The probability that in 160 tosses of a fair coin, the proportion of heads will be between 45% and 55% can be approximated using the normal distribution. This probability is approximately 0.826, indicating a high likelihood of the proportion falling within the desired range.
To calculate the probability, we can assume that the number of heads in 160 tosses of a fair coin follows a binomial distribution with parameters n = 160 (number of trials) and p = 0.5 (probability of heads). Since n is large, we can approximate the binomial distribution with a normal distribution using the Central Limit Theorem.
The mean of the binomial distribution is given by μ = np = 160 * 0.5 = 80, and the standard deviation is σ = sqrt(np(1-p)) = sqrt(160 * 0.5 * 0.5) = 6.324. Now, we standardize the range of 45% to 55% by converting it to z-scores.
To find the z-scores, we use the formula z = (x - μ) / σ, where x is the proportion in decimal form. Converting 45% and 55% to decimal form gives us 0.45 and 0.55 respectively. Plugging these values into the z-score formula, we get z1 = (0.45 - 0.5) / 0.0397 ≈ -1.26 and z2 = (0.55 - 0.5) / 0.0397 ≈ 1.26.
Next, we look up the corresponding probabilities associated with the z-scores in the standard normal distribution table. The probability of obtaining a z-score less than -1.26 is approximately 0.1038, and the probability of obtaining a z-score less than 1.26 is approximately 0.8962. Thus, the probability of the proportion of heads being between 45% and 55% is approximately 0.8962 - 0.1038 = 0.7924.
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You may need to use the appropriate appendix table or technology to answer this question. A binomial probability distribution has p-0.20 and n 100. (a) What are the mean and standard deviation? mean 20 standard deviation 4 (b) Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain, O Yes, because np z 5 and n(1 -p) 2 5. O No, because np 5 and n(1 -P) 5 O Yes, because np 5 and n(1 -P)5. O No, because np < 5 and n(1 - p)5 O Yes, because n 2 30. (e) What is the probability of exactly 23 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.) 0.0755 (a) what is the probability of 16 to 24 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.) 0.6822 (e) What is the probability of 13 or fewer successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.) 0.0308
The mean and standard deviation are 20 and 4, respectively and the probability of 13 or fewer successes is 0.0516.
Given that a binomial probability distribution has p-0.20 and n 100.
(a) The mean and standard deviation can be calculated as follows:
Mean = μ = np = 100 × 0.2 = 20
Standard deviation = σ = √(npq) = √[100 × 0.2 × 0.8] ≈ 4.00
Therefore, the mean and standard deviation are 20 and 4, respectively.
(b) To determine whether binomial probabilities can be approximated by the normal probability distribution, we can use the rule np > 5 and nq > 5.If we put p = 0.2 and q = 0.8, then:
np = 100 × 0.2 = 20,
and nq = 100 × 0.8 = 80.
So, np and nq are both greater than 5, thus we can say that this situation is one in which binomial probabilities can be approximated by the normal probability distribution.
Now, we can use the normal approximation of the binomial distribution to answer the following:
(e) To find the probability of exactly 23 successes, we can use the normal approximation of the binomial distribution as follows:
P(X = 23) = P(22.5 < X < 23.5)≈ P[(22.5 – 20)/4 < (X – 20)/4 < (23.5 – 20)/4]≈ P[0.625 < z < 1.125], where z = (X – μ)/σ = (23 – 20)/4 = 0.75
Using the standard normal table, P(0.625 < z < 1.125) = P(z < 1.125) – P(z < 0.625) = 0.8708 – 0.7953 = 0.0755
Therefore, the probability of exactly 23 successes is 0.0755.
(a) To find the probability of 16 to 24 successes, we can use the normal approximation of the binomial distribution as follows:
P(16 ≤ X ≤ 24) = P(15.5 < X < 24.5)≈ P[(15.5 – 20)/4 < (X – 20)/4 < (24.5 – 20)/4]≈ P[-1.125 < z < 1.125], where z = (X – μ)/σ = (16 – 20)/4 = –1 and z = (X – μ)/σ = (24 – 20)/4 = 1
Using the standard normal table, P(-1.125 < z < 1.125) = P(z < 1.125) – P(z < –1.125) = 0.8708 – 0.1292 = 0.6822
Therefore, the probability of 16 to 24 successes is 0.6822.
(e) To find the probability of 13 or fewer successes, we can use the normal approximation of the binomial distribution as follows:
P(X ≤ 13) = P(X < 13.5)≈ P[(X – μ)/σ < (13.5 – 20)/4]≈ P[z < –1.625], where z = (X – μ)/σ = (13 – 20)/4 = –1.75
Using the standard normal table, P(z < –1.625) = 0.0516
Therefore, the probability of 13 or fewer successes is 0.0516.
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Suppose f(x) = x^2 +1 and g(x) = x+1 . Then (f + g)(x) = ______ (f - g)(x) =______. (ƒg)(x) = _____. (f/g)(x) = _____. (fog)(x) = _____. (gof)(x) = _____.
The expressions for (f + g)(x), (f - g)(x), (f * g)(x), (f / g)(x), (f o g)(x), and (g o f)(x), we'll substitute the given functions:
f(x) = x² + 1 and g(x) = x + 1
We are to find the following: (f + g)(x), (f - g)(x), (f × g)(x), (f/g)(x), (fog)(x)
and (gof)(x).(f + g)(x) = f(x) + g(x)
=[tex]x^2 + 1 + x + 1[/tex]
=[tex]x^2+ x + 2(f - g)(x)[/tex]
= f(x) - g(x)
=[tex]x^2 + 1 - x - 1[/tex]
= [tex]x^2 - x(fg)(x)[/tex]
= f(x) × g(x)
=[tex](x^2 + 1) \times (x + 1)[/tex]
= [tex]x^3 + x^2 + x + 1(f/g)(x)[/tex]
= f(x)/g(x)
=[tex](x^2 + 1)/(x + 1)(fog)(x)[/tex]
= f(g(x))
= f(x + 1)
= [tex](x + 1)^2 + 1[/tex]
=[tex]x^2 + 2x + 2(gof)(x)[/tex]
Since the numerator and denominator cannot be simplified further, we leave it as (x^2 + 1) / (x + 1).
