Answer:
Combination, but keep in mind that if the committee had two open positions, say President and Secretary, it would be a permutation
Step-by-step explanation:
The first thing to keep in mind is the difference between combination and permutation.
The main difference is that in the combinations the order does not matter, whereas in the permutations the order does matter.
Combination example:
Choose 7 people for a project.
Example of permutation:
Choose 5 men for each specific role in a soccer team.
Therefore, "group of 5 senators is chosen to be part of a special committee" is a combination, but keep in mind that if the committee had two open positions, say President and Secretary, it would be a permutation.
Prove that If A1, A2, ... , An and B1, B2,...,Bn are sets such that Aj ⊆ Bj for j = 1, 2, 3, ... , n, then ∪j=1nAj ⊆ ∪j=1nBj .
Answer:
This is proved using Proof by induction method. There are two steps in this method
Let P(n) represent the given statement ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
1. Basis Step: This step proves the given statement for n = 1
2. Induction step: The step proves that if the given statement holds for any given case n = k then it should also be true for n = k + 1.
If the above two steps are true this means that given statement P(n) holds true for all positive n and the mathematical induction P(n): ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] is true.
Step-by-step explanation:
Basis Step:
For n = 1
∪[tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] = ∪[tex]{ {{1} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] = A₁ ⊆ B₁ = ∪[tex]{ {{1} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] = ∪[tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
We show that
∪[tex]{ {{1} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] = A₁ ⊆ B₁ = ∪[tex]{ {{1} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] for n = 1
Hence P(1) is true
Induction Step:
Let P(k) be true which means that we assume that:
for all k with k≥1, P(k): ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] is true
This is our induction hypothesis and we have to prove that P(k + 1) is also true
This means if ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] holds for n = k then this should also hold for n = k + 1.
In simple words if P(k): ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] is true then ∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] is also true
∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] = ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ∪ [tex]A_{k+1}[/tex]
⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] ∪ [tex]A_{k+1}[/tex] As ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] ∪ [tex]B_{k+1}[/tex] As [tex]A_{k+1}[/tex] ⊆ [tex]B_{k+1}[/tex]
= ∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
The whole step:
∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] = ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ∪ [tex]A_{k+1}[/tex] ⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] ∪ [tex]A_{k+1}[/tex] ⊆ ∪[tex]{ {{k} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] ∪ [tex]B_{k+1}[/tex] = ∪[tex]{ {{k+1} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
shows that the P(k+1) also holds for ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex]
hence P(k+1) is true
So proof by induction method proves that P(n) is true. This means
P(n): ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]A_{j}[/tex] ⊆ ∪ [tex]{ {{n} \atop {j=1}} \right.[/tex] [tex]B_{j}[/tex] is true
You can model that you expect a 1.25% raise each year that you work for a certain company. If you currently make $40,000, how many years should go by until you are making $120,000? (Round to the closest year.)
Answer:
94 years
Step-by-step explanation:
We can approach the solution using the compound interest equation
[tex]A= P(1+r)^t[/tex]
Given data
P= $40,000
A= $120,000
r= 1.25%= 1.25/100= 0.0125
substituting and solving for t we have
[tex]120000= 40000(1+0.0125)^t \\\120000= 40000(1.0125)^t[/tex]
dividing both sides by 40,000 we have
[tex](1.0125)^t=\frac{120000}{40000} \\\\(1.0125)^t=3\\\ t Log(1.0125)= log(3)\\\ t*0.005= 0.47[/tex]
dividing both sides by 0.005 we have
[tex]t= 0.47/0.005\\t= 94[/tex]
Find the total surface area of this triangular prism 13cm 5cm 12cm 9cm 15cm 20cm
Answer:
924 cm²
Step-by-step explanation:
The surface area is equal to the area of the two triangles + area of the three rectangles.
Area of two triangles:
12 × (9+5) × 1/2
= 84
84(2) = 168
Area of the three rectangles:
15 × 20 + 13 × 20 + 14 × 20
= 840
840 + 84
The surface area of the triangular prism is 924 cm².
Use the Remainder Theorem to determine which of the roots are roots of F(x). Show your work.
