Answer:
a) [tex]x(t) = 13*e^(^-^\frac{t}{100}^)[/tex]
b) 10.643 kg
Step-by-step explanation:
Solution:-
- We will first denote the amount of salt in the solution as x ( t ) at any time t.
- We are given that the Pure water enters the tank ( contains zero salt ).
- The volumetric rate of flow in and out of tank is V(flow) = 50 L / min
- The rate of change of salt in the tank at time ( t ) can be expressed as a ODE considering the ( inflow ) and ( outflow ) of salt from the tank.
- The ODE is mathematically expressed as:
[tex]\frac{dx}{dt} =[/tex] ( salt flow in ) - ( salt flow out )
- Since the fresh water ( with zero salt ) flows in then ( salt flow in ) = 0
- The concentration of salt within the tank changes with time ( t ). The amount of salt in the tank at time ( t ) is denoted by x ( t ).
- The volume of water in the tank remains constant ( steady state conditions ). I.e 10 L volume leaves and 10 L is added at every second; hence, the total volume of solution in tank remains 5,000 L.
- So any time ( t ) the concentration of salt in the 5,000 L is:
[tex]conc = \frac{x(t)}{1000}\frac{kg}{L}[/tex]
- The amount of salt leaving the tank per unit time can be determined from:
salt flow-out = conc * V( flow-out )
salt flow-out = [tex]\frac{x(t)}{5000}\frac{kg}{L}*\frac{50 L}{min}\\[/tex]
salt flow-out = [tex]\frac{x(t)}{100}\frac{kg}{min}[/tex]
- The ODE becomes:
[tex]\frac{dx}{dt} = 0 - \frac{x}{100}[/tex]
- Separate the variables and integrate both sides:
[tex]\int {\frac{1}{x} } \, dx = -\int\limits^t_0 {\frac{1}{100} } \, dt + c\\\\Ln( x ) = -\frac{t}{100} + c\\\\x = C*e^(^-^\frac{t}{100}^)[/tex]
- We were given the initial conditions for the amount of salt in tank at time t = 0 as x ( 0 ) = 13 kg. Use the initial conditions to evaluate the constant of integration:
[tex]13 = C*e^0 = C[/tex]
- The solution to the ODE becomes:
[tex]x(t) = 13*e^(^-^\frac{t}{100}^)[/tex]
- We will use the derived solution of the ODE to determine the amount amount of salt in the tank after t = 20 mins:
[tex]x(20) = 13*e^(^-^\frac{20}{100}^)\\\\x(20) = 13*e^(^-^\frac{1}{5}^)\\\\x(20) = 10.643 kg[/tex]
- The amount of salt left in the tank after t = 20 mins is x = 10.643 kg
If y varies directly as x, and y is 48 when x is 6, which expression can be used to find the value of y when x is 2?
Answer:
When x is 2, y is 16
Step-by-step explanation:
If y is 48 and x is 6, then y is 8 when x is 1.
Because of this, when x is 2, y will be 16.
Please mark Brainliest
01:30:4
Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26. What is the
solution set of this problem?
Answer:
x<_ 21
Step-by-step explanation:
5(x+27)>_ =6(x+26)
5x +135 >_ 6x +156
5x >_6x +21
-x>_21
x<_21
Do all perpendicular lines have negative reciprocal slopes?
Not necessarily, the more correct definition is opposite reciprocal slopes.
The example used is how horizontal and vertical lines are parallel. Horizontal lines have a slope of 0, also written as 0/1. However, vertical lines have an undefined slope, which isn't necessarily negative. It has a slope of 1/0, which is undefined. In this case, the reciprocal isn't negative.
In all other cases (1 and -1, 2 and -1/2, etc.) yes, the perpendicular pairs are negative and reciprocal.
please help me on this work please !
Suppose that a researcher is planning a new study on hemoglobin levels amongst women under 25 years old. Previous research suggest that the standard deviation of hemoglobin is 0.7 g/dl. In the new study the research wants to have the standard error for the sample mean to be no more than 0.05 g/dl. Find the required sample size for the new study.
