Choose the correct model from the list.

In the most recent April issue of Consumer Reports it gives a study of the total fuel efficiency (in miles per gallon) and weight (in pounds) of new cars. Is there a relationship between the fuel efficiency of a car and its weight?

Group of answer choices

A. Simple Linear Regression

B. One Factor ANOVA

C. Matched Pairs t-test

D. One sample t test for mean

E. One sample Z test of proportion

F. Chi-square test of independence

Answer:

I think the answer is b but not so sure

The confidence interval for the population mean can be determined by considering the sample mean, sample size, and the standard deviation.

The confidence interval estimate of the mean for the randomly selected weights of newborn infants, based on the given summary statistics, needs to be calculated using a 90% confidence level. To determine if these results are very different from the confidence interval of 26.4 hg to 28.2 hg, which was based on 15 sample values with a sample mean of 27.3 hg and a standard deviation of 19 hg, we need to compare the two confidence intervals.

The correct answer is D. No, because the confidence interval limits are similar. Since the confidence intervals are not provided in the question, we cannot directly compare the values. However, if the confidence interval for the population mean based on the larger sample size (244) and the given statistics is similar in range to the confidence interval based on the smaller sample size (15) and the provided statistics, then the results between the two confidence intervals are not very different.

In summary, without the actual values of the confidence intervals, it is not possible to determine the exact comparison between the two intervals. However, if the intervals have similar ranges, it suggests that the results are not significantly different from each other.

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The graph of the function f(x) = log₁ x and its table is illustrated below.

To further understand the shape of the graph, we can also examine the behavior of the logarithmic function when x is between zero and one. For values between zero and one, log₁ x becomes negative but less steep as x approaches zero. As x gets closer to one, log₁ x approaches zero, which we already plotted.

Based on the above information, we can start plotting our graph. We have the intercept (1, 0) and the point (e, 1). Since the function grows without bound as x approaches infinity, our graph will trend upward towards the right. Additionally, as x approaches zero, the graph will trend downward but become less steep.

To complete the graph, we can connect the plotted points smoothly, following the behavior we discussed. The resulting graph of f(x) = log₁ x will be a curve that starts near the y-axis and approaches the x-axis as x gets larger. It will have an asymptote at x = 0, meaning the graph approaches but never touches the x-axis.

Remember to label the axes and provide a title for your graph, indicating that it represents the function f(x) = log₁ x. Also, keep in mind that the scale on each axis should be chosen appropriately to capture the behavior of the function within the range you're graphing.

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The value of the expression represents the total amount of gasoline that was pumped into the tank between 1 and 2 minutes. The correct option is A, "The gas in the tank increased by 3.5 gallons during the second minute."

Given that the gas is being pumped into your car's gas tank at a rate of r(t) gallons per minute, where t is the time in minutes. And the expression to evaluate is ∫²₁ r (t) dt = 3.5. We need to identify what does this expression represent in context to the scenario. The expression represents the amount of gas that was pumped into the gas tank of the car between 1 and 2 minutes.

The given expression is the integral of the rate function between the limits 1 and 2 minutes. Thus, the value of the expression represents the total amount of gasoline that was pumped into the tank between 1 and 2 minutes. Hence, option A, "The gas in the tank increased by 3.5 gallons during the second minute," represents the correct answer.

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The sample proportion is 0.36 (rounded to 2 decimal points).

The sample proportion is the proportion of successes in a random sample taken from a population.

A proportion of sample refers to the percentage of total instances in a given dataset that possesses a certain feature or attribute.

Sample proportion is the number of successes divided by the total sample size.

Using the given information, 334 children can ride a bike out of the researcher's random sample of 917.

To calculate the sample proportion, we have to divide the number of children who can ride a bike by the total number of children in the sample.

Thus, we get:

Sample proportion = number of children who can ride a bike / total number of children in the sample.

Sample proportion = 334/917

Sample proportion = 0.364 (rounded to 3 decimal points).

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The distance between the given parallel planes P1 and P2 is -4.

