtain the most revenue.

Answer:

30.0

Step-by-step explanation:

Given our data is normally distribute with [tex]\mu=21.3[/tex] and [tex]\sigma=5.9[/tex]

-Top 7% is given by find the z-value corresponding to p=(1-0.07)=0.93

-We substitute our values in the equation below;

[tex]z=\frac{\bar X-\mu}{\sigma}\\\\\\=\frac{X-21.3}{5.9}, z_{0.035}=1.476\\\\\therefore 1.476=\frac{X-21.3}{5.9}\\\\X=5.9\times 1.476+21.3\\\\=30.0084\approx30.0[/tex]

Hence, the minimum score required for the scholarship is 30.0

The minimum ACT Reading score required for a university scholarship awarded to the top 7% is approximately 30.0.

To find the minimum ACT Reading score required for a scholarship awarded to students in the top 7%, we need to determine the z-score that corresponds to the top 7% of a normal distribution. We can then use this z-score to find the corresponding ACT score.

The z-score for the top 7% of a standard normal distribution is approximately 1.475. Since the ACT Reading scores have a mean (μ) of 21.3 and a standard deviation (σ) of 5.9, we can use the z-score formula to find the minimum score 'x' required for the scholarship: z = (x - μ) / σ.

Solving for 'x', we get: x = zσ + μ = 1.475(5.9) + 21.3 ≈ 30.0. Therefore, the minimum ACT Reading score required for the scholarship is approximately 30.0.

To prove the statement (A − B) ∪ (A ∩ B) = A, we can use the property of set difference, distribution, and identity from Theorem 6.2.2.

To construct an algebraic proof for the statement (A − B) ∪ (A ∩ B) = A, we can use the property of set difference, distribution, and identity from Theorem 6.2.2.

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Answer: The one-day SAT prep class is associated with statistically significant improvements in SAT writing performance.

Step-by-step explanation: just took the quiz

The correct conclusion about the situation is, the one-day SAT prep class produces statistically significant improvements in SAT writing performance, which is option (e).

Given that:

It is assessing the performance of the students in the SAT writing exam before and after SAT prep class.

The hypothesis is:

H₀: μ = 0

H₁: μ > 0

This is a one-tailed test.

Here, the T-test is used.

Now, the significance level is, α = 0.05

p-value = 0.028

Since, the p-value, 0.028 is less than the significance level 0.05, the null hypothesis is rejected.

So, the mean of the difference in SAT scores is greater than 0.

That is, there is a significant effect in SAT exam by the prep class.

Hence, the correct conclusion is, The one-day SAT prep class produces statistically significant improvements in SAT writing performance, which is option (e).

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Answer:

A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution. so its A

Step-by-step explanation:

i looked it up and i think this is right :)

Answer:

its A

Step-by-step explanation:

Answer:= (504, 516)

Therefore, the 90% confidence interval (a,b) = ( 504, 516)

Step-by-step explanation:

Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.

The confidence interval of a statistical data can be written as.

x+/-zr/√n

Given that;

Mean gain x = 510

Standard deviation r = 37

Number of samples n = 95

Confidence interval = 90%

z(at 90% confidence) = 1.645

Substituting the values we have;

510+/-1.645(37/√95)

510+/-1.645(3.796)

510+/-6.24

510+/-6

= (504, 516)

Therefore at 90% confidence interval (a,b) = ( 504, 516)

The mean of a distribution indicates its central tendency, with more observations clustering around this central value in a normally distributed dataset.

When examining various distributions, the mean of each distribution is a critical value that gives information about the central tendency of the data. In a normally distributed dataset, the mean is at the peak of the bell curve, suggesting that more observations cluster around this central value.

As for different types of distributions, such as binomial or normal, knowing the mean helps us compare them effectively.

For instance, if both distributions are normal with the same mean, they will overlap, but varying standard deviations will affect the spread of the data around that mean. The larger the standard deviation, the wider the distribution.

Additionally, the concept of skewness also affects the mean. In a positively skewed distribution, the mean is higher than the median, while in a negatively skewed distribution, the mean is less than the median. Considering skewness helps gauge the data's asymmetry and the mean's position relative to other central tendency measures.

