4


Consider the inequalities -1/4a > 3 a and b – 12> -3. What values, if any, make


both inequalities true? Show your work.

The value of X and y when substitution method is used to solve the given quadratic equation would be = 8 and 2 respectively.

The equations that are given is listed below:

X - 3y = 2 ---> equation 1

2x - 6y = 6 ----> equation 2

In equation 1, make X the subject of formula;

X = 2 + 3y

Substitute X = 2 + 3y into equation 2,

2( 2 + 3y) - 6y = 6

4 + 6y - 6y = 6

y = 6-4

y = 2

Substitute y = 2 into equation 1;

x - 3(2) = 2

X = 2 + 6

X= 8

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Answer:

B

Step-by-step explanation:

Every fifth people means one person from 5 people in total. So when we convert that into numbers, it becomes  [tex]\frac{1}{5}[/tex].

And in total there are 85 people involved, so the answer is

[tex]85[/tex] × [tex]\frac{1}{5}[/tex]

Answer:

Step-by-step explanation:

correct asnswer b

Answer:

The answer to your problem is, 39

Step-by-step explanation:

In order to find the area of the triangle use the formula down below:

A = [tex]\frac{h_{b} b}{2}[/tex]

Base = 13

Height = 6

Replace them equals:

= [tex]\frac{6*13}{2}[/tex] = 39

Thus the answer to your problem is, 39

The length of OP to the nearest tenth of a foot is approximately 41.5 feet

To find the length of OP, we can use the Pythagorean theorem since we have a right triangle.

OP^2 = PN^2 - ON^2

First, we need to find ON using the trigonometric ratio of tangent.

tan(39) = ON/PN

ON = PN * tan(39)
ON = 72 * tan(39)
ON ≈ 53.4 feet

Now we can plug in our values:

OP^2 = 72^2 - 53.4^2
OP^2 ≈ 1720.84
OP ≈ 41.5 feet (rounded to the nearest tenth of a foot)

Therefore, the length of OP to the nearest tenth of a foot is approximately 41.5 feet.

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The percent increase in temperature of the liquid is 61.9%. This means that the temperature increased by 61.9% of its original value.

When we want to calculate the percent increase in a value, we need to compare the new value to the old value.

In this case, the old value is the initial temperature of the liquid, which is 63 °F, and the new value is the final temperature of the liquid, which is 102 °F.

To calculate the percent increase, we use the formula I mentioned earlier, which subtracts the old value from the new value, divides the result by the old value.

And then multiplies the quotient by 100% to express the result as a percentage.

So, for this experiment, we can calculate the percent increase in temperature as:

((102 - 63) / 63) x 100% = 61.9%

This means that the temperature of the liquid increased by 61.9% of its original value. Alternatively, we can also say that the final temperature is 161.9% of the initial temperature.

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An equation representing the statement that 282828 is 121212 less than kkk is kkk = 282828 + 121212.

An equation is a mathematical statement showing that two or more mathematical or algebraic expressions share equality or equivalence.

Equations are represented using the equal symbol (=).

Unlike equations, mathematical expressions do not use the equal symbol.

282828 = kkk - 121212

kkk = 282828 + 121212

Thus, one way of representing the statement that that 282828 is 121212 less than kkk is by forming an equation like kkk = 282828 + 121212.

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The radius of the circle is 10 inches.

A sector is a part of a given circle which is made from two radii and an arc. It's area can be determined by;

area of a sector = θ/360*πr^2

where θ is the measure of its central angle, and r is its radius.

Then from the given question, we have;

area of a sector = θ/360*πr^2

162 = 180/360 *3.14*r^2

      = 1.57r^2

r^2 = 162/1.57

     = 103.1847

So that;

r = (103.1847)^1/2

 = 10.16

The radius of the circle is approximately 10 inches.

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At both the 5% and 1% significance levels, we have enough evidence to reject the null hypothesis that the proportion of US adults who have great confidence in banks is 15% or higher. Therefore, we can conclude that the economist's claim that less than 15% of US adults have great confidence in banks is supported by the data.

Null Hypothesis: The proportion of US adults who have great confidence in banks is 15% or higher.

Alternative Hypothesis: The proportion of US adults who have great confidence in banks is less than 15%.

We can use a one-tailed z-test to test the economist's claim.

The test statistic is

z = (P - p) / √(p * (1-p) / n)

where P is the sample proportion, p is the hypothesized proportion, and n is the sample size.

