Workers at a warehouse of consumer goods gather items from the warehouse to fill customer orders

Answer:  if a function's output values are all above the x-axis, then the function is positive

Step-by-step explanation:

We would choose "smaller" for the necessary sample size and "smaller" for the confidence level.

(a) We know that the margin of error E is 4.7 and the population standard deviation is o = 17.5.

The formula for the margin of error is:

E = z* (o/ sqrt(n))

where z is the critical value for the desired level of confidence, o is the population standard deviation, and n is the sample size.

We want to find n, so we can rearrange the formula to solve for n:

n = (z*o/E)^2

For a 99.8% confidence level, the critical value is z = 2.967.

Substituting the values into the formula, we get:

n = (2.967*17.5/4.7)^2

n = 157.82

Rounding up to the nearest integer, we get a sample size of 158.

Therefore, a sample size of 158 must be drawn in order to obtain a 99.8% confidence interval with a margin of error equal to 4.7.

(b) If the required confidence level were 99.5%, the necessary sample size would be smaller.

This is because the critical value for a 99.5% confidence level is smaller than the critical value for a 99.8% confidence level. As the critical value gets smaller, the margin of error also gets smaller, which means we need a smaller sample size to achieve the same margin of error.

So, we would choose "smaller" for the necessary sample size and "smaller" for the confidence level.

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Liz received $458.57 in cash after getting a deposit ticket. So the answer is (a) $458.57.

The deposit ticket provides us with information on the amounts deposited, the subtotal, and the total after cash is received. To find the amount of cash Liz received, we need to subtract the total after cash received from the subtotal.

Subtotal = $1346.75 (This is the total amount of the two deposits)

Total after cash received = $888.18 (This is the total amount of the deposits after the cash received has been deducted)

To find the amount of cash Liz received:

Therefore, Liz received $458.57 in cash.

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Answer:

Step-by-step explanation:

The experimental probability of landing on heads is 12% greater than the theoretical probability of landing on heads. The correct option is C

Flipping a fair coin, the theoretical chance of landing on heads is 0.5, or 50%. The experimental probability is the ratio of the total number of coin flips to the number of times the coin landed on heads.

The experiment's experimental probability is 62/100 = 0.62 or 62% since the student flipped the coin 100 times and it came up heads 62 times.

We can see that by comparing the experimental and theoretical probabilities, 62% - 50% = 12%

So the experimental probability is 12% greater than the theoretical probability.

Therefore, The experimental probability of landing on heads is 12% greater than the theoretical probability of landing on heads.

Therefore, The correct option is C

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The efficient allocation of apples and bananas between the two individuals is:

A1 = B1 = 50 (allocated to individual 1)

A2 = 50 and B2 = 100 (allocated to individual 2)

What is utility?

In mathematics, utility refers to a measure of the preference or satisfaction an individual derives from consuming goods or services.

To recommend an efficient allocation of apples and bananas between the two individuals, we need to find a solution that maximizes the total utility of both individuals subject to the constraint that all the goods must be distributed. In other words, we need to solve the following optimization problem:

Maximize U1(A1, B1) + U2(A2, B2) subject to A1 + A2 = 100 and B1 + B2 = 150

Let's begin by solving for individual 1's optimal allocation. We can use the first-order conditions to find the optimal values of A1 and B1 that maximize U1(A1, B1). Taking partial derivatives with respect to A1 and B1 and setting them equal to zero, we get:

∂U1/∂A1 = 0 => 0 = 0

∂U1/∂B1 = 0 => 2 = 2B1/B2

Solving for B1/B2, we get B1/B2 = 1. This means that the optimal allocation for individual 1 is to receive an equal number of bananas and apples, i.e., A1 = B1 = 50.

Next, we solve for individual 2's optimal allocation. Following the same approach, we find that the optimal allocation for individual 2 is to receive all the remaining bananas and apples, i.e., A2 = 50 and B2 = 100.

Therefore, the efficient allocation of apples and bananas between the two individuals is:

A1 = B1 = 50 (allocated to individual 1)

A2 = 50 and B2 = 100 (allocated to individual 2)

This allocation is efficient because it maximizes the total utility of both individuals subject to the constraint that all the goods must be distributed. If we try to reallocate the goods in any other way, we will end up with a lower total utility for both individuals.