= g(f(x))
= [tex]g(x^2 + 1)[/tex]
= [tex](x^2 + 1) + 1[/tex]
= [tex]x^2 + 2[/tex]
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Use a double-angle formula to find the exact value of the given expression 1 - 2 sin 2105 1 - 2 sin 2105° 0 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression)
The exact value of the given expression is (2 - √6 - √2) / 2.
We are supposed to find the exact value of the given expression 1 - 2 sin 2105° by using a double angle formula.
The double angle formula for sin2θ is given by sin2θ=2sinθcosθ.
Now, let's use this double angle formula to simplify the given expression.
Solution:Here is the given expression: 1 - 2 sin 2105°
We need to find the exact value of the given expression using the double angle formula.
Let's begin by finding sin 2θ.Let's take θ = 105°.
Then, we have: sin 2θ = 2 sin θ cos θ
Now, we know that sin 2θ = 2 sin θ cos θsin 105° = sin (45° + 60°) = sin 45° cos 60° + cos 45° sin 60°
We know that: sin 45° = cos 45° = √2 / 2and sin 60° = √3 / 2, cos 60° = 1 / 2
Now, substituting the values, we get:sin 2 x 105° = √2 / 2 × 1 / 2 + √2 / 2 × √3 / 2= (√6 + √2) / 4
Therefore, sin 210° = sin 2 x 105° / 2= (√6 + √2) / 4
Now, let's substitute this value in the given expression, we get:1 - 2 sin 2105°= 1 - 2 × (√6 + √2) / 4= 1 - (√6 + √2) / 2= (2 - √6 - √2) / 2
Therefore, the exact value of the given expression is (2 - √6 - √2) / 2.
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For a function y = (x² + 2) (x³ + x² + 1)², state the steps to find the derivative.
Using product rule and chain rule, the derivative of the function y = (x² + 2)(x³ + x² + 1)² is given by:
y' = 2x(x³ + x² + 1)² + 2(x² + 2)(x³ + x² + 1)(3x² + 2x)
What is the derivative of the function?To find the derivative of the function y = (x² + 2)(x³ + x² + 1)², we can use the product rule and the chain rule.
Let's denote the first factor (x² + 2) as u and the second factor (x³ + x² + 1)² as v.
Using the product rule (u * v)', the derivative of the function is given by:
y' = u' * v + u * v'
First, let's find the derivative of u (x² + 2):
u' = d/dx (x² + 2)
= 2x
Next, let's find the derivative of v (x³ + x² + 1)² using the chain rule:
v' = d/dx (x³ + x² + 1)²
= 2(x³ + x² + 1) * (d/dx (x³ + x² + 1))
= 2(x³ + x² + 1) * (3x² + 2x)
Now we can substitute the values of u, u', v, and v' into the derivative formula:
y' = (2x) * (x³ + x² + 1)² + (x² + 2) * [2(x³ + x² + 1) * (3x² + 2x)]
Simplifying further:
y' = 2x(x³ + x² + 1)² + (x² + 2) * 2(x³ + x² + 1) * (3x² + 2x)
y' = 2x(x³ + x² + 1)² + 2(x² + 2)(x³ + x² + 1)(3x² + 2x)
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Problem-1 Analyze the truss manually and using the software and compare your results, P is 8 kN. 60° 60 4 m 4 m
The force in each member of the truss is P/√3 = 4.62 kN, using the method of joints.
Load P = 8 kN60 degree60 degree. The length of each member is 4 mAnalysis
:Using the Method of JointsTo analyze the truss using the method of joints, we assume that all the joints are in equilibrium.
Summary: The force in each member of the truss is P/√3 = 4.62 kN, using the method of joints.
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The following function t(n) is defined recursively as: 1, n = 1 t(n) = 43, n = 2 (1) -2t(n-1) + 15t(n-2), n ≥ 3 a) Compute t(3) and t(4). b) Find a general non-recursive formula for the recurrence. c) Find the particular solution which satisfies the initial conditions t(1) = 1 and t(2) = 43.
a) t(3) = -25 and t(4) = 665.
b) General formula: t(n) = A(3^n) + B(5^n), where A and B are constants.
c) Particular solution: t(n) = (1/2)(3^n) + (1/2)(5^n) satisfies initial conditions t(1) = 1 and t(2) = 43.
a) By applying the recursive definition, we find that t(3) is obtained by substituting the values of t(1) and t(2) into the recurrence relation, giving t(3) = -2t(2) + 15t(1) = -2(43) + 15(1) = -25. Similarly, t(4) is found by substituting the values of t(2) and t(3), resulting in t(4) = -2t(3) + 15t(2) = -2(-25) + 15(43) = 665.
b) To derive a general non-recursive formula for the recurrence t(n) = -2t(n-1) + 15t(n-2), we solve the associated characteristic equation, which yields distinct roots of 3 and 5. This allows us to express the general solution as t(n) = A(3^n) + B(5^n), where A and B are constants.
c) By applying the initial conditions t(1) = 1 and t(2) = 43 to the general solution, we obtain a system of equations. Solving this system, we find A = 1/2 and B = 1/2, leading to the particular solution t(n) = (1/2)(3^n) + (1/2)(5^n).
In conclusion, t(3) = -25 and t(4) = 665. The general non-recursive formula is t(n) = A(3^n) + B(5^n), with the particular solution t(n) = (1/2)(3^n) + (1/2)(5^n) satisfying the initial conditions.
