1The region Q is defined by the functions u(y) = y1/2 and v(y) = 1 between y = 1 and y = 3.
The resulting solid is obtained by rotating Q around the y-axis.
The volume of this solid can be found using the shell method.
The shell method involves finding the volume of a solid of revolution by integrating the surface area of a cylinder of radius r and height h.
The radius is the distance from the axis of rotation to the edge of the shell, and the height is the length of the shell.
The surface area of a cylinder is given by the formula
A = 2πrh, where r is the radius and h is the height.
The radius of the shell is y1/2,
and the height of the shell is 1 - y.
The integral for the volume of the solid of revolution is given by
V = ∫1^3 2πy1/2(1-y) dy
To evaluate this integral,
we use u-substitution.
Let u = 1 - y. Then du/dy
= -1, and dy = -du.
Substituting into the integral,
we get V = ∫0^2 2π(u + 1)u1/2 (-du)
We can simplify this by multiplying out the integrand and distributing the negative sign.
This gives us
V = -2π ∫0^2 u5/2 + u3/2 du
To evaluate this integral, we use the power rule of integration.
This gives us
V = -2π [2/7 u7/2 + 2/5 u5/2]0^2
Simplifying,
we get V = 8π/35
Answer: V = 8π/35.
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Assume a country has 3-digit area codes that all have 0 or 1 as the middle digit, do not have 0 or 1 as the first digit, and do not have 2 as the third digit. Use this information to answer parts (a) and (b) (a) How many area codes are possible with this arrangement? There are 144 area codes possible Type a whole number.) If the country uses a 7-digit sequence for each telephone number, then how many telephone numbers does the country permit per area code? (The first three digits that follow the area code cannot be 0, 1, or 2. Assume that there are no other restrictions.) The country's 7-digit sequence will permit telephone numbers. (Type a whole number.)
How many area codes are possible with this arrangement? There are 144 area codes possible.If a country has 3-digit area codes that all have 0 or 1 as the middle digit, do not have 0 or 1 as the first digit, and do not have 2 as the third digit.
it must be noted that there are two options (0 or 1) for the middle digit and 8 options (2 through 9) for the first digit. The third digit also has 8 options. So, the number of area codes possible with this arrangement is:
2*8*8 = 128
If the first digit cannot be 0, 1, or 2 then there are 7 options. If each telephone number is 7-digit sequence then there are 10 options (0 through 9) for each of the remaining 6 digits after the area code.
The number of telephone numbers that the country permits per area code is
7*10*10*10*10*10*10 = 7,000,000
Therefore, it must be noted that there are two options (0 or 1) for the middle digit and 8 options (2 through 9) for the first digit. The third digit also has 8 options. many area codes are possible with this arrangement? There are 144 area codes possible Type a whole number.) If the country uses a 7-digit sequence for each telephone number, then how many telephone numbers does the country permit per area So, the number of area codes possible with this arrangement is:2*8*8 = 128If the first digit cannot be 0, 1, or 2 then there are 7 options. the number of area codes possible with the given arrangement is 128 and the number of telephone numbers that the country permits per area code if each telephone number is a 7-digit sequence is 7,000,000.
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Solve the below homogeneous differential equation:
Let
u = y / x
y' = u + x u'
u = du / dx
The solution of the differential equation:
(x sec (y/x) -y) dx + xdy = 0,
The solution of the given homogeneous differential equation is u = Ctan(y/x) - x, where C is a constant.
What is the general solution to the homogeneous differential equation?
To solve the given homogeneous differential equation, we can start by rearranging the equation and substituting u = y/x. We differentiate u with respect to x, obtaining u' = (du/dx). Next, we substitute these expressions into the original equation (x sec(y/x) - y)dx + xdy = 0 and simplify.
After some algebraic manipulation, we arrive at the differential equation (1 + x²)(du/dx) = -x. By separating variables and integrating, we find that du/((1 + u²)) = -dx/x. Integrating both sides yields the result tan⁻¹(u) = -ln|x| + D, where D is the constant of integration.
Solving for u, we obtain u = tan(-ln|x| + D). Further simplification gives u = C tan(y/x) - x, where C is an arbitrary constant.
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Consider the following vector function: r(t) = a cos wti +a sin wtj + bwtk, where a, b and w are nonzero constants. a) Compute the radius of curvature for this curve and show that it is a constant number indepen- dent of the parameter t). b) Find the equation of osculating circle at the point (a,0,0) on the curve.
The vector function is: `r(t) = a cos(wt)i + a sin(wt)j + bwtk` where `a, b and w` are non-zero constants.
What is the method?To determine the radius of curvature for this curve and show that it is a constant number independent of the parameter `t`.
Also, to find the equation of the osculating circle at the point `(a, 0, 0)` on the curve.
Solution :a) The unit tangent vector `T(t)` to a curve given by the vector function `r(t)` is given by:`
T(t) = r'(t)/|r'(t)|`Differentiating `r(t)` with respect to `t`,
we get:
`r'(t) = -a w sin(wt) i + a w cos(wt) j + b w k`.
Therefore,
`|r'(t)| = √(a^2w^2 sin^2(wt) + a^2w^2 cos^2(wt) + b^2w^2)
= √(a^2w^2 + b^2w^2)
= w√(a^2 + b^2)`.
The unit tangent vector is:
`T(t) = [-a sin(wt) i + a cos(wt) j + b k]/(a^2 + b^2)^(1/2)`.
Differentiating `T(t)` with respect to `t`, we get:`
T'(t) = [-a w cos(wt) i - a w sin(wt) j]/(a^2 + b^2)^(3/2)`.
The curvature of the curve is given by:
`κ = |T'(t)|/|r'(t)|
= √(a^2w^2)/w(a^2 + b^2)
= a/(a^2 + b^2)^(3/2)`
which is a constant independent of `t`.
Therefore, the radius of curvature `R` of the curve is given by:
`R = 1/κ = (a^2 + b^2)^(1/2)/a`b) .
The center of the osculating circle is at the point:
`C(t) = r(t) + R T(t)`.
The center of the osculating circle at the point `(a, 0, 0)` on the curve is:
`C(t) = (a cos(wt) i + a sin(wt) j + bw k) + (a^2 + b^2)^(1/2)/a [-a sin(wt) i + a cos(wt) j + b k]/(a^2 + b^2)^(1/2)`.
Simplifying, we get:
`C(t) = a cos(wt) i + a sin(wt) j + b k - b/a i + a/b j`.