Understanding the characteristics of a probability distribution, especially the normal distribution, which is symmetrical about its mean, is fundamental in statistics. The probability density functions have properties that allow us to predict the likelihood of outcomes within a range, expressed through confidence intervals or the standard deviation.

Answer:

157,286,400 bacteria.

Step-by-step explanation:

We have been given that a certain type of bacteria, given favorable growth medium, quadruples in population every 6 hours. Given that there were 150 bacteria to start with.

We will use exponential growth function to solve our given problem.

[tex]y=a\cdot b^x}[/tex], where

y = Final value,

a = Initial value,

b = Growth factor.

x = Time.

Quadruples meaning 4 at a time, so growth factor is 4.

We are also told that population becomes 4 times every 6 hours, so time would be [tex]\frac{1}{6}x[/tex].

Initial value is given as 150.

Upon substituting these values in above formula, we will get:

[tex]y=150(4)^{\frac{1}{6}x}[/tex]

Let us convert two and a half days into hours.

1 day = 24 hours.

2.5 days = 2.5*24 hours = 60 hours.

To find the bacteria population in two and half days, we will substitute [tex]x=60[/tex] in our formula as:

[tex]y=150(4)^{\frac{1}{6}(60)}[/tex]

[tex]y=150(4)^{10}[/tex]

[tex]y=150(1048576)[/tex]

[tex]y=157,286,400[/tex]

Therefore, there will be 157,286,400 bacteria in two and a half days.

Answer:

Step-by-step explanation:

We would set up the hypothesis test.

For the null hypothesis,

p = 0.5

For the alternative hypothesis,

p < 0.5

Considering the population proportion, probability of success, p = 0.5

q = probability of failure = 1 - p

q = 1 - 0.5 = 0.5

Considering the sample,

P = 42/100 = 0.42

We would determine the test statistic which is the z score

z = (P - p)/√pq/n

n = 400

z = (0.42 - 0.5)/√(0.5 × 0.5)/400 = - 3.2

Recall, population proportion, p = 0.5

We want the area to the left of 0.5 since the alternative hypothesis is lesser than 0.5. Therefore, from the normal distribution table, the probability of getting a proportion < 0.5 is 0.00069

So p value = 0.00069

Since alpha, 0.05 > than the p value, 0.00069, then we would reject the null hypothesis.

Therefore, there is significant evidence to conclude that that less than 50% of people under the age of 25 check social media sites right after they wake up.

Answer:

the interest amount is $240

Step-by-step explanation:

I hope it helps.

Answer:

all sides are congruent

Step-by-step explanation:

its talking about sides

I believe A is correct

Good luck!

Answer:

The probability that the mean amplifier output would be greater than 364.8 watts in a sample of 52 amplifiers is 0.3156

Step-by-step explanation:

Mean output of amplifiers = 364

Standard deviation = [tex]\sigma[/tex] = 12

We have to find the probability that the mean output for 52 randomly selected amplifiers will be greater than 364.8. Since the population is Normally Distributed and we know the value of population standard deviation, we will use the z-distribution to solve this problem.

We will convert 364.8 to its equivalent z-score and then finding the desired probability from the z-table. The formula to calculate the z-score is:

[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]

x=364.8 converted to z score for a sample size of n= 52 will be:

[tex]z=\frac{364.8-364}{\frac{12}{\sqrt{52} } }=0.48[/tex]

This means, the probability that the output is greater than 364.8 is equivalent to probability of z score being greater than 0.48.

i.e.

