Z = (23.8 - 25.505) / 3.25987972
To compute the Z-scores, we will use the formula:
Z = (X - μ) / σ
where:
X = individual data point (points per game)
μ = population mean (mean points per game)
σ = population standard deviation (sample standard deviation)
Given the mean (μ) of 25.505 and the sample standard deviation (σ) of 3.25987972, we can compute the Z-scores for the following players:
a) Westbrook, Russell: X = 22.9
Z = (22.9 - 25.505) / 3.25987972
b) Durant, Kevin: X = 26
Z = (26 - 25.505) / 3.25987972
c) Harden, James: X = 36.1
Z = (36.1 - 25.505) / 3.25987972
d) Irving, Kyrie: X = 23.8
Z = (23.8 - 25.505) / 3.25987972
To compute the Z-scores for each player, substitute the respective X values into the formula and calculate the result.
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Question 6 Assume the experiment is to roll a 6-sided die 4 times. a. The probability that all 4 rolls come up with a 6. b. The probability you get at least one roll that is not a 6 is (4 decimal places) 6 pts (4 decimal places)
The probability of getting at least one roll that is not a 6 is given by:
which is approximately 0.9988 (rounded to 4 decimal places).
a. The probability that all 4 rolls come up with a 6 is (1/6)4 = (1/1296) which is approximately 0.0008.
b. The probability you get at least one roll that is not a 6 is 1 - probability of getting all 4 rolls as 6 which is 1 - (1/1296) = 1295/1296, which is approximately 0.9988 (rounded to 4 decimal places).
Explanation:
Given that the experiment is to roll a 6-sided die 4 times.There are 6 equally likely outcomes for each roll, i.e. 1, 2, 3, 4, 5, or 6.
The probability that all 4 rolls come up with a 6 is obtained as follows:
P(rolling a 6 on the first roll) = 1/6P(rolling a 6 on the second roll) = 1/6P(rolling a 6 on the third roll) = 1/6P(rolling a 6 on the fourth roll)
= 1/6
The probability of getting all 4 rolls as 6 is the product of the probabilities of getting a 6 on each roll, i.e.P(getting all 4 rolls as 6) = (1/6)4 = 1/1296
Therefore, the probability that all 4 rolls come up with a 6 is 1/1296, which is approximately 0.0008.
To find the probability that at least one roll is not a 6, we use the complement rule which states that:
P(event A does not occur) = 1 - P(event A occurs P(getting at least one roll that is not a 6) = 1 - P(getting all 4 rolls as 6) = 1 - 1/1296 = 1295/1296,
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3. Calculating the mean when adding or subtracting a constant A professor gives a statistics exam. The exam has 50 possible points. The s 42 40 38 26 42 46 42 50 44 Calculate the sample size, n, and t
The sample consists of 9 exam scores: 42, 40, 38, 26, 42, 46, 42, 50, and 44. The mean when adding or subtracting a constant A professor gives a statistics exam is √44.1115 ≈ 6.6419
To calculate the sample size, n, and t, we need to follow the steps below:
Find the sum of the scores:
42 + 40 + 38 + 26 + 42 + 46 + 42 + 50 + 44 = 370
Calculate the sample size, n, which is the number of scores in the sample:
n = 9
Calculate the mean, μ, by dividing the sum of the scores by the sample size:
μ = 370 / 9 = 41.11 (rounded to two decimal places)
Calculate the deviations of each score from the mean:
42 - 41.11 = 0.89
40 - 41.11 = -1.11
38 - 41.11 = -3.11
26 - 41.11 = -15.11
42 - 41.11 = 0.89
46 - 41.11 = 4.89
42 - 41.11 = 0.89
50 - 41.11 = 8.89
44 - 41.11 = 2.89
Square each deviation:
[tex](0.89)^2[/tex] = 0.7921
[tex](-1.11)^2[/tex] = 1.2321
[tex](-3.11)^2[/tex] = 9.6721
[tex](-15.11)^2[/tex] = 228.6721
[tex](0.89)^2[/tex] = 0.7921
[tex](4.89)^2[/tex] = 23.8761
[tex](0.89)^2[/tex] = 0.7921
[tex](8.89)^2[/tex] = 78.9121
[tex](2.89)^2[/tex] = 8.3521
Find the sum of the squared deviations:
0.7921 + 1.2321 + 9.6721 + 228.6721 + 0.7921 + 23.8761 + 0.7921 + 78.9121 + 8.3521 = 352.8918
Calculate the sample variance, [tex]s^2[/tex], by dividing the sum of squared deviations by (n-1):
[tex]s^2[/tex] = 352.8918 / (9 - 1) = 44.1115 (rounded to four decimal places)
Calculate the sample standard deviation, s, by taking the square root of the sample variance:
s = √44.1115 ≈ 6.6419 (rounded to four decimal places)
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I think it's c but not sure
Given the following function and the transformations that are taking place, choose the most appropriate statement below regarding the graph of f(x) = 5 sin[2 (x - 1)] +4 Of(x) has an Amplitude of 5. a
The function can be graphed by first identifying the midline, which is the vertical shift of 4 units up from the x-axis, and then plotting points based on the amplitude and period of the function.