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Solve the following inequality problem and choose the interval notation of the solution: -8 < -5x + 2 <-3 2 a. (2,1] b. (-0,0) c. (0,+0) d. [0,+0) e. (1,2) f. [2,1) g. (-00,0] h. (1,2]
The interval notation of the solution: -8 < -5x + 2 <-3 2 is (1, 2).Therefore, option e. (1,2) is the correct answer. Given inequality is -8 < -5x + 2 < -3. We need to find the solution of the inequality and choose the interval notation of the solution.
To solve the given inequality, we will solve both inequalities separately.
-8 < -5x + 2
⇒ -8-2 < -5x
⇒ -10 < -5x
⇒ -10/-5 > x
⇒ 2 > x i.e x < 2.
So, the first part of the solution is -infinity
< x < 2.-5x + 2 < -3
⇒ -5x + 2 + 3 < 0
⇒ -5x + 5 < 0
⇒ -5(x - 1) < 0
⇒ x - 1 > 0
⇒ x > 1.
So, the second part of the solution is x > 1.
Now, we will combine the two solutions. -infinity < x < 2 and x > 1.
If we combine these solutions, then the solution will be 1 < x < 2.
As the solution is including 1 and 2. The solution will be (1, 2).
Therefore, option e. (1,2) is correct.
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3. Although it is not needed for navigation purposes, the crewmembers would like to find the
distance between Dothan City and Lemont using only the information they have calculated. Find
this distance to the nearest tenth of a mile. (2 points)
The distance between Dothan City and Lemont is 95.4 miles.
From the given figure, the distance between Lemont and Buoy is 44.6 miles.
Let the distance between Ship and Buoy be x.
Now tan36°=44.6/x
0.7265=44.6/x
x=44.6/0.7265
x=61.4 miles
Let the distance between ship and Lemont be y.
By using Pythagoras theorem, we get
y²=44.6²+61.4²
y²=5759.12
y=√5759.12
y=75.9 miles
Let the distance Dothan City and Lemont be z.
By using Pythagoras theorem, we get
z²=57.8²+75.9²
z²=9101.65
z=√9101.65
z=95.4 miles
Therefore, the distance between Dothan City and Lemont is 95.4 miles.
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ind all x-intercepts and y-intercepts of the graph of the function. f(x)=-3x³ +24x² - 45x If there is more than one answer, separate them with commas.
The x-intercepts of the graph of the function f(x) = -3x³ + 24x² - 45x are 0, 3, and 5. These are the values of x for which the function intersects or crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x. In this case, we have -3x³ + 24x² - 45x = 0. By factoring out an x from each term, we get x(-3x² + 24x - 45) = 0. The equation is satisfied when either x = 0 or -3x² + 24x - 45 = 0. Solving the quadratic equation, we find that x = 3 and x = 5 are the additional x-intercepts.
The y-intercept of a function is the value of the function when x = 0. In this case, when we substitute x = 0 into the function f(x) = -3x³ + 24x² - 45x, we get f(0) = 0. Therefore, the y-intercept is 0.
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Let X1, X2, ..., X16 be a random sample from the normal distribution N(90, 102). Let X be the sample mean and $2 be the sample variance. Fill in each of the fol- lowing blanks
Let X1, X2, ..., X16 be a random sample from the normal distribution N(90,102). Let X be the sample mean and s² be the sample variance.In the context of the given question, we are required to fill in the blanks. As per the definition of sample variance:s² = Σ(X - µ)² / (n - 1)where Σ(X - µ)² is the sum of squared deviations of sample data from the sample mean and n - 1 represents degrees of freedom.
We are given the values of sample mean and variance as:
X = (X1 + X2 + ... + X16) / 16
= (X1/16) + (X2/16) + ... + (X16/16)s²
= [(X1 - X)² + (X2 - X)² + ... + (X16 - X)²] / (16 - 1)From the given problem, we have: Mean, µ = 90Variance, σ² = 102We
(a) P(88 < X < 92) = P[-2/((2/4)(1/2)) < (X - 90)/(2/4) < 2/((2/4)(1/2))] (By using the standardization of the normal variable)
P(-4 < (X - 90) / (1/2) < 4)By using the probability table, we can write:P(-4 < Z < 4) = 0.9987P(88 < X < 92) = 0.9987(b) P(91 < X < 93) = P[(91 - 90) / (1/4) < (X - 90) / (1/2) < (93 - 90) / (1/4)] (By using the standardization of the normal variable)P(4 < (X - 90) / (1/2) < 12)By using the probability table.
P(4 < Z < 12) ≈ 0P(91 < X < 93) ≈ 0(c) P(X > 92) = P[(X - 90) / (1/4) > (92 - 90) / (1/4)] (By using the standardization of the normal variable)P(X > 92) = P(Z > 8) = 1 - P(Z < 8)By using the probability table, we can write:
P(Z < 8) = 1.00P(X > 92) = 1 - 1.00 = 0(d) P(2s < X < 6s) = P[2 < (X - 90) / (s) < 6]
(By using the standardization of the normal variable)P(2s < X < 6s) = P(4 < Z < 12)By using the probability table, we can write :
P(4 < Z < 12) ≈ 0P(2s < X < 6s) ≈ 0(e) P(X < 88) = P[(X - 90) / (1/4) < (88 - 90) / (1/4)]
(By using the standardization of the normal variable)P(X < 88) = P(Z < -8)By using the probability table, we can write:
P(Z < -8) = 0.00P(X < 88) = 0
Therefore, all the blanks have been filled correctly. Thus, the solution to the given problem has been demonstrated.
