The correct answer is: False.
As the sample size increases, the variation of the sample mean from the population mean typically becomes smaller. This is known as the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution with a mean equal to the population mean and a standard deviation that decreases as the sample size increases. Therefore, with larger sample sizes, the sample mean tends to be a more accurate estimate of the population mean, and the variation around the population mean decreases.
Without calculation, decide if each of the integrals below are positive, negative, or zero. Let W be the solid bounded by z=sqrt(x2+y2) and z=2.1. ∭(z−2)dV2. ∭e−xyzdV3. ∭(z−sqrt(x2+y2))dV
Answer:
1) ∭(z−2) dV negative.
2) ∭e^{−xyz} dV positive.
3) ∭( z-\sqrt{x²+y²}) positive.
Step-by-step explanation:
From Exercise we have:
z=\sqrt{x²+y²}
z=2
⇒2=\sqrt{x²+y²}
4=x²+y²
Therefore, we get that the solid bounded by:
\sqrt{x²+y²}≤z≤2
4=x²+y²
1) From initial condition we have that
\sqrt{x²+y²}≤z≤2
⇒ 2-z≤0
Therefore, we get that the triple integral is
∭(z−2) dV negative.
2) We know that e^{-xyz} is always positive number.
Therefore, we get that the triple integral is
∭e^{−xyz} dV positive.
3) From initial condition we have that
\sqrt{x²+y²}≤z≤2
⇒ z-\sqrt{x²+y²}>0
Therefore, we get that the triple integral is
∭( z-\sqrt{x²+y²}) positive.
The given triple integrals, when evaluated over the defined region, result in the first integral being zero, while the second and third integrals yield positive values. This determination is made based on the properties of the integrand over the given region, without resorting to complete calculation.
Explanation:Triples Integral EvaluationThe question involves the evaluation of triple integrals over a given region W bounded by the surfaces z = sqrt(x^2 + y^2) and z = 2.1. Without explicit calculations, we can determine the sign of the integrals by assessing the integrand function over the specified region.
∭(z − 2)dV: Here, the integral will be zero as over the specified region W, z varies from sqrt(x^2 + y^2) to 2.1 which implies for some portions of W, z-2<0 and for some portions, it's >0. Hence, it cancels each other.∭e^−xyz dV: It will be a positive value since exponential of a number is always positive. ∭(z − sqrt(x^2 + y^2))dV: This integral will be positive since over the region W, z ranges from sqrt(x^2 + y^2) to 2.1, and hence z - sqrt(x^2 + y^2) is always greater than 0.Therefore, the first integral would yield zero, while the second and third integrals would yield positive values.
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Complete the following statements. In general, % of the values in a data set lie at or below the median. % of the values in a data set lie at or below the third quartile (Q3). If a sample consists of 500 test scores, of them would be at or below the median. If a sample consists of 500 test scores, of them would be at or above the first quartile (Q1).
Answer:
Complete the following statements. In general, 50% of the values in a data set lie at or below the median. 75% of the values in a data set lie at or below the third quartile (Q3). If a sample consists of 500 test scores, of them 0.5*500 = 250 would be at or below the median. If a sample consists of 500 test scores, of them 0.75*500 = 375 would be at or above the first quartile (Q1).
Step-by-step explanation:
The median separates the upper half from the lower half of a set. So 50% of the values in a data set lie at or below the median, and 50% lie at or above the median.
The first quartile(Q1) separates the lower 25% from the upper 75% of a set. So 25% of the values in a data set lie at or below the first quartile, and 75% of the values in a data set lie at or above the first quartile.
The third quartile(Q3) separates the lower 75% from the upper 25% of a set. So 75% of the values in a data set lie at or below the third quartile, and 25% of the values in a data set lie at or the third quartile.
The answer is:
Complete the following statements. In general, 50% of the values in a data set lie at or below the median. 75% of the values in a data set lie at or below the third quartile (Q3). If a sample consists of 500 test scores, of them 0.5*500 = 250 would be at or below the median. If a sample consists of 500 test scores, of them 0.75*500 = 375 would be at or above the first quartile (Q1).
Two new wind-farm tower projects are proposed for a small company that installs them in south western Wisconsin. Project A will cost $250,000 to complete and is expected to have an annual net cash flow of $75,000. Project B will cost $150,000 to complete and should generate annual net cash flows of $52,000. As a small company, the owner and senior management team are very concerned about their cash flow. Use the payback period method and determine which project is better from a cash flow standpoint. Can someone show me how I can help my son solve this problem so I can guide him through it?