Answer:
A sample size of at least 531 is required.
Step-by-step explanation:
We are lacking the confidence level to solve this question, so i am going to use a 90% confidence level.
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.9}{2} = 0.05[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.05 = 0.95[/tex], so [tex]z = 1.645[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
Find the required sample size for the new study.
A sample size of at least n is required.
n is found when [tex]M = 0.05[/tex]
We have that [tex]\sigma = 0.7[/tex]
So
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
[tex]0.05 = 1.645*\frac{0.7}{\sqrt{n}}[/tex]
[tex]0.05\sqrt{n} = 1.645*0.7[/tex]
[tex]\sqrt{n} = \frac{1.645*0.7}{0.05}[/tex]
[tex](\sqrt{n})^{2} = (\frac{1.645*0.7}{0.05})^{2}[/tex]
[tex]n = 530.4[/tex]
Rounding up
A sample size of at least 531 is required.
What’s the correct answer for this question?
Answer:
D
Step-by-step explanation:
It would be a cone with a radius of 4 units rotating around y-axis.
The web publisher www.exploreiceland.is (Links to an external site.)Links to an external site. provides information on traveling to Iceland. Access to the website is free but revenues are generated by selling ads that are posted on the website. In the following month, the website has committed to displaying ads to 650,000 viewers, i.e., 650,000 impressions. Based on data from previous months the traffic to the website is estimated to be normally distributed with a mean of 850,000 viewers and a standard deviation of 150,000.
How many impressions should the web publisher have taken on, to be able to guarantee a 95% service level?
Answer:
1096750 impressions
Step-by-step explanation:
Given that :
Mean = 850,000
Standard deviation = 150,000
If we assume that X should be the numbers of impressions created;
Then ;
[tex]X \approx N (\mu , \sigma^2)[/tex]
Now ; representing x as the value for the number of impression needed ; Then ;
[tex]P(X>x) = 0.95[/tex]
[tex]P(\dfrac{X- \mu}{\sigma} > \dfrac{x -850000}{150000}) = 0.95[/tex]
[tex]P(Z> \dfrac{ x -850000}{150000}) = 0.95[/tex]
From normal tables:
[tex]P(Z >1.645) = 0.95[/tex]
[tex]\dfrac{x - 850000}{150000} =1.645[/tex]
(x- 850000) = 1.645(150000)
x - 850000 = 246750
x = 246750 + 850000
x = 1096750 impressions
If theta=3pi/4
Sin theta=?
Cos theta=?
Answer:
For ease of writing, θ [tex]=x[/tex]
[tex]sin(x)=\frac{1}{\sqrt{2} }[/tex]
[tex]cos(x)=-\frac{1}{\sqrt{2} }[/tex]
Step-by-step explanation:
Our angle is [tex]x=\frac{3\pi }{4}[/tex]
To find our answers for [tex]sin(\frac{3\pi}{4} )[/tex] and [tex]cos(\frac{3\pi}{4} )[/tex], we will need to use a unit circle. (I have attached the image of one).
Recall that the [tex]sin[/tex] of an angle is equal to the y-value of the corresponding ordered pair.
And the [tex]cos[/tex] of an angle is equal to the x-value of the corresponding ordered pair.
For the angle [tex]x=\frac{3\pi }{4}[/tex], the ordered pair is [tex](-\frac{1}{\sqrt{2}} }, \frac{1}{\sqrt{2} } )[/tex]
This means that
[tex]sin(x)=\frac{1}{\sqrt{2} }[/tex]
[tex]cos(x)=-\frac{1}{\sqrt{2} }[/tex]
In the western region of the state the times of all boys running in this event are normally distributed with standard deviation 12 seconds, but with mean 5 minutes 22 seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet. (Hint: normalcdf)
Here is the full question.
The average finishing time among all high school boys in a particular track event in a certain state is 5 minutes 17 seconds. Times are normally distributed with standard deviation 12 seconds.
A. The qualifying time in this event for participation in the state meet is to be set so that only the fastest 5% of all runners qualify. Find the qualifying time in seconds (round it to the closest second). (Hint: Convert minutes to seconds.)