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Let R be a relation on the set of ordered pairs of positive integers, (a,b) E Z* x Z. The relation R is: (a,b) R (c,d) - ad = bc. (another way to look at right side is 4) Determine whether or not this is an Equivalence Relation. If it is, ther determine/describe the equivalence classes. a b
Given R be a relation on the set of ordered pairs of positive integers, (a,b) E Z* x Z. The relation R is (a,b) R (c,d) ⇔ ad = bc.
Determine whether or not this is an Equivalence Relation. If it is, then determine/describe the equivalence classes.Step-by-step solution:
To prove that R is an equivalence relation, we need to prove that it satisfies the following three conditions:
Reflexive: (a, b) R (a, b) for all (a, b) ∈ Z* x Z.
Symmetric: (a, b) R (c, d) implies that (c, d) R (a, b) for all (a, b), (c, d) ∈ Z* x Z.Transitive: If (a, b) R (c, d) and (c, d) R (e, f), then (a, b) R (e, f) for all (a, b), (c, d), (e, f) ∈ Z* x Z.1.
Reflexive: (a, b) R (a, b) ⇔ ab = ba, which is always true.
2. Symmetric: (a, b) R (c, d) ⇔ ad = bc. We have to show that (c, d) R (a, b).
This is true because ad = bc implies cb = da. Hence, (c, d) R (a, b).3. Transitive: Suppose (a, b) R (c, d) and (c, d) R (e, f). Then ad = bc and cf = de.
Multiplying these two equations, we get adcf = bcde. Since ad = bc, we can substitute ad for bc in this equation to get adcf = adde or cf = de. Thus, (a, b) R (e, f).Therefore, R is an equivalence relation.
The equivalence class of (a, b) is {[c, d] : ad = bc}.
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The equivalence classes are as follows:For all positive integers a and b, [a, b] represents all pairs (c, d) such that ad = bc.
Let R be a relation on the set of ordered pairs of positive integers, (a,b) E Z* x Z.
The relation R is: (a,b) R (c,d) - ad = bc. (another way to look at right side is 4)
Determine whether or not this is an Equivalence Relation and find the equivalence classes.
Definition of relation:A relation is a set of ordered pairs.
The set of ordered pairs, which are related, is called the relation.
R is an equivalence relation if it is reflexive, symmetric, and transitive.
The relation is reflexive, symmetric and transitive and hence it is an equivalence relation:
Reflexive property: (a, b) R (a, b) as ab = ba
Symmetric property: If (a, b) R (c, d), then (c, d) R (a, b) as ab = cd is equivalent to cd = ab
Transitive property: If (a, b) R (c, d) and (c, d) R (e, f), then (a, b) R (e, f) as ab = cd and cd = ef implies ab = ef
Therefore, the relation R is an equivalence relation.
Equivalence Classes:Let's figure out the equivalence classes by using the definition.
The equivalence class [a,b] = {(c,d) ∈ Z* × Z | ad = bc}
We need to find all the ordered pairs (c, d) such that they are equivalent to (a, b) under the relation R.
It implies that ad = bc.Then [a,b] = {(c,d) E Z* x Z | ad = bc}
Therefore, the equivalence classes are as follows:For all positive integers a and b, [a, b] represents all pairs (c, d) such that ad = bc.
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a subjective question, hence you have to write your answer in the Text-Field giver 76261
Solve the following LP using M-method [10M]
Subject to Maximize
zx₁ + 5x₂
3x1 + 4x₂ ≤ 6
X₁ + 3x₂ ≥ 2,
X1, X2, ≥ 0.
To solve the given linear programming problem using the M-method, we begin by introducing slack variables and an artificial variable. We then convert the problem into standard form and construct the initial tableau. Next, we apply the M-method to iteratively improve the solution until an optimal solution is reached. The final tableau provides the optimal values for the decision variables.
To solve the linear programming problem using the M-method, we start by introducing slack variables to convert the inequality constraints into equations. We add variables s₁ and s₂ to the first constraint and variables a₁ and a₂ to the second constraint. This yields the following equalities:
3x₁ + 4x₂ + s₁ = 6
x₁ + 3x₂ - a₁ = 2
Next, we introduce an artificial variable, M, to the objective function to create an auxiliary problem. The objective function becomes:
z = zx₁ + 5x₂ + 0s₁ + 0s₂ + Ma₁ + Ma₂
We then convert the problem into standard form by adding surplus variables and replacing the inequality constraint with an equality. The problem is now:
Maximize z = zx₁ + 5x₂ + 0s₁ + 0s₂ + Ma₁ + Ma₂
subject to:
3x₁ + 4x₂ + s₁ = 6
x₁ + 3x₂ - a₁ + a₂ = 2
x₁, x₂, s₁, s₂, a₁, a₂ ≥ 0
Constructing the initial tableau with the given coefficients, we apply the M-method by selecting the most negative coefficient in the bottom row as the pivot element. We perform row operations to improve the solution until all coefficients in the bottom row are non-negative.
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2. Source: Levin & Fox (2003), pp. 249, no. 19 (data modified) A personnel consultant was hired to study the influence of sick-pay benefits on absenteeism. She randomly selected samples of hourly employees who do not get paid when out sick and salaried employees who receive sick pay. Using the following data on the number of days absent during a one-year period, test the null hypothesis that hourly and salaried employees do not differ with respect to absenteeism. Salary Scheme Days Absent Subject 1 Hourly 1 2 Hourly 1 3 Hourly 2 2 4 Hourly 3 - 5 Hourly 3 6 Monthly 2 7 Monthly 2 8 Monthly 4 9 Monthly 2 10 Monthly 2 11 Monthly 5 12 Monthly 6 Answer the following questions regarding the problem stated above. a. What t-test design should be used to compute for the difference? b. What is the Independent variable? At what level of measurement? c. What is the Dependent variable? At what level of measurement? d. Is the computed value greater or lesser than the tabular value? Report the TV and CV. e. What is the NULL hypothesis? f. What is the ALTERNATIVE hypothesis? 8. Is there a significant difference? h. Will the null hypothesis be rejected? WHY? i. If you are the personnel consultant hired, what will you suggest to the company with respect to absenteeism?
