(Unit 2) What makes the results of a study statistically significant?

The order of the PDEs, their linearity, and homogeneity are:

(a) Order: 2, Linear, Homogeneous
(b) Order: 1, Non-linear, Inhomogeneous
(c) Order: 3, Linear, Homogeneous


(a) yuxx + xuyy = 0
This PDE has second-order partial derivatives (uxx and uyy). It is linear because the highest power of u or its derivatives is 1. It is homogeneous because there are no terms without u or its derivatives.

(b) (x + y)u + eˣ + xyday = 0
This PDE has a first-order partial derivative (day). It is non-linear because the term xyday contains a product of u and a dependent variable (y). It is inhomogeneous because of the eˣ term, which does not contain u or its derivatives.

(c) Uyu + ux = Uxyz + xyz
This PDE has third-order partial derivatives (Uxyz). It is linear because the highest power of u or its derivatives is 1. It is homogeneous because there are no terms without u or its derivatives.

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The interval that contains 99.7% of IQ scores is between a score of 55 and 145.

Here is a rough sketch of the normal distribution curve for IQ scores with a mean of 100 and standard deviation of 15:

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------|----------------

    55  70  85 100 115 130 145

To use the empirical rule to show that 95% of IQ scores are between 70 and 130, we can start by finding the z-scores for these values:

z-score for 70 = (70 - 100) / 15 = -2

z-score for 130 = (130 - 100) / 15 = 2

According to the empirical rule, 95% of data falls within 2 standard deviations of the mean. Since the standard deviation is 15, this means that 95% of data falls between -30 and 30 points from the mean. In terms of z-scores, this means that 95% of data falls between -2 and 2. Since the z-scores for 70 and 130 are within this range, we can conclude that 95% of IQ scores are between 70 and 130.

To find the interval that contains 99.7% of IQ scores, we can use the same logic but change the number of standard deviations to 3, since 99.7% of data falls within 3 standard deviations of the mean according to the empirical rule.

z-score for lower end of interval = (100 - 3 * 15 - 100) / 15 = -3

z-score for upper end of interval = (100 + 3 * 15 - 100) / 15 = 3

So the interval that contains 99.7% of IQ scores is between a score of 55 and 145.

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Since each insertion has an amortized time complexity of O(1), the overall time complexity for filling an array of size n is O(n).

When using a dynamically resizing array, the array's capacity is doubled with each insertion. The amortized time of each insertion can be analyzed using the accounting method. In this case, let's assign a cost of 3 for each insertion operation:

1 for the actual insertion and 2 as a "token" that will be used later when the array needs to be resized. Now, let's analyze how the tokens are used: - When the array is resized the first time, it has a capacity of

2. It has 2 tokens (1 for each of the 2 elements), which are enough to pay for copying those elements to the new array. - When the array is resized again (capacity = 4), it has 4 tokens (1 for each element).

Again, there are enough tokens to pay for copying the elements to the new array. - This pattern continues as the array keeps doubling in size. Since the total cost of each insertion is 3, the amortized time complexity of each insertion is O(1).

For the overall process of filling an array of size n, we can calculate the total time complexity as the sum of the cost of individual insertions.

Since each insertion has an amortized time complexity of O(1), the overall time complexity for filling an array of size n is O(n).

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The study is an observational study.

The doctor's study can be categorized as an observational study. The patients are randomly selected into two groups, one receiving the medication and the other receiving a placebo, without any intervention or manipulation by the doctor. The patients are then observed over a period of six weeks, with the doctor monitoring their condition and recording any changes in the frequency or severity of migraines.

The study is classified as an observational study because the doctor is not actively manipulating or controlling any variables. The patients are assigned to the medication or placebo group randomly, without any interference from the doctor.

The doctor simply observes and records data on the patients' migraines over time, without intervening or changing the patients' conditions. This type of study is useful for investigating associations or correlations between variables, but it does not allow for direct causal conclusions to be drawn.

Therefore, the study is an observational study.

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The function f(x) = ln(ln2x+3) is equivalent to the function f(x) = 2x+3.

This means that the natural logarithm function is used to transform the argument ln2x+3 into the exponent 2x+3.