Using the sample data, we have

P = 149/1325 = 0.1121

p = 0.15

n = 1325

The test statistic is

z = (0.1121 - 0.15) / √(0.15 × (1-0.15) / 1325) = -3.196

Using a significance level of α = 0.05, the critical value for a one-tailed test is -1.645. Since our test statistic is less than the critical value, we reject the null hypothesis.

The p-value for this test is P(Z < -3.196) = 0.0007. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis.

At the 5% significance level, we have enough evidence to reject the null hypothesis that the proportion of US adults who have great confidence in banks is 15% or higher. Therefore, we can conclude that the economist's claim that less than 15% of US adults have great confidence in banks is supported by the data.

Using a significance level of α = 0.01, the critical value for a one-tailed test is -2.33. Since our test statistic is less than the critical value, we reject the null hypothesis.

The p-value for this test is P(Z < -3.196) = 0.0007. Since the p-value is less than the significance level of 0.01, we reject the null hypothesis.

At the 1% significance level, we have enough evidence to reject the null hypothesis that the proportion of US adults who have great confidence in banks is 15% or higher. Therefore, we can conclude that the economist's claim that less than 15% of US adults have great confidence in banks is supported by the data.

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2 cos x - 1 = 0, cos x = 1/2, and x = π/3 or 5π/3. These are the critical numbers of g(x) on (0,7). the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.

To find the critical numbers for g(x) = 2sin(r) - r on the interval (0, 7), follow these steps:
1. Find the derivative of g(x): g'(x) = 2cos(r) - 1
2. Set the derivative equal to zero: 2cos(r) - 1 = 0
3. Solve for r: r = cos^(-1)(1/2)
Now, find the absolute maximum and minimum values for g(x) on the interval (0, 7):
1. Evaluate g(x) at the critical numbers: g(cos^(-1)(1/2)) = 2sin(cos^(-1)(1/2)) - cos^(-1)(1/2)
2. Evaluate g(x) at the endpoints of the interval: g(0) = 2sin(0) - 0, g(7) = 2sin(7) - 7
3. Compare the values from steps 1 and 2 to find the maximum and minimum values.
The critical numbers for g(x) = 2sin(r) - r on the interval (0, 7) are r = cos^(-1)(1/2). The absolute maximum and minimum values for g(x) on the interval (0, 7) can be found by comparing g(cos^(-1)(1/2)), g(0), and g(7).

To find the critical numbers of g(x) = 2 sin x - x on (0,7), we first need to find its derivative:
g'(x) = 2 cos x - 1
Setting g'(x) = 0, we get:
2 cos x - 1 = 0
cos x = 1/2
x = π/3 or 5π/3
These are the critical numbers of g(x) on (0,7).

To find the absolute maximum and minimum values of g(x) on (0,7), we need to evaluate g(x) at the critical numbers and at the endpoints of the interval (0,7).
g(0) = 0
g(π/3) = 2 sin(π/3) - π/3 = √3 - π/3
g(5π/3) = 2 sin(5π/3) - 5π/3 = -√3 - 5π/3
g(7) = 2 sin(7) - 7
To determine the absolute maximum and minimum values, we compare these values:
The absolute maximum value is √3 - π/3, which occurs at x = π/3.
The absolute minimum value is -√3 - 5π/3, which occurs at x = 5π/3.
Therefore, the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.

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The limit of the given function as t approaches 0 is undefined.

To find the limit of the given function,[tex]lim (e^(5t) - 1) / (t × sin(t))[/tex] as t approaches 0, follow these steps:

Observe the given function

[tex]lim (e^(5t) - 1) / (t × sin(t)) as t → 0[/tex]

Apply L'Hopital's Rule, since the limit is of the form 0/0 as t approaches 0.

Differentiate the numerator and denominator with respect to t.

Numerator: [tex]d(e^(5t) - 1)/dt = 5e^(5t)[/tex]

Denominator: [tex]d(t × sin(t))/dt = sin(t) + t × cos(t)[/tex]

Rewrite the function with the new numerator and denominator.

lim [tex](5e^(5t)) / (sin(t) + t × cos(t)) as t → 0[/tex]

Evaluate the limit as t approaches 0.

[tex](5e^(5 × 0)) / (sin(0) + 0 × cos(0)) = 5 / 0[/tex]

Since the denominator is still 0, the limit does not exist.