Polynomial: F(x)=x^3-x^2-4x+4
Roots: 1, -2, and 2.
Answer: x1=1 x2=-2 and x3=2
Step-by-step explanation:
1st x1=1 is 1 of the roots , so
F(1)=1-1-4+4=0 - true
So lets divide x^3-x^2-4x+4 by (x-x1), i.e (x^3-x^2-4x+4) /(x-1)=(x^2-4)
x^2-4 can be factorized as (x-2)*(x+2)
So x^3-x^2-4x+4=(x-1)*(x^2-4)=(x-1)(x-2)*(x+2)
So there are 3 dofferent roots:
x1=1 x2=-2 and x3=2
What is the equation of a line passes thru the point (4, 2) and is perpendicular to the line whose equation is y = ×/3 - 1 ??
Answer:
Perpendicular lines have slopes that are opposite and reciprocal. Therefore, the line we are looking for has a -3 slope.
y= -3x+b
Now, we can substitute in the point given to find the intercept.
2= -3(4)+b
2= -12+b
b=14
Finally, put in everything we've found to finish the equation.
y= -3x+14
Answer:
y = -3x + 14
Step-by-step explanation:
First find the reciprocal slope since it is perpendicular. Slope of the other line is 1/3 so the slope for our new equation is -3.
Plug information into point-slope equation
(y - y1) = m (x-x1)
y - 2 = -3 (x-4)
Simplify if needed
y - 2 = -3x + 12
y = -3x + 14
For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. n = 24 and p = 0.6.
Answer:
Since both np > 5 and np(1-p)>5, it is suitable to use the normal distribution as an approximation.
Step-by-step explanation:
When the normal approximation is suitable?
If np > 5 and np(1-p)>5
In this question:
[tex]n = 24, p = 0.6[/tex]
So
[tex]np = 24*0.6 = 14.4[/tex]
And
[tex]np(1-p) = 24*0.6*0.4 = 5.76[/tex]
Since both np > 5 and np(1-p)>5, it is suitable to use the normal distribution as an approximation.
(a +2b)2 + 4b² - a²
Answer:
a^2+4b^2+2a+4b
Step-by-step explanation:
(a +2b)2 + 4b² - a²
=2a+4b+4b^2+a^2
=a^2+4b^2+2a+4b
Susan decides to take a job as a transcriptionist so that she can work part time from home. To get started, she has to buy a computer, headphones, and some special software. The equipment and software together cost her $1000. The company pays her $0.004 per word, and Susan can type 90 words per minute. How many hours must Susan work to break even, that is, to make enough to cover her $1000 start-up cost? If Susan works 4 hours a day, 3days a week, how much will she earn in a month.
Answer:
46.3 hours of work to break even.
$1036.8 per month (4 weeks)
Step-by-step explanation:
First let's find how much Susan earns per hour.
She earns $0.004 per word, and she does 90 words per minute, so she will earn per minute:
0.004 * 90 = $0.36
Then, per hour, she will earn:
0.36 * 60 = $21.6
Now, to find how many hours she needs to work to earn $1000, we just need to divide this value by the amount she earns per hour:
1000 / 21.6 = 46.3 hours.
She works 4 hours a day and 3 days a week, so she works 4*3 = 12 hours a week.
If a month has 4 weeks, she will work 12*4 = 48 hours a month, so she will earn:
48 * 21.6 = $1036.8
Answer:
46.3 hours of work to break even.
$1036.8 per month (4 weeks)
Step-by-step explanation:
Which statement is true about the steps that Pablo used to simplify the expression?
What is the simplified form of this expression?
(-3x^2+ 2x - 4) + (4x^2 + 5x+9)
OPTIONS
7x^2 + 7x + 5
x^2 + 7x + 13
x^2 + 11x + 1
x^² + 7x+5
Answer:
Option 4
Step-by-step explanation:
=> [tex]-3x^2+2x-4 + 4x^2+5x+9[/tex]
Combining like terms
=> [tex]-3x^2+4x^2+2x+5x-4+9[/tex]
=> [tex]x^2+7x+5[/tex]
Over the past several years, the proportion of one-person households has been increasing. The Census Bureau would like to test the hypothesis that the proportion of one-person households exceeds 0.27. A random sample of 125 households found that 43 consisted of one person. The Census Bureau would like to set α = 0.05. Use the critical value approach to test this hypothesis. Explain.