Answer:
Step-by-step explanation:
The simplified formula for payback period (assuming even cashflow) is as follows:
Payback Period (in years) = Cost / cashflow per year
Generally, the choice which gives the shorter payback period is desired
For project A,
Cost = $250,000 and cashflow per year = $75,000
Payback Period for project A
= 250,000 ÷ 75,000 = 3.333 years
For Project B
Cost = $150,000 and cashflow per year = $52,000
Payback Period for project B
= 150,000 ÷ 52,000 = 2.88 years
comparing the PP for A & B, it is clear that B has the shorter payback period, and hence choice B is more desirable
The simplified formula for the payback period (assuming even cashflow) is as follows:
Payback Period (in years) = Cost / cashflow per year
Generally, the choice which gives the shorter payback period is desired
For project A,
Cost = $250,000 and cashflow per year = $75,000
Payback Period for project A
= 250,000 ÷ 75,000 = 3.333 years
For Project B
Cost = $150,000 and cashflow per year = $52,000
Payback Period for project B
= 150,000 ÷ 52,000 = 2.88 years
comparing the PP for A & B, it is clear that B has the shorter payback period, and hence choice B is more desirable
What is an example of a word problem?Word problem usually consists of mathematical modeling questions, where statistics and information about a sure machine are given and a student is required to expand a version. for example, Jane had $five.00, then spent $2.00. How lots does she have now?
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Which of the following functions are solutions of the differential equation y+4y+4y=0? A. y(x) =e^-2x B. y(x) = e^+22 C. y(x) = xe^-2x D. y(x) = -2x E. y(x) = 0 F. g(x) = x^2e^-2x
Options A, C, and F are solutions to the differential equation.
Explanation:The given differential equation is y''+4y'+4y=0.
To check which of the given functions are solutions of the differential equation, we substitute each function into the equation and check if it satisfies the equation.
By substituting each function and simplifying, we find that options A, C, and F are solutions to the differential equation.
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Find the general solution of the given differential equation. cos(x) dy dx + (sin(x))y = 1
To solve the given differential equation, we employ the method of integrating factors, multiplying the equation by an integrating factor to simplify it. After manipulation, it involves integrating sec(x) to find the general solution, which includes trigonometric and possibly logarithmic functions.
Explanation:The differential equation given is cos(x) dy/dx + (sin(x))y = 1. To find its general solution, we will use the method of integrating factors. This method involves multiplying the entire equation by a function, called the integrating factor, that will allow the left side of the equation to be expressed as the derivative of a product of functions.
Firstly, the differential equation can be rewritten in the form dy/dx + (sin(x)/cos(x))y = 1/cos(x). Notice that the integrating factor, μ(x), can be found using the formula μ(x) = e^(∫ P(x) dx), where P(x) = sin(x)/cos(x). This evaluates to μ(x) = e^(ln|sec(x)|), which simplifies to μ(x) = sec(x).
Multiplying the entire original differential equation by sec(x) leads to sec(x)cos(x) dy/dx + sec(x)sin(x)y = sec(x). This simplifies to dy/dx + tan(x)y = sec(x). Integrating both sides with respect to x, we find the general solution involves an integration of sec(x), which may lead to a solution involving logarithmic and trigonometric functions, depending on the specifics of the integration.
Thus, the general solution to the differential equation involves integrating sec(x) on the right side after applying the integrating factor, leading to a solution of the form y(x) = C*sec(x) + f(x), where C is a constant and f(x) is an integral involving sec(x).
The formula for a buffer solution contains 1.24% w/v of boric acid. How many milli-liters of a 5% w/v boric acid solution should be used to obtain the boric acid needed in preparing 1 liter of the buffer solution?
Answer: 248mL
Step-by-step explanation:
Given:
Concentration of buffer solution Cs = 1.24%w/v of boric acid
Concentration of boric acid solution Cb = 5 % w/v boric acid
For 1 liter of buffer solution, the weight of boric acid needed is:
mb = 1 × 1.24 = 1.24 unit weight
mb = Cb × Vb .....1
Cb = concentration of boric acid solution.
Vb = volume of boric acid solution needed.
mb = weight of boric acid needed.
From equation 1.
Vb = mb/Cb
Vb = 1.24/5
Vb = 0.248L
Vb = 248mL
248 milli-liters of a 5% w / v boric acid solution should be used to obtain the boric acid needed.
Since the formula for a buffer solution contains 1.24% w / v of boric acid, to determine how many milli-liters of a 5% w / v boric acid solution should be used to obtain the boric acid needed in preparing 1 liter of the buffer solution, the following calculation must be performed:
1000 milliliters = 1 liter 5 = 1000 1.24 = X 1.24 x 1000/5 = X 1240/5 = X 248 = X
Therefore, 248 milli-liters of a 5% w / v boric acid solution should be used to obtain the boric acid needed.