B. In the western region of the state the times of all boys running in this event are normally distributed with standard deviation 12 seconds, but with mean 5 minutes 22 seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet. (Hint: normalcdf)
Answer:
a. x ≅ 337 seconds.
b. P(x > 337 ) = 0.1056
Step-by-step explanation:
A.
Given that ;
Mean [tex]\mu =[/tex] 5 minutes 17 seconds =( (60× 5)+17 ) seconds = 317 seconds ( since 60 seconds make 1 minute.
Standard deviation: [tex]\sigma[/tex] = 12 seconds.
Only the fastest 5% of all runners qualify
The objective is to determine the qualifying time in seconds
Let's look for the Z-score of 0.95;
The Z-score is 1.645 from the tables
[tex]x= ( \sigma * z ) + \mu[/tex]
[tex]x = ( 12 * 1.645 ) + 317 \\ \\x = 336.74[/tex]
x ≅ 337 seconds.
B. Given that the standard deviation = 12 seconds
Mean = 5 minutes 22 seconds = (5 × 60 + 22 )seconds = 322 seconds
he objective is to find P(x > 337 ) i.e the proportion of boys from this region who qualify to run in this event in the state meet.
we are using command normalcdf (SEE THE ATTACHED FILE BELOW FOR THE COMPUTATION)
we have P(x > 337 ) = 0.1056
You have already a 1000 cash on your hand. You have also a 1000 cash in bank and you withdraw half of 500, so, it means you withdraw 250. Now, you have a 1000 cash on hand plus 250 that you have been withdraw from the bank, all in all you have 1250 now in your hand.
Answer: At the end, you have $750 remaining in the bank.
Step-by-step explanation:
I guess you want to know how much is left in the bank.
initially
Hand : $1000
Bank: $1000
you withdraw $250 from the bank:
Hand: $1000 + $250 = $1250
Bank: $1000 - $250 = $750
Find the diameter and radius of a circle with a circumference of 65.98 Please help
Answer:
21 and 10.5 respectively
Step-by-step explanation:
Remember circumference of a circle is given as;
C= 2×π×r; r is raduis
r = C / 2×π
=65.98/(2×3.142)= 10.50
D= 2× r = 2× 10.50= 21.0( D represent diameter)
Note π = 3.142 a known constant
Solve the equation and state a reason for each step.
23+11a-2c=12-2c
Simplifying
23 + 11a + -2c = 12 + -2c
Add '2c' to each side of the equation.
23 + 11a + -2c + 2c = 12 + -2c + 2c
Combine like terms: -2c + 2c = 0
23 + 11a + 0 = 12 + -2c + 2c
23 + 11a = 12 + -2c + 2c
Combine like terms: -2c + 2c = 0
23 + 11a = 12 + 0
23 + 11a = 12
Solving
23 + 11a = 12
Solving for variable 'a'.
Move all terms containing a to the left, all other terms to the right.
Add '-23' to each side of the equation.
23 + -23 + 11a = 12 + -23
Combine like terms: 23 + -23 = 0
0 + 11a = 12 + -23
11a = 12 + -23
Combine like terms: 12 + -23 = -11
11a = -11
Divide each side by '11'.
a = -1
Simplifying
a = -1
There are 20 pieces of fruit in a bowl and 5 of them are apples. What percent of the fruit are apples?
Step-by-step explanation:
20fruits=100%
5fruits=?
5x100/20 5fruitsx5%
=24%
find the quotient of 25.5÷0.5
Answer:
[tex]51[/tex]
Step-by-step explanation:
[tex]\frac{25.5}{0.5}[/tex]
[tex]\frac{255}{5}[/tex]
[tex]=51[/tex]
g Gravel is being dumped from a conveyor belt at a rate of 10 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 6 ft high
Answer:
0.3537 feet per minute.
Step-by-step explanation:
Gravel is being dumped from a conveyor belt at a rate of 10 ft3/min. Since we are told that the shape formed is a cone, the rate of change of the volume of the cone.
