Points (-3,6 ) (-2,9 ) the equation in point slope form step by step

The pooled variance is calculated by combining the sum of squares (SS) and the degrees of freedom from two samples. Here are the steps to calculate the pooled variance:

1. Calculate the degrees of freedom for each sample (df1 = n1 - 1 and df2 = n2 - 1)
2. Calculate the pooled sum of squares (PSS = SS1 + SS2)
3. Calculate the pooled degrees of freedom (PDF = df1 + df2)
4. Calculate the pooled variance (PV = PSS / PDF)

For your samples:

Sample 1: n1 = 6, SS1 = 56, df1 = n1 - 1 = 5
Sample 2: n2 = 4, SS2 = 40, df2 = n2 - 1 = 3

Now, let's compute the pooled variance:

PSS = SS1 + SS2 = 56 + 40 = 96
PDF = df1 + df2 = 5 + 3 = 8
PV = PSS / PDF = 96 / 8 = 12

So, the pooled variance for the given samples is 12, which corresponds to answer choice (b).

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Answer:

-22

Step-by-step explanation:

This is the result of the operation, in the image, the answer is all the way at the bottom, hope this helps

Answer:

Step-by-step explanation:

banana are curved cause they grow toward the sun child

The p-value is less than 0.0001, which is smaller than our level of significance (α = 0.05). Therefore, we reject the null hypothesis and conclude that the mean area of homes built in is not equal to the mean area of homes built in at a 95% confidence level

To determine whether we can conclude that the mean area of homes built in is not equal to that of homes built in, we need to perform a hypothesis test.

Let's set up our hypotheses:

Null hypothesis (H0): The mean area of homes built in is equal to the mean area of homes built in

Alternative hypothesis (Ha): The mean area of homes built in is not equal to the mean area of homes built in

We will use a two-tailed t-test, as we are testing for inequality rather than a specific direction.

Assuming a level of significance (α) of 0.05, our critical values are ±1.96.

To calculate the test statistic, we can use the formula:

t = (sample mean - population mean) / (sample standard deviation/sqrt (sample size))

t = ( - ) / ( / sqrt())

t = -4.72

Using a t-distribution table with degrees of freedom equal to (sample size - 1), we can find the p-value associated with this t-statistic.
The p-value is less than 0.0001, which is smaller than our level of significance (α = 0.05). Therefore, we reject the null hypothesis and conclude that the mean area of homes built in is not equal to the mean area of homes built in at a 95% confidence level. In other words, there is sufficient evidence to suggest that the mean area of homes built in is different from the mean area of homes built in.

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Answer: false

Step-by-step explanation:

True. A conditional statement consists of an antecedent (the "if" statement) and a consequent (the "then" statement). The statement is false only when the consequent is true and the antecedent is false.

In other words, for a conditional statement "if A then B," if A is false and B is true, the statement is false. If A is true and B is false or if both A and B are false, the statement is still considered true.

The given statement is incorrect. A conditional statement is false only when the antecedent is true and the consequent is false. In a conditional statement (usually written as "if P, then Q"), the antecedent (P) represents the condition, and the consequent (Q) represents the result of the condition being satisfied. The statement is considered false if the condition (antecedent) holds, but the result (consequent) does not occur.

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The requried factored form of the expression 2x^2-2x+1-4x-1 is 2x(x-3/2).

Here,
To factor the expression [tex]2x^2-2x+1-4x-1[/tex], we can rearrange the terms to get:

[tex]2x^2 - 6x = 0[/tex]

Now we can factor out the common factor of 2x to get:

2x(x-3) = 0

So the solutions are:

x = 0, x = 3/2

Therefore, the factored form of the expression 2x^2-2x+1-4x-1 is 2x(x-3/2).

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The bookstore should 300 calendars in order to have the maximum expected monetary value of $252.

To determine this, we need to calculate the expected monetary value for each option.