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An ice cream cone is filled with vanilla and chocolate ice cream at a ratio of 2:1. If the diameter of the cone is 2 inches and the height is 6 inches, approximately what is the volume of vanilla ice cream in the cone? (round to nearest tenth) A) 1.0 in3 B) 2.1 in3 C) 4.2 in3 D) 6.3 in3
Answer:
D. 6.3 in^3
Step-by-step explanation:
V= 1/3 (3.14)(r^2)(h)
V= 1/3 (3.14) (1^2)(6)
V=6.3 in^3
Answer:
c
Step-by-step explanation: it was on usa test prep and the answer that was there was wrong.
The branch of Statistics concerned with using the sample data to make an inference about a large set of data is called
Answer:
Statistical Inference.
Step-by-step explanation:
Statistical Inference:
It is the procedure for approximating and estimating the parameters of a population with the help of a sample.The sample statistics are used to describe population parameters.It is the practice of forming judgement on the population based on the statistic of sample.Basically with the help of sample we describe a population.Thus,
The branch of Statistics concerned with using the sample data to make an inference about a large set of data is called statistical inference.
Suppose a parallelogram has sides of length a and b. Is this enough information to compute the area of the parallelogram? If not, what additional information is required?
Answer:
No. it's not enough
Step-by-step explanation:
1) No. it's not enough
2) To compute the area of a parallelogram it's necessary the height of the parallelogram, i.e. a perpendicular line segment from the base up to its parallel side.
3) Because a parallelogram is made up by two triangles and the area of it is calculated using the height. Then to cover the area of a parallelogram it's mandatory to calculate the height, the perpendicular distance between their horizontal sides.
No, the lengths of the sides of a parallelogram alone are not enough to compute its area. Additional information, such as the length of the perpendicular distance between the sides, is needed to calculate the area.
Explanation:No, the lengths of the sides of a parallelogram alone are not enough to compute its area. To calculate the area of a parallelogram, you need two pieces of information: the length of one of its sides and the length of the corresponding altitude (or height), which is the perpendicular distance between the side and the opposite side. With just the lengths of a and b, we don't know the height, so we can't calculate the area.
To find the area of a parallelogram, you can use the formula: Area = base x height. The base is one of the side lengths, and the height is the length of the perpendicular from the base to the opposite side.
For example, if side a is the base, the height is the distance between side a and the opposite side, which can be found by drawing a perpendicular line. Once you know the height, you can calculate the area using the formula: Area = a x height.
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In a trial of 167 patients who received 10-mg doses of a drug daily, 45 reported headache as a side effect.
(a) Obtain a point estimate for the population proportion of patients who received 10-mg doses of a drug daily and reported headache as a side effect. (round to two decimal places as needed)
(b) Verify that the requirements for constructing a confidence interval about p are satisfied
Answer:
a) [tex]\hat{p} = 0.27[/tex]
b) All the conditions are met for constructing a confidence interval.
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 167
Number of people who had side effects, x = 45
a) point estimate
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{45}{167} = 0.27[/tex]
The point estimate for the population proportion of patients who received 10-mg doses of a drug daily and reported headache as a side effect is 0.27
b) Conditions for constructing a confidence interval
The data is a random sample from the population of interest.Individual observations can be considered independent.[tex]n\hat{p}>10\\167\times 0.27 = 45.09 > 10\\n(1-\hat{p})>10\\167(1-0.27) = 121.91 > 10[/tex]
Thus, all the conditions are met for constructing a confidence interval.
The standard deviation is expressed in absolute terms (i.e. in the same unit of measure as the data: lbs;inches;dollars;etc) while the coefficient of variation is relative measure?
a. True.
b. False.
Answer:
a) True
Step-by-step explanation:
The given statement is true.
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
The standard deviation express the variation of data from the mean and therefore, have the same unit as the data like lbs, inches, dollars, etc.
Coefficient of variation on the other hand is the ratio of standard deviation and mean. It is also a measure of dispersion but since it is a ratio, the units cancel each other.
[tex]C.V = \dfrac{s}{\bar{x}}[/tex]
Thus, coefficient of variation is dimensionless.
Thus, it is a relative measure.
Consider the following sample set of scores. Assume these scores are from a discrete distribution. 21 29 32 38 38 45 50 64 72 100 For the given data, the mean is ___ , the median is , and the mode is . Suppose the score 21 in the data is mistakenly recorded as 2 instead of 21. For the sample with this error, the mean is , the median is , and the mode is __ . The mean __ , the median ___ , and the mode ___ . Suppose the score 21 in the original sample is inadvertently removed from the sample. For the sample with this score removed, the mean is ___ , the median is __ , and the mode is __ . The mean ___ , the median __, and the mode ___
The mean, median, and mode of the original set of scores are 54.9, 41.5, and 38 respectively. When the score of 21 is recorded as 2, the mean decreases but the median and mode remain unchanged. When the score of 21 is removed, both the mean and median change but the mode stays the same.
