Using the Fixed-Point Iteration Method with an initial estimate of 2, the root of the function F(x) = ex - 2x - 5 is approximately x ≈ 1.7746. Using the Fixed-Point Iteration Method with an initial estimate of π, the real root of the equation 2cos(3√x) - sin(√x) = 1/4 is approximately x ≈ 3.1416, accurate to four significant figures.
To determine a root greater than zero of the function F(x) = ex - 2x - 5 using the Fixed-Point Iteration Method, we start with an initial estimate of x0 = 2 and iterate using the formula:
xn+1 = g(xn)
where g(x) is a function that transforms the original equation into a fixed-point equation, i.e., x = g(x).
1. Let's choose g(x) = ln(2x + 5), which is derived by rearranging the original equation.
2. Using the initial estimate x0 = 2, we can compute the iterations as follows:
x1 = g(x0) = ln(2(2) + 5) = 1.7917595
x2 = g(x1) = ln(2(1.7917595) + 5) = 1.7757471
x3 = g(x2) = ln(2(1.7757471) + 5) = 1.7746891
x4 = g(x3) = ln(2(1.7746891) + 5) = 1.7746328
After four iterations, we obtain an approximation of the root as x ≈ 1.7746, accurate to five decimal places.
To solve the equation 2cos(3√x) - sin(√x) = 1/4 using the Fixed-Point Iteration Method, we start with an initial estimate of x0 = π and aim to achieve an accuracy of four significant figures.
1. Let's rewrite the equation as a fixed-point equation by adding x to both sides:
x = g(x) = 4cos(3√x) - 4sin(√x) + x
2. Using the initial estimate x0 = π, we can compute the iterations as follows:
x1 = g(x0) = 4cos(3√π) - 4sin(√π) + π = 3.073315
x2 = g(x1) = 4cos(3√3.073315) - 4sin(√3.073315) + 3.073315 = 3.150428
x3 = g(x2) = 4cos(3√3.150428) - 4sin(√3.150428) + 3.150428 = 3.141804
x4 = g(x3) = 4cos(3√3.141804) - 4sin(√3.141804) + 3.141804 = 3.141593
After four iterations, we obtain an approximation of the real root as x ≈ 3.1416, accurate to four significant figures.
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Prev Question 25 - of 25 Step 1 of 1 Find the Taylor polynomial of degree 3 near x = 0 for the following function. y = ³√4x + 1 Answer 2 Points √√4x + 1 ≈ P₃(x) = Keypad Keyboard Shortcuts Next
To find the Taylor polynomial of degree 3 near x=0 for the function y=³√4x+1,
we need to find the derivatives of y up to the third degree. The formula for the nth derivative of y is given by the following formula:nth derivative of y = n! × (4/3)^(-n) × x^(-2/3+n)
Let's find the first three derivatives of y:
First derivative of y: y' = (4/3)^(-1) × x^(-2/3) = 3/(4√x)
Second derivative of y: y'' = 2!(4/3)^(-2) × x^(1/3) = 9/(8x^(3/2))
Third derivative of y: y''' = 3!(4/3)^(-3) × x^(5/3) = 27/(16x^(5/2))
plug these values into the formula for the Taylor polynomial of degree 3:P₃(x) = y(0) + y'(0)x + (y''(0)/2!)x² + (y'''(0)/3!)x³P₃(x) = 1 + 0 + (3/2)x² + (27/16)x³Simplifying:P₃(x) = 1 + (3/2)x² + (27/16)x³
Therefore, the Taylor polynomial of degree 3 near x=0 for the function y=³√4x+1 is P₃(x) = 1 + (3/2)x² + (27/16)x³.
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Find the Laplace transform F(s) = L{f(t)} of the function f(t) = e²t-12 h(t-6), defined on the interval t > 0. F(s) = L {e²t-12 (t-6)} =
The Laplace transform of the function f(t) = e²t-12 h(t-6) is given by F(s) = L{e²t-12 (t-6)}. To compute the Laplace transform, we can apply the linearity property of the transform.
The Laplace transform of e²t is 1/(s-2), and the Laplace transform of h(t-6) is e^(-6s)/s.
Using the property of multiplication in the Laplace domain, we can multiply the individual Laplace transforms to obtain F(s) = 1/(s-2) * e^(-6s)/s.
Simplifying further, we can rewrite F(s) as (e^(-6s))/(s(s-2)).
Therefore, the Laplace transform of f(t) = e²t-12 h(t-6) is F(s) = (e^(-6s))/(s(s-2)).
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If Find the value of x+y.. Attachments (n-1)! Σ 69.70.71.....(68+n) X y
Given a series with the formula (n-1)! Σ 69.70.71.....(68+n) X y.
We need to find the value of x+y.
We are given that the sum of a series can be represented in the form of the first term multiplied by the common ratio raised to the power of the number of terms divided by the common ratio minus 1.
Mathematically, it can be represented as:
[tex]S = a(rⁿ - 1) / (r - 1)[/tex]
Where, S = Sum of seriesa = First termm = Number of termsn = m - 68r = Common ratio For the given series, we can observe that the first term is 69, and the common ratio is 1 as the difference between each consecutive term is 1.
Hence, the sum of the series can be represented as:S = a(m) = 69(m - 68)
Also, we are given that the sum of the series is equal to (n-1)! X y.
Substituting the value of S in the above equation,
we get:(n-1)! X y = 69(m - 68)
Solving the above equation,
we get:
m = (y + 68)
Putting this value of m in the equation of S,
we get:S = 69(y + n)
Therefore, the value of x + y is equal to 69.
Hence, the answer is 69 only in 100 words.
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Use the four implication rules to create proof for the following argument.
H ⊃ G
F ⊃ ~G
F /~H
We can prove that ~H is true by using the four implication rules since the argument is not valid
The argument is not valid. We have H ⊃ G, F ⊃ ~G, and F.
We have to prove that ~H is true by using the four implication rules.
Let's get started:(1) H ⊃ G (Premise)(2) F ⊃ ~G (Premise)(3) F (Premise)(4) ~G MP: 2,3(5) ~H MT: 1,4
Therefore, by using the four implication rules, we can prove that ~H is true.