Use independent samples t-test. Independent variable: Salary scheme. Dependent variable: Number of days absent.
To compute the difference in absenteeism between hourly and salaried employees, the appropriate statistical test is the independent samples t-test. The independent variable in this study is the salary scheme, categorized as either hourly or monthly.
The level of measurement for the independent variable is categorical/nominal. The dependent variable is the number of days absent during a one-year period, measured on an interval scale. The computed t-value and tabular value cannot be determined without conducting the t-test.
The null hypothesis states that there is no difference in absenteeism between hourly and salaried employees, while the alternative hypothesis suggests that a difference exists. The significance of the difference and whether the null hypothesis will be rejected depends on the results of the t-test and the chosen critical value or significance level.
As a personnel consultant, the suggestion to the company regarding absenteeism would depend on the analysis results, considering factors such as the magnitude of the difference and the practical implications for the organization.
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Compute The Area Of The Curve Given In Polar Coordinates R(θ) = Sin(θ), For Between 0 And π
The total area of the regions between the curves is 2 square units
Calculating the total area of the regions between the curvesFrom the question, we have the following parameters that can be used in our computation:
R(θ) = sin(θ)
The interval is given as
0 ≤ θ ≤ π
Using definite integral, the area of the regions between the curves is
Area = ∫R(θ) dθ
So, we have
Area = ∫sin(θ) dθ
Integrate
Area = -cos(θ)
Recall that 0 ≤ θ ≤ π
So, we have
Area = -cos(π) + cos(0)
Evaluate
Area = 3.33
Hence, the total area of the regions between the curves is 2 square units
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Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11 - α(1- S[1 - α(1-2e-x1)(1 - 2e-x₂)]ex1-x2 otherwise ,0 ≤ x₁,0 ≤ x₂. i) Find the marginal distribution of X₁. ii) Find E(X₁X₂).
To calculate this integral, we need to define the ranges of integration for x₁ and x₂. Since the given pdf is defined for 0 ≤ x₁, 0 ≤ x₂, we integrate over these ranges.
E(X₁X₂) = ∫[0,∞) ∫[0,∞) x₁x₂ * [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂)))] * e(x₁ - x₂) dx₁ dx₂
This gives us the marginal distribution of X₁.
Performing the integration over the ranges, we can evaluate the expected value E(X₁X₂).
To find the marginal distribution of X₁, we integrate the joint probability density function (pdf) over the range of X₂.
i) Marginal distribution of X₁:
To find the marginal distribution of X₁, we integrate the joint pdf f(x₁, x₂) with respect to x₂ over its range.
∫[0,∞) f(x₁, x₂) dx₂ = ∫[0,∞) [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂)))]e(x₁ - x₂)] dx₂
Simplifying the integral:
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂))])] * ∫[0,∞) e^(x₁ - x₂) dx₂
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂))])] * [-e(x₁ - x₂)] evaluated from x₂=0 to x₂=∞
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-∞))])] * [-e(x₁ - ∞)] - [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-0))])] * [-e(x₁ - 0)]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 20))])] * [0 - (-e(x₁))] - [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 21))])] * [0 - (-e(x₁))]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 0))])] * [e(x₁)] - [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2))])] * [e(x₁)]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1)])])] * [e(x₁)] - [11 - α(1 - S[1 - α(1 - 2e(-x₁))(0)])])] * [e^(x₁)]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))])]] * [e(x₁)] - [11 - α(1 - S[1 - α(1 - 0)])]] * [e(x₁)]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))])]] * [e(x₁)] - [11 - α(1 - S[1 - α(1)])]] * [e(x₁)]
= [11 - α(1 - S[1 - α(1 - 2e(-x₁))])]] * [e(x₁)] - [11 - α(1 - S[1 - α])]] * [e(x₁)]
This gives us the marginal distribution of X₁.
ii) E(X₁X₂):
To find E(X₁X₂), we need to calculate the expected value of the product X₁X₂ using the joint pdf f(x₁, x₂).
E(X₁X₂) = ∫∫ x₁x₂ * f(x₁, x₂) dx₁ dx₂
= ∫∫ x₁x₂ * [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂)))] * e(x₁ - x₂) dx₁ dx₂
To calculate this integral, we need to define the ranges of integration for x₁ and x₂. Since the given pdf is defined for 0 ≤ x₁, 0 ≤ x₂, we integrate over these ranges.
E(X₁X₂) = ∫[0,∞) ∫[0,∞) x₁x₂ * [11 - α(1 - S[1 - α(1 - 2e(-x₁))(1 - 2e(-x₂)))] * e(x₁ - x₂) dx₁ dx₂
Performing the integration over the ranges, we can evaluate the expected value E(X₁X₂).
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Evaluate the integral (x² – 2y²) dA, where R is the first quadrant region - between the circles of radius 1 and radius 2 centred at the origin. R(x² – 2y²) dA =
The value of the integral (x² – 2y²) dA over the region R, which is the first quadrant region between the circles of radius 1 and radius 2 centered at the origin, can be evaluated as 2π/3.
To evaluate the given integral, we can convert it to polar coordinates since the region R is defined in terms of circles centered at the origin. In polar coordinates, the region R can be represented as 0 ≤ r ≤ 2 and 0 ≤ θ ≤ π/2.
Converting the integral to polar coordinates, we have: R(x² – 2y²) dA = R[(r²cos²θ) – 2(r²sin²θ)] r dr dθ
Simplifying the expression inside the integral, we get: R[(r²cos²θ) – 2(r²sin²θ)] r dr dθ = R(r²cos²θ – 2r²sin²θ) r dr dθ
Expanding further, we have: R(r⁴cos²θ – 2r⁴sin²θ) dr dθ
Integrating with respect to r from 0 to 2 and with respect to θ from 0 to π/2, we evaluate the integral and obtain the result as 2π/3.