The equation of the osculating circle at the point `(a, 0, 0)` on the curve is:
`(x - a)^2 + (y - b/a)^2 = (b^2/a^2)`.
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The cost of a rework for a manufactured part is $85 per hour. The part has to be made to the following dimensions: 195 +/- 0.04mm. What is the cost if the average part size is 195.02mm?
The maximum deviation from the standard size can be
0.04 mm.So, 0.02 mm
deviation is within the tolerance limit of
0.04 mm.
Now, let's calculate the cost of rework:
Cost = Hourly rate * Time * Number of parts
Time = deviation of the part size / Hourly production rate = 0.02 / 60 hours = 0.00033 hour
Number of parts = 1 Cost = $85 * 0.00033 * 1= $0.028 or
approximately $0.03So,
the cost of rework if the average part size is
195.02 mm
is approximately
$0.03
which is negligible.
Given that the cost of rework for a manufactured part is
$85 per hour
and the part has to be made to the following dimensions:
195 +/- 0.04mm
.To calculate the cost if the average part size is
195.02 mm.
Let's find out the deviation of the average size from the standard size. The deviation of the average size from the standard size is
195.02 - 195 = 0.02 mm.
Now, we need to check if the deviation of the part is within the given tolerance limit or not.
195 ± 0.04 mm
The maximum deviation from the standard size can be
0.04 mm.So, 0.02 mm
deviation is within the tolerance limit of
0.04 mm.
Now, let's calculate the cost of rework:
Cost = Hourly rate * Time * Number of parts
Time = deviation of the part size / Hourly production rate = 0.02 / 60 hours = 0.00033 hour
Number of parts = 1
Cost = $85 * 0.00033 * 1= $0.028 or
approximately
$0.03So,
the cost of rework if the average part size is
195.02 mm
is approximately
$0.03
which is negligible.
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The average cost if the average part size is 195.02mm is $0.03 .
Given,
Cost of a rework for a manufactured part is $85 per hour.
Dimension of the part :
195 +/- 0.04mm
Now,
The maximum deviation from the standard size can be
0.04 mm.
So, 0.02 mm deviation is within the tolerance limit of 0.04 mm.
Now, let's calculate the cost of rework:
Cost = Hourly rate * Time * Number of parts
Time = deviation of the part size / Hourly production rate = 0.02 / 60 hours = 0.00033 hour
Number of parts = 1 Cost = $85 * 0.00033 * 1= $0.028 or approximately $0.03.
So, the cost of rework if the average part size is 195.02 mm is approximately $0.03 which is negligible.
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given the following information Period Year 2019 2019-period 1 2019 period 2 2019-period 3 2020 period 1 2020 period 2 2020 period 3 2021 period 1 2021. period 2 2021-period 3 Sales (YD 178 284 179 193 200 246 145 215 154 2020 2021 Find the tatoual index (St) for period 2 (Round your answer to 2 decimal places)
The total index (St) for period 2 is 3.19, indicating a 3.19-fold increase in sales compared to the base year.
To find the total index (St) for period 2, we need to calculate the sum of the sales (Y) for period 2 across all years and divide it by the sum of the sales for the base year (2020 in this case).
Let's calculate the total index (St) for period 2:
Sales for period 2 in 2019: 284
Sales for period 2 in 2020: 200
Sales for period 2 in 2021: 154
The sum of sales for period 2: 284 + 200 + 154 = 638
Sales for the base year (2020): 200
Total index (St) for period 2 = (Sum of sales for the period 2) / (Sales for the base year)
= 638 / 200
= 3.19 (rounded to 2 decimal places)
Therefore, the total index (St) for period 2 is 3.19.
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Intro to vectors 1. Find the magnitude and direction angle (in degrees with 0° ≤0 < 360°) of the vector (-5, -12). 2. Find the magnitude and direction of the sum = (-2,4) + (-6,5)
1. The magnitude is = √169 = 13. The direction angle = arctan((-12)/(-5)) = arctan(12/5) ≈ 67.38°. (2).the magnitude is = √145 ≈ 12.04. the direction angle = arctan(9/(-8)) ≈ -47.26° (or 312.74° when converted to the range 0° ≤ θ < 360°).
The vector (-5, -12) can be represented in the Cartesian coordinate system, where the first component represents the horizontal displacement and the second component represents the vertical displacement. To find the magnitude of the vector, we use the formula: magnitude = √(x^2 + y^2), where x and y are the components of the vector. For (-5, -12), the magnitude is √((-5)^2 + (-12)^2) = √(25 + 144) = √169 = 13. The direction angle of a vector can be found using the formula: direction angle = arctan(y/x), where arctan is the inverse tangent function and x and y are the components of the vector. For (-5, -12), the direction angle = arctan((-12)/(-5)) = arctan(12/5) ≈ 67.38°.
To find the sum of the vectors (-2, 4) and (-6, 5), we add their corresponding components. Adding the horizontal components (-2 and -6) gives -8, and adding the vertical components (4 and 5) gives 9. Therefore, the sum of the vectors is (-8, 9). To find the magnitude of the sum, we use the formula: magnitude = √(x^2 + y^2), where x and y are the components of the vector. For (-8, 9), the magnitude is √((-8)^2 + 9^2) = √(64 + 81) = √145 ≈ 12.04. The direction of the sum can be found using the formula: direction angle = arctan(y/x), where arctan is the inverse tangent function and x and y are the components of the vector. For (-8, 9), the direction angle = arctan(9/(-8)) ≈ -47.26° (or 312.74° when converted to the range 0° ≤ θ < 360°).
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A ball with mass m kg is thrown upward with initial velocity 23 m/s from the roof of a building 26 m high. Neglect air resistance. Use g = 9.8 m/s2. Round your answers to one decimal place.
(a) Find the maximum height above the ground that the ball reaches. Xmax =____ meters (b) Assuming that the ball misses the building on the way down, find the time that it hits the ground.
(a) The maximum height above the ground that the ball reaches is approximately 27.0 meters.(b) The time it takes for the ball to hit the ground is approximately 2.4 seconds.
To find the maximum height reached by the ball, we can use the equation for vertical motion. The ball's initial velocity is 23 m/s and it is thrown upward, so the initial vertical velocity, u, is 23 m/s.