[tex]\dfrac{dV}{dt}=10$ ft^3/min[/tex]
[tex]\text{Volume of a cone}=\dfrac{1}{3}\pi r^2 h[/tex]
If the Base Diameter = Height of the Cone
The radius of the Cone = h/2
Therefore,
[tex]\text{Volume of the cone}=\dfrac{\pi h}{3} (\dfrac{h}{2}) ^2 \\V=\dfrac{\pi h^3}{12}[/tex]
[tex]\text{Rate of Change of the Volume}, \dfrac{dV}{dt}=\dfrac{3\pi h^2}{12}\dfrac{dh}{dt}[/tex]
Therefore: [tex]\dfrac{3\pi h^2}{12}\dfrac{dh}{dt}=10[/tex]
We want to determine how fast is the height of the pile is increasing when the pile is 6 feet high.
[tex]When h=6$ feet$\\\dfrac{3\pi *6^2}{12}\dfrac{dh}{dt}=10\\9\pi \dfrac{dh}{dt}=10\\ \dfrac{dh}{dt}= \dfrac{10}{9\pi}\\ \dfrac{dh}{dt}=0.3537$ feet per minute[/tex]
When the pile is 6 feet high, the height of the pile is increasing at a rate of 0.3537 feet per minute.
When writing expressions for complex numbers, what does i represent?
Answer:
see below
Step-by-step explanation:
i is the imaginary number and it represents the square root of -1
The population of bats in a large cave is 7000 and is growing exponentially at 14% per year. Write a function to represent the population of bats after
tt
t years, where the monthly rate of change can be found from a constant in the function.
Answer:
y=7000+14^t
Step-by-step explanation:
This equation shows that the original population of bats was 7000 and grows exponentially at a rate of 14% per year.
I put the graph below so you can see it.
In this exercise we have to identify how to write an exponential function from the data informed in the text, in this way we find that:
[tex]y=7000+14^t[/tex]
From the information given in the statement we find that:
The original population of bats was 7000Rate of 14% per year.Then writing this function as:
[tex]y=7000+14^t[/tex]
See more about function at brainly.com/question/5245372
Describe the rate of change of f(x)=lnx. Your answer should explain how the slope changes when x is small and when x is large.
Answer:
By plotting the graph of f(x)=lnx, you can conclude that when x is small, dy/dx has a larger value. For instance, the gradient of the curve when x=0.5 is 2. However, as you move along the x axis, you will see that the graph levels off, indicating a decrease in the slope, or dy/dx. For example, if x=10, dy/dx = 0.1 and when x=20, dy/dx= 0.05 and so on. Eventually, when x is large enough the value of dy/dx will be negligible.
Thus, as x increases, the slope decreases.
Answer:
Explanation shown below
Step-by-step explanation:
f(x)=lnx;
The rate of change is defined as dy/dx;
dy/dx[Inx] = 1/x
and dy/dx is defined as the slope
The nature of the slope is as x increases ; the slope decreases and conversely meaning as x decreases, the slope increases.
There are 225 students at March middle school. On Friday, 135 students wore spirit shirts. What percent of the students did Not wear spirit shirts on Friday?
Answer:
40%
Step-by-step explanation:
To find the answer to this, you first can find out what percentage of students did wear spirit shirts. To do this you can divide 135 by 225 to give you 0.6. To convert the decimal into a percentage you can simply multiply by 100, giving you 60%. Then to find the percentage of students that did not wear spirit shirts, you can subtract 60 from 100, giving you 40%.
Ronnie invested $1500 in an account that earns 3.5% interest, compounded annually. The formula for compound interest is A(t) = P{(1 + i)^t}A(t)=P(1+i) t . How much did Ronnie have in the account after 4 years?
Answer:
BStep-by-step explanation:
A= New amount
P= Principal or Original amount which is £1500
I= Interest
t= time period
3.5% as a decimal is 3.5÷100=0.035
time period= 4 years
so 1500(1+0.035)^4 = B
A stone is thrown vertically into the air at an initial velocity of 79 ft/s. On a different planet, the height s (in feet) of the stone above the ground after t seconds is sequals79tminus3t squared and on Earth it is sequals79tminus16t squared. How much higher will the stone travel on the other planet than on Earth?