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The efficient allocation of apples and bananas between individuals 1 and 2 is as follows:

Individual 1 gets 60 apples and 75 bananas

Individual 2 gets 40 apples and 75 bananas

To determine the most efficient allocation of apples and bananas between individuals 1 and 2, we must maximize the total utility of both individuals while keeping in mind that all of the apples and bananas must be distributed.

From the constraint equation:

A1 + A2 = 100

B1 + B2 = 150

Now, let's write out the total utility function:

U = U1 + U2

U = 2A1 + B1 + 2A2 + B2 + 2A2B2

Using the Lagrangian method:

L = 2A1 + B1 + 2A2 + B2 + 2A2B2 - λ1(A1 + A2 - 100) - λ2(B1 + B2 - 150)

Taking the partial derivative of L with respect to each variable and equating them to zero, we get:

∂L/∂A1 = 2 - λ1 = 0

∂L/∂A2 = 2 + 4B2 - λ1 = 0

∂L/∂B1 = 1 - λ2 = 0

∂L/∂B2 = 1 + 2A2 - λ2 + 4A2B2 = 0

∂L/∂λ1 = A1 + A2 - 100 = 0

∂L/∂λ2 = B1 + B2 - 150 = 0

Solving these equations, we get:

λ1 = 2, λ2 = 1, A1 = 60, A2 = 40, B1 = 75, B2 = 75

Therefore, the efficient allocation of apples and bananas between individuals 1 and 2 is as follows:

Individual 1 gets 60 apples and 75 bananas

Individual 2 gets 40 apples and 75 bananas

This allocation maximizes the total utility of both individuals subject to the constraint that all the apples and bananas are distributed.

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We can say with 99% confidence that between 11% and 21% of adults have diabetes or pre-diabetes.

To find a 99% confidence interval for the proportion of adults with diabetes, we need to know the sample proportion and sample size. Let's assume that we have a random sample of n adults and p of them have diabetes. Then, the sample proportion is:

P = p/n

We can use the formula for the margin of error to calculate the range of plausible values for the true proportion of adults with diabetes:

margin of error = z*√(P(1-P)/n)

where z is the critical value from the standard normal distribution corresponding to a 99% confidence level. From a standard normal distribution table, we find that z = 2.576.

Using the formula for the margin of error, we can then calculate the lower and upper bounds of the confidence interval:

lower bound = P - margin of error

upper bound = P + margin of error

Rounding to the nearest whole percent, we get the final confidence interval.

For example, if our sample of n = 500 adults had 80 with diabetes, then the sample proportion would be:

P = 80/500 = 0.16

The margin of error would be:

margin of error = 2.576√(0.16(1-0.16)/500) = 0.045

The lower and upper bounds of the confidence interval would be:

lower bound = 0.16 - 0.045 = 0.115 (rounded to 11%)

upper bound = 0.16 + 0.045 = 0.205 (rounded to 21%)

Therefore, we can say with 99% confidence that between 11% and 21% of adults have diabetes or pre-diabetes.

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1. The modal number of pairs of shoes owned by the children is 3.

2. The median number of pairs of shoes owned by the children is 3.

3. The Mean is 3.

1. The modal number of pairs of shoes owned by the children is 3.

2. The median number of pairs of shoes owned by the children

= 14/2 th term

= 7 th term

= 3

3. The Mean

= (1 x 2+ 2 x 3+ 3 x 5+ 4 x 2 + 5 x 1+ 6x 1)/ (2 +3 +5 +2 + 1 +1)

= 42/14

= 3

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1) Interest amount = $16.5

2) Interest amount = $4.125

3) Interest amount = $8.36

And, After 5 1/2 years;

Interest amount = $22.99

4) Interest amount = $33.02

5) Interest amount = $45.6

Now, We can simplify as;

1) Principal amount = $550

Rate = 3%

Time = 1 year

Hence, We get;

Interest amount = 550 x 3 x 1 / 100

                         = $16.5

2)  Principal amount = $1500

Rate = 3%

Time = 1/4 year

Hence, We get;