Explanation:To answer these questions, we first need to find the mean, median, and mode of the original set of scores. The mean is the sum of the values divided by the total number of values. For the given set of numbers, the mean is 54.9. The median is the number that divides the set of values into two equal halves. In the given set, the median is 41.5. The mode is the number that appears most often, which in this case is 38.
When the score of 21 is mistakenly recorded as 2, the mean decreases to 51.3, the median remains the same, and the mode also stays the same as this change does not affect the frequency of the numbers.
When the score of 21 is removed from the set, the mean decreases to 55.3, and the median increases to 43. Still, the mode remains unchanged as removing a number 21 doesn't change the frequency distribution of the scores.
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On a run, Natasha ran for 1 1/4 hours at a speed of 6 miles per hour. She then rested for 1/4 hour before running another hour at a speed of 5 miles per hour. Which graph best represents the distance she traveled?
Answer:
The answer to your question is See the picture below
Step-by-step explanation:
Analysis of the graphs
a) In the first graph, we observe that Natasha is running after 1 h and 1/4, the comeback for to her origin, finally she continues running forward. This option is incorrect.
b) In the second graph, Natasha runs for 1 1/4 hours, after this time, she comes back to her origin, takes a rest and continues running forward. This answer is incorrect.
c) In the third graph, Natasha runs for 1 1/4 hour, takes a rest and comes back to the origin. This option is incorrect.
d) In the forth graph, Natasha runs for 1 1/4 hour, takes a rest and continues running upwards. This is the right option.
Answer:
the last one to the far right at the bottom
Step-by-step explanation:
Suppose that Britain pegs the pound to gold at six pounds per ounce, whereas the exchange rate between pounds and U.S. dollars is $5 = £1. What would an ounce of gold be worth in U.S. dollars?
Given the exchange rate of $5 = £1 and the conversion rate of 6 pounds per ounce of gold, the equivalent amount for an ounce of gold would be $30.
Explanation:In this scenario, given that the exchange rate between pounds and U.S. dollars is $5 = £1 and the conversion rate of pound to gold is six pounds per ounce, we want to find the value of an ounce of gold in U.S. dollars. Firstly, we need to establish how many pounds per ounce of gold, which is set at six pounds. With the currency exchange rate, $5 equals one pound. Therefore, to find the value in dollars, simply multiply these two rates together. That comes to 6 (pounds per ounce of gold) * 5 (dollars per pound) = $30. Hence, an ounce of gold would be worth $30.
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a caterpillar stated at point(-2.5 -5.5) on a coordinate plane.She crawled in a straight line through the origin to point (45,y)What is y-axis of this?
Answer:
99
Step-by-step explanation:
Since the caterpillar crawled through the origin, her movement can be described by a straight line equation modeled with the points (-2.5; -5.5) and (0; 0).
The slope of a linear equation is given by:
[tex]m=\frac{y-y_0}{x-x_0}\\m=\frac{-5.5-0}{-2.5-0}\\m=2.2[/tex]
For x = 45, the value of y is:
[tex]y= 2.2x\\y= 2.2*45\\y=99[/tex]
The value of the y-axis is 99.
A sample of 60 women is obtained, and their heights ( in inches ) and pulse rates ( in beats per minute ) are measured. The linear correlation coefficient is 0.234 and the equation of the regression line is ^y= 17.5 + 0.850x, where x represents height. The mean of the 60 heights is 63.4 in and the mean of the 60 pulse rates is 75.6 beats per minute. Find the best predicted pulse rate of a woman who is 74 inches tall. Use a significance level of a=0.01.
Answer:
The best predicted pulse rate of a woman who is 74 inches tall is 80.4 (beats per minute).
Step-by-step explanation:
The best predicted pulse rate of a woman who is 74 inches tall is obtained below:
Let x denotes height (in inches) and y denotes pulse rates (in beats per minute).
From the information, the regression equation is, the total number of women is 60 and the linear correlation coefficient between height and pulse rates is 0.234 and the mean of heights is 63.4 and the mean of pulse rates is 75.6. The estimated regression line is
^y= 17.5 + 0.850x
The required best predicted pulse rates is,
=17.5 + (0.850×74)
=17.5 + 62.9
= 80.4 beats per minute
Consider the system of linear equations.
x + y = 9.0 0.50 x + 0.20 y = 3.30
Find the values of x and y
Answer: x = 5.0, y = 4.0
We multiply the second equation by -2.
-2(0.50 x + 0.20 y)= -2(3.30)
-x - 0.40y = -6.60
Then we add it to x + y = 9.0.
So we get: -x - 0.40y + x + y = -6.60 + 9.0
or, 0.60y = 2.4
or, y = 4.0
From x + y = 9.0 we get:
x + 4.0 = 9.0
x = 5.0
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To solve the system of linear equations, we used the substitution method, resulting in the solution x = 5 and y = 4.
