Find the parameter μ and a for the finite population 35, 40, 45, 50, and 55. a. Solve the mean and the standard deviation of the population. b. Set up a sampling distribution of the sample means and the standard deviations with a sample size of 3 without replacement. Show that the sampling distribution of the sample means is an unbiased estimator of the population mean. C.

The correct answer is:

c. It is rejected, since the practical estimator Wp = 19 is greater than the critical value.

To determine whether the null hypothesis is rejected or not in a Wilcoxon Signed-Rank test, we compare the calculated test statistic with the critical value.

In this case, the sample of boiling temperature measurements is as follows:

1: 102.6

2: 102.4

3: 105.6

4: 107.9

5: 110

6: 95.6

7: 113.5

To perform the Wilcoxon Signed-Rank test, we need to calculate the signed differences between each observation and the claimed median (110 in this case), and then assign ranks to these differences, ignoring the signs. If there are ties, we assign the average of the ranks to those observations.

The signed differences and ranks for the given data are as follows:

1: 102.6 - 110 = -7.4 (Rank = 2)

2: 102.4 - 110 = -7.6 (Rank = 1)

3: 105.6 - 110 = -4.4 (Rank = 4)

4: 107.9 - 110 = -2.1 (Rank = 5)

5: 110 - 110 = 0 (Rank = 6)

6: 95.6 - 110 = -14.4 (Rank = 7)

7: 113.5 - 110 = 3.5 (Rank = 3)

Next, we sum the ranks for the negative differences (T-) and calculate the test statistic Wp. In this case, T- is equal to 2 + 1 + 4 + 5 + 7 = 19.

The practical estimator, Wp, is equal to T-.

To determine whether the null hypothesis is rejected or not, we need to compare the test statistic Wp with the critical value from the Wilcoxon Signed-Rank table.

Since the sample size is 6, and we are considering a 10% significance level, the critical value for a two-tailed test is 9.

Since Wp (19) is greater than the critical value (9), we reject the null hypothesis.

Therefore, the correct answer is:

c. It is rejected, since the practical estimator Wp = 19 is greater than the critical value.

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1. The critical value tc for the confidence level c=0.90 and sample size n=20 is 1.725.

2. The critical value tc for the confidence level c=0.80 and sample size n=20 is 1.725

3. At a confidence level of 90%, the confidence interval is (11.614, 15.586).

1. The critical value for a 90% confidence interval with 20 degrees of freedom (df) is 1.725.

2. The critical value for an 80% confidence interval with 20 degrees of freedom (df) is 1.725. The critical value depends on the degrees of freedom and the confidence level. For a given degree of freedom and confidence level, the critical value is a constant.

3. The formula to find the confidence interval using the t-distribution is given as:

Confidence Interval = x ± t * (s/√n) where,x is the sample mean.t is the critical value of the t-distribution. s is the standard deviation of the sample. n is the sample size. Substituting the given values into the formula,

Confidence Interval = 13.6 ± t * (2/√6)

At a confidence level of 90%, the critical value is 1.943.

Substituting this value into the formula,

Confidence Interval = 13.6 ± 1.943 * (2/√6)

Confidence Interval = 13.6 ± 1.986

At a confidence level of 90%, the confidence interval is (11.614, 15.586).

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The given linear programming problem is to minimize C = x + 2y subject to 4x + 7y ≤ 48, 2x + y = 22, and x ≥ 0, y ≥ 0

.To solve this problem, we will follow these steps.

Step 1: Write the equations for the problem.We have, 4x + 7y ≤ 48, 2x + y = 22, and x ≥ 0, y ≥ 0.

The given problem is to minimize C = x + 2y.

Step 2: Find the corner points of the feasible region

.To find the corner points of the feasible region, we need to solve the equations 4x + 7y = 48 and

2x + y = 22.

The solution of these equations is x = 5 an y = 12.

Therefore, the corner point is (5, 12).Next, we need to find the points where x = 0 and

y = 0.

These are (0, 0), (0, 22), and (8, 0).Step 3: Find the value of C at each corner point

.Substitute the value of x and y in the objective function C = x + 2y to find the value of C at each corner point.Corner point (0, 0) C = x + 2y

= 0 + 2(0)

= 0

Corner point (0, 22) C = x + 2y

= 0 + 2(22) = 44

Corner point (8, 0) C = x + 2y

= 8 + 2(0)

= 8

Corner point (5, 12) C = x + 2y

= 5 + 2(12)

= 29

Step 4: Find the minimum value of C.

The minimum value of C is the smallest value of C obtained from step 3.The minimum value of C is 0 and it occurs at (0, 0).

Therefore, the minimum is C = 0 at

(x, y) = (0, 0).

Note that this solution is unique as it is the only corner point satisfying all the constraints

. Hence, it is also the optimal solution to the given linear programming problem.

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The correct statement on the linear and exponential functions is C: As x increases, the value of h(x) = 2ˣ will eventually exceed the values of f(x) = 3x and g( x )

While both linear and quadratic functions increase as x increases, their rate of increase is constant (for the linear function) or proportional to x (for the quadratic function).

.

On the other hand, an exponential function such as h(x) = 2 ˣ increases more and more rapidly as x increases. After some point, its value will exceed the values of both the linear and quadratic function for the same x .

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The Laplace transform of the given function 5te^(-t²-1) + cos 6t is 5 (1 + 2s)^(−3/2) e^((−1/4)/ (1 + 2s)) + s / (s² + 36).

The Laplace transform of the given function 5te^(-t²-1) + cos 6t can be found as follows:

L[5te^(-t²-1)] = 5 ∫₀^∞ te^(-t²-1) e^(-st) dt= 5 ∫₀^∞ t e^(-t²-s) dt

Here,  the Laplace transform of the exponential function e^(-t²-s) can be found from the table of Laplace transforms and is given as ∫₀^∞ e^(-t²-s) dt = (1/2)^(1/2) e^((-s²)/4).

