Base case (n = 1):
• On the left side: 1/(1×2) = 1/2
• On the right side: 1/(1 + 1) = 1/2
Induction hypothesis: Assume the statement is true for n = k ; that is,
1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) = k/(k + 1)
Inductive step (n = k + 1):
1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) + 1/((k + 1) × (k + 2)))
= k/(k + 1) + 1/((k + 1) × (k + 2))
= (k × (k + 2) + 1) / ((k + 1) × (k + 2))
= (k ² + 2k + 1) / ((k + 1) × (k + 2))
= (k + 1)² / ((k + 1) × (k + 2))
= (k + 1) / (k + 2)
and this is what we wanted to show.
In an accelerated failure test, components are operated under extreme conditions so that a substantial number will fail in a rather short time. In such a test involving two types of microchips, 580 chips manufactured by an existing process were tested, and 125 of them failed. Then, 780 chips manufactured by a new process were tested, and 130 of them failed. Find a 90% confidence interval for the difference between the proportions of failures for chips manufactured by the two processes. (Round the final answers to four decimal places.) The 90% confidence interval is
Answer:
The 90% confidence interval is (0.0131, 0.0845).
Step-by-step explanation:
Before finding the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Old process:
125 out of 580, so:
[tex]p_O = \frac{125}{580} = 0.2155[/tex]
[tex]s_O = \sqrt{\frac{0.2155*0.7845}{580}} = 0.0171[/tex]
New process:
130 out of 780. So
[tex]p_N = \frac{130}{780} = 0.1667[/tex]
[tex]s_N = \sqrt{\frac{0.1667*0.8333}{780}} = 0.0133[/tex]
Distribution of the difference:
[tex]p = p_O - p_N = 0.2155 - 0.1667 = 0.0488[/tex]
[tex]s = \sqrt{s_O^2+s_N^2} = \sqrt{0.0171^2 + 0.0133^2} = 0.0217[/tex]
Confidence interval:
[tex]p \pm zs[/tex]
In which
z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower bound of the interval is:
[tex]p - zs = 0.0488 - 1.645*0.0217 = 0.0131[/tex]
The upper bound of the interval is:
[tex]p + zs = 0.0488 + 1.645*0.0217 = 0.0845[/tex]
The 90% confidence interval is (0.0131, 0.0845).
Question 26 of 58
Mr. Nguyen recorded the numbers of students in his homeroom class who
participated in spirit week.
The table shows the number of students who dressed up each day.
Day
Mon Tues. Wed. Thurs. Fri. Total
Number of students 2
2
5
5
6
20
Find the mean and the median of the data set.
Determine which of these values is greater.
O A. The mean, 5, is greater than the median, 4.
OB. The mean, 5, is greater than the median, 2.
O c. The median, 6, is greater than the mean, 2.
O D. The median, 5, is greater than the mean, 4.
Answer:
D
Step-by-step explanation:
The answer is D.
An education researcher would like to test whether 2nd graders retain or lose knowledge during the summer when they are presumably not in school. She asks nine 2nd graders to take a comprehension exam at the end of the school year (May), and then asks those same students to come back after the summer (late August) to retake a different but equivalent exam, to see if their level of comprehension has changed. Using the data below, test this hypothesis using an (alpha level of 0.05.)
May August
95 100
72 80
78 95
50 65
89 85
92 98
75 70
90 96
65 87
Required:
What is the appropirate test?
Answer:
Kindly check explanation
Step-by-step explanation:
Given the data:
May August
95 100
72 80
78 95
50 65
89 85
92 98
75 70
90 96
65 87
The appropriate test is a paired t test :
d = difference between May and August
d = (-5, -8, -17, -15, 4, -6, 5, -6, -22)
The hypothesis :
H0 : μd = 0
H1 : μd ≠ 0
The test statistic :
T = dbar / (Sdbar/√n)
Where, dbar and Sdbar are the mean and standard deviation of 'd' respectively.
Using calculator :
dbar = - 7.777 ; Sdbar = 9.052
Test statistic = - 7.777 / (9.052 /√9)
Test statistic = - 2.577
The Pvalue, df = n - 1 = 9 - 1 = 8
Pvalue(-2.577, 8) = 0.0327
At α = 0.05
Pvalue < α ; WE reject the H0 ; and conclude that there has been a change in score
Identify the pair numbers
Answer:
D. 212 degrees Fahrenheit and 100 degrees Celsius.
Step-by-step explanation:
The boiling point of water in Celsius is 100.