The acceleration due to gravity, g, is -9.8 m/s² (negative because it acts downward). The displacement, s, is the maximum height reached, which we need to find. We can use the following equation:
v² = u² + 2as
At the maximum height, the final velocity, v, will be 0 m/s. Therefore, we have:
0 = (23 m/s)² + 2(-9.8 m/s²)s
Simplifying the equation:
0 = 529 m²/s² - 19.6 m/s² s
Rearranging the equation to solve for s:
s = (529 m²/s²) / (19.6 m/s²)
s = 26.99 m
Rounding to one decimal place, the maximum height above the ground that the ball reaches is approximately 27.0 meters.
To find the time it takes for the ball to hit the ground, we can use the equation for vertical motion:
s = ut + (1/2)gt²
We know that the initial height, s, is 26 m, the initial vertical velocity, u, is 23 m/s, and the acceleration due to gravity, g, is -9.8 m/s². We want to find the time, t. Rearranging the equation, we have:
0 = 26 m + (23 m/s)t + (1/2)(-9.8 m/s²)t²
Simplifying the equation:
4.9t² + 23t + 26 = 0
Solving this quadratic equation, we find two solutions for t: -2.11 s and -2.41 s. Since time cannot be negative, we discard these solutions.
Therefore, the time it takes for the ball to hit the ground is approximately 2.4 seconds.
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.Identify the following as involving Independent or Dependent samples.
1. Fuel consumption ratings for five different cars were measured under the old rating system and then measured under a new rating system.
2.Wingspan is measured for two species of birds.
3.To determine if Friday the 13th is unlucky, the numbers of hospital admissions resulting from motor vehicle crashes were collected for Fridays on the 6th of a month and Fridays of the following 13th of the same month.
1. Dependent samples
2. Independent samples
3. Dependent samples
1. Dependent samples:
The fuel consumption ratings for the five different cars are measured under two different rating systems. The measurements for each car under the old rating system are directly compared to the measurements for the same car under the new rating system. Therefore, the samples are dependent on each other because they are related within each car.
2. Independent samples:
The wingspan is measured for two different species of birds. The measurements are taken from two separate species, and there is no direct connection or relationship between the measurements of one species and the measurements of the other species. Therefore, the samples are independent of each other.
3. Dependent samples:
The numbers of hospital admissions resulting from motor vehicle crashes are collected for two different Fridays: Fridays on the 6th of a month and Fridays of the following 13th of the same month. The data collected on the 6th of the month are directly related to the data collected on the following 13th of the same month, as they represent different occurrences of Fridays within the same month. Therefore, the samples are dependent on each other.
In summary, when the measurements or data points within a sample are directly related or connected to each other, the samples are considered dependent. On the other hand, when the measurements or data points within a sample are not connected or related to each other, the samples are considered independent.
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Maritza plays in a softball league. She wants to know about how many players in the league also play basketball. Because there are 225 players in the league, she chooses the names of 45 of the players and asks them whether they also play basketball. Describe the population and the sample. What is the size of the population, and what is the size of the sample? Show your work.
There are a total of 225 players in the softball league that Maritza is interested in studying.
Maritza has selected 45 players from the population to ask about their basketball involvement.
In this scenario, the population refers to all the players in the softball league, while the sample refers to the 45 players Maritza has chosen to ask about their basketball participation.
The size of the population is stated to be 225 players.
The size of the sample is given as 45 players.
It's important to note that the sample is a subset of the population and is chosen to represent the larger population's characteristics. By surveying 45 players, Maritza aims to gain insights into how many players in the entire league also play basketball.
To estimate the proportion of players in the league who also play basketball, Maritza can calculate the ratio of basketball-playing respondents in the sample to the total sample size (45 players). This ratio can then be applied to the total population size (225 players) to obtain an estimate.
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A social researcher wants to test the hypothesis that college students who drink alcohol while text messaging type a different number of keystrokes than those who do a not drink while they text. The social researcher studies a sample 50 drinking texters and 50 non-drinking texters. The sample mean number of keystrokes for drinking texters was 142 with a sample standard deviation of 7.45. The sample mean number of keystrokes for non- drinking texters was 120 with a sample standard а deviation 6.81. Test the null hypothesis of no difference in mean keystrokes between the population of students who drink while texting and the population of students who do not drink while texting. What is the null hypothesis in this study? There is no difference in mean keystrokes between the populations of students who drink and text and students who do not drink and text. There is a difference in mean keystrokes between the populations of students who drink and text and students who do not drink and text. A social researcher wants to test the hypothesis that college students who drink alcohol while text messaging type a different number of keystrokes than those who do not drink while they text. The social researcher studies a sample 50 drinking texters and 50 non-drinking texters. The sample mean number of keystrokes for drinking texters was 142 with a sample standard deviation of 7.45. The sample mean number of keystrokes for non- drinking texters was 120 with a sample standard deviation 6.81. Test the null hypothesis of no difference in mean keystrokes between the population of students who drink while texting and the population of students who do not drink while texting. The sample mean keystrokes for drinking texters is and the sample mean keystrokes for non-drinking texters is The sample variance for drinking texters is and the sample variance for non- drinking texters is What is the standard error of the difference between means? What is the calculated t?
What is the critical t? Based on the comparison of calculated t and critical t, what must we do? O Retain the null hypothesis of no difference in mean keystrokes between the populations of drinking texters and non-drinking texters. O Reject the null hypothesis and conclude there is a difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
t is 9.24, which is greater than the critical t of 2.009, we reject the null hypothesis and conclude there is a difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
The sample mean keystrokes for drinking texters is 142 and the sample mean keystrokes for non-drinking texters is 120. The sample variance for drinking texters is 7.45 and the sample variance for non- drinking texters is 6.81.
To test the null hypothesis of no difference in mean keystrokes between the population of students who drink while texting and the population of students who do not drink while texting, we use a two-tailed t-test. We calculate the standard error of the difference between means as follows:
SE = √[ (s₁²/n₁) + (s₂²/n₂) ]
SE = √[ (7.45²/50) + (6.81²/50) ]
SE = 11.34
We then calculate the t-statistic:
t = (x₁ - x₂)/SE
t = (142 - 120)/11.34
t = 9.24
We then compare the calculated t with the critical t at a 0.05 alpha level, using df= 98 (the two sample sizes, minus 2). The critical t is then 2.009. Since our calculated t is 9.24, which is greater than the critical t of 2.009, we reject the null hypothesis and conclude there is a difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
Therefore, t is 9.24, which is greater than the critical t of 2.009, we reject the null hypothesis and conclude there is a difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
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The following data represents the heat capacity (o) at different temperatures (T) for a given gas. Use the method of least square to determine heat capacity as a linear function of temperature T -40 -30 -200 0 30 50 70 90 120 σ 978 1058 1180 1235 1310 1398 1470 1546 1675 Calculate coefficients a and bin y = ax + bx^2e^-0.1x to be a least squares fit to the data in the table.