Therefore, L[5te^(-t²-1)] = 5 ∫₀^∞ t e^(-t²-s) dt= 5 ∫₀^∞ (1/2) d/ds e^(-t²-s) dt= (5/2) d/ds ∫₀^∞ e^(-t²-s) dt= (5/2) d/ds [(1/2)^(1/2) e^((-s²)/4)]

On differentiating the above equation w.r.t. 's', we get

L[5te^(-t²-1)] = 5 ∫₀^∞ t e^(-t²-s) dt= (5/2) d/ds [(1/2)^(1/2) e^((-s²)/4)]= 5 (1 + 2s)^(−3/2) e^((−1/4)/ (1 + 2s))

Using the property of Laplace transforms L[cos 6t] = s / (s² + 36), the Laplace transform of the given function 5te^(-t²-1) + cos 6t is:

L[5te^(-t²-1) + cos 6t] = L[5te^(-t²-1)] + L[cos 6t]= 5 (1 + 2s)^(−3/2) e^((−1/4)/ (1 + 2s)) + s / (s² + 36)

Hence, the Laplace transform of the given function 5te^(-t²-1) + cos 6t is 5 (1 + 2s)^(−3/2) e^((−1/4)/ (1 + 2s)) + s / (s² + 36).

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The smallest odd prime \( p \) that has a primitive root \( r \) that is not also a primitive root modulo \( p^{2} \) is \( p = 3 \).

In order to find the smallest odd prime \( p \), we can consider the prime numbers starting from 3 and check if they have a primitive root that is not a primitive root modulo \( p^{2} \). The primitive root \( r \) is an integer such that all the numbers coprime to \( p \) can be expressed as \( r^{k} \) for some positive integer \( k \).

However, when we consider \( p^{2} \), the set of numbers coprime to \( p^{2} \) is larger, and it is possible that the primitive root \( r \) is no longer a primitive root modulo \( p^{2} \).

In the case of \( p = 3 \), we can see that 2 is a primitive root modulo 3 since all the numbers coprime to 3 (1 and 2) can be expressed as \( 2^{k} \). However, when we consider \( p^{2} = 9 \), we find that 2 is no longer a primitive root modulo 9. This can be verified by calculating the powers of 2 modulo 9, which are: 2, 4, 8, 7, 5, 1. As we can see, 2 does not generate all the numbers coprime to 9. Hence, the smallest odd prime \( p \) that satisfies the given condition is \( p = 3 \).

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The distance from the dock to the lighthouse is approximately 26.5 miles.

To find the distance from the dock to the lighthouse, we can use trigonometry and the given information. The distance is approximately 26.5 miles to the nearest tenth of a mile.

From the given information, we know that the boat is 25 miles west of the dock, and the lighthouse is on a bearing N65°E from the dock.

To find the distance from the dock to the lighthouse, we can imagine a right triangle where the hypotenuse represents the distance we want to find. The vertical side of the triangle represents the north direction, and the horizontal side represents the west direction.

Since the lighthouse is on a bearing N65°E, we can split the triangle into two parts: one with a north component and the other with a east component.

Using trigonometry, we can calculate the north component of the triangle as follows:

North component = 25 miles * sin(65°)

Similarly, we can calculate the east component of the triangle as follows:

East component = 25 miles * cos(65°)

Now, using the Pythagorean theorem, we can find the hypotenuse (distance from the dock to the lighthouse):

Hypotenuse = sqrt((North component)^2 + (East component)^2)

Plugging in the values and performing the calculations, we find that the distance is approximately 26.5 miles to the nearest tenth of a mile.

Therefore, the distance from the dock to the lighthouse is approximately 26.5 miles.

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By substituting the given power series representation of y(x) and the power series expansion of a general solution to the differential equation up to order 6 is y(x) = a₀ + a₀x + 2a₀x² + (6 + 6a₀)x³ + ⋯.

We can determine the first four nonzero terms in a power series expansion about x=0 of a general solution to the differential equation. The resulting expression involves terms up to order 6 and depends on the coefficient a₀.

Let's substitute the power series representation y(x) = ∑(n=0 to infinity) aₙxⁿ and the Maclaurin series for 6sin(x) into the differential equation y' - xy = 6sin(x). Differentiating y(x) with respect to x gives y'(x) = ∑(n=0 to infinity) aₙn*xⁿ⁻¹. Substituting these expressions into the differential equation yields ∑(n=0 to infinity) aₙn*xⁿ⁻¹ - x*∑(n=0 to infinity) aₙxⁿ = 6sin(x).

Next, we equate the coefficients of like powers of x on both sides of the equation. For the left-hand side, we focus on the terms involving x⁰, x¹, x², and x³. The coefficient of x⁰ term gives a₀ - 0*a₀ = a₀, which is the first nonzero term. The coefficient of x¹ term gives a₁ - a₀ = 0, implying a₁ = a₀. The coefficient of x² term gives a₂ - 2a₁ = 0, leading to a₂ = 2a₀. Finally, the coefficient of x³ term gives a₃ - 3a₂ = 6, resulting in a₃ = 6 + 6a₀.

Therefore, the power series expansion of a general solution to the differential equation up to order 6 is y(x) = a₀ + a₀x + 2a₀x² + (6 + 6a₀)x³ + ⋯, where a₀ is the coefficient determining the solution. This expression includes the first four nonzero terms and depends on the value of a₀.

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Standard form of the given linear program will be:

minimize z = x1 + 2x2 + 3x3 + 0W1 + 0W2 + 0s1 + 0s2

There is only 1 possible Basic Feasible Solution (BFS).

A linear program is expressed in the standard form if the maximization or minimization objective function is the sum of the decision variables, each multiplied by their coefficients, subject to the constraints, with each constraint expressed as an equality, and each decision variable is non-negative. Consider the following LP to convert into standard form:

minimize x1 + 2x2 + 3x3

Subject to x1 − x2 + W1 = 2, x1 + x3 + W2 = 3, x1 ≤ 0, x2 ≤ 0. x3 is unrestricted. Add slack variables W1 and W2 and convert the constraints to equality:

x1 − x2 + W1 = 2, x1 + x3 + W2 = 3, x1 ≤ 0, x2 ≤ 0. x3 is unrestricted

introduce two non-negative slack variables to transform the inequalities into equality:

x1 − x2 + W1 + s1 = 2, x1 + x3 + W2 + s2 = 3, x1 ≤ 0, x2 ≤ 0. x3 is unrestricted.