Explanation:The system of linear equations provided is:
x + y = 9.00.50 x + 0.20 y = 3.30To find the values of x and y, we can use the method of substitution or elimination. In this case, we'll use substitution:
Rearrange the first equation to isolate y: y = 9 - x.Substitute y in the second equation: 0.50x + 0.20(9 - x) = 3.30.Simplify the second equation: 0.50x + 1.8 - 0.20x = 3.30.Combine like terms: 0.30x + 1.8 = 3.30.Isolate x: 0.30x = 3.30 - 1.8.Solve for x: 0.30x = 1.50.Find x: x = 1.50 / 0.30.x = 5.Substitute x back into y = 9 - x: y = 9 - 5.Find y: y = 4.Therefore, the solution to the system of equations is x = 5 and y = 4.
A study was done to determine the age, number of times per week and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed. 'Duration (amount of time)' is what type of data?
The 'Duration (amount of time)' in the study is an example of quantitative data. Quantitative data is numerical data that can be measured or counted.
Explanation:
The 'Duration (amount of time)' in the study is an example of quantitative data.
Quantitative data is numerical data that can be measured or counted. In this case, the duration of resident use of the local park is being measured in terms of time (amount of time).
Examples of other quantitative data include age and number of times per week.
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The table shows the depth
of a submarine over a 2
hour period. Did the sub rise
or fall in the water? By how
much?
| Time
Depth
10 am
410 ft
12 am
289 ft please help show work or give answer will mark brainest
Answer:
The sub rose 121 feet.
Step-by-step explanation:
The sub was less deep at 12 pm than at 10 am, so rose in the water. The amount it rose was ...
410 -289 = 121 . . . . feet
__
If you like, you can think of the sub's elevation relative to the surface as changing from -410 feet to -289 feet. The difference is then an increase in elevation (rise) of ...
-289 -(-410) = -289 +410 = 121 . . . feet
_____
12 am is midnight, 14 hours after 10 am. The time 2 hours after 10 am is 12 pm, noon.
Seventeen percent of the plants in the greenhouse are broad leaf plants and the rest are grasses, What percentage of the plants are grasses?
Answer:The percentage of the plants that are grasses is 83%
Step-by-step explanation:
Seventeen percent of the plants in the greenhouse are broad leaf plants and the rest are grasses
The total number of grasses and broad leaf plants in the greenhouse is 100%. Therefore,
The percentage of the plants that are grasses would be
100% - 17% = 83%
Final answer:
By subtracting the percentage of broad-leaf plants (17%) from the total (100%), we find that 83% of the plants in the greenhouse are grasses.
Explanation:
If seventeen percent of the plants in the greenhouse are broad-leaf plants, then the remaining percentage of plants must be grasses. Since the total percentage must add up to 100%, we can subtract the percentage of broad-leaf plants from the total to find out the percentage of grasses.
To calculate the percentage of grasses, we use the following steps:
Start with the total percentage which is 100%.
Subtract the percentage of broad-leaf plants which is 17%.
The result is the percentage of grasses.
100% - 17% = 83%
Therefore, 83% of the plants in the greenhouse are grasses.
What function do you know from calculus is such that its first derivative is a constant multiple k of itself
Answer:
Exponential function if the form [tex]e^{Kx}[/tex] is the constant multiple K of itself if.
[tex]y'=\frac{dy}{dx}=cKe^{Kx} \\y'=Ky[/tex]
Step-by-step explanation:
The exponential function of the form [tex]e^x[/tex] is the function which is the derivative of itself.
Where:
x is independent variable
Now if we talk about constant then again exponential function if the form [tex]e^{Kx}[/tex] is the constant multiple K of itself if we take the dervative.
Mathmatical Prove:
Consider the general equation
let [tex]y=ce^{Kx}[/tex]
Where:
K is a constant
c is the cofficient(could be any number)
Now:
Taking derivative of above equation w.r.t x:
[tex]y'=\frac{dy}{dx}=cKe^{Kx} \\y'=Ky[/tex]
Hence proved exponential function is a constant multiple K of it self.
The function is [tex]f(x) = Ae^kx[/tex]. This function is a solution of the differential equation y' = ky.
We know that the derivative of the exponential function is the function itself. Consider the function [tex]f(x) = Ae^kx[/tex], where A is a constant. Then, take the derivative of [tex]f(x) = Ae^kx[/tex], we get:
[tex]f'(x) = Ake^{kx} = k (Ae^{kx}) = k f(x)[/tex]
Thus, a function whose first derivative is a constant multiple k of itself is[tex]f(x) = Ae^{kx}[/tex], where A is a constant.
Since [tex]f'(x) = k f(x)[/tex] for the function [tex]f(x) = Ae^{kx}[/tex], this function must be the solution of the differential equation y' = ky.