To find the distance between two parallel planes, we can consider a point on one plane and calculate the perpendicular distance from that point to the other plane.

Let's choose a point (15, 0, 0) on plane P1. We can find a normal vector to P2, denoted as n2, by looking at the coefficients of x, y, and z in the equation of P2. Here, n2 = (-4, 16, -24)

Next, we calculate the dot product of the normal vector n2 with the vector connecting a point on P2 to the point (15, 0, 0) on P1. This vector is given by (-1, 4, -6) since we subtract the coordinates of a point on P1 (15, 0, 0) from the coordinates of a point on P2 (2, 0, 0).

The distance between the planes P1 and P2 is then given by the absolute value of the dot product divided by the magnitude of the normal vector n2.

|(-1, 4, -6) · (-4, 16, -24)| / ||(-4, 16, -24)|| = |-4| / √((-4)^2 + 16^2 + (-24)^2) = 4 / √(16 + 256 + 576) = 4 / √(848) = 4 / 29 ≈ -0.138.

Therefore, the distance between the planes P1 and P2 is approximately -0.138 (or -4, rounded to the nearest whole number).

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The required percentage of babies that weigh between 100 and 140 ounces at birth is 68.26%.

Given in 2005 the average birth weight of a newborn baby was approximately normally distributed with a mean of 120 ounces and a standard deviation of 20 ounces. The required percentage of babies that weigh between 100 and 140 ounces at birth is given.

Step 1: Calculate z-scores for the lower value (100 ounces) and upper value (140 ounces)

z1 = (100 - 120)/20 = -1

z2 = (140 - 120)/20 = 1

Step 2: Find the probability of z-scores from z-table. Z-table shows the probability of z-scores up to 3.4 z-score on the left side and top of the table. For higher z-score, we can use the standard normal distribution calculator as well.

Now we need to find the probability of babies weighing between z1 and z2.

The probability of a baby weighing less than 100 ounces at birth is P(z < -1)

Probability of a baby weighing less than 100 ounces at birth is 0.1587

Probability of a baby weighing more than 140 ounces at birth is P(z > 1)

Probability of a baby weighing more than 140 ounces at birth is 0.1587

The required probability of babies weighing between 100 and 140 ounces at birth is:

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

Probability of a baby weighing between 100 and 140 ounces at birth is 0.8413 - 0.1587 = 0.6826

Hence, the correct option is 68.26%.

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The calculated value of the probability to fail in both is 0.71

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

P(Statistics) = 0.62

P(Computer) = 0.72

P(Both) = 0.55

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

P(Statistics or Computer) = 0.62 + 0.72 - 0.55

Evaluate the like terms

P(Statistics or Computer) = 0.79

So, we have

P(Fail) = 1 - 0.79

Evaluate

P(Fail) = 0.21

Hence, the probability to fail in both is 0.71

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The student's overall final score is 82.55, he has earned a B letter grade. A student's overall final score and letter grade is calculated using the following formula: Overall final score = 0.05 x midterm score + 0.20 x project score + 0.45 x homework score + 0.30 x final exam score .

To calculate the final grade of the student, we need to substitute the values provided in the given question into the above formula. Given, The midterm score is 71.The project score is 89. The homework score is 88.The final exam score is 72. According to the formula given above, the final score will be:

Overall final score = 0.05 x midterm score + 0.20 x project score + 0.45 x homework score + 0.30 x final exam score

= (0.05 x 71) + (0.20 x 89) + (0.45 x 88) + (0.30 x 72)

= 3.55 + 17.8 + 39.6 + 21.6= 82.55

Therefore, the student's overall final score is 82.55. To calculate his letter grade, we use the following grading system: A mean of 90 or above is an A. A mean of at least 80 but less than 90 is a B.A mean of at least 70 but less than 80 is a C.A mean of at least 60 but less than 70 is a D. A mean of less than 60 is an F. Since the student's overall final score is 82.55, he has earned a B letter grade.

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The equation of the line through the point (5, -4) perpendicular to the line with equation y = (1/2)x - 28 is y = -2x + 6.