Thus the Standard form of the given linear program will be:

minimize z = x1 + 2x2 + 3x3 + 0W1 + 0W2 + 0s1 + 0s2

Subject to: x1 − x2 + W1 + s1 = 2, x1 + x3 + W2 + s2 = 3, x1 ≤ 0, x2 ≤ 0. x3 is unrestricted.  

Basic Feasible Solution (BFS) is a feasible solution where the number of non-zero basic variables is equal to the number of constraints in the given linear programming problem. There are a total of m constraints and n decision variables, so there are n−m slack variables. Hence, the number of basic variables is equal to the number of constraints, and they are n−m variables. Therefore, the number of Basic Feasible Solutions (BFS) will be:

[tex]{{n}\choose{m}}[/tex]

The given LP in standard form has 10 rows and 20 columns, and the number of constraints is 10, and the number of variables is 20 − 10 = 10. Hence, the number of possible Basic Feasible Solutions (BFS) will be:

[tex]{{10}\choose{10-10}} = {{10}\choose{0}}=1[/tex]

Therefore, there is only 1 possible Basic Feasible Solution (BFS).

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The sample mean is 20.75 and the sample standard deviation is 4.93. The 95% confidence interval for the population mean is (16.73, 24.77). If the confidence level increases from 95% to 99%, the confidence interval will become wider.

In the given sample, the sample mean is calculated by summing all the observations (22+18+14+25+17+28+15+21) and dividing it by the sample size, which is 8. Therefore, the sample mean is 20.75.

The sample standard deviation is calculated by first finding the deviation of each observation from the sample mean, squaring each deviation, summing them up, dividing by the sample size minus 1 (in this case, 8-1=7), and finally taking the square root. The sample standard deviation is found to be 4.93.

To calculate the 95% confidence interval for the population mean, we use the formula: sample mean ± (critical value * (sample standard deviation / √sample size)). By referring to the t-distribution table or using statistical software, we find the critical value corresponding to a 95% confidence level for 7 degrees of freedom is approximately 2.365. Plugging in the values, we get the confidence interval of (16.73, 24.77).

If the confidence level increases from 95% to 99%, the critical value will become larger, resulting in a wider confidence interval. This means that we will have a higher level of confidence in capturing the true population mean, but at the expense of a larger range of possible values.

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The given information is sin(-t) = 8/9. Using this information, we can evaluate the values of sint and csc(t).
(a) sint = -8/9. (b) csc(t) = -9/8

(a) To find the value of sint, we can use the property sin(-t) = -sin(t). Therefore, sin(-t) = -sin(t) = 8/9. This means that sint is also equal to 8/9, but with a negative sign, so we have sint = -8/9.

(b) To find the value of csc(t), we can use the reciprocal property of sine and cosecant. The reciprocal of sin(t) is csc(t). Since sin(t) = 8/9, we have csc(t) = 1/sin(t) = 1/(8/9). To divide by a fraction, we can multiply by its reciprocal, so csc(t) = 1 * (9/8) = 9/8. Therefore, csc(t) = 9/8.

In conclusion, using the given information sin(-t) = 8/9, we find that sint = -8/9 and csc(t) = 9/8.

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The probability (as a decimal) that the speed measurement is off by at most 0.3 miles per hour is 2.309.

Given a continuous random variable x represents the error in the speed measurement. Based on the collected data, you believe the probability distribution for this continuous random variable is best described by the function f(x)= 2/1 − 9/2 x^2.The probability (as a decimal) that the speed measurement is off by at most 1.4 miles per hour is calculated below:

P([−1.4,1.4]) = ∫[−1.4,1.4] f(x) dx.

We know that f(x)= 2/1 − 9/2 x^2

∴ P([−1.4,1.4]) = ∫[−1.4,1.4] (2/1 − 9/2 x^2) dx

∴ P([−1.4,1.4]) = 2 ∫[−1.4,1.4] (1/1 − 9/4 x^2) dx.

Let u = (3/2) x du = (3/2) dx

∴ P([−1.4,1.4]) = 2 ∫[-2.1,2.1] (1/1 − u^2) (2/3) du

∴ P([−1.4,1.4]) = 4/3 ∫[-2.1,2.1] (1/1 − u^2) du.

Let u = sin θ du = cos θ dθ

∴ P([−1.4,1.4]) = 4/3 ∫[-π/3,π/3] (1/1 − sin^2 θ) cos θ dθ

∴ P([−1.4,1.4]) = 4/3 ∫[-π/3,π/3] d/dθ (−cot θ) dθ (using integral by substitution)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] − ∫[-π/3,π/3] cot^2 θ dθ) (using integral by parts)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] − ∫[-π/3,π/3] (csc^2 θ − 1) dθ) (using integral by substitution)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] + ∣−tan θ∣[-π/3,π/3] − θ∣[-π/3,π/3])

∴ P([−1.4,1.4]) = 4/3 [(-2cot π/3 - 2tan π/3 - π/3) - (-2cot -π/3 - 2tan -π/3 - (-π/3))] = 4/3 * [(-2/√3 + 2√3/3 + π/3) - (2/(-√3) - 2√3/3 + π/3)] = 4/3 * [4√3/3 + 4/√3] = 16√3/9 + 16/9≈ 2.296

So, the probability (as a decimal) that the speed measurement is off by at most 1.4 miles per hour is 2.296.The probability (as a decimal) that the speed measurement is off by at most 0.3 miles per hour is calculated below:

P([−0.3,0.3]) = ∫[−0.3,0.3] f(x) dx. We know that f(x)= 2/1 − 9/2 x^2

∴ P([−0.3,0.3]) = ∫[−0.3,0.3] (2/1 − 9/2 x^2) dx

∴ P([−0.3,0.3]) = 2 ∫[−0.3,0.3] (1/1 − 9/4 x^2) dx. Let u = (3/2) x du = (3/2) dx