For ordering 100 calendars:
- Expected revenue: $12 x 100 = $1200
- Expected cost: $4 x 100 = $400
- Probability of selling all calendars: 0.35
- Probability of selling some calendars back to a supplier: 0.65
- Expected revenue from selling some calendars back to supplier: $2 x (100 - sales) = $2 x (100 - 35) = $130
- Expected monetary value: (0.35 x $1200) - $400 + (0.65 x $130) = $370

For ordering 200 calendars:
- Expected revenue: $12 x 200 = $2400
- Expected cost: $4 x 200 = $800
- Probability of selling all calendars: 0.25
- Probability of selling some calendars back to the supplier: 0.75
- Expected revenue from selling some calendars back to a supplier: $2 x (200 - sales) = $2 x (200 - 50) = $300
- Expected monetary value: (0.25 x $2400) - $800 + (0.75 x $300) = $350

For ordering 300 calendars:
- Expected revenue: $12 x 300 = $3600
- Expected cost: $4 x 300 = $1200
- Probability of selling all calendars: 0.40
- Probability of selling some calendars back to a supplier: 0.60
- Expected revenue from selling some calendars back to a supplier: $2 x (300 - sales) = $2 x (300 - 120) = $360
- Expected monetary value: (0.40 x $3600) - $1200 + (0.60 x $360) = $252

Based on these calculations, we can see that ordering 300 calendars gives the highest expected monetary value of $252.

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The function F(x)= ¹/₄(⁵/₂)ˣ , is a growth function.

The given function is;

F(x)= ¹/₄(⁵/₂)ˣ

To determine if the function is growth function or decay function, we will compare it to the general form of the function.

So the given function is an exponential function of the form;

f(x) = a(b)ˣ

Where;

Since the base (b) is greater than 1, we can conclude that function is a growth function.

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Answer:

5s + 0.65 = 14.4

      - 0.65  - 0.65

______________

5s = 13.75

___.   ___

5          5

s = 2.75 meters of string for each pennant

Explanation:

s represents the length of string required for each pennant.

We know that Ingrid is using a total of 14.4 meters of string to hang a banner and 5 pennants.

We also know that out of that 14.4 meters, 0.65 meters is used to hang the banner. The rest of the string left is equally distributed among the 5 pennants.

So, therefore, we have a constant of 0.65 and a coefficient of 5 with variable, s.

Next, we solve this by first isolating the constant by subtracting 0.65 on both sides.

Then. we isolate the variable by dividing 5 into both sides.

So, s = 2.75 meters of string for each pennant

The sum of all these probabilities is equal to 1, which is the total probability of all possible outcomes.

To solve this problem, we need to consider all the possible outcomes that result in at most four successes. These outcomes are:

0 successes

1 success

2 successes

3 successes

4 successes

The probability of each of these outcomes can be calculated using the binomial distribution formula:

P(k) = (n choose k) [tex]* p^k * (1-p)^(n-k)[/tex]

where n is the number of trials, k is the number of successes, and (n choose k) is the binomial coefficient, which is equal to n!/(k!*(n-k)!).

To find the probability of at most four successes, we need to add up the probabilities of all these outcomes. So the expression that would provide the answer is:

P(0 successes) + P(1 success) + P(2 successes) + P(3 successes) + P(4 successes)

This expression includes all the possible outcomes that correspond to having at most four successes in five trials. Note that it does not include the probability of having five successes, since we are only interested in the probability of at most four successes.

For example, if the probability of success in each trial is 0.3, then the probability of having zero successes is:

P(0 successes) = (5 choose 0)[tex]* 0.3^0 * 0.7^5 = 0.168[/tex]

Similarly, the probability of having one success is:

P(1 success) = (5 choose 1) [tex]* 0.3^1 * 0.7^4 = 0.360[/tex]

We can calculate the probabilities of having two, three, and four successes in a similar way. Then, we can add up all these probabilities to get the probability of at most four successes:

P(at most 4 successes) = P(0 successes) + P(1 success) + P(2 successes) + P(3 successes) + P(4 successes)

In this example, we get:

P(at most 4 successes) = 0.168 + 0.360 + 0.308 + 0.132 + 0.032 = 1.000

Note that the sum of all these probabilities is equal to 1, which is the total probability of all possible outcomes.