Interest amount = 1500 x 3 x 1 / 100 x 4

                         = $4.125

3) Principal amount = $418

Rate = 2%

Time = 1 year

Hence, We get;

Interest amount = 418 x 2 x 1 / 100

                         = $8.36

And, After 5 1/2 years;

Interest amount = 418 x 11 x 1 / 100 x 2

                         = $22.99

4)  Principal amount = $825.5

Rate = 4%

Time = 1 year

Hence, We get;

Interest amount = 825.5 x 4 x 1 / 100

                         = $33.02

5) Principal amount = $1140

Rate = 6%

Time = 1 year

Hence, We get;

Interest amount = 1140 x 6 x 1 / 100

                         = $68.4

And, After 5 4 years;

Interest amount = 1140 x 4 x 1 / 100

                         = $45.6

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The true mean ages for cars owned by students and faculty is (−0.25, 1.27).

Rounding to two decimal places, the 98% confidence interval is (-0.25, 1.27).

Step 1:

The point estimate for the true difference between the population means is:

x1 - x2 = 7.41 - 6.9 = 0.51

Step 2:

The margin of error can be calculated as:

ME = z*(σ1²/n1 + σ2²/n2)^(1/2)

where z is the critical value for a 98% confidence level, n1 and n2 are the sample sizes, and σ1 and σ2 are the population standard deviations for the two groups.

For a 98% confidence level, the critical value is 2.33 (from a standard normal distribution table).

Substituting the given values, we get:

ME = 2.33*(3.72²/215 + 2.26²/252)^(1/2) = 0.758282

Rounding to six decimal places, the margin of error is 0.758282.

Step 3:

The 98% confidence interval can be calculated as:

(x1 - x2) ± ME

Substituting the values, we get:

0.51 ± 0.76

Therefore, the 98% confidence interval for the difference between the true mean ages for cars owned by students and faculty is (−0.25, 1.27).

Rounding to two decimal places, the 98% confidence interval is (-0.25, 1.27).

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Answer:

Step-by-step explanation:

If the squares of the two shorter sides add up to the square of the hypotenuse, the triangle contains a right angle.

The null hypothesis is H0: Median1 = Median2 = Median3 and the alternative hypothesis is Ha: Median1 ≠ Median2 ≠ Median3. The test statistic is H = 9.73. The p-value is 0.007. Reject H0. There is sufficient evidence to conclude that there is a significant difference in the time required by the three methods.

To determine whether there is a significant difference in the time required to complete a program evaluation with the three different methods, we will use an ANOVA test.

1. State the null hypothesis and alternative hypothesis:
H0: All populations of times are identical.
Ha: Not all populations of times are identical.

2. Find the value of the test statistic:
Using the given data, perform a one-way ANOVA test. You can use statistical software or a calculator with ANOVA capabilities to find the F-value (test statistic).

3. Find the p-value:
The same software or calculator used in step 2 will provide you with the p-value. Remember to round your answer to three decimal places.

4. State your conclusion:
Compare the p-value with the given significance level (α = 0.05).
- If the p-value is less than α, reject H0. There is sufficient evidence to conclude that there is a significant difference in the time required by the three methods.
- If the p-value is greater than or equal to α, do not reject H0. There is not sufficient evidence to conclude that there is a significant difference in the time required by the three methods.

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Answer:

they jumped

Step-by-step explanation:

They jump because they want to escape Mario and Javier. Paula is very nervous because there are many people, it is not possible to escape quickly

I hope I’m right if not I’m sorry

The volume of the piñata that Mrs. Powell is making for her son's birthday, would be 2, 016 in ³

The piñata that Mrs. Powell is making, has a composite shape which means that you can find the volume by first finding the volume of the two composite shapes.

The volume of the cube is:

= Length x Width x Height

= 12 x 12 x 12

= 1, 728 in ³

Then the volume of the triangular prism :

= 1 / 2 x base x height x width

= 1 / 2 x 8 x 12 x 6

= 288 in ³

The volume of the pinata is:

= 1, 728 + 288

= 2, 016 in ³

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The number of possibilities to make three-digit numbers from 1,4,5,6,3 that the first digit is even and the third digit is odd is 24.