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Calculate Dz In Terms Of U And Y Using The Dv 2 X² + Y² Chain Nule, For I = Where X = E-Usinzi X+Y And Y=E E-4 COS2V
Using the given equations, X = e^(-U*sin(Z)) + Y and Y = e^(e^(-4*cos(2V))), and applying the chain rule, we can express dZ in terms of dU and dY as dZ = (-U*cos(Z)*e^(-U*sin(Z))) * dU + (-8*sin(2V)*e^(-4*cos(2V))*e^(e^(-4*cos(2V)))) * dY.
To calculate dZ in terms of dU and dY, we first differentiate the equations with respect to their respective variables. The derivative of X with respect to Z, denoted as dX/dZ, is obtained by applying the chain rule. Similarly, the derivative of Y with respect to V, denoted as dY/dV, is also computed.
Substituting these derivatives into the chain rule formula, we obtain the expression for dZ. By multiplying dU with the derivative of X with respect to Z and dY with the derivative of Y with respect to V, we can compute the respective contributions to the change in Z.Hence, the final expression for dZ in terms of dU and dY is given by dZ = (-U*cos(Z)*e^(-U*sin(Z))) * dU + (-8*sin(2V)*e^(-4*cos(2V))*e^(e^(-4*cos(2V)))) * dY. This expression allows us to determine how changes in U and Y affect the change in Z.
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Data was collected on the amount of time that a random sample of 8 students spent studying for a test and the grades they earned on the test. A scatter plot and line of fit were created for the data.
scatter plot titled students' data, with the x-axis labeled study time in hours and the y-axis labeled grade percent. Points are plotted at 1 comma 70, 2 comma 60, 2 comma 70, 2 comma 80, 3 comma 70, 3 comma 90, 4 comma 80, and 4 comma 88, and a line of fit drawn passing through the points 0 comma 60 and 2 comma 70
Determine the equation of the line of fit.
y = 5x + 60
y = 5x + 70
y = 10x + 60
y = 10x + 70
For the scattered plot, The equation of the line of fit is y = 5x + 60. Option A
How do we identify the best equation for the line of best fit?The equation for the line of best fit is often written in the form y = mx + b, wher m is the slope of the line and b is the y-intercept.
In scenaro presented, two points have been provided that the line of fit passes through, (0,60) and (2,70).
The slope (m) of the line can be determined by taking the difference in the y-values and dividing by the difference in the x-values, i.e., m = (70-60) / (2-0) = 10 / 2 = 5.
The y-intercept (b) is the value of y when x=0, which from the point (0,60), we can see is 60.
So the equation of the line of fit would be y = 5x + 60.
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British researchers recently added genes from snapdragon flowers to tomatoes to increase the tomatoes' levels of antioxidant pigments called anthocyanins. Tomatoes with the added genes ripened to an almost eggplant purple. The modified tomatoes produce levels of anthocyanin about on a par with blackberries,blueberries, and currants, which recent research has touted as miracle fruits. Because of the high cost and infrequent availability of such berries,tomatoes could be a better source of anthocyanins. Researchers fed mice bred to be prone to cancer one of two diets. The first group was fed standard rodent chow plus 10% tomato powder.The second group was fed standard rodent chow plus 10% powder from the genetically modified tomatoes.Below are the data for the life spans for the two groups. Data are in days. GroupI GroupII n 20 20 347 days 451 days 48 days 32days longer than the group receiving the unmodified tomato powder?
The group receiving the modified tomato powder lived longer than the group receiving the unmodified tomato powder. However, more research is needed to understand the impact of consuming genetically modified foods on human health and the environment.
The researchers added genes from snapdragon flowers to tomatoes to increase the tomatoes' levels of antioxidant pigments called anthocyanins
.Tomatoes with the added genes ripened to an almost eggplant purple.
The modified tomatoes produce levels of anthocyanin about on a par with blackberries, blueberries, and currants, which recent research has touted as miracle fruits
.Researchers fed mice bred to be prone to cancer one of two diets.
The first group was fed standard rodent chow plus 10% tomato powder.The second group was fed standard rodent chow plus 10% powder from the genetically modified tomatoes.
The group receiving the modified tomato powder lived longer than the group receiving the unmodified tomato powder.
Group I
n = 20,
mean = 347,
SD = 48.
Group II
n = 20,
mean = 451,
SD = 32.
Group II is longer than Group I by (451 - 347) = 104 days. The data imply that the modified tomato powder lengthened the lifespan of the mice. However, more research is needed to understand the impact of consuming genetically modified foods on human health and the environment.
The group receiving the modified tomato powder lived longer than the group receiving the unmodified tomato powder. However, more research is needed to understand the impact of consuming genetically modified foods on human health and the environment.
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c. Last week April worked 44 hours. She is paid $11.20 per hour for a regular workweek of 40 hours and overtime at time and one-half regular pay. i. What were April's gross wages for last week? ii. What is the amount of the overtime premium
i) April's gross wages for last week were $515.20.
ii) The overtime premium is $67.20.
To calculate April's gross wages for last week, we need to consider the regular pay for 40 hours and the overtime pay for the additional hours worked.
i. Gross wages for last week:
Regular pay = 40 hours * $11.20 per hour = $448
Overtime pay:
April worked 44 hours in total, which means she worked 4 hours of overtime (44 - 40).
Overtime rate = 1.5 * regular pay rate = 1.5 * $11.20 = $16.80 per hour
Overtime pay = 4 hours * $16.80 per hour = $67.20
Total gross wages = Regular pay + Overtime pay = $448 + $67.20 = $515.20
Therefore, April's gross wages for last week were $515.20.
ii. Overtime premium:
The overtime premium refers to the additional amount paid for the overtime hours worked.