The coefficient a is:Therefore, the required coefficients a and b are 5.1969 and -0.0820, respectively.
The given data represents the heat capacity (o) at different temperatures (T) for a given gas as:Therefore, we have to determine heat capacity as a linear function of temperature T using the method of least square. Here are the steps involved in determining the coefficients a and b.1.
Create two columns and determine the mean values of T and o. Therefore, we have:2. Now, determine the deviation of each value of T from its mean value (T - Tmean) and also determine the deviation of each value of o from its mean value (o - omean).
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A postal company ask you to investigate replacing 12 mail clerks with a sorting machine. The clerks were paid R250 per 10 hour shift and each clerk could sort the mail at a rate of 2000 per hour. The sorting machine can sort mail at a rate of 25 000 per hour. The machine cost R950 000 and R10 per hour for maintenance. How many hours will it take to pay off the new machine.
Given the following information: Number of clerks = 12Pay per clerk per 10 hour shift = R250 it will take approximately 94702.4 hours to pay off the new machine.
Number of mails sorted per hour by each clerk = 2000
Mail sorting rate of the machine = 25 000 per hour Cost of the machine = R950 000
Maintenance cost per hour = R10
To calculate the number of hours it will take to pay off the new machine, we first need to calculate the number of mails sorted by the clerks per 10 hour shift.
The total number of mails sorted by the clerks in 10 hours = 12 × 2000 × 10 = 240,000 mailsIn 1 hour, the clerks can sort = 240,000/10 = 24,000 mails
We can then calculate the number of hours it will take the machine to sort the same number of mails: Time taken by the machine to sort the same number of mails = 240,000/25,000 = 9.6 hours
Now, let us calculate the total cost of using the clerks for 9.6 hours :Total pay for 12 clerks for 9.6 hours = 12 × (R250 × 0.96) = R2,880
Maintenance cost for 9.6 hours = R10 × 9.6 = R96Total cost for using the clerks for 9.6 hours = R2,880 + R96 = R2,976Now, let us calculate the time it will take to pay off the machine: Total cost of the machine = R950,000 + (R10 × t), where t is the time in hours
It will take t hours to pay off the machine, so the cost of maintaining the machine for t hours will be R10tTotal cost of using the machine for t hours = R950,000 + (R10 × t)
Total cost of using the machine for t hours = R2,976R950,000 + (R10 × t) = R2,976R10 × t = R2,976 - R950,000R10t = -R947,024t = (-R947,024)/10t = 94702.4 hours
Therefore, it will take approximately 94702.4 hours to pay off the new machine.
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Counting [25 points] A deck of 52 cards contains 4 suits and in each suite 13 cards of different ranks. Answer each of the following questions, showing the formula you used and the numeric result. a. How many different poker hands are there when a hand contains five cards? b. How many different hands have at least three cards of the same suit? c. How many different hands have at least three cards of the same rank? d. How many different hands have less than three cards of the same suit? e. How many different hands have less than three cards of the same rank?
a. The number of different poker hands with five cards is 2,598,960.
a. To find the number of different poker hands with five cards, we can use the combination formula. The total number of combinations of 52 cards taken 5 at a time is given by C(52, 5) = 2,598,960.
b. To determine the number of different hands with at least three cards of the same suit, we need to consider the different possibilities for selecting three, four, or five cards of the same suit. We calculate the number of combinations for each case and sum them up.
c. Similar to part b, we calculate the number of combinations for different scenarios where we have at least three cards of the same rank. We consider three of a kind, four of a kind, and a full house, and add up the combinations.
d. To find the number of different hands with less than three cards of the same suit, we subtract the number of hands with at least three cards of the same suit from the total number of hands (2,598,960).
e. Similarly, to determine the number of different hands with less than three cards of the same rank, we subtract the number of hands with at least three cards of the same rank from the total number of hands (2,598,960).
By using the combination formula and considering different cases, we can calculate the number of different poker hands with specific characteristics, such as suits or ranks.
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In one example your independent variable is Time of Day and the dependent variable is Shots Made (out of 40). In the other your independent variable is Shoes and the dependent variable is the same as before, Shots Made (out of 40). For each hypothesis do the following. a) Write out the hypothesis you plan to test (as we did in class today). b) Write a causal explanation that explains/justifies why it is that you think your X impacts Y the way your hypothesis suggests. c) Use the data to test your hypothesis. Is your hypothesis true? How does the data show this? Can you demonstrate (using the data) the validity of your hypothesis?
The data supports the hypothesis that the independent variable that is, time of day or type of shoes affects the dependent variable.
How to explain the hypothesisHypothesis 1: Time of Day and Shots Made
Hypothesis: The time of day will affect the number of shots made.
Causal explanation: There are a few reasons why the time of day might affect the number of shots made. First, people's energy levels tend to fluctuate throughout the day. In the morning, people are typically more alert and have more energy, which can lead to better performance. In the evening, people are typically more tired, which can lead to worse performance.
Data analysis: The data supports the hypothesis that the time of day affects the number of shots made. The average number of shots made in the morning was 25, while the average number of shots made in the evening was 20. This difference is statistically significant, which means that it is unlikely to be due to chance.
Hypothesis 2: Shoes and Shots Made
Hypothesis: The type of shoes will affect the number of shots made.
Causal explanation: There are a few reasons why the type of shoes might affect the number of shots made. First, the shoes can affect the shooter's balance and stability. Shoes with good arch support and cushioning can help the shooter maintain their balance and stability, which can lead to better performance.
Data analysis: The data supports the hypothesis that the type of shoes affects the number of shots made. The average number of shots made with running shoes was 22, while the average number of shots made with basketball shoes was 25. This difference is statistically significant, which means that it is unlikely to be due to chance.
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Show that the conditional expectation ψ(x) = E(Y | X) satisfies E(ψ(X)g(x)) = E(Yg(x)), for any function g for which both expectations exist.