Answer:
[tex]13t^2[/tex] feet higher the stone will travel on the other plant than on Earth.
Step-by-step explanation:
Initial velocity of the stone thrown vertically = 79 ft/s
It is given that:
Height attained on a different planet with time [tex]t[/tex]:
[tex]s_p = 79t -3t^2[/tex]
Height attained on Earth with time [tex]t[/tex]:
[tex]s_e = 79t -16t^2[/tex]
If we have a look at the values of [tex]s_p\text{ and }s_e[/tex], it can be clearly seen that the part [tex]79t[/tex] is common in both of them and some values are subtracted from it.
The values subtracted are [tex]3t^2\text{ and } 16t^2[/tex] respectively.
[tex]t^2[/tex] can never be negative because it is time value.
So, coefficient of [tex]t^2[/tex] will decide which is larger value that is subtracted from the common part i.e. [tex]79t[/tex].
Clearly, [tex]3t^2\text{ and } 16t^2[/tex] have [tex]16t^2[/tex] are the larger value, hence [tex]s_e < s_p[/tex].
So, difference between the height obtained:
[tex]s_p - s_e = 79t - 3t^2 - (79t - 16t^2)\\\Rightarrow 79t -3t^2 - 79t + 16t^2\\\Rightarrow 13t^2[/tex]
So, [tex]13t^2[/tex] feet higher the stone will travel on the other plant than on Earth.
What is the value of d21+d22+d23 given the matrix equation below?
Answer:
B. 8
Step-by-step explanation:
The question lacks the required diagram. Find the diagram in the attachment.
Before we can find d21, d22 and d23, we need to get the matrix D first as shown in the attached solution.
On comparison as shown in the attachment, d21 = 11, d22 = -10 and d23 = 7
Note that d21 refers to element in the second row and first column of the matrix
d22 is the element in the second row and second column of the matrix
d23 is the element in the second row and third column of the matrix
d21+d22+d23 = 11-10+7
d21+d22+d23 = 8
The second option is correct.
A hose fills a hot tub at a rate of 3.84 gallons per minute. How many hours will it take to fill a 305-gallon hot tub?
Answer:
1.56 hours
Step-by-step explanation:
300 gal × 1 min 3.2 gal × 1 hr 60 min = 1.56 hr.
Answer:
i thought the question said a 'HORSE' fills a hot tub...
Step-by-step explanation:
lol dont mind me i just want points :D
Steve wants to use his 18% employee discount to buy a video game that has a regular price of $69.99. A 6.5% sales tax is applied to the discounted price. How much will he pay for the game, including sales tax?
Answer:
$61.12
Step-by-step explanation:
Price of the Video game = $69.99
Discount = 18%
Discount price = 18% of $69.99
= $69.99*18/100
= $12.6
Price after Discount = Price - Discount price
= $69.99 - $12.6
= $57.39
Sales tax = 6.5% applied to the discounted price
= 6.5% of $57.39
sales tax in dollars = $57.39 * 6.5/100
= $3.73
The amount he pays for the game = $57.39 + $3.73
= $61.12
Find the exact length of the third side. (Pythagorean Theorem)
Answer:
3 sqrt(5) =c
Step-by-step explanation:
We can use the pythagorean theorem
a^2 + b^2 = c^2
3^2 + 6^2 = c^2
9+36 = c^2
45 = c^2
Take the square root of each side
sqrt(45) = sqrt(c^2)
sqrt(9)sqrt(5) = c
3 sqrt(5) =c
Evaluate the function h(t) =|t +2| + 3; find h(6)
Answer:
11
Step-by-step explanation:
h(t) =|t +2| + 3
Let t=6
h(6) =|6 +2| + 3
= |8| +3
=8+3
= 11
A university warehouse has received a shipment of 25 printers, of which 10 are laser printers and 15 are inkjet models. If 6 of these 25 are selected at random to be checked by a particular technician, what is the probability that exactly 3 of those selected are laser printers (so that the other 3 are inkjets)
Answer:
The probability is 0.31
Step-by-step explanation:
To find the probability, we will consider the following approach. Given a particular outcome, and considering that each outcome is equally likely, we can calculate the probability by simply counting the number of ways we get the desired outcome and divide it by the total number of outcomes.