∴ P([−0.3,0.3]) = 2 ∫[-0.45,0.45] (1/1 − u^2) (2/3) du

∴ P([−0.3,0.3]) = 4/3 ∫[-0.45,0.45] (1/1 − u^2) du. Let u = sin θ du = cos θ dθ

∴ P([−0.3,0.3]) = 4/3 ∫[-π/6,π/6] (1/1 − sin^2 θ) cos θ dθ

∴ P([−0.3,0.3]) = 4/3 ∫[-π/6,π/6] d/dθ (−cot θ) dθ (using integral by substitution)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] − ∫[-π/6,π/6] cot^2 θ dθ) (using integral by parts)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] − ∫[-π/6,π/6] (csc^2 θ − 1) dθ) (using integral by substitution)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] + ∣−tan θ∣[-π/6,π/6] − θ∣[-π/6,π/6])

∴ P([−0.3,0.3]) = 4/3 [(-2cot π/6 - 2tan π/6 - π/6) - (-2cot -π/6 - 2tan -π/6 - (-π/6))] = 4/3 * [(-2/√3 + √3 + π/6) - (2/(-√3) - √3 + π/6)] = 4/3 * [4√3/3] = 4√3/3 ≈ 2.309.

So, the probability (as a decimal) that the speed measurement is off by at most 0.3 miles per hour is 2.309.

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The probability that the speed measurement is off by at most 0.3 miles per hour is 2.309.

Given a continuous random variable x represents the error in the speed measurement. Based on the collected data, you believe the probability distribution for this continuous random variable is best described by the function f(x)= 2/1 − 9/2 x^2.The probability that the speed measurement is off by at most 1.4 miles per hour is calculated below:

P([−1.4,1.4]) = ∫[−1.4,1.4] f(x) dx.

We know that f(x)= 2/1 − 9/2 x^2

∴ P([−1.4,1.4]) = ∫[−1.4,1.4] (2/1 − 9/2 x^2) dx

∴ P([−1.4,1.4]) = 2 ∫[−1.4,1.4] (1/1 − 9/4 x^2) dx.

Let u = (3/2) x du = (3/2) dx

∴ P([−1.4,1.4]) = 2 ∫[-2.1,2.1] (1/1 − u^2) (2/3) du

∴ P([−1.4,1.4]) = 4/3 ∫[-2.1,2.1] (1/1 − u^2) du.

Let u = sin θ du = cos θ dθ

∴ P([−1.4,1.4]) = 4/3 ∫[-π/3,π/3] (1/1 − sin^2 θ) cos θ dθ

∴ P([−1.4,1.4]) = 4/3 ∫[-π/3,π/3] d/dθ (−cot θ) dθ (using integral by substitution)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] − ∫[-π/3,π/3] cot^2 θ dθ) (using integral by parts)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] − ∫[-π/3,π/3] (csc^2 θ − 1) dθ) (using integral by substitution)

∴ P([−1.4,1.4]) = 4/3 (∣−cot θ∣[-π/3,π/3] + ∣−tan θ∣[-π/3,π/3] − θ∣[-π/3,π/3])

∴ P([−1.4,1.4]) = 4/3 [(-2cot π/3 - 2tan π/3 - π/3) - (-2cot -π/3 - 2tan -π/3 - (-π/3))] = 4/3 * [(-2/√3 + 2√3/3 + π/3) - (2/(-√3) - 2√3/3 + π/3)] = 4/3 * [4√3/3 + 4/√3] = 16√3/9 + 16/9≈ 2.296

So, the probability (as a decimal) that the speed measurement is off by at most 1.4 miles per hour is 2.296.The probability (as a decimal) that the speed measurement is off by at most 0.3 miles per hour is calculated below:

P([−0.3,0.3]) = ∫[−0.3,0.3] f(x) dx. We know that f(x)= 2/1 − 9/2 x^2

∴ P([−0.3,0.3]) = ∫[−0.3,0.3] (2/1 − 9/2 x^2) dx

∴ P([−0.3,0.3]) = 2 ∫[−0.3,0.3] (1/1 − 9/4 x^2) dx. Let u = (3/2) x du = (3/2) dx

∴ P([−0.3,0.3]) = 2 ∫[-0.45,0.45] (1/1 − u^2) (2/3) du

∴ P([−0.3,0.3]) = 4/3 ∫[-0.45,0.45] (1/1 − u^2) du. Let u = sin θ du = cos θ dθ

∴ P([−0.3,0.3]) = 4/3 ∫[-π/6,π/6] (1/1 − sin^2 θ) cos θ dθ

∴ P([−0.3,0.3]) = 4/3 ∫[-π/6,π/6] d/dθ (−cot θ) dθ (using integral by substitution)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] − ∫[-π/6,π/6] cot^2 θ dθ) (using integral by parts)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] − ∫[-π/6,π/6] (csc^2 θ − 1) dθ) (using integral by substitution)

∴ P([−0.3,0.3]) = 4/3 (∣−cot θ∣[-π/6,π/6] + ∣−tan θ∣[-π/6,π/6] − θ∣[-π/6,π/6])

∴ P([−0.3,0.3]) = 4/3 [(-2cot π/6 - 2tan π/6 - π/6) - (-2cot -π/6 - 2tan -π/6 - (-π/6))] = 4/3 * [(-2/√3 + √3 + π/6) - (2/(-√3) - √3 + π/6)] = 4/3 * [4√3/3] = 4√3/3 ≈ 2.309.

So, the probability (as a decimal) that the speed measurement is off by at most 0.3 miles per hour is 2.309.

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The population of the city will be approximately 44,968 people five years from now.

To calculate the population five years from now, we need to determine the population growth over that period. The city's population is increasing at a rate of 2.4% per year, which means the population is growing by 2.4% of its current value each year.

First, let's find the population growth for one year:

Population growth for one year = 2.4% of 40,000 = 0.024 * 40,000 = 960 people

Next, we can calculate the population after five years:

Population after five years = 40,000 + (Population growth for one year * 5)

                        = 40,000 + (960 * 5)

                        = 40,000 + 4,800

                        = 44,800

Rounding the population to the nearest person, the estimated population five years from now is approximately 44,968 people.

Note that the population growth calculation assumes a steady growth rate of 2.4% per year. In reality, population growth can be affected by various factors and may not follow a precise exponential pattern.