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Find det (A) given that A has p(A) as its characteristic polynomial. p(A) = 13 - 412 + +8 det (A) = i Hint: See the proof of Theorem 7.1.4. (lf given det (11 - A) = 1" + C21n-1 + ... + C, then, on setting A = 0, det (-A) = Cnor (- 1)"det (A) = Cn)
The determinant of matrix A, det(A), is equal to 8i.
To find the determinant of matrix A, we are given its characteristic polynomial p(A) = 13 - 412 + 8 det(A) = i. According to Theorem 7.1.4, if we set A = 0 in the polynomial p(A), we can obtain the determinant of -A.
Setting A = 0 in the polynomial p(A), we get p(0) = 13 - 412 + 8 det(0) = i. Simplifying this equation, we have 13 - 412 + 8 det(0) = i. Since det(0) is the determinant of a zero matrix, which is always zero, we can rewrite the equation as 13 - 412 = i. Solving for i, we find that i = -399.
Now, using the result from Theorem 7.1.4, we have det(-A) = C(-1)^n det(A). Plugging in the given value det(11 - A) = 1 + C21n-1 + ... + C, we can set A = 0 to find det(-A). By substituting A = 0 into the equation, we get det(11 - 0) = 1 + C21n-1 + ... + C, which simplifies to det(11) = 1 + C21n-1 + ... + C. Since det(11) is the determinant of matrix 11, which is just 11, we have 11 = 1 + C21n-1 + ... + C. Simplifying further, we get 10 = C21n-1 + ... + C.
Finally, we can substitute det(A) = 8i (from the given characteristic polynomial) into the equation det(-A) = C(-1)^n det(A). Since we found i = -399, we have det(-A) = C(-1)^n * 8 * -399 = -3192C(-1)^n.
In conclusion, the determinant of matrix A, det(A), is equal to 8i, which simplifies to -3192C(-1)^n.
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What is the standard error of the estimate? A. A measure of the variation of the X variable B. A measure of explained variation C. A measure of the variation around the sample regression line D. A measure of total variation of the Y variable
The standard error of the estimate is a measure of the variation around the sample regression line.What is standard error of the estimate? The standard error of the estimate is defined as a measure of the deviation around the sample regression line. It's also known as the mean square error. In simple words, it represents the average difference between the real and the predicted value of Y.
The formula for calculating standard error of the estimate is: $S_{yx}=\sqrt{\frac{\sum{(Y-\hat Y)}^2}{n-2}}$Where,Syx = Standard error of estimateY = Observed data valueŶ = Predicted data value using regression equation = Number of observations in the sample The standard error of the estimate is used in regression analysis to measure how well the regression equation approximates the actual values of the response variable.
The standard error of the estimate is used to assess the precision of the estimates and the goodness of fit of the model.
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For the curve g(x) = 2 (-)-4 [8] a) Circle whether the function is increasing or decreasing ✓ b) Using a series of transformations on the grid, accurately graph g(x). Ensure all the important poi
a) The function g(x) = 2x - 4 is increasing. b) To graph g(x), we start with the linear function y = 2x and apply a transformation by subtracting 4 from the y-values. This shifts the entire graph downwards by 4 units. The important points to plot on the graph are the y-intercept at (0, -4) and the slope, which is 2.
a) The function g(x) = 2x - 4 is increasing because the coefficient of x is positive (2). This means that as x increases, the corresponding y-values will also increase, resulting in an upward trend.
b) To graph g(x), we consider the original linear function y = 2x, which has a slope of 2 and a y-intercept of (0, 0). By subtracting 4 from the y-values, we shift the entire graph downwards by 4 units. The y-intercept of the transformed function g(x) = 2x - 4 is therefore at (0, -4).
To find other points, we can choose any x-values and calculate the corresponding y-values. For example, when x = 1, y = 2(1) - 4 = -2. Thus, we have the point (1, -2). Similarly, when x = -1, y = 2(-1) - 4 = -6, giving us the point (-1, -6). By plotting these points and drawing a straight line through them, we obtain the graph of g(x).
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4. (1 point) Show that for each bilinear form b, b (u,0) = b (0, u)=0.
We have proved that b(u, 0) = b(0, u) = 0 for each bilinear form b.
Given that b is a bilinear form, and u is a vector in V (a vector space). We need to prove that b(u, 0) = b(0, u) = 0. Here, 0 refers to the zero vector in the vector space V.
Let's start with the first one:
b(u, 0) = b(u, 0+0) [adding zero vector to 0 gives 0]
b(u, 0) = b(u, 0) + b(u, 0) [bilinear property: b(u, v+w) = b(u,v) + b(u,w)]
b(u, 0) - b(u, 0) = b(u, 0) + b(u, 0) - b(u, 0)b(u, 0) - b(u, 0) = 0 => b(u, 0) = 0
Now let's look at the second one: b(0, u) = b(0+0, u) [adding zero vector to 0 gives 0]
b(0, u) = b(0, u) + b(0, u) [bilinear property: b(u+v, w) = b(u,w) + b(v,w)]
b(0, u) - b(0, u) = b(0, u) + b(0, u) - b(0, u)b(0, u) - b(0, u) = 0 => b(0, u) = 0
Hence, we have proved that b(u, 0) = b(0, u) = 0 for each bilinear form b.
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The volume, L liters, of paint in a plastic tub may be assumed to be normally distributed with mean 10.25 and variance σ^2.
(a) assuming that variance = 0.04, determine P(L<10).
(b) Find the value of standard deviation so that 98% of tubs contain more than 10 liters of paint.
Assuming a variance of 0.04, determine the probability P(L < 10) and find the standard deviation that ensures 98% of tubs contain more than 10 liters of paint, we need to calculate the appropriate value.
(a) To determine the probability P(L < 10), we need to calculate the cumulative distribution function (CDF) of the normal distribution with a mean of 10.25 and a variance of 0.04. By standardizing the variable using the z-score formula and looking up the corresponding value in the standard normal distribution table, we can find the probability.