P( X > 364.8 ) = P( z > 0.48 )

From the z-table:

P( z > 0.48) = 1 - P(z < 0.48)

= 1 - 0.6844

= 0.3156

Since, P( X > 364.8 ) = P( z > 0.48 ), we can conclude that:

The probability that the mean amplifier output would be greater than 364.8 watts in a sample of 52 amplifiers is 0.3156

Answer:

a = 1/3

b = -3

c = 26/3

d = -6

e = 0

Step-by-step explanation:

Given the quartic polynomial

p(x)=ax⁴+bx³+cx²+dx+e and

p(i) =2i when i=0,1,2,3,4

If i = 0:

p(0) = 2(0)

p(0) = 0

0 = 0+0+0+0+0++e

e = 0

When i = 1

p(1) = 2(1) = 2

2 = a(1)⁴+b(1)³+c(1)²+d(1)+e

2 = a+b+c+d+0

a+b+c+d = 0... (1)

When i = 2, p(2) = 2(2)

p(2) = 4

4 = a(2)⁴+b(2)³+c(2)²+d(2)+e

4 = 16a+8b+4c+2d+0

16a+8b+4c+2d = 4

8a+4b+2c+d = 2... (2)

When i = 3

p(3) = 8

8 = a(3)⁴+b(3)³+c(3)²+d(3)+0

8 = 81a+27b+9c+3d..(3)

When i = 4

p(4) =16

16 = a(4)⁴+b(4)³+c(4)²+d(4)+0

16 = 256a+64b+16c+4d

64a+16b+4c+d = 4...(4)

Solving equation 1 to 4 simultaneously.

Check the attachment for solution.

The problem here is to determine the coefficients of a quartic polynomial to match the given conditions. This results in a system of linear equations which can be solved to find the desired coefficients.

This question is a

polynomial problem

and involves finding the coefficients of a

quartic polynomial

, and for that we form a system of linear equations. Using the given conditions, we get the following equations:

By solving the above system of equations, we can find the values of a, b, c, d and e that satisfy those equations simultaneously.

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Answer:

P(Y and R) = P(Y)*P(R) + P(R)*P(Y)

P(Y and R) = 16/125 = 0.128 = 12.8%

Step-by-step explanation:

There are 8 Yellow marbles in the bag

There are 9 Green marbles in the bag

There are 3 Purple marbles in the bag

There are 5 Red marbles in the bag

The total number of marbles in the bag are

Total marbles = 8 + 9 + 3 + 5 = 25

We want to find the probability of selecting two marbles that is one Yellow marble and one Red marble from the bag.

The probability of selecting a Yellow marble is given by

P(Y) = number of Yellow marbles/total number of marbles

P(Y) = 8/25

The probability of selecting a Red marble is given by

P(Y) = number of Red marbles/total number of marbles

P(Y) = 5/25

P(Y) = 1/5

It is possible that the first marble selected is Yellow and the second is Red, and it is also possible that first marble selected is Red and the second is Yellow.

P(Y and R) = P(Y)*P(R) + P(R)*P(Y)

P(Y and R) = (8/25)*(1/5) + (1/5)*(8/25)

P(Y and R) = 16/125

P(Y and R) = 0.128

P(Y and R) = 12.8%

Answer:

probability of selecting one yellow and one red = 2/15

Step-by-step explanation:

We are told there are;

8 yellow marbles

9 green marbles

3 purple Marbles

5 red Marbles

Since two Marbles are selected,

Number of ways of selecting one yellow and one red is:

C(8,1) x C(5,1) = 8!/(1!(8 - 1)!) x 5!/(1!(5 - 1)!)

This gives 40

Now, the total number of Marbles in the question will be;

8 + 9 + 3 + 5 = 25 Marbles

Thus, number of ways to select any two Marbles from the total is;

C(25,2) = 25!/(2!(25 - 2)!) = 300

Thus; probability of selecting one yellow and one red = 40/300 = 2/15

Answer:

  0

Step-by-step explanation:

The slope formula is ...

  m = (y2 -y1)/(x2 -x1)

Filling in the given point values, we find the slope to be ...

  m = (-4 -(-4))/(9 -5) = 0/4 = 0

The slope is 0.

_____

The y-values are the same at -4, the equation of the line is y = -4. It is a horizontal line with zero slope.

Answer:

a) True

Step-by-step explanation:

Repeated samples are a type of samples that are used to determine the features or characteristics or a given set of data.

In repeated samples, statistical techniques are applied whereby two samples that have similar characteristics are tested or analysed under different conditions.

Repeated samples can also be called matched or paired samples.