The amplitude of the function f(x) = 5 sin[2 (x - 1)] + 4 is 5.
This is because the amplitude of a function is the absolute value of the coefficient of the trigonometric function.
Here, the coefficient of the sine function is 5, and the absolute value of 5 is 5.
The transformation that is taking place in this function is a vertical shift up of 4 units.
Therefore, the appropriate statement regarding the graph of the function is that it has an amplitude of 5 and a vertical shift up of 4 units.
The function can be graphed by first identifying the midline, which is the vertical shift of 4 units up from the x-axis, and then plotting points based on the amplitude and period of the function.
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Given f(x)=x^2-6x+8 and g(x)=x^2-x-12, find the y intercept of (g/f)(x)
a. 0
b. -2/3
c. -3/2
d. -1/2
The y-intercept of [tex]\((g/f)(x)\)[/tex]is (c) -3/2.
What is the y-intercept of the quotient function (g/f)(x)?To find the y-intercept of ((g/f)(x)), we first need to determine the expression for this quotient function.
Given the functions [tex]\(f(x) = x^2 - 6x + 8\)[/tex] and [tex]\(g(x) = x^2 - x - 12\)[/tex] , the quotient function [tex]\((g/f)(x)\)[/tex]can be written as [tex]\(\frac{g(x)}{f(x)}\).[/tex]
To find the y-intercept of ((g/f)(x)), we need to evaluate the function at (x = 0) and determine the corresponding y-value.
First, let's find the expression for ((g/f)(x)):
[tex]\((g/f)(x) = \frac{g(x)}{f(x)}\)[/tex]
[tex]\(f(x) = x^2 - 6x + 8\) and \(g(x) = x^2 - x - 12\)[/tex]
Now, let's substitute (x = 0) into (g(x)) and (f(x)) to find the y-intercept.
For [tex]\(g(x)\):[/tex]
[tex]\(g(0) = (0)^2 - (0) - 12 = -12\)[/tex]
For (f(x)):
[tex]\(f(0) = (0)^2 - 6(0) + 8 = 8\)[/tex]
Finally, we can find the y-intercept of ((g/f)(x)) by dividing the y-intercept of (g(x)) by the y-intercept of (f(x)):
[tex]\((g/f)(0) = \frac{g(0)}{f(0)} = \frac{-12}{8} = -\frac{3}{2}\)[/tex]
Therefore, the y-intercept of [tex]\((g/f)(x)\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex], which corresponds to option (c).
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If you are testing hypotheses and you find p-value which gives you an acceptance of the alternative hypotheses for a 1% significance level, then all other things being the same you would also get an acceptance of the alternative hypothesis for a 5% significance level.
True
False
The statement give '' If you are testing hypotheses and you find p-value which gives you an acceptance of the alternative hypotheses for a 1% significance level, then all other things being the same you would also get an acceptance of the alternative hypothesis for a 5% significance level '' is False.
The significance level, also known as the alpha level, is the threshold at which we reject the null hypothesis. A lower significance level indicates a stricter criteria for rejecting the null hypothesis.
If we find a p-value that leads to accepting the alternative hypothesis at a 1% significance level, it does not necessarily mean that we will also accept the alternative hypothesis at a 5% significance level.
If the p-value is below the 1% significance level, it means that the observed data is very unlikely to have occurred by chance under the null hypothesis. However, this does not automatically imply that it will also be unlikely under the 5% significance level.
Accepting the alternative hypothesis at a 1% significance level does not guarantee acceptance at a 5% significance level. The decision to accept or reject the alternative hypothesis depends on the specific p-value and the chosen significance level.
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HELPP Write the equation of the given line in slope-intercept form:
Answer:
y = -3x - 1
Step-by-step explanation:
The slope-intercept form is y = mx + b
m = the slope
b = y-intercept
Slope = rise/run or (y2 - y1) / (x2 - x1)
Point (-1, 2) (1, -4)
We see the y decrease by 6 and the x increase by 2, so the slope is
m = -6 / 2 = -3
Y-intercept is located at (0, - 1)
So, the equation is y = -3x - 1
Consider the given density curve.
A density curve is at y = one-third and goes from 3 to 6.
What is the value of the median?
a. 3
b. 4
c. 4.5
d. 6
The median value in this case is:(3 + 6) / 2 = 4.5 Therefore, the correct answer is option (c) 4.5.
We are given a density curve at y = one-third and it goes from 3 to 6.
We have to find the median value, which is also known as the 50th percentile of the distribution.
The median is the value separating the higher half from the lower half of a data sample. The median is the value that splits the area under the curve exactly in half.
That means the area to the left of the median equals the area to the right of the median.
For a uniform density curve, like we have here, the median value is simply the average of the two endpoints of the curve.
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find the volume of the solid whose base is bounded by the circle x^2 y^2=4
the volume of the solid whose base is bounded by the circle x²y² = 4 is 0.
The equation of a circle in the coordinate plane can be written as(x - a)² + (y - b)² = r², where the center of the circle is (a, b) and the radius is r.