The way to calculate Celsius to Fahrenheit:
F = (C × 9/5) + 32
So we plug in 100 for C.
F = (100 × 9/5) + 32
F = 180 + 32
F = 212
Therefore, the numbered pair is 212 degrees F and 100 degrees C.
A person's email for one day contained a total of 78 messages. The number of spam
messages was two less than four times the number of other messages. How many of
the email messages were spam?
Answer:
62 of the email messages were spam
Step-by-step explanation:
Let the number of spam and other messages be s and o respectively.
Total number of messages= 78
s +o= 78 -----(1)
s= 4o -2 -----(2)
Substitute (2) into (1):
4o -2 +o= 78
Simplify:
5o -2= 78
+2 on both sides:
5o= 78 +2
5o= 80
Divide both sides by 5:
o= 80 ÷5
o= 16
Since s +o= 78, s= 78 -o.
s= 78 -16
s= 62
Find the area of the circle. Use 3.14 for it. E d = 10 cm A = [?] cm2 A=7tr2
Answer:
A=(78.5)cm²
Step-by-step explanation:
d=10
r=10/2=5
A=πr²
A=3.14*5²
A=3.14*25
A=78.5cm²
Answer: d=10cm
According to the formula i.e. A=πr²
first we need 'r'
as r=d/2
hence, r= 10cm/2
r=5cm
put r=5 in formula
=3.14(5cm)²
=3.14×25cm²
=78.5cm²
Land surveyors outlined a park as shown. What is the area of the park?
la cuadra se llama 6minutos
For the z test, the critical region for rejection of H0 _________. Group of answer choices depends on N is determined only by alpha and N allows us to accept the null hypothesis is determined only by alpha
Answer:
allows us to accept the null hypothesis
Explanation:
The z test(in a normal distribution) score for the critical region determines whether we reject the null hypothesis(H0) or accept the null hypothesis(reject or fail to reject the null hypothesis). If we fail to reject the null hypothesis, then we have accepted the alternative hypothesis (H1). The critical region rejection for z test is calculated using alpha and z score, if z score is greater or less than alpha(positive or negative), we reject the null hypothesis.
= 10 + 3
=13
Example 6
If f(x) = 2x + 1, g(x) = 3x - 2 and fg(x) = 5, find the value of x,
Answer:
f(x) = 2x + 1, g(x) = 3x - 2 and fg(x) = 5
Step-by-step explanation:
Simplify: –3(y + 2)2 – 5 + 6y.
Answer:
-17
Step-by-step explanation:
–3(y + 2)2 – 5 + 6y
(–3y -6)2 – 5 + 6y
-6y -12 - 5 + 6y
-17
The point (7, 8) is the solution of which of the following systems of equations?
1. 8x - y = 48
9x + 10y = 142
2. 9x - y = 55
8x + 10y = 140
3. x-6y=-41
8x + 9y = 127
4. x-7y=-49
10x + 9y = 142
Option 4
Verification:-
[tex]\\ \sf\longmapsto x-7y=-49[/tex]
[tex]\\ \sf\longmapsto 7-7(8)=-49[/tex]
[tex]\\ \sf\longmapsto 7-56=-49[/tex]
[tex]\\ \sf\longmapsto -49=-49[/tex]
And
[tex]\\ \sf\longmapsto 10x+9y=142[/tex]
[tex]\\ \sf\longmapsto 10(7)+9(8)=142[/tex]
[tex]\\ \sf\longmapsto 70+72=142[/tex]
[tex]\\ \sf\longmapsto 142+142[/tex]
Hence verified
Answer:
4.
Step-by-step explanation:
1. 8x - y = 48
9x + 10y = 142
8(7) - 8 = 48
9(7) + 10(8) = 143
No
2. 9x - y = 55
8x + 10y = 140
9(7) - 8 = 55
9(7) - 10(7) = -24
No
3. x-6y=-41
8x + 9y = 127
7 - 6(8) = -41
8(7) + 9(8) = 128
No
4. x-7y=-49
10x + 9y = 142
7 - 7(8) = 49
10(7) + 9(8) = 142
Yes
Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use absolute values where appropriate.)
f(x) = 45−5x, x>0 .