The z-score is given by (10 - 10.25) / sqrt(0.04) = -1.25. Looking up -1.25 in the standard normal distribution table, we find that the probability is approximately 0.1056. Therefore, P(L < 10) is approximately 0.1056.
(b) To find the standard deviation that ensures 98% of tubs contain more than 10 liters of paint, we need to calculate the corresponding z-score. We want to find the z-score such that the area to the right of it in the standard normal distribution is 0.98. Looking up the value 0.98 in the standard normal distribution table, we find that the z-score is approximately 2.05.
Now we can set up an equation using the z-score formula: (10 - 10.25) / σ = 2.05. Solving for σ, we have σ ≈ (10.25 - 10) / 2.05 ≈ 0.121. Therefore, a standard deviation of approximately 0.121 would ensure that 98% of tubs contain more than 10 liters of paint.
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A multinational company operates factories around the world. Assume that the total number of serious accidents that take place per week follows a Poisson distribution with mean 2. We assume that the accidents occur independently of one another.
(a) Calculate the probability that there will be two or fewer accidents during one week. [2 marks]
(b) Calculate the probability that there will be two or fewer accidents in total during a period of 2 weeks. [3 marks]
(c) Calculate the probability that there will be two or fewer accidents each week during a period of 2 weeks. [2 marks]
(d) The company is shut for two weeks for seasonal celebrations and therefore, over a whole year, the number of accidents follows a Poisson distribution with mean 100. Using a suitable approximation, calculate the probability that there will be more than 120 accidents in one year. [3 marks]
(a) The probability of having two or fewer accidents during one week can be calculated using the Poisson distribution with a mean of 2.
(b) The probability of having two or fewer accidents in total during a period of 2 weeks can be calculated by considering the sum of two independent Poisson random variables with a mean of 2.
(c) The probability of having two or fewer accidents each week during a period of 2 weeks can be calculated by multiplying the probabilities of having two or fewer accidents in each week, which are obtained from the Poisson distribution.
(d) To calculate the probability of having more than 120 accidents in one year, we can approximate the Poisson distribution with a normal distribution using the Central Limit Theorem and calculate the cumulative probability.
(a) To calculate the probability of having two or fewer accidents during one week, we can use the Poisson distribution formula. P(X ≤ 2) = e^(-λ) * (λ^0/0!) + e^(-λ) * (λ^1/1!) + e^(-λ) * (λ^2/2!), where λ is the mean, which in this case is 2. Plugging in the values, we get P(X ≤ 2) ≈ 0.6767.
(b) To calculate the probability of having two or fewer accidents in total during a period of 2 weeks, we consider the sum of two independent Poisson random variables.
Let Y be the total number of accidents in 2 weeks. Since the mean of a Poisson distribution is additive, the mean of Y is 2 + 2 = 4. Using the Poisson distribution formula, P(Y ≤ 2) = e^(-λ) * (λ^0/0!) + e^(-λ) * (λ^1/1!) + e^(-λ) * (λ^2/2!). Plugging in λ = 4, we get P(Y ≤ 2) ≈ 0.2381.
(c) To calculate the probability of having two or fewer accidents each week during a period of 2 weeks, we multiply the probabilities of having two or fewer accidents in each week. Since the accidents occur independently, we can use the results from part (a) twice. P(X ≤ 2 each week) = P(X ≤ 2 in week 1) * P(X ≤ 2 in week 2) ≈ 0.6767 * 0.6767 ≈ 0.4577.
(d) To calculate the probability of having more than 120 accidents in one year, we can approximate the Poisson distribution with a normal distribution using the Central Limit Theorem. The mean of the Poisson distribution is 100, and the variance is also 100.
Approximating the Poisson distribution as a normal distribution with a mean of 100 and a standard deviation of √100 = 10, we can calculate the z-score for 120. The z-score is (120 - 100) / 10 = 2. Using a standard normal distribution table or a calculator, we find that the cumulative probability of having more than 120 accidents is approximately 0.0228.
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f(x) = x³ = 7+2, x>0 (a) Show that f(x) = 0 has a root a between 1.4 and 1.5. (2 marks) (b) Starting with the interval [1.4, 1.5], using twice bisection method, find an interval of width 0.025 that contains a. (8 marks) (c) Taking 1.4 as a first approximation to a, (i) conduct three iterations of the Newton-Raphson method to compute f(x) = x³. - + 2; (9 marks) (ii) determine the absolute relative error at the end of the third iteration; and (3 marks) (iii) find the number of significant digits at least correct at the end of the third iteration. (3 marks)
By evaluating f(x) at the given interval, it is shown that f(x) = 0 has a root between 1.4 and 1.5. Using the bisection method twice on the interval [1.4, 1.5], an interval of width 0.025 containing the root is found.
a) To show that f(x) = 0 has a root between 1.4 and 1.5, we can substitute values from this interval into f(x) = x³ - 7 + 2 and observe that the function changes sign. This indicates the presence of a root within the interval.
b) The bisection method involves repeatedly dividing the interval in half and narrowing down the interval containing the root. By applying this method twice on the initial interval [1.4, 1.5], an interval of width 0.025 is found that contains the root.
c) (i) To conduct three iterations of the Newton-Raphson method, we start with the first approximation of a as 1.4 and repeatedly apply the formula xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ), where f(x) = x³ - 7 + 2 and f'(x) is the derivative of f(x).
(ii) After three iterations, we can determine the absolute relative error by comparing the value obtained from the third iteration with the true root.
(iii) The number of significant digits at least correct at the end of the third iteration can be determined by counting the number of decimal places in the approximation obtained.
Overall, by applying the given methods, we can establish the presence of a root, narrow down the interval containing the root, and compute approximations using the Newton-Raphson method while assessing the error and significant digits.
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Find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn't exist. f(x)=3x² +9x-5 CIT The
The values of x where the function f(x) = 3x² + 9x - 5 is discontinuous are determined, along with their corresponding limits as x approaches those points.