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The length of the hypotenuse is 9.5 kilometers

From the question, we have the following parameters that can be used in our computation:

Legs = 8,5 and 4.2

The length of the hypotenuse is calculated as

Hyp^2 = Leg 1^2 + Leg 2^2

substitute the known values in the above equation, so, we have the following representation

Hypotenuse^2 = (8.5)^2 + 4.2^2

Evaluate

Hypotenuse^2 = 89.89

So, we have

Hypotenuse = 9.5

Hence, the hypotenuse = 9.5

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The probability of getting 2 twice in a row is 1/25 or 0.04.

Since the spinner has five equal parts labeled from 1 to 5, the probability of getting a 2 on any single spin is 1/5.

Since the spinner is spun twice, and we want to know the probability of getting 2 twice in a row, we need to multiply the probability of getting a 2 on the first spin by the probability of getting a 2 on the second spin, assuming that a 2 was already spun on the first spin.

Therefore, the probability of getting 2 twice in a row is (1/5) x (1/5) = 1/25, or 0.04, or 4%.

So, the probability of getting 2 twice in a row is 1/25 or 0.04.

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There is no value of x in the set {4, 7, 41} that makes the equation 7(x + 37) = 287 true.

To solve for x, we can use substitution. We substitute 7 for x in the equation and see if both sides are equal:

7(x + 37) = 287

7(7 + 37) = 287

7(44) = 287

308 = 287

Since 308 does not equal 287, we can conclude that 7 is not the value of x that makes the equation true. We can try the other values in the set and see if they work:

4(x + 37) = 287

4(4 + 37) = 287

4(41) = 287

164 = 287

Again, 164 does not equal 287, so 4 is not the value of x that makes the equation true.

Finally, we can try the last value in the set:

41(x + 37) = 287

41(41 + 37) = 287

41(78) = 287

3198 = 287

This time, we get an equation that is not true.

Therefore, there is no value of x in the set {4, 7, 41} that makes the equation 7(x + 37) = 287 true.

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To address the farm manager's concern about the effectiveness of fertilizer A compared to fertilizer B, formulate the null hypothesis (H0) and alternative hypothesis (H1):

Based on the experiment conducted by the farm manager, it was determined that there was no significant difference in the effectiveness of the two fertilizers. This was determined through a statistical test conducted at the 5 percent significance level. The experiment involved randomly selecting twenty identical plots of strawberries and fertilizing half with fertilizer A and half with fertilizer B. The yields were recorded and analyzed using statistical methods to determine if there was a significant difference in the effectiveness of the two fertilizers. The manager was able to conclude that the manufacturer's claim of cheaper fertilizer A being at least as effective as more expensive fertilizer B was supported by the statistical results of the experiment.

To address the farm manager's concern about the effectiveness of fertilizer A compared to fertilizer B, we can follow these steps:

1. Formulate the null hypothesis (H0) and alternative hypothesis (H1):
H0: Fertilizer A is at least as effective as fertilizer B (Yield_A ≥ Yield_B)
H1: Fertilizer A is less effective than fertilizer B (Yield_A < Yield_B)

2. Conduct the experiment: Randomly select 20 identical plots of strawberries, with 10 plots receiving fertilizer A and the other 10 receiving fertilizer B.

3. Record the yields for each plot and calculate the average yield for both fertilizer groups.

4. Perform a statistical test (such as a t-test) at the 5 percent significance level (α = 0.05) to compare the average yields of the two fertilizer groups.

5. Based on the p-value obtained from the statistical test, make a decision:
- If the p-value ≤ α (0.05), reject the null hypothesis (H0) and conclude that fertilizer A is less effective than fertilizer B.
- If the p-value > α (0.05), fail to reject the null hypothesis (H0) and conclude that there is insufficient evidence to suggest that fertilizer A is less effective than fertilizer B.

6. Report the results of the experiment to the farm manager, including the average yields for both fertilizers and the conclusion based on the statistical test. This will help the manager make an informed decision about which fertilizer to use.

