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let p(A)=prob Anna solves the problem
let p(C)=prob Cathy solves the problem
let p(nA)=prob Anna does not solve the problem
let p(nC)=prob Cathy does not solve the problem
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Given P(A)=0.6, P(C)=0.3 and P(A), P(C) are independent
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P(nA)=1-P(A)=0.4 and P(nC)=0.7
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Find the probability that only one of them will be able to solve it.
=P(A and nC)+P(nA + C)
=P(A)P(nC)+P(nA)P(C) by independence
=0.6*0.7+0.4*0.3=0.42+0.12=0.54

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y intercept represent the value of y when the independent x=0. If a word problem deals with a variable x such that it begins with 0, such as money or time and it cannot be negative, the y intercept represent the y value at the start or beginning when x=0.

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true
this is true whether the distribution is normal or not, the median is the point where 50% of the distribution is below it.

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Given: Her office is square
let s=one side of her square office
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Given: new office is s-2 by s+4
a. write expression for the dimenstions of LaTanya's new office
(s-2) by (s+4)
b. Write a polynomial expression for the area of her new office.
Area=(s-2)*(s+4)=s^2+2s-8
c.suppose her office is presently 9 feet by 9 feet. will her new office be bigger or smaller then her old office and by how much? explain.
Current office: area=9x9=81 sq ft
new office: area=9^2+2*9-8=91 sq ft
new office is 10 sq ft bigger

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square both sides
x+1=(x-1)^2=x^2-2x+1
0=x^2-3x=x(x-3)
x=0 or x=3
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validate by checking each value on orginal equation
x=0:
1=-1 this is not true so x=0 is not a solution
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x=3
2=3-1=2 check
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so x=3 is the solution

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let B=total boys, G=total girls
boy's criteria: equal number of brothers and sisters
girl's criteria: twice as many brothers as sisters
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if B=1 and G=1, this does not satify the conditions
if B=2 and G=1 satisfies boy's criteria but not sister's
if B=2 and G=2 does not satify the boy's criteria but satisfies sister's
if B=3 and G=2 satisfies boy's criteria but not sister's
if B=3 and G=3 niether condition satisfied
if B=4 and G=2 neither condition satisfied
if B=4 and G=3 satisfies both conditions

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Apples are collected in a basket for six people. One third, one fourth, one eighth, and one fifth are given to four people. The fifth Person gets ten apples, leaving one for the sixt. Find the original number of apples in the basket.
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let x=number of apples
x=x/3+x/4+x/8+x/5+10+1
x-x/3-x/4-x/8-x/5=11
120x-40x-30x-15x-24x=11*120
11x=11*120
x=120

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Problem was solved already 394690

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let A=Marley is called during 3pm to 9pm
let B=Sarah calls someone during 6pm to 6:30pm
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Given P(A)=0.15
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P(B)=30/360 =1/12 since there are 30 minutes during the 6hours that he can be called.
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Need to know P(AnB) if we assume A & B are independent
then P(AnB)=P(A)*P(B)=0.15*1/12=0.0125

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known: Area=L*W Perimeter =2*(L+W)
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given : A=2P+19 or A=4*(L+W)+19 and L=2+W
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set both expression of area equal to each other
L*W=4L+4W+19
(2+W)*W=4*(2+W)+4W+19 substitute L=2+W
2W+W^2=8+8W+19
W^2-6W-27=0
(W-9)(W+3)=0
so W=9 or W=-3 measurements must be positive so eliminatte W=-3
solution, W=9
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L=2+W=2+9+11
==
validate the answer by substituting back into Area equations
A=L*W=11*9=99
A=4*(11+9)+19=80+19=99 check

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you have two unknowns: C=gallons used in city driving and H=gallons used in highway driving
so you will need two equations to solve the two unknowns
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eq1 gallons used: C+H=51
eq2 miles driven: 25*C+40*H=1800
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solve eq1 for H: H=51-C and substitute in eq2
25*C+40*(51-C)=1800 solve for C
25C-40C+2040=1800
2040-1800=15C
C=240/15=16gallons used
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they want to know how many miles were driven in the city
25*C=25*16=400 miles

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you have two unknowns d=number of dimes and q=number of quarters
so you need two equations to find the two unknowns
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given
eq1: d=1/4q or 4d=q
eq2: 0.1d+0.25d=2.20
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multiply eq2 by 100 to eliminate decimals
10d+25d=220
10d+25(4d)=220 substitute 4d=q
110d=220
d=220/110=2 or 2 dimes
since q=4d then 8 quarters

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the domain = all possible input values of x
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Any values of x that make the function, p(x) undefined are not allowed.
for instance if p(x)=1/(x-3) then x=3 is not allowed because x=3 makes P(3) undefined.
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since no values of x, makes the function. p(x) undefined, then all real values of x are allowed as the domain.