To find the values of x where the function is discontinuous, we need to identify any points where there are breaks or jumps in the graph of f(x). However, the function f(x) = 3x² + 9x - 5 is a polynomial, and polynomials are continuous for all real numbers. Therefore, there are no values of x where the function is discontinuous.
As a polynomial, the limit of f(x) as x approaches any value a is simply f(a). In other words, the limit of f(x) as x approaches a is equal to the value of f(a) for all real numbers a.
So, for any value of x = a, the limit of f(x) as x approaches a is f(a) = 3a² + 9a - 5. The limit exists for all real numbers a.
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Consider an experiment with four groups,with two values in each a. How many degrees of freedom are there in determining the among-group variation? b.How many degrees of freedom are there in determining the within-group variation c.How many degrees of freedom are there in determining the total variation? a.There is/are degree(s) of freedom in determining the among-group variation. (Simplify your answer.) b.There is/are degree(s) of freedom in determining the within-group variation. (Simplify your answer.) c.There is/are degree(s)of freedom in determining the total variation. (Simplify your answer.)
There are three types of degrees of freedom, among-group, within-group, and total variation, in a four-group experiment with two values in each group.
Degrees of freedom (df) are used in hypothesis testing to determine the critical value of the test statistic. It is the number of observations that are free to vary after estimating the parameters in a statistical model. It is the number of independent pieces of information that are used to estimate a statistic.
The degrees of freedom are determined by the number of observations and the number of parameters estimated in the model.
For example, if there are n observations and k parameters, the degrees of freedom will be n-k.The experiment has four groups, with two values in each group.
Therefore, the total number of observations is 8.
There are three types of degrees of freedom, among-group, within-group, and total variation. The degrees of freedom for each type are calculated as follows: Degree of freedom for among-group variation = k-1= 4-1 = 3
Degree of freedom for within-group variation = N - k = 8 - 4 = 4 Degree of freedom for total variation = N-1= 8-1 = 7 .
The degrees of freedom for among-group variation are calculated by subtracting 1 from the number of groups. Therefore, there are 3 degrees of freedom for among-group variation.
The degrees of freedom for within-group variation are calculated by subtracting the number of groups from the total number of observations. Therefore, there are 4 degrees of freedom for within-group variation.
The degrees of freedom for total variation are calculated by subtracting 1 from the total number of observations. Therefore, there are 7 degrees of freedom for total variation.
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The formula for finding a number that's the square root of the sum of another number n and 6 is A. x = √n + 6. B. x = √n + 6. C.x = √n6. D. x = √n + √6.
The correct formula for finding a number that's the square root of the sum of another number n and 6 is B. x = √(n+6).
Let the number that is the square root of the sum of another number n and 6 be "x".Thus, x = √(n+6).Therefore, option B. x = √(n+6) is the correct formula for finding a number that's the square root of the sum of another number n and 6.Let "x" be the quantity that is equal to the square root of the product of another number n and six.Therefore, x = (n+6).So, go with option B. The proper formula to determine a number that is the square root of the sum of two numbers is x = (n+6).
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The formula for finding a number that's the square root of the sum of another number n and 6 is x = √(n + 6). Therefore, the correct answer is option A.
A square root is a mathematical expression that represents the value that should be multiplied by itself to get the desired number. A perfect square is a number that can be expressed as the square of an integer; 1, 4, 9, 16, and so on are all perfect squares. A square root is a number that, when multiplied by itself, produces a perfect square.
The formula to be used is x = √(n + 6).
Here, x is the number whose square root is to be found. The given number is n. The given number is to be added to 6.The square root of the resulting number is to be found, and the solution is x. Using the above formula: x = √(n + 6)Therefore, the answer is option A, x = √n + 6.
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SUCHE To test the hypothesis that the population mean mu-17.4, a sample size n-11 yields a sample mean 18.641 and sample standard deviation 1.905. Calculate the P value and choose the correct conclusion Yanıtınız: The P-value 0.009 is not significant and so does not strongly suggest that mu-17.4. The P-value 0.009 is significant and so strongly suggests that mu>17.4 The P-value 0.022 is not significant and so does not strongly suggest that mu-17.4. The P-value 0.022 is significant and so strongly suggests that mu-17.4 The P-value 0.004 is not significant and so does not strongly suggest that mu>17.4. The P-value 0.004 is significant and so strongly suggests that mu-17.4. The P-value 0.028 is not significant and so does not strongly suggest that mu-17 A. The P-value 0.028 is significant and so strongly suggests that mu-17.4. The P-value 0,003 is not significant and so does not strongly suggest that mu>17.4. The P-value 0.003 is significant and so strongly suggests that mu-17.4.
The correct conclusion is the P-value 0.028 is not significant and so does not strongly suggest that μ > 17.4
How to determine the P-valueFrom the information given, we have that;
Population mean, μ = 17.4,
sample mean = 18.641
Standard deviation (s = 1.905)
Sample size , n = 11
Using the the formula is given as;
t = (x - μ) / (s / √n)
Substitute the values, we have;
t = (18.641 - 17.4) / (1.905 / √11
t = 1.241/0.5743
Divide the values
t ≈ 2.161
Now, we have the degree of freedom as;
degree of freedom = 11 - 1 = 10
Using the t-distribution table or a statistical calculator, we have P-value as
P(0. 2151) = 0.028.
Then, we have to reject the hypothesis.
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Exponent word problem
the half-life of plutonium-239 is about 25,000 years. what
percentage of a given sample will remain after 2000 years?
The percentage of plutonium-239 remaining after 2000 years is 91.43%
The half-life of Plutonium-239 is 25,000 years. Half-life refers to the time required for a radioactive substance to decay to half its original value.
The initial amount of the radioactive substance is denoted by ‘P0’.The formula to calculate the amount of radioactive substance remaining after a given time, ‘t’ is given by:P = P0 (1/2)^(t/h) Where:P = Amount of substance remaining after time ‘t’P0 = Initial amount of the substanceh = Half-life of the substancet = Time passed
Therefore, to find the amount of plutonium-239 remaining after 2000 years, we can substitute the given values in the formula:P = P0 (1/2)^(t/h)P = P0 (1/2)^(2000/25000)P = P0 (0.918)P = 0.918 P0To find the percentage of plutonium-239 remaining, we can divide the remaining amount by the initial amount and multiply by 100.% remaining = (remaining amount/initial amount) x 100%
Remaining amount = 0.918 P0Initial amount = P0% remaining = (0.918 P0/P0) x 100% = 91.43%Therefore, the percentage of plutonium-239 remaining after 2000 years is 91.43%.