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One error-detection method that can compensate for burst errors is the cyclic redundancy check (CRC).

This method involves adding extra bits to the data being transmitted, which can detect and correct errors in the data. and has error detection. CRC is particularly effective in detecting and correcting burst errors, which occur when a group of consecutive bits are corrupted in a data transmission.

An error-detection method that can compensate for burst errors is the "Reed-Solomon code". Reed-Solomon codes are block-based error correcting codes that can detect and correct multiple errors in data transmissions, making them highly effective in handling burst errors. These codes compensate for burst errors by using redundant information added to the original data, allowing the receiver to accurately reconstruct the original data even in the presence of errors.

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Based on the mentioned informations and provided values, the angle at which the sun's rays hit the ground is calculated to be approximately 35.87 degrees.

We can use the concept of trigonometry to solve this problem. Let's suppose A represents the top of the tower, B represents the bottom of the tower, and the line connecting A and B represents the shadow cast by the tower. Let's assume that the angle between the sun's rays and the ground is θ.

We can use the tangent function to relate the angle θ to the dimensions of the triangle ABP:

tan(θ) = opposite / adjacent = AB / BP

We know that AB = 22 and BP = 33, so:

tan(θ) = 22/33

Taking the arctangent of both sides, we get:

θ = arctan(22/33)

Using a calculator, we find:

θ ≈ 35.87°

Therefore, the angle at which the sun's rays hit the ground is approximately 35.87 degrees.

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We need to use the exponential distribution formula and the concept of the sum of exponential variables. Let X be the time between two consecutive breakdowns and Y be the time to repair a broken plane.

Both X and Y follow an exponential distribution with mean 1/2 and 1/7, respectively.
The time T until a plane is available again after a breakdown can be expressed as T=X+Y. The distribution of T is the convolution of the distributions of X and Y, which is also an exponential distribution with mean 9/14.
Let Z be the number of planes available after all breakdowns have occurred and repairs have been made. Z is a binomial variable with parameters n=100 and p=P(T>1), where P(T>1) is the probability that a breakdown occurs and is repaired in more than one week.
We want to find the minimum number of repairmen, denoted by k, such that P(Z≥90)≥0.95. Using the complement rule, we can rewrite this as P(Z<90)≤0.05.
Since Z follows a binomial distribution, we can use the normal approximation to compute P(Z<90). The mean of Z is np=100p, and the variance of Z is np(1-p). Therefore, Z can be approximated by a normal distribution with mean μ=100p and standard deviation σ=sqrt(np(1-p)).
To find p, we can use the fact that P(T>1)=exp(-λT), where λ=1/mean(X+Y)=14/9. Thus, p=1-exp(-λ). Plugging in the values, we get p=0.432.
Now, we can standardize Z by subtracting its mean and dividing by its standard deviation: Z'=(Z-μ)/σ. Then, we can use a standard normal table or calculator to find the corresponding probability: P(Z'<(90-μ)/σ).
Finally, we can solve for k using the inverse standard normal function: (90-μ)/σ=invNorm(0.05). Plugging in the values, we get k=30.
Therefore, we need at least 30 repairmen to ensure that there is at least a 95% chance that 90 or more planes are available.

To determine the number of repairmen needed for Allbest Airlines to ensure there is a 95% chance that 90 or more planes are available, we'll use the following terms: airlines, breakdowns, and exponential.
Step 1: Determine the required probability.
Since we want at least 90 planes available, it means that no more than 10 planes can be under repair at a time. Thus, we need to calculate the probability that 10 or fewer planes are under repair.
Step 2: Calculate the probability of a single plane being under repair.
Planes break down twice a year and take one week to fix. Therefore, the probability of a single plane being under repair in any given week is 2 breakdowns per year * 1 week per breakdown / 52 weeks per year = 1/26.
Step 3: Use the exponential distribution.
We can model the probability of a certain number of planes being under repair using the exponential distribution.
Step 4: Calculate the probability of 10 or fewer planes being under repair.
Using a cumulative distribution function (CDF) for the exponential distribution, we can calculate the probability of 10 or fewer planes being under repair to be at least 95%. This can be done using statistical software or a calculator with the appropriate functionality.
Step 5: Determine the number of repairmen needed.
If the probability calculated in step 4 is 95% or higher, the current number of repairmen is sufficient. If not, we need to increase the number of repairmen until the probability is at least 95%. This can be done by reducing the average repair time per plane, which in turn reduces the probability of a plane being under repair at any given time.
In conclusion, to ensure that there is a 95% chance that 90 or more planes are available for Allbest Airlines, we need to determine the number of repairmen needed based on the exponential distribution and probability calculations as outlined in the steps above.