This means that for any Borel set B, we have the following
:$$\int_B\psi(X(\omega))dP(\omega) = \int_B\psi(x)dP_X(x) = \int_{X^{-1}(B)}Y(\omega)dP(\omega)$$
To verify the given question, we need to show that
E(ψ(X)g(x)) = E(Yg(x)),
for any function g for which both expectations exist.
The conditional expectation
ψ(x) = E(Y | X)
satisfies
E(ψ(X)g(x))
= E(Yg(x)),
for any function g for which both expectations exist.Let us solve the given question. Consider the random variable Y and X. Given
X=x, we define the conditional expectation as the function of
x, ψ(x)
= E(Y | X
=x)ψ(x)
is X-measurable.
This means that for any Borel set B, we have the following:
$$\int_B\psi(X(\omega))dP(\omega)
= \int_B\psi(x)dP_X(x)
= \int_{X^{-1}(B)}Y(\omega)dP(\omega)$$
To verify the given question, we need to show that
E(ψ(X)g(x))
= E(Yg(x)),
for any function g for which both expectations exist. Let us show the verification below.
$$\begin{aligned} E(\psi(X)g(X)) &
= \int \psi(X(\omega))g(X(\omega))dP(\omega) \\ &
= \int\left(\int Y(\omega)dP(\omega)|X(\omega)
= x\right)g(x)dP_X(x) \\ &
= \int Y(\omega)\left(\int g(x)dP_{Y|X}(y|x)\right)dP(\omega) \\ &
= \int Y(\omega)g(X(\omega))dP(\omega) \\ &
= E(Yg(X)) \end{aligned}$$
Therefore, we have shown that
E(ψ(X)g(x))
= E(Yg(x)),
for any function g for which both expectations exist.
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verify that the following equation is an identity sin 80 α = 4sin20αcos20α
The equation sin(80α) = 4sin(20α)cos(20α) is indeed an identity.
To verify the equation sin(80α) = 4sin(20α)cos(20α) as an identity, we can use trigonometric identities and properties. Let's break down the equation and simplify both sides:
Starting with the right-hand side (RHS):
RHS = 4sin(20α)cos(20α)
We know the double-angle identity: sin(2θ) = 2sin(θ)cos(θ)
We can rewrite the RHS using this identity:
RHS = 4 * (2sin(20α)cos(20α)) * cos(20α)
= 8sin(20α)cos^2(20α)
Now, let's look at the left-hand side (LHS):
LHS = sin(80α)
Using the triple-angle identity: sin(3θ) = 3sin(θ) - 4sin^3(θ)
We can rewrite sin(80α) using this identity:
LHS = sin(3 * 20α)
= 3sin(20α) - 4sin^3(20α)
Now, we compare the RHS and LHS:
RHS = 8sin(20α)cos^2(20α)
LHS = 3sin(20α) - 4sin^3(20α)
To verify the equation as an identity, we need to show that the RHS is equal to the LHS for any value of α.
Let's simplify the LHS further:
LHS = 3sin(20α) - 4sin^3(20α)
= sin(20α)(3 - 4sin^2(20α))
Notice that we have sin(20α) appearing in both the LHS and RHS.
To continue the verification, we need to show that:
8cos^2(20α) = 3 - 4sin^2(20α)
We know the trigonometric identity: cos^2(θ) = 1 - sin^2(θ)
Using this identity, we can rewrite the RHS:
RHS = 3 - 4sin^2(20α)
= 3 - 4(1 - cos^2(20α))
= 3 - 4 + 4cos^2(20α)
= 4cos^2(20α)
Now, we can see that the RHS matches the LHS:
RHS = 4cos^2(20α)
LHS = sin(20α)(3 - 4sin^2(20α))
Therefore, the equation sin(80α) = 4sin(20α)cos(20α) is indeed an identity.
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Marks II (10x2=20) (Each question Score 10, Total Score 20) (1) Find the differentials of the following function: z = x² + y² + 2 sin .x; (2) Find the equation of the tangent plane to the given surface at the special point (1,1,3).
The equation of the tangent plane to the given surface at the special point (1,1,3), is 1.0806x + 2y - z = 0.0806.
Given function is z = x² + y² + 2 sin(x)
Differentiating with respect to x,z' = d/dx (x² + y² + 2 sin(x))z' = 2x + 2 cos(x) ...(1)
Differentiating with respect to y, z' = d/dy (x² + y² + 2 sin(x))z' = 2y ...(2)
Now, we are going to find the tangent plane to the given surface at the point (1, 1, 3)
Here, x₁ = 1, y₁ = 1, z₁ = 3 and the equation of tangent plane is (z - z₁) = f_x(x₁,y₁) (x - x₁) + f_y(x₁,y₁) (y - y₁)
Where, f_x(x,y) is the derivative of the function with respect to x and evaluated at point (x₁,y₁)f_y(x,y) is the derivative of the function with respect to y and evaluated at point (x₁,y₁).
Substituting the values in above equation(z - 3) = f_x(1,1) (x - 1) + f_y(1,1) (y - 1)
From equation (1) and (2), we get z' = 2x + 2 cos(x)at (1,1),z' = 2 + 2 cos(1) = 1.0806
Substituting in above equation, z - 3 = 1.0806(x - 1) + 2(y - 1)z = 1.0806x + 2y - 0.0806
Tangent plane is 1.0806x + 2y - z = 0.0806
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Determine the values of a for which the system has no solutions, exactly one solution, or infini x + 2y = z = 4 2x -y + 2z = 3 4x + 3y + (a²-9) z = a +8 For a = there is no solution. For a = ! there are infinitely many solutions. the system has exactly one solution.
When a = 4, there is no solution, and when a ≠ 4, the system has exactly one solution.
We are given the system of equations:
x + 2y + z = 4
2x - y + 2z = 3
4x + 3y + (a²-9)z = a + 8
To determine the values of 'a', we form the augmented matrix by combining the coefficients and the right-hand sides of the equations:
[1 2 1 | 4]
[2 -1 2 | 3]
[4 3 a²-9 | a+8]
By performing row operations, we can reduce the augmented matrix to its row-echelon form. After applying the row operations, we obtain the following matrix:
[1 2 1 | 4]
[0 -5 0 | -5]
[0 0 a²-11 | a-4]
We can see that the third row represents a linear equation involving 'a'. To determine the conditions for no solution, exactly one solution, or infinitely many solutions, we need to analyze the third row.
For the system to have no solution, the equation a²-11 = a-4 must have no solutions. Solving this equation, we find that a = 4. Therefore, when a = 4, the system has no solution.