The equation x²y² = 4 can be rewritten as:y² = 4/x².
Therefore, the graph of x²y² = 4 is the graph of the following two functions:
y = 2/x and y = -2/x.
The line connecting the points where y = 2/x and y = -2/x is the x-axis.
We can use the washer method to find the volume of the solid obtained by rotating the area bounded by the graph of y = 2/x, y = -2/x, and the x-axis around the x-axis.
The volume of the solid is given by the integral ∫(from -2 to 2) π(2/x)² - π(2/x)² dx
= ∫(from -2 to 2) 4π/x² dx
= 4π∫(from -2 to 2) x⁻² dx
= 4π[(-x⁻¹)/1] (from -2 to 2)
= 4π(-0.5 + 0.5)
= 4π(0)
= 0.
Therefore, the volume of the solid whose base is bounded by the circle x²y² = 4 is 0.
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Given that the sum of squares for error (SSE) for an ANOVA F-test is 12,000 and there are 40 total experimental units with eight total treatments, find the mean square for error (MSE).
To ensure that all the relevant information is included in the answer, the following explanations will be given.
There are different types of ANOVA such as one-way ANOVA and two-way ANOVA. These ANOVA types are determined by the number of factors or independent variables. One-way ANOVA involves a single factor and can be used to test the hypothesis that the means of two or more populations are equal. On the other hand, two-way ANOVA involves two factors and can be used to test the effects of two factors on the population means. In the question above, the type of ANOVA used is not given.
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The equation, with a restriction on x, is the terminal side of an angle 8 in standard position. -4x+y=0, x20 www. Give the exact values of the six trigonometric functions of 0. Select the correct choi
The values of the six trigonometric functions of θ are:
Sin θ = 4/√17Cos θ = √5Cot θ = 1/4Tan θ = 1/5Cosec θ = √17/4Sec θ = √(17/5)
Therefore, the correct answer is option A.
Given, the equation with a restriction on x is the terminal side of an angle 8 in standard position.
The equation is -4x+y=0 and x≥20.
The given equation is -4x+y=0 and x≥20
We need to find the trigonometric ratios of θ.
So, Let's first find the coordinates of the point which is on the terminal side of angle θ. For this, let's solve the given equation for y.
-4x+y=0y= 4x
We know that the equation x=20 is a vertical line at 20 on x-axis.
Therefore, we can say that the coordinates of point P on terminal side of angle θ will be (20,80)
Substituting these values into trigonometric functions we get the following:
Sin θ = y/r
= 4x/√(x²+y²)= 4x/√(x²+(4x)²)
= 4x/√(17x²) = 4/√17Cos θ
= x/r = x/√(x²+y²)= 20/√(20²+(4·20)²)
= 20/√(400+1600)
= 20/√2000 = √5Cot θ
= x/y = x/4x
= 1/4Tan θ = y/x
= 4x/20
= 1/5Cosec θ
= r/y = √(x²+y²)/4x
= √(17x²)/4x = √17/4Sec θ
= r/x
= √(x²+y²)/x= √(17x²)/x
= √17/√5 = √(17/5)
The values of the six trigonometric functions of θ are:
Sin θ = 4/√17
Cos θ = √5
Cot θ = 1/4
Tan θ = 1/5
Cosec θ = √17/4
Sec θ = √(17/5)
Therefore, the correct answer is option A.
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test the series for convergence or divergence using the alternating series test. [infinity] (−1)n 7nn n! n = 1
The given series is as follows:[infinity] (−1)n 7nn n! n = 1We need to determine if the series is convergent or divergent by using the Alternating Series Test. The Alternating Series Test states that if the terms of a series alternate in sign and are decreasing in absolute value, then the series is convergent.
The sum of the series is the limit of the sequence formed by the partial sums.The given series is alternating since the sign of the terms changes in each step. So, we can apply the alternating series test.Now, let’s calculate the absolute value of the series:[infinity] |(−1)n 7nn n!| n = 1Since the terms of the given series are always positive, we don’t need to worry about the absolute values. Thus, we can apply the alternating series test.
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A triangular pyramid is pictured below. Select the type of cross-section formed when the figure is cut by a plane containing its altitude and perpendicular to its base.
a. Triangle
b. Rectangle
c. Hexagon
d. Circle
The figure is cut by a plane containing its altitude and perpendicular to its base, the cross-section formed is (A) Triangle.
Which geometric shape is formed by the cross-section?When a triangular pyramid is cut by a plane containing its altitude and perpendicular to its base, the resulting cross-section will be a triangle.
To understand why, let's visualize the pyramid. A triangular pyramid has a base that is a triangle and three triangular faces that converge at a single point called the apex.
The altitude of the pyramid is a line segment that connects the apex to the base, perpendicular to the base.
When we cut the pyramid with a plane containing its altitude and perpendicular to its base, the plane will intersect the pyramid along its height.
This means that the resulting cross-section will be a slice that is perpendicular to the base and parallel to the other two triangular faces.
Since the base of the pyramid is a triangle, and the plane cuts through it perpendicularly, the resulting cross-section will also be a triangle.