The given function is:

f(x) = ln(ln2x+3)

The natural logarithm function ln(x) is the inverse of the exponential function [tex]e^x[/tex].

It takes a positive input x and returns the exponent y such that [tex]e^y[/tex] = x.

The argument of the natural logarithm function is ln2x+3, which means that we need to find the value of y such that [tex]e^y[/tex] = ln2x+3.

To do this, we can exponentiate both sides of the equation with the base e:

[tex]e^y[/tex]= ln2x+3

[tex]e^{(e^y)[/tex]= [tex]e^{(ln2x+3)[/tex]

[tex]e^{(e^y)[/tex]= 2x+3

Now, we can express the original function in terms of this new expression:

f(x) = ln (ln2x+3)

f(x) =[tex]ln(e^y)[/tex]

f(x) = y

Substituting the expression we found earlier for y, we get:

f(x) = [tex]e^y[/tex]

f(x) = 2x+3

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Answer:

4

Step-by-step explanation:

Answer:3

Step-by-step explanation:

The probability that a randomly selected lemur has a tail shorter than 1.7 feet is: 0.266

We can solve this using the standard normal distribution by first standardizing the value of 1.7 feet:

z = (1.7 - 1.95) / 0.2 = -1.25


To find the probability that a randomly selected lemur has a tail shorter than 1.7 feet, we need to calculate the z-score first:

z = (X - μ) / σ
z = (1.7 - 1.95) / 0.2
z = -0.25 / 0.2
z = -1.25

Now, use a z-table to find the probability corresponding to the z-score of -1.25. The probability is approximately 0.211. However, this value is not listed among the given options.

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y = [tex]2^x +e^4x - cos(e^3x)[/tex]  b. y =[tex]3e^2x[/tex] / √2x+1.

dy/dx = d/dx ([tex]2^x)[/tex] + d/dx ([tex]e^4x)[/tex] - d/dx [tex](cos(e^3x))[/tex]

= [tex]2^x[/tex]ln(2) + 4[tex]e^4x[/tex] + sin[tex](e^3x) (3e^3x)[/tex]

= [tex]2^x[/tex] ln(2) + 4[tex]e^4x[/tex] + [tex]3e^3x sin(e^3x)[/tex]

Therefore, the derivative of y with respect to x is

 [tex]2^x[/tex] ln(2) + 4[tex]e^4x[/tex] + [tex]3e^3x sin(e^3x)[/tex]

dy/dx = [3([tex]e^2x[/tex])(√(2x+1))' - (√(2x+1))(3[tex]e^2x[/tex])'] / (2x+1)]

= [3([tex]e^2x[/tex])/(2√(2x+1))) - (3[tex]e^2x[/tex])(1/[tex](2(2x+1/2)^(3/2)[/tex]))] / (2x+1)]

= [3[tex]e^2x([/tex][tex]2(2x+1/2)^(3/2)[/tex] - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

= [3[tex]e^2x[/tex](4x+2) - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

Therefore, the derivative of y with respect to x is

= [3[tex]e^2x[/tex](4x+2) - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

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The dimensions of the base are [tex]25 \sqrt{2} by 100/ \sqrt{2}[/tex], and the height of the box is [tex]5 \sqrt{2}[/tex]

Let's start by defining the variables we need:

L: length of the base

W: width of the base

H: height of the box

From the problem statement, we know that:

L = 4W (the length of the base is 4 times larger than the width)

V = LWH = 25 (the volume of the box is 25 cubic feet)

We want to minimize the cost of construction, which is composed of two

parts: the cost of the base and the cost of the four sides.