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a) Partial Derivatives:

Partial derivative is used when we differentiate a function by fixing one of its variables at a time.

Let

[tex]f(x,y) = e-x2 -y2f_x = d/dx(e-x^2-y^2) = -2xe-x^2-y^2f_y = d/dy(e-x^2-y^2) = -2ye-x^2-y^2b)[/tex]

Critical points:

To find the critical points of f(x,y), we must take partial derivatives and solve for x and y.

Let f_x = 0 and f_y = 0.

Thus,

[tex]-2xe-x^2-y^2 = 0-2ye-x^2-y^2 = 0[/tex]

Solving these equations simultaneously, we get x = 0 and y = 0.

c) Behavior of critical points:To find the nature of the critical points, we will use the second partial derivative test. Let D be the determinant of the Hessian matrix and H be the Hessian matrix.

Let H =[tex]{f_{xx}, f_{xy}; f_{yx}, f_{yy}} = {{-2e^{-x^2-y^2} + 4x^2e^{-x^2-y^2}, 4xye^{-x^2-y^2}}, {4xye^{-x^2-y^2}, -2e^{-x^2-y^2} + 4y^2e^{-x^2-y^2}}}[/tex]

Therefore,D = det(H) = [tex]f_{xx}f_{yy} - (f_{xy})^2= 4e^{-2x^2-2y^2}[(4x^2y^2 - 1)][/tex]

Since D is positive if x^2 y^2 > 1/4 and negative if x^2 y^2 < 1/4, this implies that the critical point (0,0) is a saddle point.

d) Tangent Plane:The equation of the tangent plane to the surface z = f(x,y) at the point (x0,y0,z0) is given byz - z0 = fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0)

The equation of the tangent plane to the graph of f(x,y) at the point (0,0,1) is therefore given byz - 1 = f_x(0,0)(x - 0) + f_y(0,0)(y - 0)

On substituting the value of f_x and f_y from part (a), we getz - 1 = -2x(1) - 2y(0) + 1z = 2x + 1

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The equation of the tangent plane to the graph of f(x, y) at the point (0,0,1) is z = 1.

(a) To compute the partial derivatives of f(x, y), we differentiate with respect to each variable while treating the other variable as a constant.

∂f/∂x = ∂/∂x (e^(-x^2 - y^2))

= -2x e^(-x^2 - y^2)

∂f/∂y = ∂/∂y (e^(-x^2 - y^2))

= -2y e^(-x^2 - y^2)

(b) To find the critical points, we set the partial derivatives equal to zero and solve the resulting system of equations:

-2x e^(-x^2 - y^2) = 0

-2y e^(-x^2 - y^2) = 0

From these equations, we can see that the critical points occur when either x = 0 or y = 0. So, the critical points lie on the x-axis (y = 0) and the y-axis (x = 0).

(c) To determine the behavior of the critical points, we can use the second partial derivative test. We compute the second partial derivatives and evaluate them at the critical points.

∂²f/∂x² = (∂/∂x) (-2x e^(-x^2 - y^2))

= -2 e^(-x^2 - y^2) + 4x^2 e^(-x^2 - y^2)

∂²f/∂y² = (∂/∂y) (-2y e^(-x^2 - y^2))

= -2 e^(-x^2 - y^2) + 4y^2 e^(-x^2 - y^2)

∂²f/∂x∂y = (∂/∂y) (-2x e^(-x^2 - y^2))

= 4xy e^(-x^2 - y^2)

At the critical point (0,0), the second partial derivatives become:

∂²f/∂x² = -2

∂²f/∂y² = -2

∂²f/∂x∂y = 0

Using the second partial derivative test, we can determine the behavior of the critical point:

If the second partial derivatives are both positive at the critical point, it is a local minimum.

If the second partial derivatives are both negative at the critical point, it is a local maximum.

If the second partial derivatives have different signs or if the determinant of the Hessian matrix is zero, it is a saddle point.

Since the second partial derivatives are both negative at the critical point (0,0), it is a local maximum.

(d) To compute the tangent plane to the graph of f(x, y) at the point (0,0,1), we can use the equation of a plane:

z = f(0,0) + ∂f/∂x(0,0)(x - 0) + ∂f/∂y(0,0)(y - 0)

Substituting the values:

z = 1 - 2x(0) - 2y(0)

z = 1

Therefore, the equation of the tangent plane to the graph of f(x, y) at the point (0,0,1) is z = 1.

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The solution to the given initial value problem is y(x) = tan(x + π/4) - 1.  The explicit solution to the initial value problem is y(x) = ([tex]e^{2x}[/tex] - 1)/2 - 1, which can be further simplified to y(x) = ([tex]e^{2x}[/tex] - 1)/2 - 1.

To solve the initial value problem y' = 1 + 2yx, we can use the method of separation of variables. First, we rewrite the equation as y' - 2yx = 1. Rearranging, we have y' = 2yx + 1.

Next, we separate the variables by moving all the terms involving y to one side and all the terms involving x to the other side. This gives us y'/(2y + 1) = x.

Now, we integrate both sides with respect to x. The integral of y'/(2y + 1) with respect to x can be simplified by using the substitution u = 2y + 1. This leads to du = 2dy, which gives us dy = (1/2)du. Thus, the integral becomes ∫(1/2)du/u = (1/2)ln|u| + C, where C is the constant of integration.

Substituting back u = 2y + 1, we have (1/2)ln|2y + 1| + C = x + D, where D is another constant of integration.

To find the particular solution, we can apply the initial condition y(-1) = 0. Substituting x = -1 and y = 0 into the equation, we get (1/2)ln|2(0) + 1| + C = -1 + D. Simplifying, we find (1/2)ln|1| + C = -1 + D, which gives us (1/2)(0) + C = -1 + D. Therefore, C = -1 + D.

The final solution is (1/2)ln|2y + 1| + C = x + D, where C = -1 + D. Simplifying further, we obtain ln|2y + 1| + 2C = 2x + 2D. Exponentiating both sides, we have |2y + 1|[tex]e^{2C}[/tex] = [tex]e^{2x + 2D}[/tex]. Considering the absolute value, we get 2y + 1 = ±e^(2x + 2D - 2C).