The shape of the cross-section will be similar to the base triangle of the pyramid, with the same number of sides and angles.
Therefore, the correct answer is a. Triangle.
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please solve
If P(A) = 0.2, P(B) = 0.3, and P(AUB) = 0.47, then P(An B) = (a) Are events A and B independent? (enter YES or NO) (b) Are A and B mutually exclusive? (enter YES or NO)
a) Are events A and B independent? (enter YES or NO)To find if the events A and B are independent or not we need to check the condition of independence of events.
The formula for independent events is given as follows:[tex]P(A ∩ B) = P(A) × P(B)If the value of P(A ∩ B) = P(A) × P(B)[/tex] holds, the events are independent.
So, we have [tex]P(A) = 0.2, P(B) = 0.3,[/tex] and [tex]P(AUB) = 0.47[/tex]
Now, [tex]P(AUB) = P(A) + P(B) - P(A ∩ B)0.47 = 0.2 + 0.3 - P(A ∩ B)P(A ∩ B) = 0.03[/tex]As the value of [tex]P(A ∩ B[/tex]) is not equal to P(A) × P(B), events A and B are not independent.b) Are A and B mutually exclusive? (enter YES or NO)The events A and B are mutually exclusive if their intersection is null set.
We can say that if events A and B are mutually exclusive, then [tex]P(A ∩ B) = 0[/tex].
So, we have [tex]P(A ∩ B) = 0.03[/tex]
As the value of[tex]P(A ∩ B)[/tex] is not equal to 0, events A and B are not mutually exclusive.Conclusion:
We can say that events A and B are not independent as their intersection is not equal to the product of their probabilities. Similarly, we can say that events A and B are not mutually exclusive as their intersection is not equal to the null set.
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Which of the following is a required condition for a discrete
probability function?
Σf(x) < 0 for all values of x
f(x) ≤ 0 for all values of x
Σf(x) > 1 for all values of x
f(x) ≥ 0 for al
The answer is f(x) ≥ 0 for all values of x.
The required condition for a discrete probability function is that f(x) ≥ 0 for all values of x. A discrete probability function is one that assigns each point in the range of X a probability. This is defined by the probability mass function, which is abbreviated as pmf. The probability of x can be calculated using the following formula: P(X = x) = f(x), where X is a random variable. If a function is a discrete probability function, then it must follow a few important rules. One of those rules is that f(x) ≥ 0 for all values of x. The rule f(x) ≥ 0 for all values of x is significant because it ensures that the function is non-negative. The probability of an event cannot be negative. The event has either occurred or not, and it cannot have occurred negatively. Therefore, it makes sense that the function that describes the probability of the event should also be non-negative. Any function that does not satisfy this condition is not a probability function.
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Given the values of the linear functions f (x) and g(x) in the tables, where is (f – g)(x) positive?
(–[infinity], –2)
(–[infinity], 4)
(–2, [infinity])
(4, [infinity])
x -8 -5 -2 1 4
f(x) -4 -6 -8 -10 -12
g(x) -14 -11 -8 -5 -2
The obtained values are where (f – g)(x) is above the x-axis, i.e., (f – g)(x) is positive.The interval where this occurs is (–2, [infinity]). The correct option is (–2, [infinity]).
Given the linear functions f (x) and g(x) in the tables, the solution to the expression (f – g)(x) is positive where x is in the interval (–2, [infinity]).
The table has the following values:
x -8 -5 -2 1 4
f(x) -4 -6 -8 -10 -12
g(x) -14 -11 -8 -5 -2
To find (f – g)(x), we have to subtract each element of g(x) from its corresponding element in f(x) and substitute the values of x.
Therefore, we have:(f – g)(x) = f(x) - g(x)
Now, we can complete the table for (f – g)(x):
x -8 -5 -2 1 4
f(x) -4 -6 -8 -10 -12
g(x) -14 -11 -8 -5 -2
(f – g)(x) 10 5 0 -5 -10
To find where (f – g)(x) is positive, we only need to look at the values of x such that (f – g)(x) > 0.
These values are where (f – g)(x) is above the x-axis, i.e., (f – g)(x) is positive.
The interval where this occurs is (–2, [infinity]).
Therefore, the correct option is (–2, [infinity]).
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This question is from Introduction to Multivariate
Methods
Question 1 a) Let x₁,x2,...,x,, be a random sample of size n from a p-dimensional normal distribution with known but Σ unknown. Show that i) the maximum likelihood estimator for E is 72 1 Σ = S Σ
The estimator is obtained by calculating the sample mean, which is given by (1/n) Σᵢ xᵢ, where n is the sample size and xᵢ represents the individual observations.
Let's denote the p-dimensional normal distribution as N(μ, Σ), where μ represents the mean vector and Σ represents the covariance matrix. Since we are interested in estimating E, the mean vector, we can rewrite it as μ = (E₁, E₂, ..., Eₚ).