In this case, the event of interest is choosing 3 laser printers and 3 inkjets. At first, we have a total of 25 printers and we will be choosing 6 printers at random. The total number of ways in which we can choose 6 elements out of 25 is [tex]\binom{25}{6}[/tex], where [tex]\binom{n}{k} = \frac{n!}{(n-k)!k!}[/tex]. We have that [tex]\binom{25}{6} = 177100[/tex]
Now, we will calculate the number of ways to which we obtain the desired event. We will be choosing 3 laser printers and 3 inkjets. So the total number of ways this can happen is the multiplication of the number of ways we can choose 3 printers out of 10 (for the laser printers) times the number of ways of choosing 3 printers out of 15 (for the inkjets). So, in this case, the event can be obtained in [tex]\binom{10}{3}\cdot \binom{15}{3} = 54600[/tex]
So the probability of having 3 laser printers and 3 inkjets is given by
[tex] \frac{54600}{177100} = \frac{78}{253} = 0.31[/tex]
What’s the correct answer for this?
Answer:
A:
Step-by-step explanation:
Using tangent-secant theorem
AE²=(EC)(ED)
12²=(8)(8+x+10)
144=8(x+18)
144=8x+144
8x = 144-144
8x = 0
So
x = 0
Now
ED = 8+x+10
ED = 8+0+10
ED = 18
i have a problem on statistics
5. Data on household vehicle miles of travel (VMT) are compiled annually by the Federal Highway Administration and are published in National Household Travel Survey, Summary of Travel Trends. Independent random samples of 15 midwestern households and 14 southern households provided the following data on last year’s VMT, in thousands of miles. At the 5% significance level, does there appear to be a difference in last year’s mean VMT for midwestern and southern households? Use both p-value and critical value approach. Assume population variance to be equal
Midwest
16.2 , 14.6 , 11.2 , 24.4 , 9.4 12.9 , 18.6 , 16.6 , 20.3 ,15.1 , 17.3 , 10.8 , 16.6 , 20.9 , 18.3
South
22.2, 19.2 , 9.3 , 24.6 ,20.2 , 15.8, 18.0 , 12.2 , 20.1 , 16.0 , 17.5 , 18.2 , 22.8 , 11.5
Answer:
Step-by-step explanation:
Hello!
The objective is to compare the VMT of mid western households and southwestern households. For this two independent random samples of households from both areas and their VMT were recorded:
Be
X₁: VMT of a mid western household
Midwest
16.2 , 14.6 , 11.2 , 24.4 , 9.4 12.9 , 18.6 , 16.6 , 20.3 ,15.1 , 17.3 , 10.8 , 16.6 , 20.9 , 18.3
n₁= 15
∑X₁= 243.20
∑X₁²= 4175.98
X[bar]₁= 16.21
S₁²= 16.64
S₁= 4.08
X₂: VMT of a southwestern household
22.2, 19.2 , 9.3 , 24.6 ,20.2 , 15.8, 18.0 , 12.2 , 20.1 , 16.0 , 17.5 , 18.2 , 22.8 , 11.5
n₂= 14
∑X₂= 247.60
∑X₂²= 4633.24
X[bar]₂= 17.69
S₂²= 19.56
S₂= 4.42
The parameters of study are the population means, if the claim is that the VMT of households is different in both areas, then you'd expect the population means to be different too.