Let's write expressions for these costs:

Cost of the base: 5LW

Cost of the four sides: 4WH + 2LH

The total cost is then:

C = 5LW + 3(4WH + 2LH)

Substituting L = 4W and V = LWH = 25, we get:

C = 5(4W)W + 3(4W)(25/4W) + 3(2H)(25/W)

Simplifying and factoring out 25, we get:

C = 75 + 30W + 150/H

To minimize C, we need to find the values of W and H that minimize this expression. We can use calculus for that:

[tex]dC/dW = 30 - 150/H^2 = 0[/tex]

[tex]dC/dH = -150W/H^2 = 0[/tex]

From the second equation, we can see that either W = 0 or H = 0, which is not physically meaningful. So we must have:

W = 5H

Substituting this into the first equation, we get:

[tex]30 - 150/H^2 = 0[/tex]

Solving for H, we get:

[tex]H = \sqrt{ (150/3)} = 5 \sqrt{2}[/tex]

Substituting this into W = 5H, we get:

[tex]W = 25 \sqrt{2}[/tex]

Finally, we can use L = 4W and V = LWH = 25 to find:

[tex]L = 100/ \sqrt{2} \\H = 5 \sqrt{2} \\W = 25 \sqrt{2}[/tex]

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The mouse traveled 4 meters while pushing the block of cheese with 4 N of force if she did 16 J of work.

An equation is a mathematical statement that shows that two expressions are equal. Equations typically consist of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

We know that work (W) is equal to force (F) times distance (d) in the direction of the force, so we can use the formula:

W = F x d

To find the distance traveled (d), we need to rearrange the formula:

d = W / F

Plugging in the values we have:

d = 16 J / 4 N

d = 4 meters

Therefore, the mouse traveled 4 meters while pushing the block of cheese with 4 N of force, if she did 16 J of work.

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Answer:

Step-by-step explanation:

the answer is 2 because if you subtract and add you will get your and and good luck on the state test 5th 6th and 7th and younger kids.

The economist can conclude that the mean FICO score is less than 720 with a 95% confidence level.

To answer this question, we can use a one-sample t-test with a significance level of α=0.05.

First, we need to calculate the t-statistic:

t = (sample mean - hypothesized mean) / (standard deviation / √(sample size))

t = (695 - 720) / (65 / sqrt(60))
t = -2.81

Next, we need to find the critical t-value using a t-distribution table with 59 degrees of freedom (sample size - 1) and a significance level of α=0.05.

The critical t-value is -1.67 (one-tailed test).

Since the calculated t-statistic (-2.81) is less than the critical t-value (-1.67), we can reject the null hypothesis and conclude that the mean FICO score is significantly lower than 720. In other words, based on the sample data, the economist can conclude that the mean FICO score is less than 720 with a 95% confidence level.

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By binomial distribution ,0.9794 = 97.94% probability that at least 1 of the people selected will always take medicine as prescribed.

What is binomial distribution?

In probability theory and statistics, the binomial distribution is the discrete probability distribution which gives only two possible outcomes in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possibility: heads or tails. This type of distribution is said to be a binomial probability distribution.

In a large population, the probability that a person will always take medicine as prescribed  is 0.54.

So there are two chances. Either they take medicine or not.

Let us assume that the people taking medicines are considered as success and those people who are not taking medicines are  considered as failure.

The problem can be solved by binomial distribution.

By binomial distribution the formula is:

[tex]P(X=x) = C_{n,x} p^{x} q^{n-x}[/tex]  ---------------(1)

Where x= number of success

          n= number of trials

          p= probability of success in one trial

          q= 1-p = probability of failure in one trial.

and [tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex] -------------(2)

In the given problem, the probability that a person will always take medicine as prescribed  is 0.54. So p= 0.54

5 people are selected at random from the population.

so n= 5

The probability that at least 1 of the people selected will always take medicine as prescribed can be written in the format is

P(X≥1)= P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5) -------------(3)

Now, we have to find each value of equation (3) using equation (1) and equation (2).

P(X=1):-

P(X=1)= C₅,₁ (0.54)¹ (1-0.54)⁵⁻¹

         = 5× 0.54×(0.46)⁴

         = 0.12089

P(X=2):-

P(X=2)= C₅,₂ (0.54)² (1-0.54)⁵⁻²

         = 0.28383

P(X=3):-

P(X=3)= C₅,₃ (0.54)³ (1-0.54)⁵⁻³

          = 0.33319

P(X=4):-

P(X=4)= C₅,₄ (0.54)⁴ (1-0.54)⁵⁻⁴

          = 0.19557

P(X=5):-

P(X=5)= C₅,₅ (0.54)⁵ (1-0.54)⁵⁻⁵

          = 0.04592

Now putting all the values in equation (3) we get,

P(X≥1)= 0.12089+ 0.28383 + 0.33319 +0.19557+ 0.04592

         = 0.9794

Hence, by binomial distribution 0.9794 = 97.94% probability that at least 1 of the people selected will always take medicine as prescribed.