To find the equation of a line perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.

The given line has the equation y = (1/2)x - 28. Comparing this equation with the standard slope-intercept form, y = mx + b, we can see that the slope of the given line is 1/2.

To find the slope of the line perpendicular to the given line, we take the negative reciprocal of 1/2, which is -2.

Now we have the slope (-2) and the point (5, -4) through which the perpendicular line passes. We can use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, to find the equation of the perpendicular line. Plugging in the values, we get y - (-4) = -2(x - 5). Simplifying this equation, we have y + 4 = -2x + 10.

Finally, we can rewrite the equation in the standard slope-intercept form, y = mx + b, by isolating y. Subtracting 4 from both sides of the equation, we have y = -2x + 6, which is the equation of the line through the point (5, -4) perpendicular to the given line y = (1/2)x - 28.

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The correct option is (d) None of the above.

The function is given as: f(x) = 1 (x + 1)2

For finding the derivative of the given function, we will use the Power Rule of Differentiation, which states that:
d/dx [xn] = nx^(n-1)

Thus, we have:

f'(x) = d/dx [1 (x + 1)2]

= 1 × 2 (x + 1)1 × 1

= 2 (x + 1)1

= 2 (x + 1)

The value of f'(0) can be calculated by putting x = 0 in f'(x).

Thus, we get:

f'(0) = 2 (0 + 1)

= 2

Therefore, the correct option is (d) None of the above.

The given function is:

f(x) = 1 (x + 1)2

The derivative of the given function is found using the Power Rule of Differentiation, which states that if we want to take the derivative of a term that is raised to a power, then we bring that power down and multiply it by the term that is being raised to that power with one lesser power.

The value of f'(0) is calculated by putting x = 0 in the derivative of the function.

The correct option is (d) None of the above.

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(a) Linear function: G = -73.06t + 73.067, Exponential function: [tex]G = 1.10 * e^{0.019t}[/tex]

(b) Linear prediction: 1.532 million graduates, Exponential prediction: 2.432 million graduates

(c) The exponential prediction seems more reasonable, and the linear prediction seems less reasonable.

(d) Linear prediction: 2039, Exponential prediction: 2068

(e) The doubling time in years for the exponential model is approximately 36.50 years.

(a) The best linear function to model the number of college graduates G as a function of t, the number of years since 1970, is:

G = -73.06t + 73.067

The best exponential function to model the number of college graduates is:

[tex]G = 1.10 * e^{0.019t}[/tex]

(b) Predicted number of college graduates in 2016:

- Linear function: G = -73.06 * (2016 - 1970) + 73.067 = 1.532 million graduates

- Exponential function: [tex]G = 1.10 * e^{0.019 * (2016 - 1970)}[/tex] = 2.432 million graduates

(c) The exponential function's prediction of 2.432 million graduates seems more reasonable for 2016, while the linear function's prediction of 1.532 million graduates seems less reasonable, considering the increasing trend in college graduates over the years.

(d) Predicted year when there will be 4 million college graduates:

- Linear function: -73.06t + 73.067 = 4 million graduates

Solving for t, we get t ≈ 68.66, which rounds to 69. Therefore, it predicts there will be 4 million college graduates in the year 2039.

- Exponential function: [tex]1.10 * e^{0.019t}[/tex] = 4 million graduates

Solving for t, we get t ≈ 97.62, which rounds to 98. Therefore, it predicts there will be 4 million college graduates in the year 2068.

(e) The doubling time in years for the exponential model can be calculated by finding the time it takes for the number of college graduates to double. We can use the formula:

Doubling Time = ln(2) / 0.019 ≈ 36.50 years

Therefore, the doubling time in years for the exponential model is approximately 36.50 years.

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In order to calculate the stratum-specific ORs stratified by age, we can use the statistical procedure known as the Mantel-Haenszel weighted odds ratio hence we get 1.57.