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The measures of the angles JIK and JIL are 67 degrees and 157 degrees

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

The inscribed angle opposite to the same arc is half of the external angle

Using the above as a guide, we have the following:

JIK = 90 - 46/2

JIK = 67 degrees

Also, we have

JIL = 90 + JIK

So, we have

JIL = 90 + 67

JIL = 157 degrees

Hence, the measure of the angle JIL is 157 degrees

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The area of the sector is approximately 23.24 m^2.

To find the area of the sector, we first need to find the central angle of the sector.

The entire circumference of the circle is given by 2πr, where r is the radius of the circle. In this case, the circumference is 2π(3) = 6π m.

The arc length given is 47 m, which we can use to find the central angle of the sector:

central angle = (arc length / circumference) × 360°

central angle = (47 / 6π) × 360°

central angle ≈ 299.02°

Now that we have the central angle, we can use the formula for the area of a sector:

area of sector = (central angle / 360°) × πr^2

area of sector = (299.02 / 360) × π(3)^2

area of sector ≈ 7.43π m^2

Rounding to two decimal places, the area of the sector is approximately 23.24 m^2.

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The amount of total and per-unit overhead allocated to each of the three products using overhead rate is equal to,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530   $1,060,000

Clarinets $795    $1,192,500

Oboes $397.5     $695,625

Total $1,722.5    $2,948,125

Budgeted factory overhead cost = $2,948,125

The single plantwide overhead rate

= Dividing the budgeted factory overhead cost by the total budgeted direct labor hours.

For this,

Flutes= 2,000×2

         = 4,000 hours

Clarinets= 1,500×3

              = 4,500 hours

Oboes= 1,750×1.5

          = 2,625 hours

Total direct labor hours = 11,125

Substitute the value we have,

⇒ Single plantwide overhead rate = $2,948,125 / (2,000 x 2.0 + 1,500 x 3.0 + 1,750 x 1.5)

= $2,948,125 / 11,125

= $265 per direct labor hour

To allocate overhead to each product

=Multiply the overhead rate by the budgeted direct labor hours per unit for each product.

Substitute the value we have,

Flutes,

$265 x 2.0 = $530 total overhead cost, $265 per unit

Clarinets,

$265 x 3.0 = $795 total overhead cost, $265 per unit

Oboes,

$265 x 1.5 = $397.5 total overhead cost, $265 per unit

And

Total factory overhead cost allocated = Estimated manufacturing overhead rate×  Actual amount of allocation base

For,

Flutes

= 4,000× 265

= $1,060,000

Clarinets

= 4,500×265

= $1,192,500

Oboes

= 2,625×265

= $695,625

This implies,

The total and per-unit overhead allocated to each product, rounded to the nearest dollar is,

Total Per Unit Factory Overhead ,  Cost Factory Overhead Cost

Flutes $530 $1,060,000

Clarinets $795 $1,192,500

Oboes $397.5  $695,625

Total $1,722.5 $2,948,125

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The above question is incomplete, the complete question is:

Bach Instruments Inc. makes three musical instruments: flutes, clarinets, and oboes. The budgeted factory overhead cost is $2,948,125. Overhead is allocated to the three products on the basis of direct labor hours. The products have the following budgeted production volume and direct labor hours per unit:

Budgeted Production Volume Direct Labor Hours Per Unit

Flutes 2,000 units 2.0

Clarinets 1,500 3.0

Oboes 1,750 1.5

a. Determine the single plantwide overhead rate.

$ per direct labor hour

b. Use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.

Total Per Unit

Factory Overhead Cost Factory Overhead Cost

Flutes $ $

Clarinets

Oboes

Total $

Maria's average quiz score is 9.6.

B. The standard deviation of her quizzes is approximately 0.5 (rounded to the nearest tenth).

(a) The average of Maria's quizzes can be found by adding up all her scores and dividing by the total number of quizzes:

Average = (10 + 9 + 10 + 9 + 10) / 5 = 9.6

Therefore, Maria's average quiz score is 9.6.

(b) To calculate the standard deviation of her quizzes, we first need to find the variance. We can do this by finding the average of the squared differences between each score and the mean:

[(10 - 9.6)² + (9 - 9.6)² + (10 - 9.6)² + (9 - 9.6)² + (10 - 9.6)²] / 5 = 0.24

So the variance is 0.24. To find the standard deviation, we take the square root of the variance:

√0.24 ≈ 0.5

So the standard deviation of her quizzes is approximately 0.5 (rounded to the nearest tenth).