Answer:
F(x) = 45*x - (5/2)*x^2 + C
Step-by-step explanation:
Here we want to find the antiderivative of the function:
f(x) = 45 - 5*x
Remember the general rule that, for a given function:
g(x) = a*x^n
the antiderivative is:
G(x) = (a/(n + 1))*a*x^(n + 1) + C
where C is a constant.
Then for the case of f(x) we have:
F(x) = (45/1)*x^1 - (5/2)*x^2 + C
F(x) = 45*x - (5/2)*x^2 + C
Now if we derivate this, we get:
dF(x)/dx = 1*45*x^0 - 2*(5/2)*x
dF(x)/dx = 45 - 5*x
I'm stuck. Can anyone help please?
log₉(x - 7) + log₉(x - 7) = 1
2 log₉(x - 7) = 1
log₉(x - 7) = 1/2
Take the base-9 antilogarithm of both sides; in other words, make both sides powers of 9:
[tex]9^{\log_9(x-7)} = 9^{1/2}[/tex]
[tex]9^{1/2}[/tex] can also be written as √9 = 3, and [tex]b^{\log_b(a)}=a[/tex], so the equation reduces to
x - 7 = 3
Solve for x :
x = 10
Please helps fill in the charts
A and b
With order of pairs
Answer:
...
Step-by-step explanation:
seeee the above picture
Select the correct answer.
What is the value of this expression when x = -6 and ?
4(x2 + 3) − 2y
Answer:
D. 157
Step-by-step explanation:
4(x^2+3)-2y
4(6^2+3)-2(-1/2) add in given values
4(39)+1. start with parentheses
156+1. combine like terms
157. answer
Answer:
D. 157
Step-by-step explanation:
Hi there!
We want to find the value of the expression 4(x²+3)-2y is when x=-6 and y=-1/2
Let's first simplify the expression, as that will likely make it easier
Distribute 4 to both x² and 3
4x²+12-2y
That's the expression
Substitute -6 as x into the expression
4(-6)²+12-2y
Raise (-6) to the second power
4*36+12-2y
Multiply 36 by 4
144+12-2y
Add 12 and 144 together
156-2y
Now the expression is 156-2y
But remember that we know that y=-1/2, and we haven't substituted it into the expression yet
Substitute -1/2 as y into the expression
156-2(-1/2)
Multiply
156+2/2
Simplify
156+1
Add
157
Hope this helps!
perimeter of a circle 8 centimeter wide
Answer:
25.13 cm
Step-by-step explanation:
Perimeter ( circumference ) of a circle = 2πr
Given,
The circle is 8 cm wide
which means,
The diameter (d) of the circle is 8 cm.
Radius (r) of the circle = d/2
= 8/2
= 4
Radius = 4 cm
Putting the value in the formula;
2πr
= 2 (22/7) (4)
= 25.13 cm (approx)
Which describes the correlation shown in the scatterplot?
On a graph, points are grouped closely together and increase.
There is a positive correlation in the data set.
There is a negative correlation in the data set.
There is no correlation in the data set.
More points are needed to determine the correlation.
Answer:
More points are needed to determine the correlation.
Answer:
its d
Step-by-step explanation:
;)
what number increased by 130% is 69
Answer:
The number is 30.
Step-by-step explanation:
Let the number be x
so
x + (130% of x) = 69
x + 13x/10 = 69
or, (10x + 13x)/10 = 69
or, 23x = 690
or, x = 690/23
so, x = 30
Answer: 30
Step-by-step explanation:
its correct on rsm
Solve the system of equations.
4x + 3y + 5z = 6
6x + 8y + 6z = 4
4x + 2y + z = 8
(x = 1, y = -1,2 = 1)
b. (x = 3, y = -3,2 = 3)
a.
C. (x = 0, y = 0, 2 = 2)
d. (x - 2, y --2, z = 0)
which graph correctly represents this 2y+x=-5and y+3x=0
Answer:
you didn't feature any graphs for me to choose from
A concert promoter sells tickets and has a marginal-profit function given below, where P'(x) is in dollars per ticket. This means that the rate of
change of total profit with respect to the number of tickets sold, x, is P'(x). Find the total profit from the sale of the first 380 tickets, disregarding any
fixed costs.
P'(x) = 2x - 1198
The total profit is $
Answer:
The answer is "-310840".