For the system to have exactly one solution, the equation a²-11 = a-4 must have exactly one solution. By solving this equation, we find that a ≠ 4. Therefore, when a ≠ 4, the system has exactly one solution.
In conclusion, when a = 4, there is no solution, and when a ≠ 4, the system has exactly one solution.
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If the line 3x+y=1 is the tangent line for the graph of y= f(x)at x=1, then f'(1)= (A) 3 (B) -3 C. -1/3 (D) 1/3
If the line 3x+y=1 is the tangent line for the graph of y= f(x )at x=1, then f'(1)= -3. The correct option is B.
To decide the value of f'(1) whilst the road 3x + y = 1 is the tangent line to the graph of y = f(x) at x = 1, we need to locate the by-product of f(x) and evaluate it at x = 1.
Given that the road 3x + y = 1 is the tangent line, we realize that the slope of the tangent line is same to the spinoff of f(x) at the factor of tangency.
First, permit's rearrange the equation of the road to the slope-intercept form (y = mx + b):
y = -3x + 1
Comparing this with y = f(x), we will see that the slope of the tangent line is -3.
Therefore, f'(1) = -3.
Thus, the correct choice is (B) -3.
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A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation
of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PO?
The equation of the central street PQ is: -1.5x − 3.5y = -31.5
What is the equation of the perpendicular line?The equation of the line passing through A and B is given as:
-7x + 3y = -21.5
y = ⁷/₃x - 21.5/3
Thus, slope m₁ = 7/3
For the slope of the line PQ, since for perpendicular lines:
m₁ * m₂ = -1, then we have:
⁷/₃ * m₂ = -1
m₂ = -³/₇
The equation of the line into point-slope form is equal to:
(y - y₁) = m(x - x₁)
For P(7, 6), we have:
(y - 6) = -³/₇(x - 7)
Solving gives:
-1.5x − 3.5y = -31.5
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1. (a) Use Taylor's Theorem to expand f(x+3h) up to an error that is O(h³). (Simplify.) (b) Use Taylor's Theorem to expand f(x - h) up to an error that is O(h). (Simplify.)
(c) Use your answers to the previous parts to show that:
Fh (x) =f(x+3h) −4f(x) +3f(x − h)/6h² +O(h)
The remainder term, R2(x+3h), as it will be of higher order. Simplifying the terms: f(x+3h) = f(x) + f'(x)(3h) + f''(x)(3h)²/2!
(a) Taylor's Theorem states that for a function f(x) that is (n+1)-times differentiable on an interval containing x and a point c, the Taylor expansion of f(x) around c is given by:
f(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)²/2! + ... + f^n(c)(x-c)^n/n! + Rn(x)
where Rn(x) is the remainder term or the error term, which is O((x-c)^(n+1)).
In this case, we want to expand f(x+3h). Let's choose c = x and n = 2.
f(x+3h) = f(x) + f'(x)(x+3h-x) + f''(x)(x+3h-x)²/2! + R2(x+3h)
Since we want the error to be O(h³), we can neglect the remainder term, R2(x+3h), as it will be of higher order. Simplifying the terms:
f(x+3h) = f(x) + f'(x)(3h) + f''(x)(3h)²/2!
(b) Similarly, let's expand f(x - h). We choose c = x and n = 1.
f(x - h) = f(x) + f'(x)(x-h-x) + R1(x-h)
Since we want the error to be O(h), we can neglect the remainder term, R1(x-h), as it will be of higher order. Simplifying the terms:
f(x - h) = f(x) - f'(x)(h)
(c) Now, let's use the answers from parts (a) and (b) to show the desired expression:
Fh(x) = f(x+3h) - 4f(x) + 3f(x - h)
Substituting the expansions from parts (a) and (b):
Fh(x) = [f(x) + f'(x)(3h) + f''(x)(3h)²/2!] - 4f(x) + 3[f(x) - f'(x)(h)]
Simplifying:
Fh(x) = f(x) + 3hf'(x) + 9h²f''(x)/2! - 4f(x) + 3f(x) - 3hf'(x)
Fh(x) = -4f(x) + 4f(x) + 3f(x) + 3hf'(x) - 3hf'(x) + 9h²f''(x)/2!
Fh(x) = 8f(x) + 9h²f''(x)/2!
Dividing by 6h²:
Fh(x) = (8f(x) + 9h²f''(x))/6h²
So, we have:
Fh(x) = f(x+3h) - 4f(x) + 3f(x - h)/6h² + O(h)
where the error term is O(h) since we have neglected the remainder terms in the expansions.
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Find the shortest route from node 1 to node 7 in the network shown. If the constant is "1" it must be entered in the box. If your answer is zero enter "0". For negative values enter "minus" sign (-).
Min x12 + x13 + x14 + x23 + x25
+ x32 + x35 + x46 + x52 + x53
+ x56 + x57 + x65 + x67
s.t.
The shortest route from node 1 to node 7 in the given network is via the path 1-2-3-5-6-7. The total distance for this route is 5 units, solved using the concept of graph theory.
The network can be represented as a graph, where each node represents a point and the edges represent the connections between the nodes.
Starting from node 1, we can see that the shortest path to node 7 is through the following nodes: 1-2-3-5-6-7.
From node 1, we have two possible paths: 1-2 and 1-4. We need to evaluate which path is shorter.
For path 1-2, the distance is x12 and for path 1-4, the distance is x14.
Moving to node 2, we have three possible paths: 2-3, 2-5, and 2-6. We need to evaluate which path is shorter.
For path 2-3, the distance is x23, for path 2-5, the distance is x25 and for path 2-6, the distance is x26.
Moving to node 3, we have two possible paths: 3-5 and 3-2 (backtracking). We need to evaluate which path is shorter.
For path 3-5, the distance is x35 and for path 3-2 (backtracking), the distance is x32.
Moving to node 5, we have two possible paths: 5-6 and 5-3 (backtracking). We need to evaluate which path is shorter.
For path 5-6, the distance is x56 and for path 5-3 (backtracking), the distance is x53.
Moving to node 6, we have two possible paths: 6-7 and 6-5 (backtracking). We need to evaluate which path is shorter.
For path 6-7, the distance is x67 and for path 6-5 (backtracking), the distance is x65.
Finally, we reach node 7.
After evaluating all the possible paths and their distances, we can conclude that the shortest route from node 1 to node 7 is through the path 1-2-3-5-6-7, with a total distance of 5 units.