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A Pearson's product-moment correlation can result in a coefficient ranging from -1 to 1, with 0 indicating no correlation and positive or negative values indicating the direction and strength of the correlation.

A coefficient close to 1 or -1 indicates a strong correlation, while a coefficient close to 0 indicates a weak correlation. It is important to note that correlation does not imply causation and that the results should be interpreted with caution. Additionally, the validity of the results depends on the quality of the data and the content loaded into the analysis.

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Answer:

Step-by-step explanation:

Sorry It looks so blurry, find the radius then square it, times it by pie/3.14

The central angle measurement of the section that makes up 45% of the circle graph is 162 degrees. Answer: 162°

It is the center of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

If one section makes up 45% of the circle graph, then the other three sections combined make up the remaining 55% of the graph. Since the circle graph represents a full circle, which has a total central angle measurement of 360 degrees, we can set up the following proportion:

45/100 = x/360

where x is the central angle measurement of the section that makes up 45% of the graph. To solve for x, we can cross-multiply and simplify:

45 * 360 = 100 * x

x = 16,200/100

x = 162

Therefore, the central angle measurement of the section that makes up 45% of the circle graph is 162 degrees. Answer: 162°

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The conjugate of √8 - √9 is as follows:

(√8 + √9).

A pair of entities connected together is referred to as being conjugate. For instance, the two smileys—smiley and sad—are identical save from one set of characteristics that is essentially the complete opposite of the other. These smileys are identical, but you'll see if you look closely that they have the opposite facial expressions: one has a smile, and the other has a frown. Similar to this, the term "conjugate" in mathematics designates either the conjugate of a complex number or the conjugate of a surd when the number only undergoes a sign change with respect to a few constraints.

Here in the question,

The binomial is given as:

√8 - √9

The negative of this or when the operation sign is changed in the binomial, we get the conjugate as:

√8 + √9

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The representation shows the amount of money Cory earned at this rate is f(x) = $35x

One possible representation to show the amount of money Cory earned at the rate of $35 per lawn is:

Let "n" be the number of lawns mowed by Cory.

The amount of money he earned would then be:

$35n

Another possible representation is using a function:

Let "f(x)" be the amount of money Cory earned after mowing "x" lawns.

Then: f(x) = $35x

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The y-intercept of the sample regression line is approximately 1.9854.

To find the y-intercept of the sample regression line, we can use the following formula:

y-intercept (b₀) = (Σy - b₁ * Σx) / n

where b₁ is the slope of the regression line, n is the number of observations, Σx and Σy are the sums of the x and y values respectively. To find b₁, we use the formula:

b₁ = (n * Σ(xy) - Σx * Σy) / (n * Σ(x²) - (Σx)²)

We are given:
n = 27
Σy = 57.2
Σx = 1253.4
Σ(xy) = 3133.7
Σ(x²) = 73296.4

First, let's find b₁:

b₁ = (27 * 3133.7 - 1253.4 * 57.2) / (27 * 73296.4 - 1253.4²)
b₁ ≈ -0.0236

Now, we can find the y-intercept (b₀):

b₀ = (57.2 - (-0.0236) * 1253.4) / 27
b₀ ≈ 1.9854

Therefore, we can state that the y-intercept of the sample regression line is approximately 1.9854.

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The best measure of variability for the data is the standard deviation the player that more consistent is Option B: Player B.

What is standard deviation?

The standard deviation is a metric that reveals how much variance from the mean there is, including spread, dispersion, and spread. A "typical" variation from the mean is shown by the standard deviation. Because it uses the data set's original units of measurement, it is a well-liked measure of variability.

The best measure of variability for this data would be the standard deviation.

To determine which player was more consistent, we need to calculate the standard deviation for each player's data set.

For Player A, the mean is 3 and the standard deviation is 2.

For Player B, the mean is 2.44 and the standard deviation is 1.41.

Since Player B has a lower standard deviation, they are more consistent than Player A.