Summary:To find the percentage of plutonium-239 remaining after 2000 years, we can use the formula:P = P0 (1/2)^(t/h)By substituting the given values, we get:P = 0.918 P0Therefore, the percentage of plutonium-239 remaining is: % remaining = (0.918 P0/P0) x 100% = 91.43%
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Mortgage rates: Following are interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia for one day. It is reasonable to assume that the population is approximately normal.
4.754 4.373 4.174 4.678 4.426 4.229 4.124 4.250 3.952 4.195 4.296
(a) Construct an 80% confidence interval for the mean rate. Round the answer to at least four decimal places. An 80% confidence interval for the mean rate is
The 80% confidence interval for the mean rate is approximately 4.1243 to 4.5177.
Answers to the questionsGiven the interest rates (annual percentage rates) for the sample of lenders in Macon, Georgia for one day:
4.754, 4.373, 4.174, 4.678, 4.426, 4.229, 4.124, 4.250, 3.952, 4.195, 4.296.
The sample mean:
xbar = (4.754 + 4.373 + 4.174 + 4.678 + 4.426 + 4.229 + 4.124 + 4.250 + 3.952 + 4.195 + 4.296) / 11
xbar ≈ 4.321
The sample standard deviation:
[tex]s = √[(∑(xi - xbar)^2) / (n - 1)][/tex]
s ≈ √[(0.10012 + 0.03872 + 0.08132 + 0.12652 + 0.00772 + 0.01432 + 0.06072 + 0.00952 + 0.11872 + 0.03492 + 0.02412) / 10]
s ≈ √(0.63661 / 10)
s ≈ √0.063661
s ≈ 0.2523
The margin of error:
Margin of Error = t * (s / √n)
Margin of Error ≈ 1.812 * (0.2523 / √11)
Margin of Error ≈ 0.1967
The confidence interval:
Confidence Interval = xbar ± Margin of Error
Confidence Interval = 4.321 ± 0.1967
The 80% confidence interval for the mean rate is approximately 4.1243 to 4.5177.
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Compare the "Prop. contained" value from part to the confidence level associated with the simulation in one sentence h) Write a long-run interpretation for your confidence interval method in context in one sentence. Think about what would happen if you took many, many more samples
The "Prop. contained" value from part h can be compared to the confidence level associated with the simulation to assess the accuracy and reliability of the confidence interval.
A long-run interpretation for the confidence interval method means that if we were to repeat the sampling process and construct confidence intervals using the same method many, many times, the proportion of those intervals that contain the true population parameter (such as the mean or proportion) would approach the specified confidence level.
For example, if we construct 95% confidence intervals, we expect that in the long run, approximately 95% of those intervals would capture the true population parameter and only about 5% would not. This interpretation is based on the concept of repeated sampling and the idea that as the number of samples increases, the accuracy of the estimates improves.
By using the same method consistently and increasing the number of samples, we can gain more confidence in the accuracy of our estimates. This long-run interpretation provides a measure of the reliability and precision of the confidence interval approach in estimating population parameters.
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Solve the system with the addition method.
6x+4y= -4
-2x+5y= 4
Therefore, the solution to the system of equations 6x + 4y = -4 and -2x + 5y = 4 is (x, y) = (-178/57, 8/19).
To solve the system with the addition method, follow the steps below:
Step 1: Rewrite the system so that the x and y variables are lined up vertically and the constant terms are lined up vertically.
Step 2: Choose a variable to eliminate from one of the equations. In this case, x is a good choice because the coefficients of x in each equation are opposites. So, add the two equations together to eliminate x. The new equation will only have y as a variable.
Step 3: Solve the new equation for y.
Step 4: Substitute the value of y into either one of the original equations and solve for x.
Step 5: Check the solution in both original equations to make sure it is correct.
The system of equations is:
6x + 4y = -4 ........(1)
-2x + 5y = 4 ........(2)
Multiply equation 2 by 3:3(-2x + 5y = 4)
=> -6x + 15y = 12
Add equation 1 and 2:
(6x + 4y = -4) + (-6x + 15y = 12) => 19y
= 8
Divide both sides by 19: y = 8/19
Now substitute the value of y = 8/19 into equation 1:6x + 4(8/19) = -4
Simplify and solve for x:6x + 32/19 = -4 => 6x =
-4 - 32/19
=> x = -178/57
In mathematics, there are many methods to solve the system of equations. The addition method is one of them. The addition method is a way of eliminating one variable in a system of equations by adding two equations. In this method, we add two equations to eliminate one variable and then solve the resulting equation for the other variable. This method is also called the elimination method.The system of equations can be solved by substitution, graphing, and elimination methods. The addition method is a type of elimination method. In this method, we choose a variable to eliminate from one of the equations.
We add the two equations together to eliminate one variable. Then we solve the new equation for the other variable. In the given system of equations 6x + 4y = -4 and -2x + 5y = 4, we can eliminate x by adding the two equations. So, we add equation 1 and 2 and get 19y = 8. Then we solve this new equation for y and get y = 8/19. Now we substitute this value of y into equation 1 and get x = -178/57. So, the solution to the system of equations is (x, y) = (-178/57, 8/19).
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\The following table presents the result of the logistic regression on data of students y = eBo+B₁x1+B₂x₂ 1+ eBo+B₁x1+B₂x2 +€ . y: Indicator for on-time graduation, takes value 1 if the student graduated on time, 0 of not; X₁: GPA; . . x₂: Indicator for receiving scholarship last year, takes value 1 if received, 0 if not. Odds Ratio Intercept 0.0107 X₁: gpa 4.5311 X₂: scholarship 4.4760 1) (1) What is the point estimates for Bo-B₁. B₂, respectively? 2) (1) According to the estimation result, if a student's GPA is 3.5 but did not receive the scholarship, what is her predicted probability of graduating on time?