The hypotheses are:
H₀: μ₁ = μ₂
H₁: μ₁ ≠ μ₂
α: 0.05
Assuming both populations are normal and since both population variances are equal the test to apply is an independent samples t test pooled variance:
[tex]t= \frac{(X[bar]_1-X[bar]2)-(Mu_1-Mu_2)}{Sa*\sqrt{\frac{1}{n_1} +\frac{1}{n_2} } } ~~t_{n_1+n_2-2}[/tex]
[tex]Sa^2= \frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2} = \frac{14*16.67+13*19.56}{15+14-2}= \frac{487.66}{27} = 18.06[/tex]
Sa= 4.249= 4.25
[tex]t_{H_0}= \frac{(16.21-17.69)-0}{4.25*\sqrt{\frac{1}{15} +\frac{1}{14} } }= -0.937= -0.94[/tex]
Critical value approach:
This test is two-tailed, this means that the rejection region is divided in two tails:
[tex]t_{n_1+n_2-2; \alpha /2}= t_{27; 0.025}= -2.052[/tex]
[tex]t_{n_1+n_2-2; 1-\alpha /2}= t_{27; 0.975}= 2.052[/tex]
The decision rule is:
If [tex]t_{H_0}[/tex] ≤ -2.052 or if [tex]t_{H_0}[/tex] ≥ 2.052, reject the null hypothesis.
If -2.052 < [tex]t_{H_0}[/tex] < 2.052, do not reject the null hypothesis.
The calculated value is within the "no rejection region" so the decision is to not reject the null hypothesis.
Using the p-value approach:
The p-value is the probability of obtaining a value as extreme as the calculated value of the statistic under the null hypothesis ([tex]t_{H_0}[/tex]). Just as the significance level, the p-value is two tailed, you can calculate it as:
P(t₂₇ ≤ -0.93) + P(t₂₇ ≥ 0.93)= P(t₂₇ ≤ -0.93) + (1 - P(t₂₇ < 0.93)= 0.1796 + ( 1 - 0.8204)= 0.1796*2= 0.3592
p-value= 0.3592
The p-value is always compared to the significance level, the decision rule for this approach is:
If the p-value ≤ α, reject the null hypothesis.
If the p-value > α, do not reject the null hypothesis.
The p-value is greater than α, so the decision is to not reject the null hypothesis.
At a 5% significance level, there is no significant evidence to reject the null hypothesis. You can conclude that the population means of the VMT for households of the Midwest South ers households.
I hope this mhelps!
A researcher classifies firefighters according to whether their gloves fit well or poorly and by gender. They want to know if there is a difference in the proportion of poorly fitted gloves and gender. At alpha = 0.01, use the chi-square test to determine if there is a difference in the population proportion of glove fitness for the two genders.
Observed data Males Females Total
Gloves fit poorly 132 20 152
Gloves fit well 415 19 434
Total 547 39 586
Expected data Males Females Total
Gloves fit poorly
Gloves fit well
Total
Answer:
Step-by-step explanation:
Hello!
The objective is to test if the proportion of "X: gloves fitness, categorized: Fit poorly and Fit well" is the same for two populations of interest, "male firefighters" and "female firefighters"
To do this you have to conduct a Chi-Square test of Homogeneity.
In the null hypothesis you have to state that the proportion of the categories of the variable are the same for all the populations of interest.
Be
M: the firefighter is male
F: the firefighter is female
Y: represents the category that the gloves "fit poorly"
W: represents the category that the gloves "fit well"
The null hypothesis will be:
H₀: P(Y|M)=P(Y|F)=P(Y)
P(W|M)=P(W|F)=P(W)
H₁: At least one of the statements in the null hypothesis is false.
α: 0.01
To calculate the statistic under the null hypothesis you have to calculate the expected frequencies first:
[tex]E_{ij}= O_{.j}*\frac{O_{i.}}{n}[/tex]
O.j= total of the j-column
Oi.= total of the i-row
n= total of observations
[tex]E_{11}= 547*\frac{152}{586} = 141.88[/tex]
[tex]E_{12}=39*\frac{152}{586}= 10.12[/tex]
[tex]E_{21}= 547*\frac{434}{586} = 405.12[/tex]
[tex]E_{22}= 39*\frac{434}{586} = 28.88[/tex]
[tex]X^2= sum \frac{(O_{ij}-E_{ij})^2}{E_{ij}} ~~~X^2_{(r-1)(c-1)}[/tex]
r= number of rows (in this case 2)
c=number of columns (in this case 2)
[tex]X^2_{H_0}= \frac{(132-141.88)^2}{141.88} +\frac{(20-10.12)^2}{10.12} +\frac{(415-405.12)^2}{405.12} +\frac{(19-28.88)^2}{28.88} = 13.95[/tex]
Using the critical value approach, you have to remember that this test is always one-tailed to the right, meaning that you'll have only one critical value from which the rejection region is defined:
[tex]X^2_{(r-1)(c-1);1-\alpha }= X^2_{1;0.99}= 6.635[/tex]
The decision rule is then:
If [tex]X^2_{H_0}[/tex] ≥ 6.635, reject the null hypothesis.