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The coordinate system with x-coordinates between -180 and 180 is likely to be the geographic coordinate system, specifically longitude values. This system represents positions on the Earth's surface using two angles: longitude (x-coordinates) and latitude (y-coordinates).

Longitude values range from -180 degrees (180 degrees West) to 180 degrees (180 degrees East) along the Earth's equator. Latitude values, on the other hand, range from -90 degrees (90 degrees South) to 90 degrees (90 degrees North). The origin (0, 0) of this coordinate system is the intersection of the Prime Meridian (Greenwich Meridian) and the Equator.

The units for these coordinates are typically expressed in decimal degrees (DD). These values can be converted into other units such as degrees-minutes-seconds (DMS) or radians, depending on the application or preference. When working with geographic coordinate systems, it is essential to consider the Earth's curvature and potential distortions when calculating distances or areas.

In summary, if a data set has x-coordinates between -180 and 180, the coordinate system is most likely the geographic coordinate system, representing longitude values. The coordinates are typically expressed in decimal degrees, and their positions relate to the Earth's surface.

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Step-by-step explanation:

2x-3y=7 (-3,-3)

2x-7=3y

3y=2x-7

divide both sides by 3

y = 2/3x - 7/3

m1= 2/3

m1=m2 for parallel

2/3 = y-(-3)/x-(-3)

2/3= y+3/x+3

2 = y+3

– —

3 x +3

then you cross multiply

2(x+3)=3(y+3)

2x+6= 3y+9

then move everything to one side

2x-3y+6-9 =0

2x-3y-3=0

Answer:

[tex]\textsf{Slope-intercept form:} \quad y=\dfrac{2}{3}x-1[/tex]

[tex]\textsf{Standard form:} \quad 2x-3y=3[/tex]

Step-by-step explanation:

Parallel lines have the same slope.

Therefore, in order to find the equation of the line that is parallel to 2x - 3y = 7, we must first find the slope of this line by rearranging it in the form y = mx + b.

[tex]\begin{aligned}2x-3y&=7\\2x-3y+3y&=7+3y\\2x&=3y+7\\2x-7&=3y+7-7\\2x-7&=3y\\3y&=2x-7\\\dfrac{3y}{3}&=\dfrac{2x-7}{3}\\y&=\dfrac{2}{3}x-\dfrac{7}{3}\end{aligned}[/tex]

The equation y = mx + b is the slope-intercept form of a straight line, where m is the slope and b is the y-intercept. Therefore, the slope of the line is m = 2/3.

To find the equation of the line has a slope m = 2/3 and contains the point (-3, -3), we can use the point-slope form of a straight line.

[tex]\begin{aligned}y-y_1&=m(x-x_1)\\\\\implies y-(-3)&=\dfrac{2}{3}(x-(-3))\\\\y+3&=\dfrac{2}{3}(x+3)\\\\y+3&=\dfrac{2}{3}x+2\\\\y+3-3&=\dfrac{2}{3}x+2-3\\\\y&=\dfrac{2}{3}x-1\end{aligned}[/tex]

Therefore, the equation of the line that is parallel to the graph of 2x - 3y = 7 and contains the point (-3, -3) in slope-intercept form is:

[tex]\boxed{y=\dfrac{2}{3}x-1}[/tex]

If you want the equation in standard form, rearrange the equation to Ax + By = C (where A, B and C are constants and A must be positive):