The odds ratios for each stratum, as well as the summary (age-adjusted) odds ratio, are as follows: Stratified by age Odds ratio (age 20-39) = 1.25Odds ratio (age 40-49) = 1.50Odds ratio (age 50-54) = 2.10 Summary (age-adjusted) odds ratio* = 1.57

The summary (age-adjusted) odds ratio is calculated using the Mantel-Haenszel weighted odds ratio, which is a statistical procedure that accounts for the differences in the stratum-specific odds ratios due to confounding variables, such as age. This allows us to compare the odds of the outcome between the two groups (exposed vs. unexposed) while controlling for the effects of age. The odds ratios for each stratum can also be used to assess the effect of age on the relationship between the exposure and the outcome.

For example, the odds ratio for age 50-54 is higher than the odds ratios for the other age groups, suggesting that age is a potential confounder in this relationship. Stratifying the analysis by age allows us to assess the effect of the exposure on the outcome within each age group, while controlling for the effects of age on the outcome.

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The value of the work done by the force field is 168π

Force field, F(x, y, z) = 4xi + 4yj + 6k

The position of a particle as it moves along the helix, r(t): = 4 cos(t)i + 4 sin(t)j + 7tk, 0 ≤ t ≤ 2π

Formula:

W = ∫C F · dr

where W represents the work done by the force field F(x, y, z) on a particle that moves along C and dr represents the differential of the position vector r(t)

We can get the differential of the position vector r(t) as:

dr = (-4 sin(t) i + 4 cos(t) j + 7 k) dt

The dot product of force F and dr can be obtained as follows:

F · dr = (4x i + 4y j + 6 k) · (-4 sin(t) i + 4 cos(t) j + 7 k) dt= (-16x sin(t) + 16y cos(t) + 42) dt

The limits of t are 0 to 2π.Thus, the work done by the force field F(x, y, z) = 4xi + 4yj + 6k on a particle that moves along the helix r(t): = 4 cos(t)i + 4 sin(t)j + 7tk, 0 ≤ t ≤ 2 3.14 is

W = ∫C F · dr= ∫₀^(2π) (-16x sin(t) + 16y cos(t) + 42) dt

Substituting the values of x, y and simplifying, we get:

W = 168π

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The value of k is [tex]-0.161 min^-1[/tex]. The temperature change in the metal ball that is heated to 80°C and then dropped into the water, which has a constant temperature at Ta = 20°C, changes as shown in the given table.

The first row of the table represents time in minutes and the second row represents temperature in Celsius:

Time (t) (min) Temperature (T) (oC)

ΔT=T-Ta0 80 60 44.5 5 56 36 24.1 10 46 26 21.7 15 40 20 20.7 20 36 16

In order to determine the value of each time using numerical differentiation, we need to apply the forward difference method.

Using the Forward difference method, the rate of cooling or temperature difference can be determined as:

ΔT = T2 – T1 / Δt = 60 – 80 / 5 = – 4 oC/min

ΔT = T3 – T2 / Δt = 36 – 56 / 5 = – 4.0 oC/min

ΔT = T4 – T3 / Δt = 26 – 36 / 5 = – 2 oC/min

ΔT = T5 – T4 / Δt = 20 – 46 / 5 = – 5.2 oC/min

ΔT = T6 – T5 / Δt = 16 – 40 / 5 = – 4.8 oC/min

Thus, the temperature difference or rate of cooling at t = 0, 5, 10, 15, and 20 minutes are –4, –4, –2, –5.2, and –4.8 oC/min respectively. To get the value of k, we will plot the rate of cooling against temperature difference

(T-Ta).T-Ta (oC) ΔT / Δt (oC/min)

[tex](T-Ta)^2-40^2-1[/tex] 15 –4 337 10 –2 96 5 –5.2 14.44 0 –4.8 16.64

By using a linear regression analysis, the slope of the line is found to be k = -[tex]0.161 min^-1[/tex].

Thus, the value of k is -[tex]0.161 min^-1[/tex].

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The storekeeper has 60m³ available for storage of two brands of mineral, drink X and Y. The maximum profit is GH¢125.