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A. Main Answer:

The hypothesis that the variability in both populations is the same against the alternative that it is larger in the second population, we can perform a permutation test using two different test statistics:

(i) the difference of sample ranges and

(ii) the ratio of sample variances. By comparing the results of both test statistics, we can draw conclusions about the variability in the populations.

(i) Difference of Sample Ranges:

1. Calculate the sample range for each sample. The sample range is the difference between the maximum and minimum values in the sample.

  Sample 1 Range = Maximum value - Minimum value for Sample 1

  Sample 2 Range = Maximum value - Minimum value for Sample 2

2. Calculate the observed difference of sample ranges, which is the difference between the sample range of Sample 2 and Sample 1.

3. Pool the data from both samples and shuffle them randomly, keeping the same sample sizes.

4. Calculate the difference of sample ranges for the shuffled data.

5. Repeat steps 3 and 4 many times (e.g., 1000 permutations) to obtain a distribution of the difference of sample ranges under the null hypothesis (where variability is the same in both populations).

6. Compare the observed difference of sample ranges from step 2 with the distribution obtained from the permutations.

(ii) Ratio of Sample Variances:

1. Calculate the sample variance for each sample.

2. Calculate the observed ratio of sample variances, which is the ratio of the sample variance of Sample 2 to the sample variance of Sample 1.

3. Pool the data from both samples and shuffle them randomly, keeping the same sample sizes.

4. Calculate the ratio of sample variances for the shuffled data.

5. Repeat steps 3 and 4 many times (e.g., 1000 permutations) to obtain a distribution of the ratio of sample variances under the null hypothesis.

6. Compare the observed ratio of sample variances from step 2 with the distribution obtained from the permutations.

By comparing the results of both test statistics, we can assess whether the variability in the second population is significantly larger than the first population. If both test statistics consistently indicate larger variability in the second population, it provides evidence against the null hypothesis and suggests that the variability in the second population is indeed larger.

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in the given triangle, using SOH CAH TOA, the value of x is 16.48

From the question, we are to determine the value of x in the given diagram

In the given diagram,

We are given a right triangle.

70° is the included angle

6 is the adjacent

and x is the opposite

Using SOH CAH TOA,

sin (angle) = Opposite / Hypotenuse

cos (angle) = Adjacent / Hypotenuse

tan (angle) = Opposite / Adjacent

Then, we can write that

tan (angle) = Opposite / Adjacent

Then,

tan (70°) = x/6

x = 6 × tan(70°)

x = 16.48

Hence, the value of x is 16.48

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The geometric term that can be used to describe the crossing sign shown is intersecting lines.

In geometry, intersecting lines are two lines that cross one another at a location known as the point of intersection. It is possible to use the point of intersection to solve issues concerning angles, segments, and geometric shapes because it is the sole point that both lines share.

Two pairs of opposite angles that are equal to one another are formed when two lines connect, giving rise to four angles.

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We know that the length of AB on the scale drawing is 10 inches

Using the scale of 1 inch: 250 feet, we can find the length of AB on the scale drawing by multiplying the actual length of AB by the scale factor.

The actual length of AB is the sum of the height of the tower (800 ft) and the length of the support wire (1,700 ft), which is 2,500 ft.

Multiplying 2,500 ft by the scale factor of 1 inch: 250 feet, we get:

2,500 ft ÷ 250 ft/inch = 10 inches

Therefore, the length of AB on the scale drawing is 10 inches.

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To find the area of a parallelogram, multiply the base length by the height. Therefore, the area of the parallelogram is 8 cm * 24 cm = 192 cm².

To find the area of a parallelogram, you can use the formula A = base × height. In this case, the given measurements are 8 cm for the base, 9 cm for the height, and 24 cm for one of the sides of the parallelogram.

First, identify the base and height of the parallelogram. In this case, the base is 8 cm and the height is 9 cm.

Next, substitute the values into the formula for the area of a parallelogram: A = base × height.

A = 8 cm × 9 cm

Multiply the base and height:

A = 72 cm²

Therefore, the area of the parallelogram is 72 square centimeters. It's important to note that the area is not drawn to scale, so the measurements given are solely used for calculation purposes.

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Answer:

$450

Step-by-step explanation:

Let's use the variables C, I, and F to represent the number of chocolate fudge bars, ice cream sandwiches, and frozen fruit bars, respectively, that you will sell.