Complete question:
What function do you know from calculus is such that its first derivative is a constant multiple k of itself? (Do not use the function f(x)=0.)
f(x)=____
The above function is a solution of which of the following differential equations?
[tex]y'=y^k[/tex]
[tex]y'=y+k[/tex]
[tex]y'=ky[/tex]
[tex]y'=e^{ky}[/tex]
[tex]y'=k[/tex]
33. The standard deviation divided by the square root of the sample size n is equal to the _______________ of the mean. Sample error Sampling error Critical value Standard error
Answer:
Standard error
Step-by-step explanation:
Standard error: In statistics, the term standard error is also denoted as SE, and is reffered to as the estimate of a particular parameter.
It is defined as the standard deviation in statistics related to its sampling distribution or else the the standard deviation's estimate.
Therefore, if a particular parameter or statistic is considered as a mean then it would be denoted as the standard error of that mean.
The standard error of an estimate may also be defined as the square root of the estimated error variance √[tex]\sigma^{2}[/tex] of the quantity,
[tex]S_{e}[/tex] =√[tex]\sigma^{2}[/tex]
In the given question, the appropriate answer would be standard error.
The names of 50 contestants are written on 50 cards. The cards are placed in a bag, and three names are picked from the bag. Identify which sampling technique is used?
The sampling technique described is called 'sampling without replacement'. In this case, names are drawn from the bag and not returned, decreasing the pool for subsequent draws. It's a type of 'simple random sampling'.
Explanation:The sampling technique used when 50 contestants’ names are written on 50 cards, placed in a bag, and three are chosen, is a method called sampling without replacement. In this process, once a name is chosen, it is not returned to the bag, reducing the pool of options for subsequent selections. As each card is drawn, the remaining pool of names shrinks, meaning the probability of drawing any remaining name changes with each draw. This process is often used when the desire is to avoid repetition in the selection.
In contrast, sampling with replacement would mean that after a card is drawn, it is put back into the bag before the next draw, ensuring that the same name could potentially be drawn more than once. Both methods are types of simple random sampling, a fundamental sampling method in statistics.
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Under what circumstances is the median likely to produce a better measure of central tendency than the mean?
Answer:
With a skewed distribution and data with outliers.
Step-by-step explanation:
There are three measures of central tendency.
Mean: It is the average value of data and affected by presence of outliers.Median: It is the value that divides the data into two equal parts. It is a position based measure of tendency.Mode: It is the most frequent observation in the data.For a symmetrical distribution for continuous data, the mean, median, and mode are equal. For such case mean is a better measure of central tendency because it includes all of the data in the calculations. Median is the best measure of central tendency when the data is not symmetrical because the median is position based.Advantage of the median:The median is not affected by outliers and skewed data as compared to the mean.
Thus, median likely to produce a better measure of central tendency than the mean with a skewed distribution and data with outliers.
Shortly after September 11th 2001, a researcher wanted to determine if the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq. In a sample of 73 females, 28 favored war with Iraq. In a sample of 54 males, 29 favored war with Iraq.
a) Let pF represent the proportion of females that favor the war, pM represent the proportion of males that favor the war. What are the proper hypotheses?
A.)H0: pF = pM versus Ha: pF > pM
B.)H0: pF < pM versus Ha: pF = pM
C.)H0: pF = pM versus Ha: pF < pM
D.)H0: pF = pM versus Ha: pF ≠ pM
b) What is the test statistic? Compute the statistic using male statistics subtracted from female statistics. Give your answer to four decimal places.
c) What is the P-value for the test? Give your answer to four decimal places.
d) Using a 0.01 level of significance, what conclusion should be reached?
A.)The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is less than 0.01.
B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
C.)The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is greater than 0.01.
D.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is less than 0.01.
e) What is the lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.
f) What is the upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.
Answer:
a) D.)H0: pF = pM versus Ha: pF ≠ pM
b) [[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
c) [tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
d) B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
e) [tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
f) [tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
Step-by-step explanation:
1) Data given and notation
[tex]X_{M}=28[/tex] represent the number of men that favored war with Iraq
[tex]X_{W}=29[/tex] represent the number of women that favored war with Iraq
[tex]n_{M}=73[/tex] sample of male selected
[tex]n_{W}=54[/tex] sample of female selected
[tex]p_{M}=\frac{28}{73}=0.384[/tex] represent the proportion of men that favored war with Iraq
[tex]p_{W}=\frac{29}{54}=0.537[/tex] represent the proportion of women that favored war with Iraq
[tex]\alpha=0.01[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
Part a
We need to conduct a hypothesis in order to checkif the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq , the system of hypothesis would be:
Null hypothesis:[tex]p_{M} = p_{W}[/tex]
Alternative hypothesis:[tex]p_{M} \new p_{W}[/tex]
The best option is:
D.)H0: pF = pM versus Ha: pF ≠ pM
Part b
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{W}-p_{M}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{28+29}{73+54}=0.449[/tex]
Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
Part c
We have a significance level provided [tex]\alpha=0.01[/tex], and now we can calculate the p value for this test.