Finally, solving for y, we have two possible solutions: [tex]2y + 1 = e^{2x + 2D - 2C}\ or\ 2y + 1 = -e^{2x + 2D - 2C}[/tex]. Simplifying each equation, we obtain y = ([tex]e^{2x + 2D - 2C}[/tex] - 1)/2 or y = ([tex]-e^{2x + 2D - 2C}[/tex] - 1)/2.

Since we have the initial condition y(-1) = 0, we can substitute x = -1 into the equations and solve for the constants. We find that D - C = 1 for the first equation and D - C = -1 for the second equation. Solving these equations simultaneously, we get D = 0 and C = -1.

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There is approximately a 99.39% probability that the mean gain for a random sample of 32 years from the Dow Jones Industrial Average population falls between 200 and 500.

To find the probability that the mean gain for the sample was between 200 and 500, we need to calculate the z-scores corresponding to these values and use the standard normal distribution.

Given that the population mean gain of the Dow Jones Industrial Average is 456, we can assume that the sampling distribution of the sample mean follows a normal distribution with a mean equal to the population mean (456) and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Since we don't have the population standard deviation, we cannot determine the exact probability. However, we can make use of the Central Limit Theorem, which states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed.

The standard deviation of the sample mean can be estimated by the standard deviation of the population divided by the square root of the sample size. If we assume a standard deviation of 100 for the population, we can calculate the standard deviation of the sample mean as follows:

Standard deviation of the sample mean = 100 / √(32) ≈ 17.68

Now, we can calculate the z-scores for the values 200 and 500:

z₁ = (200 - 456) / 17.68 ≈ -12.48

z₂ = (500 - 456) / 17.68 ≈ 2.49

Using a standard normal distribution table or a calculator, we can find the area under the curve between these z-scores:

P(-12.48 < Z < 2.49) ≈ P(Z < 2.49) - P(Z < -12.48)

Therefore, the probability that the mean gain for the sample was between 200 and 500 is approximately:

P(200 < [tex]\bar X[/tex] < 500) ≈ P(-12.48 < Z < 2.49) ≈ 0.9939 - 0.0000 ≈ 0.9939

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The probability that neither die shows a four is 25/36.

The possible results when two 6-sided dice are rolled is 36, and the sample space is the set of all possible outcomes:

S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.

Let A be the event where neither die shows 4, that is,

A = {(1,1), (1,2), (1,3), (1,5), (1,6), (2,1), (2,2), (2,3), (2,5), (2,6), (3,1), (3,2), (3,3), (3,5), (3,6), (5,1), (5,2), (5,3), (5,5), (5,6), (6,1), (6,2), (6,3), (6,5), (6,6)}.

Now consider the complement of A. The complement of A is the set of outcomes not in A, that is,

{ (4,1), (4,2), (4,3), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4), (4,4) }.

Therefore, the probability of A is the complement of the probability of not A.

P(not A) = 11/36

P(A) = 1 - P(not A)

= 1 - 11/36

= 25/36

Hence, the probability that neither die shows a four is 25/36.

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A graph of the square root function [tex]f(x)=x^{\frac{1}{2} } +3[/tex] is shown in the image attached below.

In Mathematics and Geometry, a square root function refers to a type of function that typically has this form f(x) = √x, which basically represent the parent square root function i.e f(x) = √x.

In this scenario and exercise, we would use an online graphing tool to plot the given square root function [tex]f(x)=x^{\frac{1}{2} } +3[/tex] as shown in the graph attached below.

In conclusion, we can logically deduce that the transformed square root function [tex]f(x)=x^{\frac{1}{2} } +3[/tex] was created by translating the parent square root function f(x) = √x upward by 3 units.

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Complete Question:

Sketch the graph of [tex]f(x)=x^{\frac{1}{2} } +3[/tex] if x ≥ 0.

The volume of the triangular prism is 32 units³ which we calculated using triangular prism formula. So correct answer is option A.

To find the volume of a triangular prism, we need to multiply the base area of the triangular base by the height of the prism. The formula for the volume of a triangular prism is given by V = (1/2) * b * h * H, where b is the base length of the triangular base, h is the height of the triangular base, and H is the height of the prism.

Since the options provided do not specify the base length or heights, we cannot calculate the volume directly. However, based on the given options, option A, which states the volume as 32 units³, is the most reasonable choice. It is important to note that without further information, we cannot confirm the accuracy of the answer.

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The breakaway point of the given system is approximately -1.56. This point is determined by finding the value of s at which the characteristic equation of the system, obtained by setting the denominator of the transfer function equal to zero, has a double root.

To find the breakaway point, we set the denominator of the transfer function equal to zero:

s(s+4)(s+4∓j4)=0

Expanding this equation, we get:

s^3 + (8∓j4)[tex]s^2[/tex] + 16(s∓j4) = 0

Since the characteristic equation has a double root at the breakaway point, we can approximate the value of s by neglecting the higher-order terms. By doing so, we can solve the quadratic equation:

(8∓j4)[tex]s^2[/tex] + 16(s∓j4) = 0

Solving this quadratic equation gives us the value of s, which is approximately -1.56. Therefore, option (b) is the correct answer for the breakaway point.

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The product of the complex numbers z1 and z2 =cos π/24 +isin π/24 in polar form.

To find the product of the complex numbers z1 and z2 and express it in polar form, we can multiply their magnitudes and add their arguments. Given: z1=cosπ/4+isin π/4 and z2= cos π/6+isinπ/6. Let's calculate the product: z1.z2=(cos π/4+isin π/4)(cos π/6+isinπ/6.)

Using the formula for the product of two complex numbers: z1.z2=cosπ/4.cos π/6-sin π/4.sinπ/6+i(sin π/4cosπ/4+cosπ/4sin π/4). Simplifying the expression: z1.z2= cosπ/24+isinπ/24. Now, to express the product in polar form, we can rewrite it as: z1.z2=sqrt(cos^2 π/24+sin^2 π/24)(cosθ +isinθ), where θ s the argument of z1z2.

Since the magnitude is equal to 1 (due to the trigonometric identities cos^2θ +sin^2θ=1, the polar form of the product is:  z1z2=cosθ ++isinθ. Therefore, the product of the complex numbers z1 and z2 =cos π/24 +isin π/24 in polar form.