The cost of each chocolate fudge bar is $0.75, the cost of each ice cream sandwich is $0.85, and the cost of each frozen fruit bar is $0.50. Therefore, the total cost of the frozen treats that you buy will be:

Total cost = 0.75C + 0.85I + 0.50F

We want to make sure that this total cost is no more than $450. Therefore, we can write the following inequality:

0.75C + 0.85I + 0.50F ≤ 450

This inequality represents the number of chocolate fudge bars, C, the number of ice-cream sandwiches, I, and the number of frozen fruit bars, F, that will cost you no more than $450.

Answer:15)5760.1 6)78. 17)582.4 18)112

Step-by-step explanation:

volume =πr^2h

=π(9)^2*12

=π81*12

=972πcm^3

The students will need $74 on the third day.

Let x be the cost of one item at Store A and y be the cost of one item at Store B. Then the system of equations to model the situation is:

4x + 7y = 30
3x + 5y = 22

To find the cost on Day 3, we need to solve for x and y, and then use those values to calculate:

10x + 17y = ?

Part B: Solve the system of equations to find x and y
To solve the system of equations, we can use elimination or substitution. Here, we'll use substitution.

From the first equation, we can solve for x:

4x + 7y = 30
4x = 30 - 7y
x = (30 - 7y)/4

Substitute this expression for x into the second equation:

3x + 5y = 22
3((30 - 7y)/4) + 5y = 22
(90 - 21y)/4 + 5y = 22
90 - 21y + 20y = 88
-y = -2
y = 2

Now that we know y = 2, we can substitute this value back into either equation to find x:

4x + 7y = 30
4x + 7(2) = 30
4x + 14 = 30
4x = 16
x = 4

So x = 4 and y = 2.

Part C: Calculate the amount of money needed on Day 3
Finally, we can use these values to calculate the amount of money needed on Day 3:

10x + 17y = 10(4) + 17(2) = 40 + 34 = 74

Therefore, the students will need $74 on the third day.

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Saturn is approximately 9.52 times further from the sun than Earth.

To find out how many times further from the sun Saturn is compared to Earth, we need to divide Saturn's distance from the sun by Earth's distance from the sun.

First, let's correct the distances given:
- Earth's distance from the sun: 1.496 x 10^8 km
- Saturn's distance from the sun: 1.4246 x 10^9 km (I assume you missed the exponent)

Now, let's calculate the ratio:

Ratio = (Saturn's distance) / (Earth's distance)
Ratio = (1.4246 x 10^9 km) / (1.496 x 10^8 km)

To make the calculation easier, let's factor out the common exponent (10^8):

Ratio = (1.4246 x 10) / (1.496)
Ratio ≈ 9.52

So, Saturn is approximately 9.52 times further from the sun than Earth.

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[tex](\text{x}+2)^{1/2}-5 = -2\\\\\sqrt{\text{x}+2}-5 = -2\\\\\sqrt{\text{x}+2} = -2+5\\\\\sqrt{\text{x}+2} = 3\\\\(\sqrt{\text{x}+2})^2 = 3^2\\\\\text{x}+2= 9\\\\\text{x}= 9-2\\\\\text{x}= 7\\\\[/tex]

-----------------------

Check:

[tex](\text{x}+2)^{1/2}-5 = -2\\\\\sqrt{\text{x}+2}-5 = -2\\\\\sqrt{7+2}-5 = -2\\\\\sqrt{9}-5 = -2\\\\3-5 = -2\\\\-2 = -2 \ \ \ \checkmark\\\\[/tex]

The answer is confirmed.

-----------------------

To find the Laplace transform of f(t), we use the formula:

L{f(t)} = ∫[0,∞) [tex]e^(-st)[/tex] f(t) dt

where L{f(t)} denotes the Laplace transform of f(t) and u(t) is the unit step function.

Using the linearity of the Laplace transform, we can find the Laplace transform of each term separately and add them up.

L{6u(t-2)} = [tex]6e^(-2s)[/tex] / s (applying the time-shift property)

L{3u(t-5)} = [tex]3e^(-5s)[/tex] / s (applying the time-shift property)

L{-4u(t-6)} = -[tex]4e^(-6s[/tex]) / s (applying the time-shift property)

Therefore, the Laplace transform of f(t) is:

F(s) = L{f(t)} = 6[tex]e^(-2s)[/tex] / s + [tex]3e^(-5s)[/tex] / s - [tex]4e^(-6s)[/tex]/ s

= [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s[/tex]

Hence, the Laplace transform of f(t) is F(s) = [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s.[/tex]

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Answer:

Step-by-step explanation:

50.24n2

Answer: B

Step-by-step explanation:

The cardboard box with the largest volume can hold the greatest number of 1 in x 2 in x 4 in sponges.