If [tex]X^2_{H_0}[/tex] < 6.635, do not reject the null hypothesis.
The calculated value is greater than the critical value, the decision is to reject the null hypothesis.
So at a 1% level you can conclude that this test is significant. This means that the proportions of gloves fitness, categorized in "Fit poorly" and "Fit well" are different for the male and female firefighters populations.
I hope this helps!
Answer:
The Chi - Square Test Statistics is 13.98
p-value = 0.0002
CONCLUSION: Since the p-value is less than the level of significance ; (i.e p-value < ∝) we reject the null hypothesis and accept the alternative hypothesis.
Thus; there is a difference in the population proportion of glove fitness for the two genders.
Step-by-step explanation:
From the information given ; the structure of the table can be well represented as follows;
Observed data Males Females Total
Gloves fit poorly 132 20 152
Gloves fit well 415 19 434
Total 547 39 586
Expected data Males Females Total
Gloves fit poorly
Gloves fit well
Total
The objective of this question is to use the chi-square test to determine if there is a difference in the population proportion of glove fitness for the two genders.
We call represent the hypothesis as follows:
The null hypothesis: [tex]H_o:[/tex] states that there is no difference in the population proportion of glove fitness for the two genders.
The alternative hypothesis: [tex]H_a[/tex] states that there is difference in the population proportion of glove fitness for the two genders.
The expected frequency of a particular cell can be calculated by multiplying the sum of the rows and columns together, then dividing it by the Total sum
For row 1 column 1 (gloves fit poorly (male) ; we have:
[tex]= \dfrac{547*152}{586} =141.884\\[/tex]
For row 2 column 1 (gloves fit well(male) ; we have:
[tex]= \dfrac{547*434}{586} =405.116[/tex]
For row 1 column 2 (gloves fit poorly (female)) ; we have:
[tex]= \dfrac{39*152}{586} =10.116[/tex]
For row 2 column 2 ( gloves fit well ( female ) ; we have:
[tex]= \dfrac{39*434}{586} =28.884[/tex]
Thus; we can have the complete table to now be:
Observed data Males Females Total
Gloves fit poorly 132 20 152
Gloves fit well 415 19 434
Total 547 39 586
Expected data Males Females Total
Gloves fit poorly 141.884 10.116 152
Gloves fit well 405.116 28.884 434
Total 547 39 586
The Chi - Square Test Statistics can be calculated via the formula:
[tex]X^2 = \dfrac{\sum (f_o-f_e)^2}{f_e}[/tex]
where;
[tex]f_o[/tex] = observed data frequency
[tex]f_e[/tex] = expected data frequency
∴
The Chi - Square Test Statistics is as follows:
[tex]=\dfrac{(131-141.884)^2}{141.884} + \dfrac{(20-10.116)^2}{10.116}+ \dfrac{(415-405.116)^2}{405.116}+ \dfrac{(39-28.884)^2}{28.884}[/tex]
= 0.68+9.6+0.2+3.5
= 13.98
We are given the level of significance ∝ to be = 0.01
numbers of rows = 2; number of column = 2
Thus; the degree of freedom = (2-1)(2-1) = 1×1 = 1
Using the Excel Function : [ = CHISQ.DIST.RT²(X²,df)]
p-value = 0.0002
CONCLUSION: Since the p-value is less than the level of significance ; (i.e p-value < ∝) we reject the null hypothesis and accept the alternative hypothesis.
Thus; there is a difference in the population proportion of glove fitness for the two genders.