Therefore, the correct answer is -

Option B: Standard deviation; Player B was more consistent.

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The 98% confidence interval for the population mean (μ) is approximately (2.711, 3.009).

1: Identify the given data

Sample size (n) = 150 students

Sample mean (x) = 2.86

Population standard deviation (σ) = 0.78

Confidence level = 98%

2: Find the critical z-value (z*) for a 98% confidence level

Using a z-table or calculator, you'll find that the critical z-value for a 98% confidence level is 2.33 (approximately).

3: Calculate the standard error (SE)

SE = σ / √n

SE = 0.78 / √150 ≈ 0.064

4: Calculate the margin of error (ME)

ME = z* × SE

ME = 2.33 × 0.064 ≈ 0.149

5: Construct the confidence interval

Lower limit = x - ME = 2.86 - 0.149 ≈ 2.711

Upper limit = x + ME = 2.86 + 0.149 ≈ 3.009

The 98% confidence interval is approximately (2.711, 3.009).

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The lateral area of the given cone is 13.5π cm².

What is the lateral area of a cone?

The lateral area of a cone is the total area of the curved surface of the cone, excluding the area of the circular base. It is the area of the lateral or side surface of the cone.

The formula for the lateral area of a cone is LA = πrℓ, where r is the radius of the base of the cone, and ℓ is the slant height of the cone.

To find the lateral area of a cone, we use the formula LA = πrℓ, where r is the radius of the base of the cone, and ℓ is the slant height.

Given that the slant height ℓ = 15 cm and the radius of the base r = 9 mm = 0.9 cm.

Therefore, the lateral area LA = πrℓ = π(0.9)(15) = 13.5π cm² (rounded to one decimal place)

Hence, the lateral area of the given cone is 13.5π cm².

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The scenario presented highlights the importance of emergency response times for accident victims, particularly those with life-threatening injuries. The fact that emergency personnel in one city took more than 8 minutes to arrive on 22% of all calls involving such injuries underscores the need for improvement.

To assess whether there has been any improvement after 6 months, the city manager selects a sample of 400 calls involving life-threatening injuries and examines the response times. She then performs a test at the ag = 0.05 level, with the null hypothesis (H0) being that the true proportion of calls for which emergency personnel took more than 8 minutes to arrive is 0.22, and the alternative hypothesis (Ha) being that the true proportion is less than 0.22. This test will help determine whether there has been a significant improvement in emergency response times over the past 6 months. It is crucial that emergency response times are monitored and improved upon to ensure that accident victims receive the care they need in a timely manner.

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False. For dependent events, the probability of B may not be equal to the probability of B, given A.

The probability of B given A takes into account the knowledge that A has occurred and may therefore be different from the probability of B without any knowledge of A. The formula for conditional probability is P(B|A) = P(A and B)/P(A), where P(A and B) is the probability that both A and B occur, and P(A) is the probability that A occurs. In general, the probability of B given A may be greater or smaller than the probability of B without any knowledge of A

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The reasonable total amount spend by Mr. Habib is $32 under the condition that the total number of gifts was 8 which ranged from $2 and $5 on each gift. Then the  required  correct option is Option C.

To evaluate the following question we have to implement basic multiplication of numbers

In case of spending $2 for each gift
Amount Spend = 2× 8 = $16
In case of spending $5 for each gift
Amount Spend = 5×8 = $40

So when we compare the amounts generated after choosing any one of the given cases, the in between option that is suitable and meets the criteria is $32.

The reasonable total amount spend by Mr. Habib is $32 under the condition that the total number of gifts was 8 which ranged from $2 and $5 on each gift. Then the correct option is Option C.

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The odds ratio of 11.25 indicates that individuals who ate lettuce in their submarine sandwiches were approximately 11 times more likely to become ill compared to those who did not eat lettuce.

To calculate the odds ratio in this case-control study, we need to use the formula:

Odds ratio = (a/c) / (b/d)

Where:
a = number of case-patients who ate lettuce
b = number of case-patients who did not eat lettuce
c = number of controls who ate lettuce
d = number of controls who did not eat lettuce

Plugging in the values given in the question, we get:

Odds ratio = (53/1) / (33/7) = 184.67

The odds ratio in this case is 184.67. This means that those who ate lettuce in their submarine sandwich were 184.67 times more likely to become ill with vomiting and diarrhea than those who did not eat lettuce.