[tex]\begin{aligned}y&=\dfrac{2}{3}x-1\\\\3 \cdot y&=3 \cdot \left(\dfrac{2}{3}x-1\right)\\\\3y&=2x-3\\\\3y+3&=2x-3+3\\\\3y+3&=2x\\\\3y+3-3y&=2x-3y\\\\3&=2x-3y\\\\2x-3y&=3\end{aligned}[/tex]

Therefore, the equation of the line that is parallel to the graph of 2x - 3y = 7 and contains the point (-3, -3) in standard form is:

[tex]\boxed{2x-3y=3}[/tex]

Answer: 6 turns

Step-by-step explanation:

The sum of t+4/t+5 and 9/t-1 with a common denominator is (t²+12t+41)/[(t+5)(t-1)].

To find a common denominator for the given expressions t+4/t+5 and 9/t-1, we need to determine the least common multiple (LCM) of the denominators (t+5) and (t-1):

The prime factorization of t+5 is (t+5).

The prime factorization of t-1 is (t-1).

Therefore, the LCM is (t+5)(t-1).

To convert t+4/t+5 into an equivalent fraction with the denominator (t+5)(t-1), we multiply both the numerator and denominator by (t-1):

t+4/t+5 = (t+4)(t-1)/[(t+5)(t-1)] = (t²+3t-4)/[(t+5)(t-1)]

To convert 9/t-1 into an equivalent fraction with the denominator (t+5)(t-1), we multiply both the numerator and denominator by (t+5):

9/t-1 = 9(t+5)/[(t+5)(t-1)] = (9t+45)/[(t+5)(t-1)]

Now both fractions have the same denominator, so we can add them:

(t²+3t-4)/[(t+5)(t-1)] + (9t+45)/[(t+5)(t-1)] = (t²+3t-4+9t+45)/[(t+5)(t-1)]

Simplifying the numerator gives:

t²+12t+41

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Answer:

99

Step-by-step explanation:

132 x ¾ = 396/4

396 divided by 4

99

Using the combined results gives a better probability estimate than using only one person's results.

The probability of each of them landing a 4 is calculated using the formula below:

The probability of each of them landing a 4 is given below:

Emma: 25/70 = 0.36 or 36%

Kyran: 20/50 = 0.4 or 40%

Polly: 23/80 = 0.29 or 29%

b) The combined estimated probability of the spinner landing on 4 is calculated as follows:

Total number of times spinner landed on 4 = 25 + 20 + 23 = 68

Total number of spins = 70 + 50 + 80 = 200

Combined estimated probability = 68/200

The combined estimated probability = 0.34 or 34%

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The value of length of AC is,

⇒ AC = 8.25

We have to given that;

CD = 6.6 cm, DE = 3.4 cm, CE = 4.2 cm, and BC = 5.25 cm

Now, We can formulate;

BC / AC = EC / CD

Substitute the values we get;

5.25 / AC = 4.2 / 6.6

Solve for AC;

5.25 x 6.6 / 4.2 = AC

AC = 8.25

Thus, The value of length of AC is,

⇒ AC = 8.25

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Based on the recommendation of having at least 10 observations per predictor in the training data set, with 20 predictor variables based on the 10 observations per predictor rule:20 predictor variables * 10 observations per predictor = 200 records for the training set.

In this study, the minimum number of records needed for the training set  alone would be 200 (10 x 20).

If the training set is planned to be 50% of the total data, then the total minimum number of records needed for the entire study would be 400 (200 / 0.5). To break it down further, the validation set would require a minimum of 120 records (10 x 20 x 0.3 = 120) and the testing set would require a minimum of 80 records (10 x 20 x 0.2 = 80).
Therefore, the total minimum number of records needed for this study would be 400 (200 for training + 120 for validation + 80 for testing).

Since the training set represents 50% of the total data, we can use this information to calculate the total number of records needed for the entire study:200 records (training set) / 0.5 (50% proportion of training set) = 400 total records

Therefore, a minimum of 400 records should be available for this study to ensure stable results in the predictive analytics model with 20 predictor variables.