In repeated samples , we have what we refer to as confidence intervals. These are intervals whereby the true and correct value of certain parameters such as mean, the standard deviation of a given data or distribution is determined. We have confidence interval levels of 90%, 95% and 99%.

In repeated samples, approximately 99% of all differences in sample means will fall within the bounds of the interval already computed.

Answer: C

Step-by-step explanation:

Answer:

f( - 2) =12

Step-by-step explanation:

[tex]f(x) = x^2-3x+2 \\ plugging \: x = - 2 \\ f( - 2) = ( - 2)^2-3( - 2)+2 \\ f( - 2) =4 + 6+2 \\ f( - 2) =12 \\ [/tex]

Answer:

a) P(x=108)=0.0020

b) P(x≥108)=0.9980

c) P(x<124)=0.7794

d) P(124≤x≤128)=0.1925

Step-by-step explanation:

We know the population proportion, that is p=0.84.

We take a sample of size n=143.

We will use the normal approximation to the binomial distribution to model this problem.

The mean and standard deviation of the normal approximation to the binomial distribution will be:

[tex]\mu=np=143*0.84=120.12\\\\\sigma=\sqrt{np(1-p)}=\sqrt{143*0.84*0.16}=\sqrt{19.22}=4.38[/tex]

a) We have to calculate the probability that exactly 108 flights are on time.

As the normal distribution considers the random variable to be continous, we have to apply the continuity correction factor.

In this case, the probability of 108 flights on time can be calculated as P(107.5<x<108.5):

[tex]P(x=108)=P(107.5<x<108.5)=P(x<108.5)-P(x<107.5)\\\\\\ z_1=(x_1-\mu)/\sigma=(107.5-120.12)/4.38=-12.62/4.38=-2.88\\\\z_2=(x_2-\mu)/\sigma=(108.5-120.12)/4.38=-11.62/4.38=-2.65\\\\\\P(x<108.5)-P(x<107.5)=P(z<-2.65)-P(z<-2.88)\\\\P(x<108.5)-P(x<107.5)=0.0040-0.0020=0.0020[/tex]

b) Now we have to calculate that at least 108 flights are on time.

As the probability includes 108, the continuity factor will indicates that we calculate P(x>107.5). The z-value for x=107.5 has been already calculated in point a:

[tex]P(x\geq108)=P(x>107.5)=P(z>-2.88)=0.9980[/tex]

c) We have to calculate the probability that fewer than 124 flights are on time. According to the continuity factor, we have to calculate the probability P(x<123.5), as the flight number 124 is not included in the interval.

[tex]P(x<124)=P(x<123.5)=P(z<0.77)=0.7794\\\\\\z=(x-\mu)/\sigma=(123.5-120.12)/4.38=0.77[/tex]

d) We have to calculate the probability that between 124 and 128 flights, inclusive, are on time.

This interval corresponds to the probability P(123.5<x<128.5)

[tex]P(123.5<x<128.5)=P(x<128.5)-P(x<123.5)\\\\\\ z_1=(x_1-\mu)/\sigma=(128.5-120.12)/4.38=8.38/4.38=1.91\\\\z_2=(x_2-\mu)/\sigma=(123.5-120.12)/4.38=0.77\\\\\\P(x<128.5)-P(x<123.5)=P(z<1.91)-P(z<0.77)\\\\P(x<128.5)-P(x<123.5)=0.9719-0.7794=0.1925[/tex]

Answer:

Circle

Step-by-step explanation:

I don't know if there is a more "professional" way to solve this, but I wrote out the pattern until I got to the twentieth shape and it ended up being a circle :)

The 20th shape in the pattern is a circle.

To determine the 20th shape in the pattern of triangle, circle, circle, we need to analyze the pattern. The pattern starts with a triangle and is followed by two circles. This sequence repeats - triangle, circle, circle. To find the 20th shape, we need to determine how many times this sequence repeats within the first 20 shapes. Each complete sequence consists of 3 shapes (triangle, circle, circle), so we divide 20 by 3 to get 6 complete sequences. The 6th complete sequence ends with a circle, so the 20th shape in the pattern is also a circle.

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Answer:

i need points.