1.Point estimates for Bo, B₁, and B₂:
Bo (intercept): The point estimate is 0.0107.
B₁ (coefficient for GPA): The point estimate is 4.5311.
B₂ (coefficient for scholarship): The point estimate is 4.4760.
2.The predicted probability of a student with a GPA of 3.5 and no scholarship graduating on time is approximately 0.972 or 97.2%.
Based on the given table, the logistic regression equation is as follows:
y = e^(Bo + B₁x₁ + B₂x₂) / (1 + e^(Bo + B₁x₁ + B₂x₂))
Point estimates for Bo, B₁, and B₂:
Bo (intercept): The point estimate is 0.0107. This indicates the estimated log-odds of on-time graduation when both GPA (x₁) and scholarship (x₂) are zero.
B₁ (coefficient for GPA): The point estimate is 4.5311. This suggests that for every unit increase in GPA, the log-odds of on-time graduation increase by approximately 4.5311, assuming all other variables are held constant.
B₂ (coefficient for scholarship): The point estimate is 4.4760. This indicates that students who received a scholarship (x₂ = 1) have approximately 4.4760 times higher log-odds of on-time graduation compared to those who did not receive a scholarship (x₂ = 0), assuming all other variables are held constant.
2. To calculate the predicted probability of graduating on time for a student with a GPA of 3.5 and no scholarship (x₁ = 3.5, x₂ = 0), we substitute the values into the logistic regression equation:
y = e^(0.0107 + 4.53113.5 + 4.47600) / (1 + e^(0.0107 + 4.53113.5 + 4.47600))
Simplifying the equation:
y = e^(0.0107 + 4.53113.5) / (1 + e^(0.0107 + 4.53113.5))
Using a calculator or software to perform the calculations:
y ≈ 0.972
Therefore, the predicted probability of a student with a GPA of 3.5 and no scholarship graduating on time is approximately 0.972 or 97.2%.
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1. Match the definition to the correct vocabulary word. ____1. a statistical tool that shows the observed frequencies of two variables; one variable is listed in a row and another variable is listed in columns ___2 the ratio of the sum of the joint frequencies in a row of a column over the total number of data values
____3. the ratio of a frequency of a particular category to the entire set of data ___4. the ratio of individual occurrences over the total occurrences * 5 when a relative frequency is determined by a row or column
a conditional relative frequency
b. marginal frequency - c two-way table d. joint frequency e relative frequency
1. Match the definition to the correct vocabulary word.
1. Two-way table: a statistical tool that shows the observed frequencies of two variables; one variable is listed in a row and another variable is listed in columns.
2. Conditional relative frequency: the ratio of the sum of the joint frequencies in a row of a column over the total number of data values.
3. Relative frequency: the ratio of a frequency of a particular category to the entire set of data.
4. Joint frequency: the ratio of individual occurrences over the total occurrences.
5. Marginal frequency: when a relative frequency is determined by a row or column.
1. Two-way table: A two-way table is a statistical tool that shows the observed frequencies of two variables. It is also known as a contingency table, cross-tabulation, or a contingency matrix.
One variable is listed in a row and another variable is listed in columns. Two-way tables are often used to summarize categorical data and to investigate the relationship between two variables.
2. Conditional relative frequency: Conditional relative frequency is the ratio of the sum of the joint frequencies in a row of a column over the total number of data values. It is used to analyze the association between two categorical variables. It helps in determining the relationship between two variables when one variable is conditioned by another.
3. Relative frequency: Relative frequency is the ratio of a frequency of a particular category to the entire set of data. It helps to find out the proportion of each category in the whole dataset. It is often expressed as a percentage and is a useful tool in data analysis and statistics.
4. Joint frequency: Joint frequency is the ratio of individual occurrences over the total occurrences. It is used in probability theory and statistics to determine the probability of two or more events occurring simultaneously.
5. Marginal frequency: Marginal frequency is when a relative frequency is determined by a row or column. It is the sum of a row or column in a two-way table.
Marginal frequency is used to calculate the probability of an event occurring by considering all possible outcomes. It is useful in probability theory and data analysis.
it is clear that two-way tables, conditional relative frequency, relative frequency, joint frequency, and marginal frequency are all statistical tools that are used to analyze data and to determine the relationship between variables.
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"
For the subspace below, (a) find a basis, and (b) state the dimension. 6a + 12b - 2c 12a - 4b-4c - : a, b, c in R -9a + 5b + 3C - - 3a + b + c a. Find a basis for the subspace.
Using Gaussian Elimination,{[3 6 -1 -3], [0 2 -6 -9], [0 0 -16 32]}So we can have a maximum of 3 linearly independent vectors.
The basis of the subspace is {(3, 6, -1, 0, 0, 0), (-9, 5, 3, 0, 0, 0), (2, -2, 3, 0, 0, 0)}.The dimension of the subspace is 3.
Given subspace is as follows.
6a + 12b - 2c12a - 4b-4c-9a + 5b + 3C-3a + b + c
We will first write the above subspace in terms of linear combination of its variables a,b,c as shown below:
6a + 12b - 2c + 0d + 0e + 0f
= 2(3a + 6b - c + 0d + 0e + 0f) + 0(-9a + 5b + 3c + 0d + 0e + 0f) + (-3a + b + c + 0d + 0e + 0f)12a - 4b-4c + 0d + 0e + 0f
= 0(3a + 6b - c + 0d + 0e + 0f) + 2(-9a + 5b + 3c + 0d + 0e + 0f) + 3(-3a + b + c + 0d + 0e + 0f)-9a + 5b + 3C + 0d + 0e + 0f
= -3(3a + 6b - c + 0d + 0e + 0f) + 0(-9a + 5b + 3c + 0d + 0e + 0f) + (2a - 2b + 3c + 0d + 0e + 0f)-3a + b + c + 0d + 0e + 0f
= -1(3a + 6b - c + 0d + 0e + 0f) + 1(-9a + 5b + 3c + 0d + 0e + 0f) + (2a - 2b + 3c + 0d + 0e + 0f)
The above subspace can also be written as linearly independent vectors as follows:
{(3, 6, -1, 0, 0, 0), (-9, 5, 3, 0, 0, 0), (2, -2, 3, 0, 0, 0), (-3, 1, 1, 0, 0, 0)}These are the four vectors of the subspace, out of which we can select a maximum of 3 linearly independent vectors to form a basis of the subspace.The first vector is a multiple of the fourth vector.