Since is a one two sided test the p value would be:
[tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
Part d
If we compare the p value and the significance level given [tex]\alpha=0.01[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and the best conclusion would be:
B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
Part e
The confidence interval for the difference of two proportions would be given by this formula
[tex](\hat p_W -\hat p_M) \pm z_{\alpha/2} \sqrt{\frac{\hat W_A(1-\hat p_W)}{n_W} +\frac{\hat p_M (1-\hat p_M)}{n_M}}[/tex]
For the 99% confidence interval the value of [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=2.58[/tex]
And replacing into the confidence interval formula we got:
[tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
Part f
[tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
The proper hypotheses are H0: pF = pM and Ha: pF ≠ pM. The test statistic is -0.087. The p-value is 0.1651.
Explanation:a) The proper hypotheses for this situation are:
H0: pF = pM
Ha: pF ≠ pM
Hence, the correct option is D.
b) The test statistic is computed by subtracting the male statistic from the female statistic:
Test statistic = pF - pM = 28/73 - 29/54 = -0.087
c) The p-value for the test is computed using the test statistic and the appropriate test statistic distribution. Based on the given values, the p-value is found to be 0.1651.
d) Using a 0.01 level of significance, the conclusion should be: The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the p-value is greater than 0.01. Hence, the correct option is D.
e) The lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is -0.2302.
f) The upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is 0.0566.
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Systematic Sampling Technique is used to select 5 numbers between 1 and 200. The first selected number is 12. What should be next four numbers?
Answer: 52, 92 , 132, 172
Step-by-step explanation:
A systematic random sampling is a random sampling technique in which a sample from a large population are chosen according to a random beginning point and a fixed periodic interval (k).Given : Systematic Sampling Technique is used to select 5 numbers between 1 and 200.
⇒ Population size : N=200
Required Sample size : n=5
Since , Sampling interval is given by :-
[tex]k=\dfrac{N}{n}[/tex]
⇒ [tex]k=\dfrac{200}{5}[=40/tex]
If the first selected number is 12, then the next four numbers would be :
(12+K) , (12+2k) , (12+3k) , (12+4k)
Put value of k , we get
(12+40) , (12+2(40)) , (12+3(40)) , (12+4(40)) =(52, 92 , 132, 172)
Hence, the next four numbers are : 52, 92 , 132, 172
Identify the following measures as either quantitative or qualitative:
a. The 30 high-temperature readings of the last 30 days.
b. The scores of 40 students on an English test.
c. The blood types of 120 teachers in a middle school.
d. The last four digits of social security numbers of all students in a class.
e. The numbers on the jerseys of 53 football players on a team.
Answer:
a. Quantitative
b. Quantitative
c. Qualitative
d. Qualitative
e. Qualitative
Step-by-step explanation:
a. The temperature readings can be quantify and can be represented numerically so, it is quantitative variable.
b. The scores in English test can be quantify and can be represented numerically so, it is quantitative variable.
c. The blood type can be categorized as O plus, O negative, A plus, A negative, AB plus, Ab negative, B plus, B negative and cannot be represented numerically so, it is qualitative variable.
d. The last four digit of social security number take numerical values, yet they are qualitative because these are used as identifiers and mathematical operations of social security don't have meaningful interpretation.
e. The number on the jerseys of 53 football players also takes numerical values yet they are qualitative because these are used as identifiers for players and mathematical operations of number on the jerseys of football players don't have meaningful interpretation.
The measures in question are classified as either quantitative or qualitative.
Explanation:a. The 30 high-temperature readings of the last 30 days - Quantitative. These temperatures are numerical values that can be measured and compared.
b. The scores of 40 students on an English test - Quantitative. The scores represent numerical values that can be added, averaged, and compared.
c. The blood types of 120 teachers in a middle school - Qualitative. Blood types are descriptive categories or labels that cannot be measured numerically.
d. The last four digits of social security numbers of all students in a class - Qualitative. The last four digits are specific identifiers that cannot be measured or compared numerically.
e. The numbers on the jerseys of 53 football players on a team - Qualitative. The numbers are labels or identifiers, not measurable quantities.
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how many different license plates are possible if each contains 2 letters(out of the alphabet’s 26 letters) followed by 5 digits (from 0 to 9)? how many of these license plates contain no repeated letters and no repeated digits?
The total number of different license plates possible with 2 letters and 5 digits is 67,600,000. If no letter or digit can be repeated, the number reduces to 32,760,000.