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Answer:

Step-by-step explanation:

ratio a:b

1:2      > Each of the sides can be multiplied by 2 on rect A

         >1(2)=2

         >5(2) = 10

Ratio Area(A): Area(B)

1²:2²         is for the lengths but area is squared so the lengths get squared for area

1:4

YOu can check:

Area(A) = (1)(5) = 5

Area(B) = (2)(10)  = 20

You can see B is 4 times as big as A for Area so 1:4 is right

The 6th percentile of a normally distributed random variable with a mean of µ = -47 and a standard deviation of σ = 7 is -53.24.

To find the 6th percentile, we can use the standard normal distribution table or a statistical calculator. The 6th percentile represents the value below which 6% of the data falls. In other words, there is a 6% probability of observing a value less than the 6th percentile.

Using the standard normal distribution table, we look for the closest value to 0.06 (6%) in the cumulative probability column. The corresponding Z-value is approximately -1.56. We can then use the formula for converting Z-values to raw scores:

X = µ + Zσ

Substituting the given values, we have:

X = -47 + (-1.56) * 7 = -53.24

Therefore, the 6th percentile of the distribution is -53.24.

This means that approximately 6% of the data will be less than -53.24 when the random variable X follows a normal distribution with a mean of -47 and a standard deviation of 7.

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A data point of 240 is a typical data point. False, a Z-score of -1.5 is not considered a typical Z-score.

Explanation: In statistics, a Z-score measures how many standard deviations a data point is away from the mean of a distribution. A Z-score of -1.5 indicates that the data point is 1.5 standard deviations below the mean. In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, a Z-score of -1.5 would be considered somewhat typical as it falls within the range of approximately 34% of the data. However, the question does not specify the distribution of the data, so we cannot assume it to be a standard normal distribution.

Regarding the second question, a data point of 240 in a population with an average of 120 and a standard deviation of 40 would not be considered a typical data point. To determine whether a data point is typical or not, we can use the concept of Z-scores. Calculating the Z-score for this data point, we get:

Z = (240 - 120) / 40 = 120 / 40 = 3

A Z-score of 3 indicates that the data point is 3 standard deviations above the mean. In a normal distribution, data points that are several standard deviations away from the mean are considered atypical or outliers. Therefore, a data point of 240 would be considered atypical in this context.

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The values of all integration function have been obtained.

Q2.  x²/2 tan²x - x² tan x/2 + x tan x - ln|cosx| + C

Q3.  32 ln|x + 1| + 23/15 ln|x - 2| - 41/8 ln|x + 3| + C.

Q2.

To find the integration of the given function i.e.

∫x tan²xdx,

Using the integration by parts, we use the following formula:

∫u dv = uv - ∫v du

Let us consider u = tan²x and dv = x dx.

So, du = 2 tan x sec²x dx and v = x²/2.

Using these values in the formula we get:

∫x tan²xdx = ∫u dv

                 = uv - ∫v du

                 = x²/2 tan²x - ∫x²/2 * 2 tan x sec²x dx

                 = x²/2 tan²x - x² tan x/2 + ∫x dx     (integration of sec²x is tanx)

                 = x²/2 tan²x - x² tan x/2 + x tan x - ∫tan x dx

                                                                (using integration by substitution)

                 = x²/2 tan²x - x² tan x/2 + x tan x - ln|cosx| + C

So, the integration of the given function using integration by parts is

x²/2 tan²x - x² tan x/2 + x tan x - ln|cosx| + C.

Q3.

To find the integration of the given function i.e.

∫(2x² + 9x - 35)/[(x + 1)(x - 2)(x + 3)] dx,

Using partial fraction, we have to first factorize the denominator.

Let us consider (x + 1)(x - 2)(x + 3).

The factors are (x + 1), (x - 2) and (x + 3).

Hence, we can write the given function as

A/(x + 1) + B/(x - 2) + C/(x + 3),

Where A, B and C are constants.

To find these constants A, B and C, let us consider.

(2x² + 9x - 35) = A(x - 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x - 2).

Putting x = -1, we get

-64 = -2A,

So, A = 32 Putting x = 2, we get

23 = 15B,

So, B = 23/15 Putting x = -3, we get

41 = -8C,

So, C = -41/8

So, we can write the given function as

∫(2x² + 9x - 35)/[(x + 1)(x - 2)(x + 3)] dx = ∫32/(x + 1) dx + ∫23/15(x - 2) dx - ∫41/8/(x + 3) dx

Now, we can integrate these three terms separately using the formula: ∫1/(x + a) dx = ln|x + a| + C

So, we get

= ∫(2x² + 9x - 35)/[(x + 1)(x - 2)(x + 3)] dx

= 32 ln|x + 1|/1 + 23/15 ln|x - 2|/1 - 41/8 ln|x + 3|/1 + C

= 32 ln|x + 1| + 23/15 ln|x - 2| - 41/8 ln|x + 3| + C

So, the integration of the given function using partial fraction is 32 ln|x + 1| + 23/15 ln|x - 2| - 41/8 ln|x + 3| + C.

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To find the equivalent nominal interest rate with daily compounding, we can use the formula: Nominal Interest Rate =[tex](1 + Effective Annual Rate)^(1/n) - 1[/tex] Where: Effective Annual Rate is the rate quoted by the bank (in decimal form) n is the number of compounding periods per year

In this case, the quoted rate by the bank is 15.6% (or 0.156 in decimal form), and the compounding is done annually, so there is one compounding period per year.

Plugging in these values into the formula, we get:

Nominal Interest Rate = (1 + 0.156)^(1/365) - 1

Calculating the expression inside parentheses:

Nominal Interest Rate = (1.156)^(1/365) - 1

Calculating the exponent:

Nominal Interest Rate = 1.000427313 - 1

Subtracting:

Nominal Interest Rate = 0.000427313

Converting to a percentage to 2 decimal places:

Nominal Interest Rate = 0.0427%

Therefore, the equivalent nominal interest rate with daily compounding is approximately 0.0427%.