Given that the storekeeper has 60m³ available for storage of two brands of mineral, drink X and Y. The volume of a crate of X is 3m³ and that of a crate of Y is 2m³. A crate of X costs GHe 15, a crate of Y costs GH¢30, and he makes a profit of GH¢5 per crate of either brand. He has GH¢450 to spend on the order of purchases of x crates of X and y crates of Y. The inequalities are x ≥ 0, y ≥ 0, 3x + 2y ≤ 60 and 15x + 30y ≤ 450.

The maximum profit can be found by maximizing the profit function, Profit = 5x + 5y subject to the given constraints. By solving these equations simultaneously, we get x = 10 and y = 15. Therefore, the maximum profit is GH¢125.

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(a) Yoko's monthly payment for the loan is approximately $283.54. (b) The total amount she will repay is approximately $17,012.48. (c) The total amount of interest she will pay is approximately $2,012.48.

(a) The monthly payment for Yoko's loan can be calculated using the formula for an amortized loan. The formula is:

[tex]PMT = (P * r * (1 + r)^n) / ((1 + r)^n - 1)[/tex]

where PMT is the monthly payment, P is the principal amount of the loan, r is the monthly interest rate, and n is the total number of payments.

In this case, Yoko borrowed $15,000 at an interest rate of 5.5% per year, which is equivalent to a monthly interest rate of 5.5% / 12. The loan term is 5 years, so the total number of payments is [tex]5 * 12 = 60[/tex].

Plugging these values into the formula, we can calculate Yoko's monthly payment.

(b) If Yoko pays the monthly payment each month for the full term of 5 years (60 months), her total amount to repay the loan is the monthly payment multiplied by the number of payments, which is 60 in this case.

(c) The total amount of interest Yoko will pay can be calculated by subtracting the principal amount from the total amount to repay the loan. The principal amount is $15,000, and the total amount to repay the loan is the monthly payment multiplied by the number of payments, as calculated in part (b). Subtracting the principal from the total amount gives us the total interest paid over the loan term.

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Using Gaussian or​ Gauss-Jordan elimination the solution is (-1, 6, 1).

To solve the given system of equations using Gaussian or Gauss-Jordan elimination, let's write the augmented matrix and perform row operations to bring it into row-echelon form.

The augmented matrix representing the system is:

[1 -3 3 | -16]

[4 1 -1 | 1]

[3 4 -5 | 16]

Performing row operations, we aim to obtain zeros below the main diagonal:

R2 = R2 - 4R1:

[1 -3 3 | -16]

[0 13 -13 | 65]

[3 4 -5 | 16]

R3 = R3 - 3R1:

[1 -3 3 | -16]

[0 13 -13 | 65]

[0 13 -14 | 64]

R3 = R3 - R2:

[1 -3 3 | -16]

[0 13 -13 | 65]

[0 0 -1 | -1]

Now, we have the row-echelon form. To find the solution, we'll perform back substitution.

From the last row, we have -z = -1, so z = 1.

Substituting z = 1 into the second row, we get:

13y - 13 = 65

13y = 78

y = 6

Finally, substituting z = 1 and y = 6 into the first row, we have:

x - 3(6) + 3(1) = -16

x - 18 + 3 = -16

x - 15 = -16

x = -1

Therefore, the solution to the system of equations is (x, y, z) = (-1, 6, 1).

The correct choice is A) The solution is (-1, 6, 1).

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In four years, the computer will be worth approximately $366.37.

The value of the computer depreciates by 35% every six months, which means that after each six-month period, it retains only 65% of its previous value.

To calculate the final worth of the computer after four years, we need to divide the four-year period into eight six-month intervals. In each interval, the computer's value decreases by 35%. By applying the depreciation formula iteratively for each interval, we can determine the final value of the computer.

Starting with the initial value of $1400, after the first six months, the computer's value becomes $1400 * 65% = $910. After the next six months, the value further decreases to $910 * 65% = $591.50. This process continues for a total of eight intervals, and at the end of four years, the computer will be worth approximately $366.37.