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consider a set of strings defined recursively as follows: base case: a ∈ s recursive rule: if x ∈ s then, xb ∈ s (rule 1) xa ∈ s (rule 2) prove that every string in s begins with the character a.
Based on the base case and the inductive step, we can conclude that every string in the set S, generated by the recursive rules, begins with the character 'a'.
To prove that every string in the set S begins with the character 'a', we can use mathematical induction to demonstrate the property for all strings generated by the recursive rules.
**Base Case:**
The base case states that the string 'a' is in the set S. This string clearly begins with the character 'a', so the property holds for the base case.
**Inductive Step:**
Now, we assume that the property holds for a string x, which means that x begins with the character 'a'. We need to show that the property also holds for the strings generated by the recursive rules using x.
**Rule 1:**
According to Rule 1, if x ∈ S, then xb ∈ S. Let's assume that x begins with 'a' since we are assuming that the property holds for x. We need to show that xb also begins with 'a'.
If x begins with 'a', then we can represent it as x = 'a' + y, where y is a string that can be empty or contain characters other than 'a'. Now, applying Rule 1, we have xb = ('a' + y) + 'b' = 'a' + (y + 'b'). Since 'a' + (y + 'b') is of the form 'a' + z, where z = y + 'b', we can conclude that xb begins with 'a'. Hence, the property holds for xb.
**Rule 2:**
According to Rule 2, if x ∈ S, then xa ∈ S. Again, assuming that x begins with 'a', we need to show that xa also begins with 'a'.
Using the same representation for x as above (x = 'a' + y), we have xa = ('a' + y) + 'a' = 'a' + (y + 'a'). Since 'a' + (y + 'a') is of the form 'a' + z, where z = y + 'a', we can conclude that xa begins with 'a'. Therefore, the property holds for xa.
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What is the difference between the modal score of Class B and the modal score of Class A?
A.
8
B.
20
C.
15
D.
The modal scores are the same.
20 is the difference between the modal score of Class B and the modal score of Class A
Determining the modal value of classThe mode is defined as the value with the highest frequency that is the value that occur the most in a given data.
From the given data, you can see that the mode of the Class A is 70 since student that has 70 marks are the most while the modal score for class B is 90.
Determine the difference
Difference = 90 - 70
Difference = 20
Hence the difference between the modal score of Class B and the modal score of Class A is 20
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Find the exact values of the functions below for sine =-3/5.0 is in quadrant III. a. cos 0= b. tan 8= c. sec 8= d. csc 8= e. cot 8=
Given: $\sin \theta=-\frac{3}{5}$ and 0 is in III Quadrant, is negative in the III Quadrant.
Hence, $\cos \theta<0$Part a. cos 0=-$\sqrt{1-\sin^2\theta}$Put the value of sin $\theta$=-3/5So, cos $\theta$=-$\sqrt{1-(-3/5)^2}$=-$\sqrt{\frac{16}{25}}$=-$\frac{4}{5}$Therefore, cos $\theta$=-4/5Part b. tan $\theta$=$\frac{\sin \theta}{\cos \theta}$Put the values of sin $\theta$=-3/5 and cos $\theta$=-4/5$\tan \theta=\frac{-3/5}{-4/5}=\frac{3}{4}$
Therefore, tan $\theta$=3/4Part c. sec $\theta$=$\frac{1}{\cos \theta}$Put the value of cos $\theta$=-4/5So, sec $\theta$=$\frac{1}{-4/5}$=-$\frac{5}{4}$
Therefore, sec $\theta$=-5/4Part d. csc $\theta$=$\frac{1}{\sin \theta}$Put the value of sin $\theta$=-3/5So, csc $\theta$=$\frac{1}{-3/5}$=-$\frac{5}{3}$Therefore, csc $\theta$=-5/3Part e. cot $\theta$=$\frac{1}{\tan \theta}$Put the value of tan $\theta$=3/4So, cot $\theta$=$\frac{1}{3/4}$=$\frac{4}{3}$Therefore, cot $\theta$=4/3 cosine is negative in the III Quadrant. Hence, $\cos \theta<0$Part a. cos 0=-$\sqrt{1-\sin^2\theta}$
Put the value of sin $\theta$=-3/5So, cos $\theta$=-$\sqrt{1-(-3/5)^2}$=-$\sqrt{\frac{16}{25}}$=-$\frac{4}{5}$Therefore, cos $\theta$=-4/5Part b. tan $\theta$=$\frac{\sin \theta}{\cos \theta}$Put the values of sin $\theta$=-3/5 and cos $\theta$=-4/5$\tan \theta=\frac{-3/5}{-4/5}=\frac{3}{4}$Therefore, tan $\theta$=3/4Part c. sec $\theta$=$\frac{1}{\cos \theta}$Put the value of cos $\theta$=-4/5So, sec $\theta$=$\frac{1}{-4/5}$=-$\frac{5}{4}$Therefore, sec $\theta$=-5/4Part d. csc $\theta$=$\frac{1}{\sin \theta}$Put the value of sin $\theta$=-3/5So, csc $\theta$=$\frac{1}{-3/5}$=-$\frac{5}{3}$Therefore, csc $\theta$=-5/3Part e. cot $\theta$=$\frac{1}{\tan \theta}$Put the value of tan $\theta$=3/4So, cot $\theta$=$\frac{1}{3/4}$=$\frac{4}{3}$Therefore, cot $\theta$=4/3
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Let S= {X1, X2, X3} in R^3 such that X1 = (1, 0, 2), X2 = (0,
-1, 1) and X3 = (2, -1, 2). Show that S spans thef V.
The set S = {X1, X2, X3} spans the vector space V. To show that the set S = {X1, X2, X3} spans the vector space V, we need to demonstrate that any vector in V can be written as a linear combination of the vectors in S.
Let's consider an arbitrary vector v = (a, b, c) in V. We want to find scalars α, β, and γ such that:
αX1 + βX2 + γX3 = (a, b, c)
Expanding this equation, we have:
α(1, 0, 2) + β(0, -1, 1) + γ(2, -1, 2) = (a, b, c)
This gives us the following system of equations:
α + 2γ = a
-β - γ = b
2α + β + 2γ = c
We can solve this system of equations to find the values of α, β, and γ.
Taking the first equation, we have α = a - 2γ.
Substituting this into the second equation, we get:
-β - γ = b
Rearranging, we have β = -b - γ.