Step-by-step explanation:

Answer:

a) Null hypothesis: [tex]\mu_d= 0[/tex]

Alternative hypothesis: [tex]\mu_d \neq 0[/tex]

b) [tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]

The next step is calculate the degrees of freedom given by:

[tex]df=n-1=42-1=41[/tex]

Now we can calculate the p value, since we have a left tailed test the p value is given by:

[tex]p_v =2*P(t_{(41)}>4.905) =0.000015[/tex]

So the p value is lower than any significance level given, so then we can conclude that we can reject the null hypothesis that the difference between he two groups are equal.

Step-by-step explanation:

Assuming the following questions:

We assume the following data: [tex] n = 42 ,\bar d= 850 , s_d = 1123[/tex]

Previous concepts

A paired t-test is used to compare two population means where you have two samples in  which observations in one sample can be paired with observations in the other sample. For example  if we have Before-and-after observations (This problem) we can use it.  

a. Formulate the null and alternative hypotheses to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.

The system of hypothesis for this case are:

Null hypothesis: [tex]\mu_2- \mu_1 = 0[/tex]

Alternative hypothesis: [tex]\mu_2 -\mu_1 \neq 0[/tex]

Or equivalently

Null hypothesis: [tex]\mu_d= 0[/tex]

Alternative hypothesis: [tex]\mu_d \neq 0[/tex]

b. Use a .05 level of significance. Can you conclude that the population means differ? What is the p-value?

[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]

The next step is calculate the degrees of freedom given by:

[tex]df=n-1=42-1=41[/tex]

Now we can calculate the p value, since we have a left tailed test the p value is given by:

[tex]p_v =2*P(t_{(41)}>4.905) =0.000015[/tex]

So the p value is lower than any significance level given, so then we can conclude that we can reject the null hypothesis that the difference between he two groups are equal.

Answer:

a)  3.6%

Step-by-step explanation:

The given question mixed up, below is the correct question:

The bumper car ride at the state fair has 2 red cars, 4 green cars, and 2 blue cars. Garth is first in line for the ride and is assigned a car at random.  Patty is next in line and is randomly assigned a car. Find the probability that both events A and B occur. Express your answer as a percent. If necessary, round your answer to the nearest tenth.

Calculation:

Given that the state fair has 2 red cars, 4 green cars and 2 blue cars.

There are therefore 2+4+2 = 8 cars in total.

Probability that Events A occurs P(A) = [tex]\frac{2}{8}[/tex] = 4

Probability that Events B occurs P(B) = [tex]\frac{1}{7}[/tex]

Probability that Events A and B occur P(A ∩ B) = [tex]\frac{2}{8}[/tex] × [tex]\frac{1}{7}[/tex] = [tex]\frac{2}{56}[/tex] = 0.0357 = 3.57%  ≈ 3.6%

Therefore, the probability that both events A and B occur is 3.6%

Final answer:

The probability that both Garth and Patty will drive a red bumper car is found by multiplying the probability of Garth picking a red car (1/4) by the probability of Patty picking a red car after Garth (1/7), resulting in 1/28 or approximately 3.6%.

Explanation:

To solve the problem, we need to calculate the probability that both events A and B happen, which involves Garth and Patty both getting a red bumper car. Initially, there are 2 red cars, 4 green cars, and 2 blue cars, totaling 8 cars.

Event A: Garth picks a red car. The probability of this happening is the number of red cars over the total number of cars. So P(A) = 2/8 = 1/4.

After Garth picks a red car, there is 1 red car, 4 green cars, and 2 blue cars left, totaling 7 cars.

Event B: Patty picks a red car after Garth has already picked one. The probability of this happening is the number of remaining red cars over the total number of remaining cars. So P(B after A) = 1/7.

The probability that both A and B occur is the product of the probability of A and the probability of B given A has occurred. So P(A and B) = P(A) × P(B after A) = (1/4) × (1/7).

P(A and B) = 1/28. To express this as a percent, we multiply by 100%: (1/28) × 100% ≈ 3.6%.

Therefore, the probability that both Garth and Patty will drive a red bumper car is approximately 3.6%, which corresponds to option d.