1) To find the number of possibilities to make three-digit numbers from 1, 4, 5, 6, 3 where the first digit is even and the third digit is odd, follow these steps:

Identify the even numbers (for the first digit) - 4 and 6.
Identify the odd numbers (for the third digit) - 1, 3, and 5.
Calculate the possibilities for the second digit. Since we're using the remaining digits, there are 3 options left for each combination.
Multiply the possibilities together: 2 (even numbers) x 3 (second digit options) x 3 (odd numbers) = 18 possibilities.

2) To find the number of ways 5 students can seat in a circle, use the formula (n-1)!. Where n is the number of students.

For 5 students, there are (5-1)! = 4! = 4 x 3 x 2 x 1 = 24 ways for them to sit in a circle.

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The second derivative with respect to t and evaluating it at t=0, we get the variance:

Var(X+Y) = Mx+y''(0) - [Mx+y'(0)]^2 = [-0.18(4e^-2t) + 0

Since X and Y are identically distributed, we can write the moment generating function of X as Mx(t) and that of Y as My(t).

Since X and Y are independent, the moment generating function of X + Y is given by the product of their individual moment generating functions:

Mx+y(t) = Mx(t)My(t)

We are given the moment generating function of X + Y as:

Mx+y(t) = 0.09e^-2t + 0.24e^t + 0.34 + 0.24e^t + 0.09e^2t

We can rewrite this as:

Mx+y(t) = 0.09(e^-2t + e^2t) + 0.48e^t + 0.34

Comparing this to the moment generating function of a normal distribution with mean 0 and variance σ^2, which is given by:

M(t) = e^(μt + σ^2t^2/2)

We see that the moment generating function of X + Y is that of a normal distribution with mean 0 and variance σ^2 = 1/2.

Thus, X + Y ~ N(0, 1/2).

Since X and Y are identically distributed, X ~ N(0, 1/4) and Y ~ N(0, 1/4).

Therefore,

P(X ≤ 0) = P(X - Y ≤ -Y) = P(Z ≤ -Y/√(1/2)),

where Z ~ N(0,1).

Since X and Y are identically distributed, we have

P(X - Y ≤ -Y) = P(Y - X ≤ X) = P(-Y + X ≤ X) = P(X ≤ Y)

So,

P(X ≤ 0) = P(X ≤ Y) = P(X - Y ≤ 0)

= P[(X+Y) - 2Y ≤ 0]

= P[Z ≤ 2(Y - X)/√2]

where Z ~ N(0,1).

Now, let's find the mean and variance of X + Y:

E[X + Y] = E[X] + E[Y] = 2E[X]

Since X and Y are identically distributed, we have E[X] = E[Y].

Thus, E[X + Y] = 2E[X] = 2E[Y]

And,

Var(X + Y) = Var(X) + Var(Y) = 2Var(X)

Since X and Y are identically distributed, we have Var(X) = Var(Y).

Thus, Var(X + Y) = 2Var(X)

Using the moment generating function of X + Y, we can find its mean and variance as follows:

Mx+y(t) = E[e^(t(X+Y))]

Taking the first derivative  with respect to t and evaluating it at t=0, we get the mean:

E[X+Y] = Mx+y'(0) = [0.09(-2e^-2t) + 0.48e^t + 0.24e^t + 0.18(2e^2t)]|t=0

= -0.18 + 0.24 + 0.18 = 0.24

Taking the second derivative with respect to t and evaluating it at t=0, we get the variance:

Var(X+Y) = Mx+y''(0) - [Mx+y'(0)]^2 = [-0.18(4e^-2t) + 0.

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It is true that, If (a, b) = 1 then if a and b are odd numbers, then (a + b, a − b) = 2. Otherwise, (a + b, a − b) = 1

GCD, or Greatest Common Divisor, is a fundamental concept in number theory. It is defined as the largest positive integer that divides both two or more integers without leaving a remainder.

In other words, the GCD of two numbers is the largest positive integer that divides both of them evenly.

Here we have

Let's consider two cases:

Case 1: a and b are odd numbers

In this case, we can express a and b as:

a = 2k+1

b = 2m+1

where k and m are integers.

Then,

a+b = (2k+1) + (2m+1) = 2(k+m+1)

a-b = (2k+1) - (2m+1) = 2(k-m)

We can see that both a+b and a-b are even.