Therefore, the first vector can be excluded. Let's examine the remaining three vectors to check whether they are linearly independent or not using Gaussian Elimination.
Using Gaussian Elimination,{[3 6 -1 -3], [0 2 -6 -9], [0 0 -16 32]}So we can have a maximum of 3 linearly independent vectors.
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the y-intercept of the line x=2y +5 is (0,5).
True
False
Answer:
False.
Step-by-step explanation:
To find the y-intercept of a line, we set x = 0 and solve for y. In the given equation, x = 2y + 5. Let's substitute x = 0:
0 = 2y + 5
Subtracting 5 from both sides:
-5 = 2y
Dividing both sides by 2:
-5/2 = y
Therefore, the y-intercept is (0, -5/2), not (0, 5). Hence, the statement "The y-intercept of the line x=2y +5 is (0,5)" is false.
x² a. The revenue (in dollars) from the sale of x units of a certain product is given by the function The cost (in dollars) of producing x units is given by the function C(x) = 15x + 40000. Find the profit on sales of x units. R(x) = 60x - 100 b. Suppose that the demand x and the price p (in dollars) for the product are related by the function x = f(p) = 5000-50p for 0 ≤ps 100. Write the profit as a functyion of demand p. c. Use a graphing calculator to plot the graph of your profit function from (b). Then use this graph to determine the price that would yield the maximum profit and determine what this maximum profit is. Include a screen shot of your graph.
a. The profit on sales of x units can be calculated by subtracting the cost function from the revenue function Profit(x) = Revenue(x) - Cost(x)
Profit(x) = R(x) - C(x)
Profit(x) = (60x - 100) - (15x + 40000)
Profit(x) = 45x - 40100
b. To express the profit as a function of demand p, we need to substitute the value of x in terms of p from the demand function into the profit function.
From the given demand function x = f(p) = 5000 - 50p, we can solve for p in terms of x:
x = 5000 - 50p
50p = 5000 - x
p = (5000 - x)/50
Now, substitute this expression for p into the profit function:
Profit(p) = 45x - 40100
Profit(p) = 45(5000 - 50p) - 40100
Profit(p) = 225000 - 2250p - 40100
Profit(p) = -2250p + 184900
c. Using a graphing calculator, we can plot the graph of the profit function Profit(p) = -2250p + 184900. The graph will show the relationship between the price (p) and the corresponding profit.
By analyzing the graph, we can determine the price that would yield the maximum profit and the maximum profit itself.
Here is a step-by-step procedure to plot the graph of the profit function using a graphing calculator:
Enter the equation Profit(p) = -2250p + 184900 into the graphing calculator.
Set the viewing window appropriately to display the range of prices that are relevant to the problem (0 ≤ p ≤ 100).
Plot the graph of the profit function.
Analyze the graph to identify the price that corresponds to the maximum profit. This will be the x-coordinate of the vertex of the graph.
Read the maximum profit from the y-coordinate of the vertex.
The graph will provide a visual representation of the profit function and allow us to determine the price that maximizes profit and the value of the maximum profit.
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(20 points) Consider the nonlinear system x' = x(1 - x - y) y = y(2-y-3x) (a) Find all equilibrium points. There are four of them. (b) Linearize the system around each equilibrium point and determine their stability. (c) Does the linearized system accurately describe the local behavior near the equilibrium points? (d) Sketch the x- and y- nullclimes. Locate the equilibrium points and sketch the phase portrait to describe the global behavior.
The equilibrium points are the points where the two functions intersect, therefore to find all the equilibrium points, we need to solve for when x' and y are zero. The solution is given below:Equilibrium points: (0, 0), (1, 0), (0, 2), (−1, 1)b) Linearize the system around each equilibrium point and determine their stability.
Linearization of a nonlinear system is the process of approximating a nonlinear system at a particular operating point by a linear system. In this case, we use the Jacobian matrix to calculate the linearization. The linearized system accurately describes the local behavior near the equilibrium points for (0, 2) and (−1, 1). However, for (0, 0) and (1, 0), the linearization is not informative and does not describe the local behavior.d) Sketch the x- and y- nullclines. Locate the equilibrium points and sketch the phase portrait to describe the global behavior. Nullclines are the lines where the vector field is horizontal or vertical, and hence the vector field is tangent to these lines. Then the nullclines are given by y = x(1 − x) and y = 2 − y − 3x respectively. We can use these to sketch the nullclines as shown below Nullclines and equilibrium points:Now we can sketch the phase portrait by considering the signs of x' and y' in each quadrant.
The global behavior of the system has two equilibrium points (0, 2) and (−1, 1) which are both sinks, and two saddle points (0, 0) and (1, 0). The separatrices separate the phase plane into four regions. In regions I and III, all solutions approach the equilibrium point (−1, 1). In regions II and IV, all solutions approach the equilibrium point (0, 2).
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The manufacturer of a new chewing gum claims that at least 80% of dentists surveyed prefer their type of gum andrecommend it for their patients who chew gum. An independent consumer research firm decides to test their claim. The findings in a sample of 200 dentists indicate that 74.1% of the respondents do actually prefer their gum. A. What are the null and alternative hypotheses for the test? B. What is the decision rule? C. The value of the test statistic is:
The null hypothesis (H0) is that the proportion of dentists who prefer the new chewing gum is 80% or greater. The alternative hypothesis (H1) is that the proportion is less than 80%. The decision rule depends on the significance level chosen for the test. If the significance level is α, a common choice is α = 0.05, the decision rule would be: Reject H0 if the test statistic is less than the critical value obtained from the appropriate distribution.