Step-by-step explanation:
[tex]\begin{array}{l} P'(x) = 2x - 1198\\ \\ \frac{{dP}}{{dx}} = 2x - 1198\\ \\ dP = \left( {2x - 1198} \right)dx\\ \\ \int {dP} = \int\limits_0^{380} {\left( {2x - 1198} \right)dx} \\ \\ P = \left( {2\frac{{{x^2}}}{2} - 1198x} \right)_0^{380}\\ \\ P = {380^2} - 1198 \times 380 = 144400-455240=-310840\\ \end{array}[/tex]
A height of 2.5 cm represents 100 goats. What should be the height for 170 goats?
Answer: 4.25
Step-by-step explanation:
2.5/100 = 0.025
0.025 × 170 = 4.25
but the question there is any goats in 2.5 cm ??
that is impossible
find the missing length indicated
Answer:
192
Step-by-step explanation:
geometric mean theorem :
with p and q being the segments of the Hypotenuse, then
h = x = sqrt(p×q)
p = 144
q = 400-144 = 256
h = x = sqrt(144×256) = 12×16 = 192
Which one is the intersection point of
f(x) = x3 + 3x and
g(x) = x2 + 3
A) (0,0)
B) (0,3)
C) (1,4)
D) (-1,4)
I URGENTLY NEED HELP PLEASE , I WOULD ALSO MARK AS BRAINLIEST!!
Answer: C) (1,4)
Step-by-step explanation:
The intersection point is where f(x) = g(x)
x³ + 3x = x² + 3
x³ - x² +3x - 3 = 0
A. (0, 0) → x = 0 → (0)³ - (0)² +3(0) - 3 = -3 ≠ 0B. (0, 3) → x = 0 → (0)³ - (0)² +3(0) - 3 = -3 ≠ 0C. (1, 4) → x = 1 → (1)³ - (1)² +3(1) - 3 = 0 + 0 = 0D. (-1, 4) → x = -1 → (-1)³ - (-1)² +3(-1) - 3 = 0 - 3 - 3 = -6 ≠ 0Help me out with these 2 questions for 15 points.
Step-by-step explanation:
The time dilation formula is given by
[tex]F(t) = \dfrac{t}{\sqrt{1-v^2}}[/tex]
where t is the time measured by the moving observer and F(t) is the time measured by the stationary earth-bound observer and v is the velocity of the moving observer expressed as a fraction of the speed of light.
a) If the observer is moving at 80% of the speed of light and observes an event that lasts for 1 second, a stationary observer will see the same event occurring over a time period of
[tex]F(t) = \dfrac{1\:\text{s}}{\sqrt{1-(0.8)^2}}[/tex]
[tex]\:\:\:\:\:\:\:=\dfrac{1\:\text{s}}{\sqrt{1-(0.8)^2}} =\dfrac{1\:\text{s}}{\sqrt{1-(0.64)}}[/tex]
[tex]\:\:\:\:\:\:\:=1.67\:\text{s}[/tex]
This means that any event observed by this moving observer will be seen by a stationary observer to occur 67% longer.
b) Given:
t = 1 second
F(t) = 2 seconds
We need to find the speed of the observer such that an event seen by this observer will occur twice as long as seen by a stationary observer. Move the term containing the radical to the left side so the equation becomes
[tex]\sqrt{1-v^2} = \dfrac{t}{F(t)}[/tex]
Take the square of both sides, we get
[tex]1 - v^2 =\dfrac{t^2}{F^2(t)}[/tex]
Solving for v, we get
[tex]v^2 = 1 - \dfrac{t^2}{F^2(t)}[/tex]
or
[tex]v = \sqrt{1 - \dfrac{t^2}{F^2(t)}}[/tex]
Putting in the values for t and F(t) we get
[tex]v = \sqrt{1 - \dfrac{(1\:\text{s})^2}{(2\:\text{s})^2}}[/tex]
[tex]v = \sqrt{1 - \dfrac{1}{4}} = \sqrt{0.75}[/tex]
[tex]\:\:\:\:=0.866[/tex]
This means that the observer must moves at 86.6% of the speed of light.
identify an equation in point slope form for the line perpendicular to y=5x=2 that passes through (-6,-1)
Answer:
y=-1/5x-11/5
Step-by-step explanation:
perpendicular, product of both gradients = -1
hence, slope = -1/5
y=-1/5x+c
sub y=-1, x=-6
-1=-1/5(-6)+c
c = -1-6/5=-11/5
y=-1/5x-11/5
4. (08.02 MC)
Factor completely 2x3 + 10x2 + 14x + 70. (5 points)
(2x^2 + 14)(x + 5)
(x^2 + 7)(2x + 10)
2(x^3 + 5x2 + 7x + 35)
2[(x^2 + 7)(x + 5)]
Answer:
2[(x^2 + 7)(x + 5)]
Step-by-step explanation:
Note that the ratio of the first and second terms is the same as that between the third and fourth terms. So this cubic will factor by grouping, but first we can separate out the common scalar factor 2...