Therefore, (a+b, a-b) is at least 2.

Now, let's show that (a+b, a-b) cannot be larger than 2:

Suppose, for contradiction, that (a+b, a-b) = d > 2.

Then, d divides both (a+b) and (a-b).

We can write (a+b) and (a-b) as:

=> a+b = dx

=> a-b = dy

where x and y are integers.

Adding the above two equations, we get:

2a = d(x+y)

Since a is odd, d must be odd as well.

Substituting for 'a' in terms of x and y, we get:

=> 2(2k+1) = d(x+y)

=> 4k+2 = d(x+y)

=> 2(2k+1) = 2d(x+y)/2

=> 2k+1 = d(x+y)/2

We can see that d must divide 2k+1 since x and y are integers.

However, we know that (a,b) = 1, which means that a and b do not have any common factors other than 1.

Since a is odd, 2 does not divide a.

Therefore, d cannot be greater than 2, which is a contradiction.

Hence,

(a+b, a-b) = 2 when a and b are odd numbers.

Case 2: a and b are not both odd numbers

Without loss of generality,

Let's assume that a is even and b is odd.

Then, a+b and a-b are both odd.

Since odd numbers do not have any factors of 2, (a+b, a-b) = 1.

Therefore,

(a+b, a-b) = 2 if a and b are both odd and (a+b, a-b) = 1 if a and b are not both odd.

By the above explanation,

It is true that, If (a, b) = 1 then if a and b are odd numbers, then (a + b, a − b) = 2. Otherwise, (a + b, a − b) = 1

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The correct answer is c. Whenever the sample size is less than 5% of the population size, a finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means.

This correction factor takes into account the fact that when the sample size is small relative to the population, the variability of the sample means is affected. Without the correction factor, the standard deviation of the sampling distribution would be overestimated. However, if the sample size is large enough (more than 5% of the population size), the effect of finite population correction is negligible and can be ignored. If the population is infinite, the correction factor is not necessary as the sample size can be considered as a small proportion of the infinite population.

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A X-squared goodness-of-fit test is performed on a random sample of 360 individuals to see if the number of birthdays each month is proportional to the number of days in the month. X-squared is determined to be 23.5. The P -value for this test is between d) 0.010 < P < 0.025

In this scenario, a X-squared goodness-of-fit test is performed to determine if the number of birthdays each month is proportional to the number of days in the month. With a random sample of 360 individuals and a χ-squared value of 23.5, you need to find the corresponding P-value range.

To find the P-value range, you can use a χ-squared distribution table or calculator. Since there are 12 months in a year, the degrees of freedom for this test will be 12 - 1 = 11.

Upon checking the table or using a calculator, you will find that the P-value for this test is between:

Your answer: d) 0.010 < P < 0.025

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The algebraic expression for Mr. Robinson's budget constraint if running shoes and basketball shorts cost $20 each is:

200 = 20S + 20B, where S is the number of running shoes and B is the number of basketball shorts

The algebraic expression for Mr. Robinson's budget constraint if the cost of the basketball shorts rises to $30 each is:

200 = 20S + 30B

a) If Mr. Robinson spends his entire income on running shoes and basketball shorts, which cost $20 each, we can write the budget constraint as:
200 = 20S + 20B, where S is the number of running shoes and B is the number of basketball shorts.

b) To determine the number of goods he will buy, we need more information about his preferences. Without any further information, we cannot determine the exact quantities of running shoes (S) and basketball shorts (B) he will buy.

c) If the price of basketball shorts rises to $30 each, the budget constraint becomes:
200 = 20S + 30B

d) Again, to determine the number of goods he will buy with the new prices, we need more information about his preferences.

e) To illustrate the results in parts (a) and (c), you would create a graph with running shoes on the x-axis and basketball shorts on the y-axis. The budget constraint in part (a) would be a straight line with a slope of -1 and an intercept of 10 on both axes. For part (c), the budget constraint would be a straight line with a slope of -2/3 and an intercept of 10 on the x-axis and 6.67 on the y-axis.

As for the decomposition of the income and substitution effect, this cannot be determined without more information about Robinson's preferences or the shape of his indifference curves.