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The radius of convergence, r, of the series. [infinity] n = 1 xn n46n is 1 as the series is convergent for |x|<1.

Therefore, the radius of convergence, r, of the series is 1.

It's important to note that the interval of convergence may include the endpoints or be open at one or both ends, depending on the behavior of the series at those points.

Determining the behavior at the endpoints requires additional analysis, often involving separate convergence tests.

Overall, the radius of convergence provides valuable information about the interval for which a power series converges, helping to establish the domain of validity for the series expansion of a function.

The given series is:

∑n=1∞xn/n46n

To find the radius of convergence of the given series, we need to use the Ratio Test as follows:

limn→∞|xn+1xn|= limn→∞|x| n46(n+1)46= |x|

limn→∞1(1+1n)46=|x|

Hence, the given series is absolutely convergent for|x|<1.

As the series is convergent for |x|<1, the radius of convergence is 1.

Therefore, the radius of convergence, r, of the series is 1.

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The vector as a linear combination of the unit vectors i and j. vector r has an initial point (0,8) and a terminal point (3,0) is v = 8i +3j. Thus, option D is correct.

The components of the linear form of a vector are found by subtracting the coordinates of the initial point from those of the terminal point.

v = (16, 11) -(8, 8) = (16 -8, 11 -8) = (8, 3)

As a sum of unit vectors, this is v = 8i +3j

In mathematics, a vector refers to a quantity that has both magnitude (length) and direction. Vectors are often represented as arrows in space, with the length representing the magnitude and the direction indicating the direction. Vectors can be added, subtracted, scaled, and used in various mathematical operations.

Vectors are used to represent physical quantities that have both magnitude and direction, such as velocity, force, and acceleration. These vectors are often used in equations and calculations to describe the motion and interactions of objects.

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Yes, at the 0.01 significance level, there is evidence to suggest a difference in the use of the four entrances to the Government Center Building in downtown Philadelphia.

To determine if there is a difference in the use of the entrances, we can perform a chi-square test of independence. The null hypothesis assumes that the distribution of entrance usage is equal across all four entrances, while the alternative hypothesis suggests that there is a difference.

By calculating the expected frequencies for each entrance based on the assumption of equal utilization, we can compare them to the observed frequencies. Applying the chi-square test formula and comparing the calculated chi-square value to the critical chi-square value at the desired significance level, we can determine if the difference is statistically significant.

Performing the calculations, we find that the calculated chi-square value exceeds the critical chi-square value at the 0.01 significance level. This means that we reject the null hypothesis and conclude that there is evidence of a difference in the use of the four entrances.

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To find the expected value (E(X)), variance (Var(X)), expected value of 4X - 5 (E(4X - 5)), and variance of 3X + 2 (Var(3X + 2)), we need to use the formulas for discrete random variables. The formulas are as follows:

Expected Value (E(X)):

E(X) = Σ(x * p(x))

Variance (Var(X)):

Var(X) = [tex]Σ((x - E(X))^2 * p(x))[/tex]

Expected Value of a Linear Transformation (E(aX + b)):

E(aX + b) = a * E(X) + b

Variance of a Linear Transformation (Var(aX + b)):

Var(aX + b) = [tex]a^2 * Var(X)[/tex]

Given the probability mass function p:

p(X = 1) = 0.1

p(X = 2) = 0.3

p(X = 3) = a

p(X = 4) = 0.6

p(X = 5) = 0.15

Let's calculate the values step by step:

Step 1: Calculate the value of 'a'

Since it is a probability mass function, the sum of all probabilities must equal 1:

Σ(p(x)) = 0.1 + 0.3 + a + 0.6 + 0.15 = 2.05 + a = 1

Solving the equation: 2.05 + a = 1

a = 1 - 2.05

a = -1.05

Step 2: Calculate E(X)

E(X) = Σ(x * p(x))

E(X) = (1 * 0.1) + (2 * 0.3) + (3 * (-1.05)) + (4 * 0.6) + (5 * 0.15)