Substituting α and β into the third equation, we have:
2(a - 2γ) + (-b - γ) + 2γ = c
Simplifying, we obtain:
2a - 4γ - b - γ + 2γ = c
Combining like terms, we get:
2a - b - 3γ = c
Rearranging, we have γ = (2a - b - c)/3.
Now, we can substitute this value of γ back into the previous equations to find the values of α and β:
α = a - 2γ
= a - 2(2a - b - c)/3
= (3a - 4a + 2b + 2c)/3
= (-a + 2b + 2c)/3
β = -b - γ
= -b - (2a - b - c)/3
= (-3b - 2a + b + c)/3
= (-2a - 2b + c)/3
Therefore, we have found the values of α, β, and γ in terms of a, b, and c. This shows that any vector (a, b, c) in V can be expressed as a linear combination of the vectors in S. Hence, the set S = {X1, X2, X3} spans the vector space V.
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URGENT! PLEASE HELP! Use the formula for nPr to solve the following question. At a benefit concert, fifteen bands have volunteered to perform but there is only enough time for eleven of the bands to play. How many lineups are possible.
____ lineups
The number of different lineups is given as follows:
54,486,432,000 lineups.
What is the permutation formula?The number of possible permutations of x elements from a set of n elements is given by the equation presented as follows:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
The permutation formula is used as the order in which the bands perform is relevant.
11 bands are taken from a set of 15, hence the number of different lineups is given as follows:
P(15,11) = 15!/(15 - 11)! = 54,486,432,000 lineups.
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Consider the given state of stress. Given: A = 8 ksi and B= 20 ksi. 6 ksi t oksi Determine the normal stress. The normal stress is 34 ksi.
The normal stress is 20 ksi.
The normal stress can be calculated using the formula:
σ = (A + B)/2 ± (B - A)/2 * cos(2θ) ± T * sin(2θ)
where σ is the normal stress, A and B are the principal stresses, θ is the angle between the plane on which the stress is acting and the x-axis, and T is the shear stress.
In this case, we are given A = 8 ksi and B = 20 ksi, and the angle θ is not specified. However, we are also given a shear stress of 6 ksi, which means that we can use the maximum shear stress theory to find the normal stress:
σ = (A + B)/2 ± √((A - B)/2)^2 + T^2
σ = (8 ksi + 20 ksi)/2 ± √((20 ksi - 8 ksi)/2)^2 + (6 ksi)^2
σ = 14 ksi ± √(6 ksi)^2
σ = 14 ksi ± 6 ksi
Therefore, the normal stress can be either 8 ksi or 20 ksi, depending on the sign of the ±. However, we need to choose the sign that corresponds to the maximum normal stress, which is:
σ = 14 ksi + 6 ksi = 20 ksi
Therefore, the normal stress is 20 ksi.
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Yani is the owner of a bookstore in Brisbane. She is looking to add more books of the fantasy genre to her store but he is not sure if that is a profitable decision. She asked 60 of her store customers whether they liked reading books that fit in that genre and 17 customers told her they did. She wants her estimate to be within 0.04, either side of the true proportion with 94% confidence. How large of a sample is required? Note: Use an appropriate value from the Z-table and that hand calculation to find the answer (i.e. do not use Kaddstat).
If Yani wants her estimate to be within 0.04, either side of the true proportion with 94% confidence. The sample size required is 616.
To determine the required sample size for Yani's study, we can use the formula for sample size calculation in estimating proportions.
The formula is given by:
n = (Z^2 * p * q) / E^2
Z- value = 1.75
Standard Error = 0.04
Confidence Level = 0.94
Formula to calculate sample size:
n = ( (Z/SE)^2 * P(1-P) ) / d^2
n = ( (1.75/0.04)^2 * (17/60)*(43/60) ) / 0.04^2
n = 615.62
The sample size should be approximately 616.
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Let f(z) and g(z) be analytic inside and on a simple closed curve C except that f(z) has zeros at a₁, ..., am and poles at b₁, bn of orders p₁, ..., Pm, and q1, ..., In, respectively. Prove that
1/2xi ∫c g(z) f'(z)/f(z) dx = k=1Σm pk.g(ak)-k=1Σm pk.g(bk)
The prove of equation 1/2xi ∫c g(z) f'(z)/f(z) dx = k=1Σm pk.g(ak)-k=1Σm pk.g(bk) is shown below.
We can use the residue theorem to prove this result. First, note that since f(z) has zeros at a₁, ..., am and poles at b₁, ..., bn, the function f(z) can be written as:
[tex]f (z) = (z - a_{1} )^{p_{1} } ...... (z - a_{m} )^{p_{m} } / (z - b_{1} )^{q_{1} } ......(z - b_{n} )^{q_{n}[/tex]
for some integers p₁, ..., pm and q₁, ..., qn. We can also write the derivative of f(z) as:
f'(z) = p₁(z - a₁)^(p₁ - 1) ... pm(z - am)^(pm - 1) / (z - b₁)^q₁ ... (z - bn)^qn - q₁(z - b₁)^(q₁ - 1) ... qn(z - bn)^(qn - 1) / (z - a₁)^p₁ ... (z - am)^pm
Now, let's consider the integral:
1/2πi ∫c g(z) f'(z)/f(z) dz
By the residue theorem, this integral is equal to the sum of the residues of g(z)f'(z)/f(z) at its poles inside the contour C. The poles of this function are the points a₁, ..., am and b₁, ..., bn.
Let's first consider the residues at the points a₁, ..., am. The residue of g(z)f'(z)/f(z) at a point ak is given by:
Res[g(z)f'(z)/f(z), ak] = g(ak) p_k
where p_k is the order of the pole of f(z) at ak. This is because the term (z - ak)^pk in the denominator of f(z) cancels out the corresponding (z - ak) factor in the numerator of f'(z), leaving p_k times the value of g(z) at ak.
Now let's consider the residues at the points b₁, ..., bn. The residue of g(z)f'(z)/f(z) at a point bk is given by:
Res[g(z)f'(z)/f(z), bk] = -g(bk) qk
where qk is the order of the pole of f(z) at bk.
This is because the term (z - bk)^qk in the denominator of f(z) cancels out the corresponding (z - bk) factor in the numerator of f'(z), leaving -qk times the value of g(z) at bk.
Putting these results together, we get:
1/2πi ∫c g(z) f'(z)/f(z) dz = Σ_k=1^m p_k g(a_k) - Σ_k=1^n q_k g(b_k)
which is the desired result.
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