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a. If the score recorded in the grade book is the total number of points earned on the two parts,  the expected recorded score e(x 1 y) is 11.6.

b.  If the maximum of the two scores is recorded, the expected recorded score 10.08.

a) The expected recorded score is given by:

e(x + y) = ΣΣ(x + y) * p(x, y)

So, we have:

e(x + y) = (0+0)*0.02 + (5+0)*0.04 + (10+0)*0.06 + (15+0)*0.02 + (5+10)*0.15 + (10+10)*0.20 + (15+10)*0.15 + (5+15)*0.01 + (10+15)*0.14 + (15+15)*0.01

Simplifying:

e(x + y) = 0.02(0 + 0) + 0.04(5 + 0) + 0.06(10 + 0) + 0.02(15 + 0) + 0.15(5 + 10) + 0.20(10 + 10) + 0.15(15 + 10) + 0.01(5 + 15) + 0.14(10 + 15) + 0.01(15 + 15)

e(x + y) = 11.6

So, the expected recorded score is 11.6.

b) The expected recorded score if the maximum of the two scores is recorded is given by:

e(max(x, y)) = ΣΣ(max(x, y)) * p(x, y)

So, we have:

e(max(x, y)) = max(0, 5)*0.06 + max(5, 0)*0.04 + max(10, 0)*0.06 + max(15, 0)*0.02 + max(5, 10)*0.15 + max(10, 10)*0.20 + max(15, 10)*0.15 + max(5, 15)*0.01 + max(10, 15)*0.14 + max(15, 15)*0.01

Simplifying:

e(max(x, y)) = 0.065 + 0.045 + 0.0610 + 0.0215 + 0.1510 + 0.2010 + 0.1515 + 0.0115 + 0.1415 + 0.0115

e(max(x, y)) = 10.08

So, the expected recorded score is 10.08.

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Hi! To answer your question, let's denote the total number of processors as n and the total number of jobs as n as well. Since there are n possible assignments for each job, the total number of assignment vectors is n^n.

To find the probability that exactly one processor will be idle, we can use the following steps:

1. Select the idle processor: There are n ways to choose the idle processor.
2. Assign jobs to the remaining (n-1) processors: Each of the n jobs can be assigned to any of the remaining (n-1) processors, which gives us (n-1)^n possible assignment vectors.

Now, to calculate the probability, we can divide the number of assignment vectors with exactly one idle processor by the total number of assignment vectors:

Probability = (n * (n-1)^n) / n^n

This expression gives the probability that exactly one processor will be idle when there are n jobs and n processors.

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P(t < 1.94) ≈ 0.913 (rounded to three decimal places). For 14 degrees of freedom and P(-c < t < c) = 0.95, c ≈ 2.145

(a) To compute P(t < 1.94) for a t distribution with 3 degrees of freedom, you can use a t-distribution table or statistical software. Looking up the value in a table or using software, you will find that P(t < 1.94) ≈ 0.913.
(b) To find the value of c for a t distribution with 14 degrees of freedom such that P(-c < t < c) = 0.95, you can use a t-distribution table or statistical software again. For a 0.95 probability and 14 degrees of freedom, you will find that c ≈ 2.145.
So, the answers are:
(a) P(t < 1.94) ≈ 0.913 (rounded to three decimal places)
(b) For 14 degrees of freedom and P(-c < t < c) = 0.95, c ≈ 2.145

For the first part of the question, we need to use a t-distribution table or calculator to find the probability of the t variable being less than 1.94 with 3 degrees of freedom. Using a t-distribution table, we find that the probability is 0.950 with three decimal places. Therefore, P(t < 1.94) = 0.950.
For the second part of the question, we need to find the value of c such that the probability of the t variable being less than -c with 14 degrees of freedom is 0.025. Using a t-distribution table or calculator, we find that the value of c is 2.145 with three decimal places. Therefore, P(-c < t < c) = 0.95.

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The solutions to the equation to the nearest tenth are x = 10.1 and x = 2.9.

We have,

-2x² - 13x + 20 = -3x²

Combining like terms

-2x² - 13x + 20 = -3x²

x² - 13x + 20 = 0 (adding 3x² to both sides)

Now we can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / 2a

In this case,

a = 1, b = -13, and c = 20.