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remember the relationship distance=rate*time or d=rt
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going: 80 miles at speed r and time t 80=rt
coming: 80 miles at speed 2r and time t-2 80=2r*(t-2) or 40=r(t-2)=rt-2r
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80=rt and 40=rt-2r
40=80-2r
2r=40
r=20 this is the going speed 20mph

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already answered 395247

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understand that csc(a)=1/sin(a)
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work with the left hand of the equation to get the right hand side

= substitute csc(a)=1/sin(a)
= invert
= factor out a cos^2(a)
= indetity

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set up the following equation

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next multiply both sides by the common denominator (x+1)(x+3)(x-5)
-

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at this point you can expand out all terms, then combined common terms and set correponding parts equal yielding a system of equation to solve for A,B and C
but another alternative is to select "convenient values of x and solve
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x=-1:
9+24-57=-12A or A=-24/-12=2
x=-3:
81+72-57=72B or B=96/72=4/3
x=5:
225-120-57=48C or C=48/48=1
==
if I did my math correctly

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You need to put the formula in standard form y=a(x-h)^2+k
(h,k) is the vertex
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you have three points. the origion (0,0) the end (32,0) and its middle (16,14) which is the vertex
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let's use input the vertex in the standard formula
y=a(x-16)^2+14
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lets use the origin (0,0) to solve for "a"
0=a(-16)^2+14
0=256a+14
a=-14/256= - 7/128
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substitute a into formula
y=-7/128*(x-16)^2+14

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the point (-2,5) makes a right triangle whose hypotenus is
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the six trig ratios are;
cos t = adjacent/hypotenus=-2/sqrt(29)
sin t = opposite/hypotenus = 5/sqrt(29)
tan t = opposite/adjacent= 5/-2
cot t = adjacent/opposite = -2/5
sec t = hypotenus/adjacent=sqrt(29)/2
csc t = hypotenus /opposite = sqrt(29)/5
The angle is using anyone of these and finding the arc value, for simplicity i'll use tan t=-2.5
t=arc tan(-2.5)=180-arc tan(2.5)=180-68.2=111.8

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I presume that by distribution being mound-shaped that implies that its normally distributed.
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a) If the production line is operating correctly, approximately what proportion of batches from a days' production will contain less than 1.84% of zinc phosphid?
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let x=% of zinc phosphid
Assuming Normally distribution with mean=2.0 and standard deviation=0.08
P(x<1.84)=P(z<(1.84-2)/.08)=P(z<-2.0)=0.02275 or 2.3% of batches
==
b) Suppose one batch chosen randomly actually contains 1.80% zinc phosphide. Does this indicate that there is too little zinc phosphide in today's production? Explain your reasoning.
P(x<1.80)=p(z<-2.5)=0.00621 or 0.62% This is highly unlikely from a normal distribution centered at 2.0 and std dev=0.08. This implies that the sample was obtained from a distribution centered lower than 2.0

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this was answered already
using excel =NORMSDIST(-2.6)

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the pvalue for z test statistic of -2.6 is 0.00047
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Since the pvalue is less than the significance level 0.05, then reject Ho.
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The pvalue represent the probability that the test statistic, assuming Ho is true deviated from Ho by pure chance. Since this probablity is small, you conclude that it is evidence that it does not come from Ho.

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the appropriate test "always" when the population variance is unknown is to use a t-test. although as the samples sizes get larger and larger, the t-test and Z-test will converge but this does not happen (based on my experience) until the sample size is greater than 100.
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The other wrinkle is that if the sample size is small, and the population variance is unknown, you can only use the t-test if the sample comes from a normal distribution as the central limit theorem can't help you with small samples.

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the right tail test implies that you want to reject Ho the null, if your test statistic exceeds the critical value
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c. Reject H0 if t > 2.05

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first, isolate the radical


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next square both sides to eliminate the radical


multiply both sides by 4
gather terms on one side
0=(x-5)*(x+3) factor
x=5, or x=-3
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verify by substituting the answers into original equation
x=5:

2*3-1=5 check
x=-3:

2-1=1 does not equal -3 so x=-3 is not a solution
--
the answer is x=5

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let T = number truck owners
let C=number of car owners
let B=number who own both
let N=number who own neither
===
given T=34, C=54, B=12, N=8
Total interviewed =34+54-12+8=84
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subtract B=12 because in counting T and C the B is counted twice.

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when solving these types of equations, look at what you have and what you need.
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you have an identiy 79.8 calories/1gram , this identity can be written as 1gram/79.8calories. It can be used either way depending on what you need.
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you have 1000 calories and want to find grams
so set up your expression

notice that I used this form of the identity so I could cross cancel the units and leave me with grams.
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=12.53 grams

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--
break the inside of the square root expression into perfect squares

--
factor out the perfect squares

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Ho: Ux=Uy=Uz
Ha: not all are equal
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compute the sum squares for between groups x,y,z
SSB=sum(ni*(Xi-Xbar)^2) where ni=sample size for group i, Xi=sample average for group i, and Xbar=overall average.
ni=25 for each group, Xx=5,Xy=4,Xz=6, Xbar=5
SSB=25*(0^2+(-1)^2+1^2)=50
degrees of freedom= k-1=3-1=2, k=number of groups
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Compute the sum squares for the error
SSE=sum((ni-1)*Si^2)=24*(2+1.5+2.5)=144
degrees of freedom= k(ni-1)=3(25-1)=3*24=72, k=number of groups
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Compute Mean sum of squares for between groups and error
MSB=SSB/dof=50/2=25
MSE=SSE/dof=144/72=2
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compute F statistic= MSB/MSE=25/2=12.5
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determine the critical F value for alpha 0.05, num dof=2, denom dof=72
Fcritical= 3.12
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Since Fstatistic > Fcritical then reject Ho and conclude
The universities differ in their average time socializing