The likelihood function, denoted by L(μ, Σ), is defined as the joint probability density function of the observed sample values x₁, x₂, ..., xₙ. Since the observations are independent and follow a p-dimensional normal distribution, the likelihood function can be written as:
L(μ, Σ) = f(x₁; μ, Σ) * f(x₂; μ, Σ) * ... * f(xₙ; μ, Σ)
where f(xᵢ; μ, Σ) represents the probability density function (pdf) of the p-dimensional normal distribution evaluated at xᵢ.
Since the sample values are assumed to be independent, the joint pdf can be expressed as the product of individual pdfs:
L(μ, Σ) = Πᵢ f(xᵢ; μ, Σ)
Taking the logarithm of both sides, we obtain:
log L(μ, Σ) = log(Πᵢ f(xᵢ; μ, Σ))
By using the properties of logarithms, we can simplify this expression:
log L(μ, Σ) = Σᵢ log f(xᵢ; μ, Σ)
Now, let's focus on the term log f(xᵢ; μ, Σ). For the p-dimensional normal distribution, the pdf can be written as:
f(xᵢ; μ, Σ) = (2π)⁻ᵖ/₂ |Σ|⁻¹/₂ exp[-½ (xᵢ - μ)ᵀ Σ⁻¹ (xᵢ - μ)]
Taking the logarithm of this expression, we have:
log f(xᵢ; μ, Σ) = -p/2 log(2π) - ½ log |Σ| - ½ (xᵢ - μ)ᵀ Σ⁻¹ (xᵢ - μ)
Substituting this expression back into the log-likelihood equation, we get:
log L(μ, Σ) = Σᵢ [-p/2 log(2π) - ½ log |Σ| - ½ (xᵢ - μ)ᵀ Σ⁻¹ (xᵢ - μ)]
To find the maximum likelihood estimator for E, we differentiate the log-likelihood function with respect to μ and set it equal to zero. Since we are differentiating with respect to μ, the term (xᵢ - μ)ᵀ Σ⁻¹ (xᵢ - μ) can be considered as a constant when taking the derivative.
∂(log L(μ, Σ))/∂μ = Σᵢ Σ⁻¹ (xᵢ - μ) = 0
Simplifying this equation, we obtain:
Σᵢ xᵢ - nμ = 0
Rearranging the terms, we have:
nμ = Σᵢ xᵢ
Finally, solving for μ, the maximum likelihood estimator for E is given by:
μ = (1/n) Σᵢ xᵢ
This estimator represents the sample mean of the random sample x₁, x₂, ..., xₙ and is also known as the sample average.
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En la función de la imagen la ecuación de la asíntota vertical es___
The equation for the asymptote of the graphed function is x = 7
How to identify the asymptote?The asymptote is a endlessly tendency to a given value. A vertical one is a tendency to infinity.
Here we can see that there is a vertical asymoptote, notice that in one end the function tends to positive infinity and in the other it tends to negative infinity.
The equation of the line where the asymptote is, is:
x = 7
So that is the answer.
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given the equation 4x^2 − 8x + 20 = 0, what are the values of h and k when the equation is written in vertex form a(x − h)^2 + k = 0? a. h = 4, k = −16 b. h = 4, k = −1 c. h = 1, k = −24 d. h = 1, k = 16
the values of h and k when the equation is written in vertex form a(x − h)^2 + k = 0 is (d) h = 1, k = 16.
To write the given quadratic equation [tex]4x^2 - 8x + 20 = 0[/tex] in vertex form, [tex]a(x - h)^2 + k = 0[/tex], we need to complete the square. The vertex form allows us to easily identify the vertex of the quadratic function.
First, let's factor out the common factor of 4 from the equation:
[tex]4(x^2 - 2x) + 20 = 0[/tex]
Next, we want to complete the square for the expression inside the parentheses, x^2 - 2x. To do this, we take half of the coefficient of x (-2), square it, and add it inside the parentheses. However, since we added an extra term inside the parentheses, we need to subtract it outside the parentheses to maintain the equality:
[tex]4(x^2 - 2x + (-2/2)^2) - 4(1)^2 + 20 = 0[/tex]
Simplifying further:
[tex]4(x^2 - 2x + 1) - 4 + 20 = 0[/tex]
[tex]4(x - 1)^2 + 16 = 0[/tex]
Comparing this to the vertex form, [tex]a(x - h)^2 + k[/tex], we can identify the values of h and k. The vertex form tells us that the vertex of the parabola is at the point (h, k).
From the equation, we can see that h = 1 and k = 16.
Therefore, the correct answer is (d) h = 1, k = 16.
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Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. (ETR) The indicated z score is (Round to two decimal places as needed.) 20 0.8238 O
The indicated z-score is 0.8238.
Given the graph depicting the standard normal distribution with a mean of 0 and standard deviation of 1. The formula for calculating the z-score is z = (x - μ)/ σwherez = z-score x = raw scoreμ = meanσ = standard deviation Now, we are to find the indicated z-score which is 0.8238. Hence we can write0.8238 = (x - 0)/1. Therefore x = 0.8238 × 1= 0.8238
The Normal Distribution, often known as the Gaussian Distribution, is the most important continuous probability distribution in probability theory and statistics. It is also referred to as a bell curve on occasion. In every physical science and in economics, a huge number of random variables are either closely or precisely represented by the normal distribution. Additionally, it can be used to roughly represent various probability distributions, reinforcing the notion that the term "normal" refers to the most common distribution. The probability density function for a continuous random variable in a system defines the Normal Distribution.