Overtime premium = Overtime pay - Regular pay = $67.20 - $448 = -$380.80
However, since the overtime premium is typically considered a positive value, we can interpret it as the additional amount earned for the overtime hours.
Therefore, the overtime premium is $67.20.
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f(x+h)-f(x), for h*0. 3. (10pt) Let f(x)=8x²-5x. Compute and simplify 4. (10pt) For the polynomial f(x)=x' +9x² +18x-10, find all roots algebraically, in simplest radical form.
The given functions and expressions are: f(x) = 8x² - 5xf(x + h) = 8(x + h)² - 5(x + h). The roots of the polynomial function are: x = -2, (-7 + √69) / 2, (-7 - √69) / 2.
For the polynomial function f(x) = x³ + 9x² + 18x - 10, we need to find all its roots algebraically, in the simplest radical form. We start by finding its possible rational roots using the Rational Root Theorem. The factors of the constant term (-10) are ±1, ±2, ±5, ±10, and the factors of the leading coefficient (1) are ±1.
Hence, its possible rational roots are ±1, ±2, ±5, ±10. Next, we perform synthetic division with each of the possible rational roots until we find one that results in a zero remainder. We obtain the following result with
x = -2:x³ + 9x² + 18x - 10
= (x + 2)(x² + 7x - 5)
We continue by finding the roots of the quadratic factor x² + 7x - 5 using the quadratic formula: x = [tex](-7 ± √(7² + 4(1)(5))) / 2x = (-7 ± √69) / 2[/tex]
Hence, the roots of the polynomial function are: [tex]x = -2, (-7 + √69) / 2, (-7 - √69) / 2.[/tex]
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please write neatly! thank
you!
Evaluate the integral using the methods of trig integrals. (5 pts) 5. f cos5 x dx
The integral of 5cos(5x)dx using trigonometric integrals is equal to sin(5x) + C, where C is the constant of integration.
To evaluate the integral ∫5cos(5x)dx using trigonometric integrals,
we can use the following trigonometric identity,
∫cos(ax)dx = (1/a)sin(ax) + C
Here value of a is equal to 5.
Applying this identity to our integral, we have,
∫5cos(5x)dx
= (5/5)sin(5x) + C
= sin(5x) + C
where C is the constant of integration.
Therefore, the integral of 5cos(5x)dx is sin(5x) + C, where C is the constant of integration.
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The given question is incomplete, I answer the question in general according to my knowledge:
Evaluate the integral using the methods of trig integrals.
∫5cos5 x dx
please help with this . Question 5Evaluate the following limit:3+h13limh-0hO Does not existO-1/3O-1/9< Previous
Quiz Instructions
D
Question 6
Evaluate the following limit:
lim
2-3 22
-2-6
00
09
• Previous
C
G Search or
The limit of \frac{3 + h}{1 - 3h} as h approaches 0 exists and is equal to 3. Hence, the correct option is (B) -\frac13.
Given, $\lim_{h \to 0} \frac{3 + h}{1 - 3h}
Let, $f(x) = \frac{3 + h}{1 - 3h}.
Then,
f(x) = \frac{3 + h}{1 - 3h}
= \frac{(3 + h)}{(1 - 3h)} \times \frac{(1 + 3h)}{(1 + 3h)}
= \frac{(3 + h)(1 + 3h)}{(1 - 9h^2)}
= \frac{3 + 9h + h + 3h^2}{1 - 9h^2}
= \frac{3h^2 + 10h + 3}{1 - 9h^2}
Now, putting h = 0, we get,
f(0) = \frac{3 \times 0^2 + 10 \times 0 + 3}{1 - 9 \times 0^2} = 3
Therefore, the limit of \frac{3 + h}{1 - 3h} as h approaches 0 exists and is equal to 3.
Hence, the correct option is (B) -\frac13.
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"Internet Traffic" includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μμ, x, and e that would be used in that formula? INTERNET ARRIVALS For the random variable x described in Exercise 1, what are the possible values of x? Is the value of x=4.8x=4.8 possible? Is x a discrete random variable or a continuous random variable?
The values of μ, x, and e that would be used to find the probability of exactly 2 arrivals in one thousandth of a minute are: 0.4697, 2 and 2.71828 respectively.
x cannot be 4.8 since it should be a non-negative integer according to the definition of the random variable x. In this case, x is a discrete random variable.
Probability is a measure or quantification of the likelihood or chance of an event occurring. It is a fundamental concept in statistics and probability theory, widely used to analyze and predict outcomes in various fields, including mathematics, science, economics, and everyday decision-making.
In the given scenario, the random variable x represents the number of Internet traffic arrivals in one thousandth of a minute, and it follows a Poisson distribution.
To use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, we need to identify the values of μ (mu), x, and e that are used in the formula.
In the context of a Poisson distribution, the parameter μ (mu) represents the average rate of arrivals per unit of time. In this case, since 9000 arrivals occurred over a period of 19,130 thousandths of a minute, we can calculate μ as follows:
μ = (Number of arrivals) / (Time period)
= 9000 / 19,130
= 0.4697
So, μ ≈ 0.4697.
Now, we want to find the probability of exactly 2 arrivals in one thousandth of a minute. Therefore, x = 2.
Formula 5-9 for the Poisson distribution is:
P(x) = (e^(-μ) * μ^x) / x!
In this case, the values to be used in the formula are:
μ ≈ 0.4697
x = 2
e ≈ 2.71828 (the base of the natural logarithm)
Now, let's address the additional questions:
Possible values of x: The possible values of x in this case are non-negative integers (0, 1, 2, 3, ...). Since x represents the number of Internet traffic arrivals, it cannot take on fractional or negative values.