Explanation:This problem involves combinations and permutations, which are fundamental concepts in discrete mathematics. If a license plate has 2 letters followed by 5 digits, each chosen independently, we consider the product of the number of possibilities for each space. There are 26 possible letters (A-Z) for each of the 2 letter spaces and 10 possible digits (0-9) for each of the 5 digit spaces. Therefore, the total number of different license plates is 26 * 26 * 10 * 10 * 10 * 10 * 10 = 67,600,000.
However, if no letter or digit can be repeated, the number of possibilities for the second letter space becomes 25 (since one letter has already been used), and similarly for the digits, it gets reduced by one with each additional digit. Hence, the total number of different license plates without repeated letters and digits would be 26 * 25 * 10 * 9 * 8 * 7 * 6 = 32,760,000.
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There are 67,600,000 total possible license plates with two letters and five digits. When restricting to plates with no repeated letters or digits, there are 165,765,000 possible plates.
Explanation:The possible license plates can be calculated using the principle of multiplication. This principle states that if there are m ways to do one thing, and n ways to do another, then there are m*n ways to do both. In this case, there are 26 options for the first letter, another 26 for the second letter, and 10 options for each of the 5 digits (0-9). So, we can multiply these options to get our answer.
To calculate the number of possible combinations, we use the formula 26*26*10*10*10*10*10 = 67,600,000. Therefore, there are 67,600,000 possible license plates.
For the plates with no repeated letters or numbers, we still use the principle of multiplication but reduce each subsequent option by one. It will be 26*25 (because one letter has been used already and it can't be reused) *10*9*8*7*6 (same as for digits). Our formula is 26*25*10*9*8*7*6 = 165,765,000 plates.
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A car rental company charges a one-time application fee of 20 dollars, 45 dollars per day, and 15 cents per mile for its cars. Write a formula for the cost, C, of renting a car as a function of the number of days, d, and the number of miles driven, m.
Answer:
[tex]C(d,m) = 20 + 45d + 0.15m[/tex]
Step-by-step explanation:
In this problem, we have that the cost of renting a car C is a function of the number of days d, and the number of miles driven, m.
There is also a fixed cost F.
So the equation for the cost of renting a car is:
[tex]C(d,m) = F + a*d + b*m[/tex]
In which a is the daily cost and b is the cost per mile.
A car rental company charges a one-time application fee of 20 dollars, 45 dollars per day, and 15 cents per mile for its cars.
This means that [tex]F = 20, a = 45, b = 0.15[/tex]
Write a formula for the cost, C, of renting a car as a function of the number of days, d, and the number of miles driven, m.
[tex]C(d,m) = 20 + 45d + 0.15m[/tex]
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
a. Is it possible to determine by how much the mean changes? If so, by how much does it change?
b. Is it possible to determine the value of the mean after the change? If so, what is the value?
c. Is it possible to determine by how much the median changes? If so, by how much does it changes?
d. Is it possible to determine by how much the standard deviation changes? If so, by how much does it change?
Answer:
Step-by-step explanation:
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
a. Is it possible to determine by how much the mean changes? If so, by how much does it change?
Yes. The total would decrease by 11.61. so mean would decrease by 11.61/15 =0.774
b. Is it possible to determine the value of the mean after the change? If so, what is the value?
No. new men would be old mean - 0.774
c. Is it possible to determine by how much the median changes? If so, by how much does it changes?
Though the smallest number is changed the position does not change.
Hence median will not change.
d. Is it possible to determine by how much the standard deviation changes? If so, by how much does it change?
The original variance would be average of square of deviations of the old mean.
Yes, it possible to determine by how much the mean changes. so, 0.7741 change in mean.
No, it is not possible to determine the value of the mean after the change because the value of mean is unknown before the mean
No, it possible to determine by how much the median changes. so, the value of mean will remain same.
No it is impossible to determine how much the standard deviation changes .
Given that,
There are 15 numbers on a list, and the smallest number is changed from 12.9 to 1.29.
We have to find,
All the statements about mean and median are correct or not.
According to the question,
The total would decrease by 11.61.Then,
Mean would decrease by,
[tex]= \dfrac{11.61}{15} \\\\= 0.774[/tex]1
Yes, it possible to determine by how much the mean changes. so, 0.7741 change in mean.
New men would be old mean - 0.7741.
No, it is not possible to determine the value of the mean after the change because the value of mean is unknown before the mean.
Though the smallest number is changed the position does not change.
Yes, It is possible to determine by how much the median changes because we have 15 numbers in a data set and doesn’t effect the median so the median didn’t change.
Hence median will not change.
No, it possible to determine by how much the median changes. so, the value of mean will remain same.
The original variance would be average of square of deviations of the old mean.No it is impossible to determine how much the standard deviation changes .
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