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quadrilateral cdef is inscribed in circle a. quadrilateral cdef is inscribed in circle a. if m∠cfe = (2x 6)° and m∠cde = (2x − 2)°, what is the value of x? a. 22 b. 44 c. 46 d. 89
The value of x in quadrilateral cdef inscribed in circle is (b) 44.
What is the value of x in the given scenario?To find the value of x, we can use the property that opposite angles in an inscribed quadrilateral are supplementary (their measures add up to 180°).
Given that quadrilateral CDEF is inscribed in circle A, we have:
m∠CFE + m∠CDE = 180°
Substituting the given angle measures:
(2x + 6)° + (2x - 2)° = 180°
Combining like terms:
4x + 4 = 180
Subtracting 4 from both sides:
4x = 176
Dividing both sides by 4:
x = 44
Therefore, the value of x is 44.
The correct answer is:
b. 44
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3. A random sample of 149 scores for a university exam are given in the table. Score, x 0≤x≤ 20 20 < x≤ 40 40 < x≤ 60 60 < x≤ 80 80 < x≤ 100 21 Frequency 14 32 43 39 a. Find the unbiased e
The unbiased estimate of the population mean is 13.78.The unbiased estimate of the population mean can be found using the formula:
$\overline{x} = \frac{\sum{x}}{n}$,
where $\overline{x}$ is the sample mean,
$\sum{x}$ is the sum of the sample scores, and n is the sample size.
Here, we are given the frequency distribution of the sample scores, so we first need to calculate the midpoint for each class interval.
The midpoint is found by adding the lower and upper bounds of each class interval and dividing by 2.
Using this information, we can construct a table of the frequency distribution with the class midpoints as shown below.
Score, x
FrequencyMidpoint (x)014.5 (0+29)/22114.523.5 (20+39)/234032.5 (40+59)/246039.5 (60+79)/25390.5 (80+99)/2
We can then calculate the sample mean as:$$\overline{x}=\frac{\sum{x}}{n}$$$$=\frac{(14)(14.5)+(32)(23.5)+(43)(32.5)+(39)(39.5)+(21)(90.5)}{149}$$$$=\frac{2051.5}{149}$$$$=13.78$$
Therefore, the unbiased estimate of the population mean is 13.78.
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2 cos 0 = =, tan 8 < 0 Find the exact value of sin 6. 3 O A. - √5 √√5 OB. 2 √√5 oc. 3 D. 3/2 --
The correct option is (a). Given 2 cos 0 = =, tan 8 < 0, we need to find the exact value of sin 6.3.O. According to the given information: 2 cos 0 = = ⇒ cos 0 = 2/0, but cos 0 = 1 (as cos 0 = adjacent/hypotenuse and in a unit circle, adjacent side of angle 0 is 1 and hypotenuse is also 1).
Given 2 cos 0 = =, tan 8 < 0, we need to find the exact value of sin 6.3.O. According to the given information:
2 cos 0 = = ⇒ cos 0 = 2/0, but cos 0 = 1 (as cos 0 = adjacent/hypotenuse and in a unit circle, adjacent side of angle 0 is 1 and hypotenuse is also 1).
Hence 2 cos 0 = 2 * 1 = 2tan 8 < 0 ⇒ angle 8 lies in 2nd quadrant where tan is negative. Here's the working to find the value of sin 6: We know that tan θ = opposite/adjacent where θ is the angle, then opposite = tan θ × adjacent......
(1) Since angle 8 lies in 2nd quadrant, we take the adjacent side as negative. So, we get the hypotenuse and opposite as follows:
adjacent = -1, tan 8 = opposite/adjacent ⇒ opposite = tan 8 × adjacent ⇒ opposite = tan 8 × (-1) = -tan 8Hypotenuse = √(adjacent² + opposite²) ⇒ Hypotenuse = √(1 + tan² 8) = √(1 + 16) = √17
So, the value of sin 6 can be obtained using the formula for sin θ = opposite/hypotenuse where θ is the angle. Hence, sin 6 = opposite/hypotenuse = (-tan 8)/√17
Exact value of sin 6 = - tan 8/ √17
Answer: Option A: - √5
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Suppose grades of an exam is normally distributed with the mean of 65 and standard deviation of 10. If a student's grade is randomly selected, what is the probability that the grades is
a. between 70 and 90?
b. at least 70?
c. at most 70?
a. The probability that the grade is between 70 and 90 is 0.3023.
b. The probability that the grade is at least 70 is 0.3085.
c. The probability that the grade is at most 70 is 0.1915.
Suppose grades of an exam are normally distributed with a mean of 65 and a standard deviation of 10. If a student's grade is randomly selected, then the probability that the grade is a. between 70 and 90, b. at least 70, and c. at most 70 is given by;
Probability that the grade is between 70 and 90
We can find this probability by standardizing the given values of X = 70 and X = 90 to Z-scores.