A. The null hypothesis (H0) states that the proportion of dentists who prefer the new chewing gum is 80% or greater. The alternative hypothesis (H1) contradicts the null hypothesis and states that the proportion is less than 80%. In this case, the null hypothesis is that p ≥ 0.8, and the alternative hypothesis is that p < 0.8, where p represents the true proportion of dentists who prefer the gum.
B. The decision rule depends on the significance level chosen for the test. Typically, a significance level of α = 0.05 is used, which means that the null hypothesis will be rejected if the evidence suggests that the observed proportion is significantly lower than 80%. The decision rule would be: Reject H0 if the test statistic is less than the critical value obtained from the appropriate distribution, such as the standard normal distribution or the t-distribution.
C. The value of the test statistic is not provided in the given information. To determine the test statistic, one would need to calculate the appropriate test statistic based on the sample proportion, the hypothesized proportion, and the sample size. The specific test statistic used would depend on the statistical test chosen for hypothesis testing, such as the z-test or the t-test.
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Intuitively explain how could you use the non-linear least square
technique to estimate the ARMA(1, 1) and MA(2) models.
The non-linear least square technique is a method of finding the best parameters in a non-linear model to minimize the sum of squares of the differences between the observed data and the model predictions.
ARMA(1,1) Model:An ARMA(1,1) model can be represented by the equation
y[t] = φ
y[t-1] + ε[t] + θε[t-1].
Here y[t] represents the time series at time t, ε[t] is the white noise, φ and θ are the parameters to be estimated using the non-linear least square method.
The technique involves finding the values of φ and θ that minimize the sum of squares of the differences between the observed values of y[t] and the predicted values of y[t].
The equation that needs to be minimized is:
∑t=2n(y[t] - φy[t-1] - ε[t] - θε[t-1])²
MA(2) Model:An MA(2) model can be represented by the equation
y[t] = ε[t] + θ1ε[t-1] + θ2ε[t-2].
Here y[t] represents the time series at time t, ε[t] is the white noise, θ1 and θ2 are the parameters to be estimated using the non-linear least square method.
The technique involves finding the values of θ1 and θ2 that minimize the sum of squares of the differences between the observed values of y[t] and the predicted values of y[t].
The equation that needs to be minimized is: ∑t=3n(y[t] - ε[t] - θ1ε[t-1] - θ2ε[t-2])².
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I was found that 85.6% of students at IUL worldwide are enrolling to undergraduate program. A random sample of 50 students from IUL Morocco revealed that 42 of them were enrolled in undergraduate program. Is there evidence to state that the proportion of IUL Morocco differs from the IUL Morocco proportion? Use α = 0.05
To test whether the proportion of IUL Morocco differs from the IUL worldwide proportion, we can conduct a hypothesis test using the sample data.
Null Hypothesis (H0): The proportion of IUL Morocco is equal to the IUL worldwide proportion.
Alternative Hypothesis (Ha): The proportion of IUL Morocco differs from the IUL worldwide proportion.
Given:
IUL worldwide proportion: 85.6%
Sample size (n): 50
Number of students enrolled in undergraduate program in the sample (x): 42
To test the hypothesis, we can use the z-test for proportions. The test statistic (z) can be calculated using the formula:
z = (p - P) / sqrt(P(1-P)/n)
where:
p is the proportion in the sample (x/n)
P is the hypothesized proportion (IUL worldwide proportion)
n is the sample size
First, calculate the expected number of students enrolled in undergraduate program in the sample under the null hypothesis:
Expected number = n * P
Expected number = 50 * 0.856 = 42.8
Next, calculate the test statistic:
z = (42 - 42.8) / sqrt(42.8 * (1-42.8/50))
z = -0.8 / sqrt(42.8 * 0.172)
z ≈ -0.8 / 3.117
z ≈ -0.256
To determine whether there is evidence to state that the proportion of IUL Morocco differs from the IUL worldwide proportion, we compare the test statistic (z) to the critical value at α = 0.05 (two-tailed test).
The critical value for a two-tailed test at α = 0.05 is approximately ±1.96.
Since -0.256 is not in the rejection region (-1.96 to 1.96), we fail to reject the null hypothesis. This means that there is not enough evidence to state that the proportion of IUL Morocco differs significantly from the IUL worldwide proportion at α = 0.05.
In conclusion, based on the given data and hypothesis test, we do not have evidence to conclude that the proportion of IUL Morocco differs from the IUL worldwide proportion.
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Price index numbers measure changes in: Select one: O a. Physical quantity of goods produced O b. Relative changes in prices of commodities between two periods O c. Relative changes in quantities of commodities between two periods O d. None of the above e. Single variable
Price index numbers measure changes in:O b. Relative changes in prices of commodities between two periods
What is price index?Prices of products and services are tracked and quantified over time using price index numbers which are statistical metrics.
Usually stated as a percentage or an index number they offer details regarding the relative price changes between two periods. Price indices support the tracking of living expenses, analysis of economic trends, and monitoring of inflation.
Therefore the correct option is b.
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The growing seasons for a random sample of 35 U.S. aties were recorded, yielding a sample mean of 185.3 days and the population standard deviation of 52.4 days. Estimate the true population mean of the growing season with 93% confidence. Use a graphing calculator and round the answers to one decimal place.
The 93% confidence interval for the true population mean of the growing season is given as follows:
(169.2 days, 201.3 days).
What is a z-distribution confidence interval?The bounds of the confidence interval are given by the rule presented as follows:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.z is the critical value.n is the sample size.[tex]\sigma[/tex] is the standard deviation for the population.Using the z-table, for a confidence level of 93%, the critical value is given as follows:
z = 1.81.
The parameters are given as follows:
[tex]\overline{x} = 185.3, \sigma = 52.4, n = 35[/tex]
The lower bound of the interval is given as follows:
[tex]185.3 - 1.81 \times \frac{52.4}{\sqrt{35}} = 169.2[/tex]
The upper bound of the interval is given as follows:
[tex]185.3 + 1.81 \times \frac{52.4}{\sqrt{35}} = 201.3[/tex]
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