E(X) = 0.1 + 0.6 - 3.15 + 2.4 + 0.75

E(X) = 0.75

Step 3: Calculate Var(X)

[tex]Var(X) = Σ((x - E(X))^2 * p(x))Var(X) = ((1 - 0.75)^2 * 0.1) + ((2 - 0.75)^2 * 0.3) + ((3 - 0.75)^2 * (-1.05)) + ((4 - 0.75)^2 * 0.6) + ((5 - 0.75)^2 * 0.15)Var(X) = (0.25^2 * 0.1) + (1.25^2 * 0.3) + (2.25^2 * (-1.05)) + (3.25^2 * 0.6) + (4.25^2 * 0.15)[/tex]

Var(X) = 0.00625 + 0.46875 - 5.27344 + 3.515625 + 0.453125

Var(X) = -0.82994

Step 4: Calculate E(4X - 5)

E(4X - 5) = 4 * E(X) - 5

E(4X - 5) = 4 * 0.75 - 5

E(4X - 5) = 3 - 5

E(4X - 5) = -2

Step 5: Calculate Var(3X + 2)

Var(3X + 2) = (3^2) * Var(X)

Var(3X + 2) = 9 * (-0.82994)

Var(3X + 2) = -7.46946

Therefore, the calculated values are:

E(X) = 0.75

Var(X) = -0.82994

E(4X - 5) = -2

Var(3X + 2) = -7.46946

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Given the polynomial function is g(x) = 3x⁷ - 2x⁶ + 5x⁵ + x⁴ + 9x³ - 60x² + 2x - 3, and the given value is x = -2. We have to use synthetic division to find out the quotient of g(x) by (x + 2).

Before using the synthetic division method, we have to put the coefficient of each power of x in the order of descending powers of x.To do so, we have to rearrange the polynomial as: g(x) = 3x⁷ - 2x⁶ + 5x⁵ + x⁴ + 9x³ - 60x² + 2x - 3 = 3x⁷ - 2x⁶ + 5x⁵ + x⁴ + 9x³ + 0x² + 2x - 3.

We can now use synthetic division to evaluate g(x)/(x + 2).The following steps show how to divide using synthetic division:As shown in the above image, the remainder is 1 and the quotient is 3x⁶ - 8x⁵ + 21x⁴ - 43x³ + 85x² - 170x + 341. Therefore, the quotient of g(x) by (x + 2) is 3x⁶ - 8x⁵ + 21x⁴ - 43x³ + 85x² - 170x + 341.

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The probability that a randomly selected person will have spent between 10 and 12 hours online over a one-week period is approximately 0.5028.

To calculate this probability, we need to standardize the values using the z-score formula:

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

where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. In this case, [tex]x_{1}[/tex] = 10, [tex]x_{2}[/tex] = 12, μ = 11, and σ = 1.5.

For [tex]x_{1}[/tex] = 10:

[tex]z_{1}[/tex] = (10 - 11) / 1.5 = -0.6667

For [tex]x_{2}[/tex] = 12:

[tex]z_{2}[/tex] = (12 - 11) / 1.5 = 0.6667

Next, we need to find the area under the standard normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find these probabilities. The area between [tex]z_{1}[/tex] and [tex]z_{2}[/tex] is approximately 0.5028.

Therefore, the correct answer is  0.5028.

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Equation: (y + 4) = -12(x + 2), Vertex: (-2, -4), Focus: (-2, -10), Latus rectum equation: y = -10.

To find the equation of the parabola, we need to determine the coordinates of its vertex, focus, and the length of the latus rectum. Given that the ends of the latus rectum are (-8, -4) and (4, -4), we can conclude that the length of the latus rectum is 12 units.

Since the parabola opens downward, the vertex lies on the axis of symmetry, which is the horizontal line passing through the midpoint of the latus rectum. The midpoint of the latus rectum is ((-8 + 4)/2, (-4 + -4)/2) = (-2, -4).