Substituting these values into the quadratic formula:

x = (-(-13) ± √((-13)² - 4(1)(20))) / 2(1)

x = (13 ± √(169 - 80)) / 2

x = (13 ± √(89)) / 2

So the solutions are:

x = (13 + √(89)) / 2

x ≈ 10.1

and

x = (13 - √(89)) / 2

x ≈ 2.9

Therefore,

The solutions to the equation to the nearest tenth are x ≈ 10.1 and x ≈ 2.9.

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we can extend {1+x,1} to a basis of P3 by adding x^2.

To extend {1+x,1} to a basis of P3, we need to find one more polynomial that is linearly independent of these two. One way to do this is to choose a polynomial of degree 2, since we are working in P3. Let's try x^2.

We need to check if x^2 is linearly independent of {1+x,1}. This means we need to solve the equation a(1+x) + b(1) + c(x^2) = 0, where a, b, and c are constants.

Expanding this equation gives us a + ax + b + cx^2 = 0. Since x and x^2 are linearly independent, this means that a = 0 and c = 0. Therefore, we are left with just b(1) = 0, which means that b = 0 as well.

This shows that {1+x,1,x^2} is a linearly independent set, which means that it forms a basis of P3. Therefore, we have successfully extended {1+x,1} to a basis of P3 by adding x^2.

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The radius of convergence is r = 1 in the given case.

We can start by using the power series expansion of ln(1+x):

[tex]ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...[/tex]

Now we can substitute this into the integral and use the linearity of integration to obtain:

[tex]∫ x^4 ln(1+x) dx = ∫ x^5 - x^6/2 + x^7/3 - x^8/4 + ... dx[/tex]

We can integrate each term separately to get:

∫ [tex]x^5 dx - ∫ x^6/2 dx + ∫ x^7/3 dx - ∫ x^8/4 dx[/tex]+ ...

Using the power rule for integration, we can simplify this to:

[tex]x^6/6 - x^7/14 + x^8/24 - x^9/36 +[/tex]...

We have now expressed the indefinite integral as a power series with coefficients given by the formula:

[tex]a_n = (-1)^(n+1) / n[/tex]

The radius of convergence of this power series can be found using the ratio test:

[tex]lim |a_(n+1)/a_n| = lim (n/(n+1)) = 1[/tex]

Since the limit is equal to 1, the ratio test is inconclusive, and we need to consider the endpoints of the interval of convergence.

The integral is undefined at x=-1, so the interval of convergence must be of the form (-1,r] or [-r,1), where r is the radius of convergence.

To determine the value of r, we can use the fact that the series for ln(1+x) converges uniformly on compact subsets of the interval (-1,1). This implies that the series fo [tex]x^4[/tex] ln(1+x) also converges uniformly on compact subsets of (-1,1), and hence on the interval (-r,r) for any r < 1.

Therefore, the radius of convergence is r = 1.

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10.-8.=2h

12:00-10:30=1.5h

2h-1.5h=0.5h

13km÷0.5h×2h=52km

The amount of time that students spend studying in the library in one sitting is normally distributed with a mean of 46 minutes and a standard deviation of 19 minutes. Hence the distribution of X is X ~ N (46, 19).

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(a) Letting X1, ..., Xy be independent random variables and Using the union bound, we have P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0.

(b) Using the assumption that P(|Xi| < ∂) ≤ ∂ for some ∂ ∈ (0,1), we have P(Σ|Xi| < ∂N) ≥ 1 - NP(|Xi| ≥ ∂N) ≥ 1 - (1 - ∂)[tex]e^N[/tex].

Setting t = 2N[tex]e^ε[/tex], we obtain

which is equivalent to

By setting ∂ = 2Ne**ε/N, we get

P(Σ|Xi| < ∂) ≥ 1 - e**(-ε), and therefore,

Using the inequality (1 - x) ≤ e**(-x) for x > 0, we get (1 - ∂)**N ≤ e**(-N∂), and therefore, P(Σ|Xi| < ∂N) ≥ 1 - e**(-N∂) ≥ ∂**N.

Thus, we have shown that NP(Σ|Xi| < ∂N) ≥ ∂**N for some ∂ ∈ (0,1) and P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0

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