Is x = 4.8 possible? No, x cannot be 4.8 since it should be a non-negative integer according to the definition of the random variable x.
Is x a discrete or continuous random variable? In this case, x is a discrete random variable because it can only take on a countable set of distinct values (non-negative integers) rather than a continuous range of values.
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When the What-if analysis uses the average values of variables, then it is based on: O The base-case scenario and best-case scenario. The base-case scenario and worse-case scenario. The worst-case scenario and best-case scenario. The base-case scenario only.
When the what-if analysis uses the average values of variables, then it is based on the base-case scenario only.
What-if analysis refers to the process of evaluating how different outcomes could have been influenced by different decisions in hindsight. In a model designed to determine the optimal quantity of inventory to order, what-if analysis can be done to evaluate how the total cost of inventory changes as different decisions are made concerning inventory levels.
This analysis method usually requires the creation of a hypothetical model and testing it by changing specific variables.
The results of the analysis are then observed to determine how the changes affected the overall outcome. The base-case scenario represents the likely outcome of a business decision in the absence of change, whereas the worst-case scenario represents the potential for the most disastrous outcome
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the last four months of sales were 8, 9, 12, and 9 units. the last four forecasts were 5, 6, 11, and 12 units. the mean absolute deviation (mad) is
The Mean Absolute Deviation (MAD) is 3.5.
What is the mean absolute deviation (mad)?The mean absolute deviation is designed to provide a measure of overall forecast error for the model. It does this by taking the sum of the absolute values of the individual forecast errors and dividing by the number of data periods.
The last four months sales were 8, 10, 15, and 9 units. The forecasts for these same months were 5, 6, 11, and 12 units.
Forecast errors are calculated using the equation demand - forecast.
In this case, that would be:
8 - 5 = 3;10 - 6 = 4;15 - 11 = 4;9 - 12 = -3.Therefore:
= 3+4+4+3 = 14
= 14/4
= 3.5.
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Misprints on Manuscript Pages In a 530-page manuscript, there are 250 randomly distributed misprints. Use the Poisson approximation. Part: 0/2 Part 1 of 2 Find the mean number 2 of misprints per page. Round to one decimal place as needed. λ=
The mean number 2 of misprints per page is 0.5
In a 530-page manuscript, there are 250 randomly distributed misprints.
We have to find the mean number 2 of misprints per page.
We will use the Poisson approximation formula to find the answer.
The formula is given below: `λ = (number of events/number of opportunities for an event to occur)
Find the mean number 2 of misprints per page.
We can use the above formula to calculate λ as follows:
λ=`(250/530)`= `0.4716981132`
Now, we can round this answer to one decimal place as per the requirement.
Therefore, the mean number of misprints per page is 0.5 (rounded to one decimal place)
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Let n = p1p2 .... pk where the pi are distinct primes. Show that µ(d) = (−1)^k µ (n/d)
The statement µ(d) = (−1)^k µ (n/d) relates to the Möbius function µ(d) and the prime factorization of an integer n. The Möbius function is a number-theoretic function that takes the value -1 if d is a square-free positive integer with an even number of prime factors, 0 if d is not square-free, and +1 if d is a square-free positive integer with an odd number of prime factors.
The prime factorization of n is given as n = p1p2....pk, where p1, p2, ..., pk are distinct prime numbers. The exponent of each prime pi in the factorization determines whether the number is square-free or not. If the exponent is even, the number is not square-free, and if the exponent is odd, the number is square-free.
The statement µ(d) = (−1)^k µ (n/d) can be proven by considering the cases where d is square-free and not square-free. If d is square-free, it means that the exponents of the prime factors in d are either 0 or 1. In this case, the Möbius function µ(d) will have the same value as µ(n/d), since the exponents cancel out.
On the other hand, if d is not square-free, it means that at least one of the exponents in d is greater than 1. In this case, both µ(d) and µ(n/d) will be equal to 0, as d is not a square-free positive integer.
Therefore, the statement µ(d) = (−1)^k µ (n/d) holds true, as it correctly reflects the relationship between the Möbius function and the prime factorization of an integer n. The exponent k in the equation represents the number of distinct prime factors in n.
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true or false?
Let f(x)=1+x² €Z3[x], then the extension field E=Z3[x]/(f(x)) of Z3 has 8 elements. 4
The statement is false. The extension field E=Z3[x]/(f(x)) of Z3, where f(x) = 1 + x² ∈ Z3[x], does not have 8 elements. The correct statement is that the extension field E=Z3[x]/(f(x)) of Z3 has 9 elements, not 8.
1.) To determine the number of elements in E, we need to consider the degree of the polynomial f(x). In this case, the degree of f(x) is 2. Since we are working with a finite field Z3, the extension field E will have 3² = 9 elements.
2.) The elements of E can be represented as polynomials of degree less than 2 with coefficients in Z3. However, it's important to note that not all polynomials of degree less than 2 will be distinct elements in E. The elements will be equivalence classes of polynomials modulo f(x) = 1 + x².
3.) Therefore, the correct statement is that the extension field E=Z3[x]/(f(x)) of Z3 has 9 elements, not 8.
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Use the given tormation to find the number of degrees of troom, the once values and you and the confidence interval ontmate of His manorable to astume that a simple random tampis has been selected from a population with a normal distribution.
Nicotene in menthol cigaretes 95% confidence, n=21 s=0,21mg
The calculated number of degrees of freedom is 20
How to calculate the number of degrees of freedomFrom the question, we have the following parameters that can be used in our computation:
95% confidence, n = 21 s = 0.21 mg
The number of degrees of freedom is calculated as
df = n - 1
substitute the known values in the above equation, so, we have the following representation
df = 21 - 1
Evaluate
df = 20
Hence, the number of degrees of freedom is 20
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