The formula for standardizing a normal variable X is given by;Z-score (Z) = (X - µ) / σ
Where µ = mean of the distribution and σ = standard deviation of the distribution.
For X = 70,Z = (X - µ) / σ = (70 - 65) / 10 = 0.5
For X = 90,Z = (X - µ) / σ = (90 - 65) / 10 = 2.5
Using the Z-table, we find the probability as;P(0.5 ≤ Z ≤ 2.5) = P(Z ≤ 2.5) - P(Z ≤ 0.5) = 0.9938 - 0.6915 = 0.3023
b. Probability that the grade is at least 70
To find this probability, we can standardize X = 70 and find the area to the right of the standardized value, Z.
Using the formula for Z-score,Z = (X - µ) / σ = (70 - 65) / 10 = 0.5
Using the Z-table, we can find the area to the right of Z = 0.5 as 0.3085
c. Probability that the grade is at most 70
To find this probability, we can standardize X = 70 and find the area to the left of the standardized value, Z.Using the formula for Z-score,
Z = (X - µ) / σ = (70 - 65) / 10 = 0.5
Using the Z-table, we can find the area to the left of Z = 0.5 as 0.1915
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Find the measure(s) of angle θ given that (cosθ-1)(sinθ+1)= 0,
and 0≤θ≤2π. Give exact answers and show all of your work.
The measure of angle θ is 90° and 450° (in degrees) or π/2 and 5π/2 (in radians).
Given that (cos θ - 1) (sin θ + 1) = 0 and 0 ≤ θ ≤ 2π, we need to find the measure of angle θ. We can solve it as follows:
Step 1: Multiplying the terms(cos θ - 1) (sin θ + 1)
= 0cos θ sin θ - cos θ + sin θ - 1
= 0cos θ sin θ - cos θ + sin θ
= 1cos θ(sin θ - 1) + 1(sin θ - 1)
= 0(cos θ + 1)(sin θ - 1) = 0
Step 2: So, we have either (cos θ + 1)
= 0 or (sin θ - 1)
= 0cos θ
= -1 or
sin θ = 1
The values of cosine can only be between -1 and 1. Therefore, no value of θ exists for cos θ = -1.So, sin θ = 1 gives us θ = π/2 or 90°.However, we have 0 ≤ θ ≤ 2π, which means the solution is not complete yet.
To find all the possible values of θ, we need to check for all the angles between 0 and 2π, which have the same sin value as 1.θ = π/2 (90°) and θ = 5π/2 (450°) satisfies the equation.
Therefore, the measure of angle θ is 90° and 450° (in degrees) or π/2 and 5π/2 (in radians).
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8.5 A uniformly distributed random variable has mini- mum and maximum values of 20 and 60, respectively. a. Draw the density function. b. Determine P(35 < X < 45). c. Draw the density function includi
a. The density function for a uniformly distributed random variable can be represented by a rectangular shape, where the height of the rectangle represents the probability density within a given interval. Since the minimum and maximum values are 20 and 60, respectively, the width of the rectangle will be 60 - 20 = 40.
The density function for this uniformly distributed random variable can be represented as follows:
```
| _______
| | |
| | |
| | |
| | |
|______|_______|
20 60
```
The height of the rectangle is determined by the requirement that the total area under the density function must be equal to 1. Since the width is 40, the height is 1/40 = 0.025.
b. To determine P(35 < X < 45), we need to calculate the area under the density function between 35 and 45. Since the density function is a rectangle, the probability density within this interval is constant.
The width of the interval is 45 - 35 = 10, and the height of the rectangle is 0.025. Therefore, the area under the density function within this interval can be calculated as:
P(35 < X < 45) = width * height = 10 * 0.025 = 0.25
So, P(35 < X < 45) is equal to 0.25.
c. If you want to draw the density function including P(35 < X < 45), you can extend the rectangle representing the density function to cover the entire interval from 20 to 60. The height of the rectangle remains the same at 0.025, and the width becomes 60 - 20 = 40.
The updated density function with P(35 < X < 45) included would look as follows:
```
| ___________
| | |
| | |
| | |
| | |
|______|___________|
20 35 45 60
```
In this representation, the area of the rectangle between 35 and 45 would correspond to the probability P(35 < X < 45), which we calculated to be 0.25.
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Suppose I roll two fair 6-sided dice and flip a fair coin. You do not see any of the results, but instead I tell you a number: If the sum of the dice is less than 6 and the coin is H, I will tell you
Let the first die be represented by a random hypotheses X and the second die by Y. The value of the random variable Z represents the coin flip. Let us first find the sample space of the Experimen.
t:Sample space =
{ (1,1,H), (1,2,H), (1,3,H), (1,4,H), (1,5,H), (1,6,H), (2,1,H), (2,2,H), (2,3,H), (2,4,H), (2,5,H), (2,6,H), (3,1,H), (3,2,H), (3,3,H), (3,4,H), (3,5,H), (3,6,H), (4,1,H), (4,2,H), (4,3,H), (4,4,H), (4,5,H), (4,6,H), (5,1,H), (5,2,H), (5,3,H), (5,4,H), (5,5,H), (5,6,H), (6,1,H), (6,2,H), (6,3,H), (6,4,H), (6,5,H), (6,6,H) }
Let us find the events that satisfy the condition "If the sum of the dice is less than 6 and the coin is H".