2x3+10x2+14x+70=
2(x3+5x2+7x+35)
2((x3+5x2)+(7x+35))
2(x2(x+5)+7(x+5))
2(x2+7)(x+5)
Answer:
The answer is D. 2[(x2 + 7)(x + 5)]
Step-by-step explanation:
The slide is inclined at an angle of 52.0°. Danny weighs 46.0 pounds. He is sitting in a cardboard box
with a piece of wax paper on the bottom. Stacey is at the top of the slide holding on to the cardboard
box. Find the magnitude of the force Stacey must pull with, in order to keep Danny from sliding down
the slide. (We are assuming that the wax paper makes the slide into a frictionless surface, so that the
only force keeping Danny from sliding is the force with which Stacey pulls.)
Answer:
Step-by-step explanation:
Since the slide is inclined at an angle of 52.0° to the horizontal, Danny's weight (mass, m) of 46.0 pounds has a component W = mgcos52.0° perpendicular to the incline and W' = mgsin52.0° parallel to the incline where g = acceleration due to gravity = 32 ft/s²
The perpendicular component of Danny's weight is the normal force to the incline. This normal force would determine the magnitude of the friction along the incline.
Since the wax paper makes the incline surface frictionless, so, there is no friction along the surface and thus the only horizontal force acting along the surface is the component of Danny's weight along the surface.
For Stacey to keep Danny from sliding down along the incline, the force she applies along the incline, F must be equal to the component of Danny's weight along the incline.
So, F = W'
F = mgsin52.0°
F = 46lb × 32 ft/s² × sin52°
F = 1472 lb-ft/s² 0.7880
F = 1159.95 lb-f
F ≅ 1160 lb-f
15 times a certain number plus 5 times the same number is 80 what is the number
x = 4
Every step shown. Once you become used to doing this you will almost be able to do the basic one's in your head without writing much down.
Explanation:
Let the unknown value be
x
Converting the words into numbers:
First part: "15 times a certain number" → 15 x
Second part: "plus 5 times the same number" → 15 x + 5 x
The last part: " is 80" -> 15x + 5 x = 80
We are counting x ' s . 15 of them plus another 5 of them gives a total of 20.
So 15 x + 5 x = 20 x = 80
Divide both sides by 20
20 x ÷ 20 = 80÷ 20
20/20x=80/20
x=4
Let the number be x
ATQ
[tex]\\ \sf\longmapsto 15x+5x=80[/tex]
[tex]\\ \sf\longmapsto (15+5)x=80[/tex]
[tex]\\ \sf\longmapsto 20x=80[/tex]
[tex]\\ \sf\longmapsto x=\dfrac{80}{20}[/tex]
[tex]\\ \sf\longmapsto x=4[/tex]
Two solutions of the equation Ax+By = 1 are (2, -1) and (-3,-2). Find A and B.
Answer:
Substitute in the values of both given coordinates & form 2 equations:
[tex]\left \{ {{A(2)+B(-1)=1} \atop {A(-3)+B(-2)=1}} \right. \\\\=\left \{ {{2A-B=1} \atop {-3A-2B=1}} \right.[/tex]
Find the value of B from the equation 2A - B = 1:
[tex]2A-B=1\\-B=1-2A\\B=2A-1[/tex]
Substitute in the B-value to the other equation:
[tex]-3A-2B=1\\-3A-2(2A-1)=1\\-3A-4A+2=1\\-7A=1-2\\-7A=-1\\A=\frac{-1}{-7} =\frac{1}{7}[/tex]
Find the B-value using the equation from before:
[tex]B=2A-1=2(\frac{1}{7})-1=\frac{2}{7} -\frac{7}{7} =-\frac{5}{7}[/tex]
Therefore the equation Ax + By = 1 would equal:
[tex]\frac{1}{7} x-\frac{5}{7} y=1[/tex]