In other words, the odds of getting sick after eating lettuce were nearly 185 times higher for the case-patients than for the controls. This suggests a strong association between eating lettuce and becoming ill and indicates that lettuce was likely the source of the outbreak.

To calculate the odds ratio for the association between eating lettuce and becoming ill after the company luncheon, we need to compare the odds of exposure (eating lettuce) among the case-patients (those who became ill) and the controls (those who did not become ill). First, let's create a 2x2 table based on the provided information:

```
             Ill (Cases)   Not Ill (Controls)
Lettuce       53            33
No Lettuce    1             7
```

Now, we can calculate the odds ratio (OR) using the formula: (odds of exposure in cases) / (odds of exposure in controls).

Odds of exposure in cases = 53/1 = 53
Odds of exposure in controls = 33/7 ≈ 4.71

Odds ratio (OR) = 53 / 4.71 ≈ 11.25

The odds ratio of 11.25 indicates that individuals who ate lettuce in their submarine sandwiches were approximately 11 times more likely to become ill compared to those who did not eat lettuce. This suggests a strong association between eating lettuce and the risk of becoming ill after the company luncheon, implying that lettuce might be the potential source of the outbreak.

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Assuming the homoskedastic normal regression assumption holds, the critical value can be calculated using the t-distribution.
(a) n=28, 5% significance level, one-sided test
1. Determine the degrees of freedom: df = n-2 = 28-2 = 26
2. Look up the critical value in a t-distribution table for a 5% significance level and 26 degrees of freedom (one-sided): t-critical = 1.706

(b) n=40, 1% significance level, two-sided test
1. Determine the degrees of freedom: df = n-2 = 40-2 = 38
2. Look up the critical value in a t-distribution table for a 1% significance level and 38 degrees of freedom (two-sided): t-critical = 2.712

(c) n=10, 10% significance level, one-sided test
1. Determine the degrees of freedom: df = n-2 = 10-2 = 8
2. Look up the critical value in a t-distribution table for a 10% significance level and 8 degrees of freedom (one-sided): t-critical = 1.397

(d) Since n=0 in this situation, it is impossible to calculate a critical value. The sample size should be greater than 0 for a meaningful test.

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The value of x in the equation is 9

From the information given, we have that the angles are supplementary.

Then, it is important that we note the definition of supplementary angles.

Supplementary angles are simply defined as pair of angles that sum to 180 degrees. They must be two angles.

Also, angles on a straight line is equal to 180 degrees.

From the information given, we have that;

8x + 6 and 102 degrees are supplementary.

Then,

8x + 6 + 102 = 180

collect the like terms, we get;

8x = 180 - 108

subtract the values

8x = 72

x = 9

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Question

If the angles are supplementary, What is the value of x?

(8x + 6)

102°

▸

Based on the given information, we know that the cost of a bag of chips is $52.20 more than the cost of both water bottles.

To solve this problem, we need to set up an equation based on the given information. Let's assume the cost of one water bottle is "w" and the cost of one bag of chips is "c". We also know that you picked up "d" dalions of milk bottles and "r" raise bags of Youttore.

Therefore, the equation will be:
d*w + r*c + 2*w + c = 280

Simplifying the equation, we get:
d*w + r*c + 3*w + c = 280

We also know that "c" is $52.20 more than the cost of both water bottles, so we can substitute that in the equation:
d*w + r*(w+52.20) + 3*w + (w+52.20) = 280

Simplifying again, we get:
d*w + r*w + 3*w + 53.20r + 52.20 = 280

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The range for this set of data is 4.

The distance between the largest and smallest values in a collection of data is known as the range in statistics. It can be used as a measure of variability and provides a notion of how dispersed the data is. By deducting the least value from the maximum value, the range is calculated:

Range: (Maximum Value - Minimum Value)

From the given plot we see that the highest value is 5 and the lowest value is 1.

The range is thus given as:

Range = highest value - lowest value

Range = 5 - 1

Range = 4

Hence, the range for this set of data is 4.

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