Event A = { (1,1,H), (1,2,H), (1,3,H), (1,4,H), (2,1,H), (2,2,H), (2,3,H), (3,1,H) }There are 8 elements in Event A. Let us find the events that satisfy the condition "If the sum of the dice is less than 6 and the coin is H, I will tell you". There are four possible outcomes of the coin flip, namely H, T, HH, and TT. Let us find the events that correspond to each outcome. Outcome H Event B = { (1,1,H), (1,2,H), (1,3,H), (1,4,H) }There are 4 elements in Event B.
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(1 point) let f and g be functions such that f(0)=2,g(0)=5, f′(0)=9,g′(0)=−8. find h′(0) for the function h(x)=g(x)f(x).
The given problem requires us to find h′(0) for the function h(x) = g(x)f(x), where f and g are functions such that f(0) = 2, g(0) = 5, f′(0) = 9, and g′(0) = −8.In order to find h′(0), we can use the product rule of differentiation.
The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.In other words, if we have h(x) = f(x)g(x), thenh′(x) = f(x)g′(x) + f′(x)g(x).Applying this rule to our problem, we geth′(x) = f(x)g′(x) + f′(x)g(x)h′(0) = f(0)g′(0) + f′(0)g(0)h′(0) = 2(-8) + 9(5)h′(0) = -16 + 45h′(0) = 29Therefore, h′(0) = 29.
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how to find the coordinates of the center and length of the radius of the cricle.
The equation of a circle is x^2+y^2-2x+6y+3=0.
To find the coordinates of the center and the length of the radius of a circle given its equation, we need to rewrite the equation in the standard form (x - h)^2 + (y - k)^2 = r^2.
Where (h, k) represents the center of the circle and r represents the radius.
In the given equation x^2 + y^2 - 2x + 6y + 3 = 0, we can complete the square for both the x and y terms. Let's start with the x terms:
x^2 - 2x + y^2 + 6y + 3 = 0
(x^2 - 2x + 1) + (y^2 + 6y + 9) = 1 + 9
(x - 1)^2 + (y + 3)^2 = 10
Comparing this with the standard form, we can see that the center of the circle is at (1, -3) and the radius is √10.
Therefore, the coordinates of the center of the circle are (1, -3), and the length of the radius is √10.
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Determine the open t-intervals on which the curve is concave downward or concave upward. x=5+3t2, y=3t2 + t3 Concave upward: Ot>o Ot<0 O all reals O none of these
To find out the open t-intervals on which the curve is concave downward or concave upward for x=5+3t^2 and y=3t^2+t^3, we need to calculate first and second derivatives.
We have: x = 5 + 3t^2 y = 3t^2 + t^3To get the first derivative, we will differentiate x and y with respect to t, which will be: dx/dt = 6tdy/dt = 6t^2 + 3t^2Differentiating them again, we get the second derivatives:d2x/dt2 = 6d2y/dt2 = 12tAs we know that a curve is concave upward where d2y/dx2 > 0, so we will determine the value of d2y/dx2:d2y/dx2 = (d2y/dt2) / (d2x/dt2)= (12t) / (6) = 2tFrom this, we can see that d2y/dx2 > 0 where t > 0 and d2y/dx2 < 0 where t < 0.
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Question 9 1 Poin A state highway patrol official wishes to estimate the number of drivers that exceed the speed limit traveling a certain road. How large a sample is needed in order to be 99% confide
The estimated sample size needed to be 99% confident in estimating the number of drivers that exceed the speed limit is 27.
To determine the sample size needed to estimate the number of drivers that exceed the speed limit on a certain road with 99% confidence, we need to consider the desired level of confidence, the margin of error, and the population size (if available).
Let's assume that we do not have any information about the population size. In such cases, we can use a conservative estimate by assuming a large population size or using a population size of infinity.
The formula to calculate the sample size without considering the population size is:
n = (Z * Z * p * (1 - p)) / E^2
Where:
Z is the z-score corresponding to the desired level of confidence. For 99% confidence, the z-score is approximately 2.576.
p is the estimated proportion of drivers that exceed the speed limit. Since we don't have an estimate, we can use 0.5 as a conservative estimate, assuming an equal number of drivers exceeding the speed limit and not exceeding the speed limit.
E is the margin of error, which represents the maximum amount of error we are willing to tolerate in our estimate.
Let's assume we want a margin of error of 5%, which corresponds to E = 0.05. Substituting the values into the formula, we get:
n = (2.576^2 * 0.5 * (1 - 0.5)) / 0.05^2
n = (6.640576 * 0.25) / 0.0025
n = 26.562304
Since we cannot have a fractional sample size, we need to round up to the nearest whole number. Therefore, the estimated sample size needed to be 99% confident in estimating the number of drivers that exceed the speed limit is 27.
Please note that if you have information about the population size, you can use a different formula that incorporates the population size correction factor.
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