To find the box with the largest volume, first determine the volume of each sponge: V_sponge = 1 in x 2 in x 4 in = 8 cubic inches. Next, find the volume of each box by multiplying its length, width, and height (V_box = L x W x H).

To determine how many sponges each box can hold, divide the volume of the box by the volume of the sponge (V_box / V_sponge). The box with the highest resulting quotient can hold the most 1 in x 2 in x 4 in sponges.

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a) The Pearson correlation coefficient is 0.76.

b) Null hypothesis: There is no significant correlation between the grades in English and Mathematics (H0: r = 0)

Alternative hypothesis: There is a significant correlation between the grades in English and Mathematics (Ha: r ≠ 0)

c) The regression line is: y = 0.64x + 34.18

d) Interpretation and conclusion:  The Pearson correlation coefficient (r) of 0.76 indicates a strong positive correlation between the grades in English and Mathematics.

Correlation analysis:

Using the Pearson correlation coefficient to measure the strength and direction of the linear relationship between two variables.

Hypothesis testing:

Setting up null and alternative hypotheses, and using the t-test to determine whether the correlation coefficient is statistically significant.

Linear regression:

Finding the equation of the regression line that best describes the relationship between the two variables.

Interpretation and conclusion:

Using the results of the analysis to draw meaningful conclusions about the relationship between the two variables and the sample population as a whole.

Here we have

A study was conducted to determine the relationship existing between the grade in English and the grade in mathematics. a random sample of 10 students in uc was taken and the following are the results of the sampling

Student           1     2      3     4     5     6     7     8      9    10

English          75   83   80   77   89   78   92   86   93   84

Mathematics 78   87   78   76   92   81    89   89   91   84​  

a) To compute the Pearson correlation coefficient (r), first calculate the mean, standard deviation, and covariance of the two variables:

Mean of English grades (x)

= (75+83+80+77+89+78+92+86+93+84)/10 = 83.7

Mean of Math grades (y)

= (78+87+78+76+92+81+89+89+91+84)/10 = 84.5

The standard deviation of English grades (Sx)

= √((75-83.4)²+(83-83.4)²+...+(84-83.4)²)/9) = 6.52

The standard deviation of Math grades (Sy)

= √((78-84.4)²+(87-84.4)²+...+(84-84.4)²)/9) = 5.47

Covariance of the two variables

= ((75-83.4)(78-84.4)+(83-83.4)(87-84.4)+...+(84-83.4)(84-84.4))/9 = 26.6

Using the formula, r = cov(X,Y)/(SxSy),

we can calculate the correlation coefficient as follows

r = 26.6/(6.52*5.47) = 0.76

Therefore,

The Pearson correlation coefficient is 0.76.

b) Null hypothesis: There is no significant correlation between the grades in English and Mathematics (H0: r = 0)

Alternative hypothesis: There is a significant correlation between the grades in English and Mathematics (Ha: r ≠ 0)

c) To find the equation of the regression line, we need to calculate the slope (b) and the intercept (a) of the line. The formula for the slope is:

b = r(Sy/Sx) = 0.76(5.47/6.52) = 0.64

The formula for the intercept is:

=> a = y - bx = 84.4 - 0.64(83.4) = 34.18

Therefore,

The equation of the regression line is:

y = 0.64x + 34.18

Interpretation and conclusion:

The Pearson correlation coefficient (r) of 0.76 indicates a strong positive correlation between the grades in English and Mathematics.

The p-value associated with this correlation coefficient can be used to test the null hypothesis.

The equation of the regression line shows that for every one-point increase in the English grade, the predicted increase in the Mathematics grade is 0.64 points.

Therefore,

a) The Pearson correlation coefficient is 0.76.

b) Null hypothesis: There is no significant correlation between the grades in English and Mathematics (H0: r = 0)

Alternative hypothesis: There is a significant correlation between the grades in English and Mathematics (Ha: r ≠ 0)

c) The regression line is: y = 0.64x + 34.18

d) Interpretation and conclusion:  The Pearson correlation coefficient (r) of 0.76 indicates a strong positive correlation between the grades in English and Mathematics.

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