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A. Therefore, if (x, y) and (y, x) are in R, then x = y, which means that xRy and yRx only if x = y, which is not true for distinct elements x, y in A.

A relation R on a set A is said to be symmetric if for any elements x, y in A, if xRy, then yRx. A relation R on a set A is said to be antisymmetric if for any distinct elements x, y in A, if xRy, then it is not true that yRx.

One example of a relation on a set that is both symmetric and antisymmetric is the equality relation. Let A be any set and let R be the equality relation, defined as:

R = {(x, y) | x = y for x, y in A}

Then, R is symmetric because if x = y, then y = x for any x, y in A. Therefore, if (x, y) is in R, then (y, x) is also in R. R is also antisymmetric because if x = y and y = x, then x = y for any x, y in A. Therefore, if (x, y) and (y, x) are in R, then x = y, which means that xRy and yRx only if x = y, which is not true for distinct elements x, y in A.

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A random sample is a set of observations that are chosen randomly from a larger population. Each observation in the sample is independent and identically distributed (i.e. has the same underlying probability distribution).

A probability density function (PDF) is a mathematical function that describes the likelihood of a random variable taking on a particular value or range of values. It's used to model continuous random variables (as opposed to discrete random variables, which have probability mass functions).

Now, let's apply these concepts to the problem at hand.

We have a random sample Y1, Y2, ..., Yn from a distribution with the PDF:

f(y; λ) = λe^(-λy)    for y ≥ 0

where λ > 0 is a parameter of the distribution.

We want to find the maximum likelihood estimator (MLE) of λ based on this sample. The MLE is the value of the parameter that maximizes the likelihood function, which is the joint PDF of the sample.

The joint PDF of the sample is given by:

f(y1, y2, ..., yn; λ) = λ^n * e^(-λ(y1 + y2 + ... + yn))

To find the MLE of λ, we need to maximize this function with respect to λ. However, it's easier to work with the logarithm of the likelihood function, since the logarithm is a monotonic function and will preserve the location of the maximum.

Taking the logarithm of the likelihood function, we get:

log(L) = n*log(λ) - λ(y1 + y2 + ... + yn)

To maximize this function, we take the derivative with respect to λ and set it equal to zero:

d/dλ [log(L)] = n/λ - (y1 + y2 + ... + yn) = 0

Solving for λ, we get:

λ = n / (y1 + y2 + ... + yn)

This is the MLE of λ based on the sample. Note that this estimator depends on the values of the sample observations, which makes sense since the estimator is trying to capture the underlying distribution of the population based on the observed data.

To verify that this is a maximum, we can take the second derivative of the log-likelihood function with respect to λ:

d^2/dλ^2 [log(L)] = -n/λ^2 < 0

Since the second derivative is negative, this confirms that the MLE is a maximum.

So, to summarize: given a random sample Y1, Y2, ..., Yn from a distribution with the PDF f(y; λ) = λe^(-λy) for y ≥ 0, the maximum likelihood estimator of λ is λ = n / (y1 + y2 + ... + yn). This estimator captures the underlying distribution of the population based on the observed data in the sample.

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The linear regression equation is y = 4,035.33 + 2,115x.

The correlation coefficient is 0.976321113.

The type of correlation is a positive linear correlation.

Yes, the correlation is strong because the correlation coefficient approximately equals to 1.

The amount of Krabby Patties made by Spongebob after working 10 years is $25,185.

In this scenario, the years worked (x) would be plotted on the x-axis of the scatter plot while the patties made (y) would be plotted on the x-axis of the scatter plot.

By critically observing the scatter plot (see attachment) which models the relationship between the years worked (x) and the Patties made (y), an equation for the linear regression is given by:

y = 41,461.54 + 2,714.46x

Next, we would predict the amount of Krabby Patties made by Spongebob after working 10 years as follows;

y = 4,035.33 + 2,115(10)

y = 4,035.33 + 2,1150

y = $25,185.33 ≈ $25,185.

In conclusion, there is a strong correlation between the data because the correlation coefficient (r) approximately equals to 1;

0.7＜|r| ≤ 1   (strong correlation)

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