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The cut-off weight loss value for the top 2% of dieters is approximately 20.25 pounds.

a) Probability of losing more than 12 pounds

We know that the mean of weight loss is 10 pounds and the standard deviation is 5 pounds. We want to find the proportion of dieters that lost more than 12 pounds.We have to standardize the value of 12 using the formula: z = (x - μ) / σ

So, z = (12 - 10) / 5 = 0.4.

The value 0.4 is the number of standard deviations away from the mean μ.

To find the proportion of dieters that lost more than 12 pounds, we need to find the area under the normal distribution curve to the right of 0.4, which is the z-score.

P(Z > 0.4) = 0.3446

Therefore, the proportion of dieters that lost more than 12 pounds is approximately 0.3446 or 34.46%.

b) Probability of gaining weight

The probability of gaining weight can be found by calculating the area under the normal distribution curve to the left of 0 (since gaining weight is a negative value).

P(Z < 0) = 0.5

Therefore, the proportion of dieters that gained weight is approximately 0.5 or 50%.

c) Cut-off for the top 2% weight loss

To find the cut-off for the top 2% weight loss, we need to find the z-score that corresponds to the top 2% of the distribution. We can do this using a z-score table or calculator.

The z-score that corresponds to the top 2% of the distribution is approximately 2.05.

So, we can use the formula z = (x - μ) / σ and solve for x to find the cut-off weight loss value.

2.05 = (x - 10) / 5

x - 10 = 2.05(5)

x - 10 = 10.25

x = 20.25

Therefore, the cut-off weight loss value for the top 2% of dieters is approximately 20.25 pounds.

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The magnitude of the vector sum u + v is approximately 113.7. To find the sum of vectors u and v, we can use vector addition.

The magnitude of the sum is equal to the square root of the sum of the squares of the individual vector magnitudes plus twice the product of their magnitudes and the cosine of the angle between them.

Magnitude of vector u (|u|) = 39

Magnitude of vector v (|v|) = 48

Angle between u and v (θ) = 37 degrees

Using the formula for vector addition:

|u + v| = sqrt((|u|)^2 + (|v|)^2 + 2 * |u| * |v| * cos(θ))

Substituting the given values:

|u + v| = sqrt((39)^2 + (48)^2 + 2 * 39 * 48 * cos(37°))

Calculating:

|u + v| ≈ sqrt(1521 + 2304 + 2 * 39 * 48 * cos(37°))

Since the angle is given in degrees, we need to convert it to radians:

|u + v| ≈ sqrt(1521 + 2304 + 2 * 39 * 48 * cos(37° * π/180))

|u + v| ≈ sqrt(1521 + 2304 + 2 * 39 * 48 * cos(0.645))

|u + v| ≈ sqrt(3825 + 2304 + 3744 * cos(0.645))

|u + v| ≈ sqrt(9933 + 3744 * 0.804)

|u + v| ≈ sqrt(9933 + 3010.176)

|u + v| ≈ sqrt(12943.176)

|u + v| ≈ 113.7 (rounded to the nearest tenth)

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The function has a local maximum at x = 0 with a value of -2. To find the local extremum of the function [tex]f(x) = x^2 - 2[/tex], we need to use the first derivative test.

First, let's find the derivative of f(x):

f'(x) = 2x

Now, let's set f'(x) equal to zero and solve for x to find the critical points:

2x = 0

x = 0

The critical point is x = 0.

Next, we can determine the behavior of the function around x = 0 by examining the sign of the derivative in intervals:

For x < 0:

Choose x = -1 as a test point.

f'(-1) = 2(-1) = -2 (negative)

For x > 0:

Choose x = 1 as a test point.

f'(1) = 2(1) = 2 (positive)

Based on the first derivative test, when the derivative changes sign from negative to positive, we have a local minimum. Conversely, when the derivative changes sign from positive to negative, we have a local maximum.

In this case, since the derivative changes from negative to positive at x = 0, we have a local minimum at x = 0.

To find the value of the function at this extremum, substitute x = 0 into the original function:

[tex]f(0) = (0)^2 - 2 = -2[/tex]

Therefore, the function [tex]f(x) = x^2 - 2[/tex] has a local minimum at x = 0, and the value of the function at this extremum is -2.

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3. The test statistic is approximately -0.051.

4.  The p-value is approximately 0.9597.

5. The p-value (0.9597) is greater than the significance level α (0.10).

6.  Fail to reject the null hypothesis.

1. For this study, we should use:

   - One-sample proportion test.

2. The null and alternative hypotheses would be:

   - H0: The proportion of graduates getting jobs within one year is 89% (0.89).

   - H1: The proportion of graduates getting jobs within one year is significantly different from 89%.

3. The test statistic:

   - To determine the test statistic, we can use the z-test for proportions. The formula is:

     z = (p - P) / √((P × (1 - P)) / n)

     where p is the sample proportion, P is the hypothesized proportion (0.89), and n is the sample size.

     Plugging in the values:

     p = 55/62 ≈ 0.887

     P = 0.89

     n = 62

     z = (0.887 - 0.89) / sqrt((0.89 * (1 - 0.89)) / 62)

     Calculating the value:

     z ≈ -0.051

     Rounded to 3 decimal places, the test statistic is approximately -0.051.

4. The p-value:

   - The p-value represents the probability of observing a test statistic as extreme as the one calculated (in favor of the alternative hypothesis) under the assumption that the null hypothesis is true.

   - Since this is a two-sided test, we need to find the probability in both tails of the distribution.

     Using a standard normal distribution table or a calculator, we can find the p-value associated with the test statistic z = -0.051.

     The p-value ≈ 0.9597

     Rounded to 4 decimal places, the p-value is approximately 0.9597.

5. The p-value is α:

   - The p-value (0.9597) is greater than the significance level α (0.10).

6. Based on this, we should accept the null hypothesis.

   - Since the p-value is greater than α, we fail to reject the null hypothesis.

7. As such, the final conclusion is that:

   - The sample data suggest that the population proportion is not significantly different than 89% at α = 0.10. Therefore, there is not sufficient evidence to conclude that the percentage of graduates getting jobs within one year is different from 89%.

Learn more about hypothesis here:

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