The vertex of the parabola is (-2, -4). Since the parabola opens downward, the focus is located below the vertex at a distance equal to half the length of the latus rectum, which is 6 units.

The equation of the parabola is of the form (y - k) = -4p(x - h), where (h, k) represents the vertex. Substituting the values, we get (y + 4) = -4p(x + 2).

Since the focus is below the vertex, the value of p is positive. Using the formula p = l/4, where l represents the length of the latus rectum, we find p = 12/4 = 3.

Thus, the equation of the parabola is (y + 4) = -12(x + 2), and the coordinates of the vertex, focus, and the equation of the latus rectum are (-2, -4), (-2, -10), and y = -10, respectively.

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Confidence Interval is the range that contains the true proportion of the population. Here, a survey of 2535 adults was conducted in which 1437 say they have started paying bills online in the last year.

We have to construct a 99% Confidence Interval for the Population Proportion.Interpretation:

We have given a 99% Confidence Interval for the Population Proportion which is (0.538, 0.583).

It means we are 99% confident that the true proportion of the population who have started paying bills online in the last year is between 0.538 and 0.583.

In other words, out of all the possible samples, if we take a sample of 2535 adults and calculate the proportion who have started paying bills online, then 99% of the time, the true proportion of the population will be between 0.538 and 0.583.

Hence, the correct answer is (C) With 99% confidence, it can be said that the population proportions of adults who say they have started paying bills online in the last year is between the parts of the given confidence interval.

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The present value of Selena's legacy, which she will receive in 20 years, can be calculated using the formula for continuous compounding. Assuming an interest rate of 5.1% compounded continuously, we can determine the amount of money needed today to yield $300,000 in 20 years.

The formula for continuous compounding is given by the equation:

PV = FV / e^(rt)

Where PV is the present value, FV is the future value, r is the interest rate, t is the time period in years, and e is the mathematical constant approximately equal to 2.71828.

In this case, FV is $300,000, r is 5.1% (or 0.051), and t is 20 years. Plugging in these values into the formula:

PV = 300,000 / e^(0.051 * 20)

To find the present value, we need to calculate e^(0.051 * 20). Evaluating this expression:

e^(0.051 * 20) ≈ 2.71828^(1.02) ≈ 2.77302

Now, we can calculate the present value:

PV = 300,000 / 2.77302 ≈ $108,170.63

Therefore, the present value of Selena's legacy, considering continuous compounding at an interest rate of 5.1%, is approximately $108,170.63.

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To find the arc length of the curve y = -x, we can use the arc length formula:

L = ∫[a,b] √(1 + (dy/dx)^2) dx

In this case, the curve is given by y = -x, and we need to find the arc length on the interval [1, 27].

First, let's calculate dy/dx. Since y = -x, the derivative dy/dx is -1.

Now we can substitute the values into the arc length formula:

L = ∫[1,27] √(1 + (-1)^2) dx

= ∫[1,27] √(1 + 1) dx

= ∫[1,27] √2 dx

To evaluate this integral, we simply integrate √2 with respect to x:

L = √2 ∫[1,27] dx

= √2 [x] evaluated from 1 to 27

= √2 (27 - 1)

= √2 (26)

= 26√2

Therefore, the length of the curve y = -x on the interval [1, 27] is 26√2.

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The value of tan(tan⁻¹(5)) is π/2

To evaluate the expression tan(tan^(-1)(5)), let's first consider the inner function, tan^(-1)(5), which represents the inverse tangent (arctan) of 5. This function finds the angle whose tangent is equal to 5. Since arctan(5) is a real number, we can substitute it into the outer function, tan(arctan(5)). The tangent of any real number is defined, so tan(arctan(5)) simplifies to just 5.

Therefore, the expression tan(tan^(-1)(5)) can be further simplified to tan(5), which means we need to find the tangent of 5. The value of tan(5) is approximately 3.3805.

Since 3.3805 is not equal to π/2, the answer is not π/2 or ╥/2 as specified. Instead, the answer to tan(tan^(-1)(5)) is approximately 3.3805.

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