Answer:
For this case we can find the critical value with the significance level [tex]\alpha=0.05[/tex] and if we find in the right tail of the z distribution we got:
[tex] z_{\alpha}= 1.64[/tex]
The statistic is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
Replacing we got:
[tex]z=\frac{0.344 -0.27}{\sqrt{\frac{0.27(1-0.27)}{125}}}=1.86[/tex]
Since the calculated value is higher than the critical value we have enough evidence to reject the null hypothesis and we can conclude that the true proportion of households with one person is significantly higher than 0.27
Step-by-step explanation:
We have the following dataset given:
[tex] X= 43[/tex] represent the households consisted of one person
[tex]n= 125[/tex] represent the sample size
[tex] \hat p= \frac{43}{125}= 0.344[/tex] estimated proportion of households consisted of one person
We want to test the following hypothesis:
Null hypothesis: [tex]p \leq 0.27[/tex]
Alternative hypothesis: [tex]p>0.27[/tex]
And for this case we can find the critical value with the significance level [tex]\alpha=0.05[/tex] and if we find in the right tail of the z distribution we got:
[tex] z_{\alpha}= 1.64[/tex]
The statistic is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
Replacing we got:
[tex]z=\frac{0.344 -0.27}{\sqrt{\frac{0.27(1-0.27)}{125}}}=1.86[/tex]
Since the calculated value is higher than the critical value we have enough evidence to reject the null hypothesis and we can conclude that the true proportion of households with one person is significantly higher than 0.27
Unit sales for new product ABC has varied in the first seven months of this year as follows: Month Jan Feb Mar Apr May Jun Jul Unit Sales 330 274 492 371 160 283 164 What is the (population) standard deviation of the data
Answer:
Approximately standard deviation= 108
Step-by-step explanation:
Let's calculate the mean of the data first.
Mean =( 330+ 274+ 492 +371 +160+ 283+ 164)/7
Mean= 2074/7
Mean= 296.3
Calculating the variance.
Variance = ((330-296.3)²+( 274-296.3)²+ (492-296.3)²+( 371-296.3)²+ (160-296.3)² (283-296.3)²+(164-296.3)²)/7
Variance= (1135.69+497.29+38298.49+5580.09+18577.69+176.89+17503.29)/7
Variance= 81769.43/7
Variance= 11681.347
Standard deviation= √variance
Standard deviation= √11681.347
Standard deviation= 108.080
Approximately 108
When 440 junior college students were surveyed, 200 said they have a passport. Construct a 95% confidence interval for the proportion of junior college students that have a passport.
The Confidence Interval is 0.403 < p < 0.497
What is Confidence Interval?The mean of your estimate plus and minus the range of that estimate constitutes a confidence interval. Within a specific level of confidence, this is the range of values you anticipate your estimate to fall within if you repeat the test. In statistics, confidence is another word for probability.
Given:
Sample proportion = 190/425
= 0.45
Now, [tex]\mu[/tex] = 1.96 x √[0.45 x 0.55/425]
[tex]\mu[/tex] = 0.047
So, 95% CI:
0.45-0.047 < p < 0.45+0.047
0.403 < p < 0.497
Learn more about Confidence Interval here:
https://brainly.com/question/24131141
#SPJ5
Suppose we write down the smallest positive 2-digit, 3-digit, and 4-digit multiples of 9,8 and 7(separate number sum for each multiple). What is the sum of these three numbers?
Answer:
Sum of 2 digit = 48
Sum of 3 digit = 317
Sum of 4 digit = 3009
Total = 3374
Step-by-step explanation:
Given:
9, 8 and 7
Required
Sum of Multiples
The first step is to list out the multiples of each number
9:- 9,18,....,99,108,117,................,999
,1008
,1017....
8:- 8,16........,96,104,...............,992,1000,1008....
7:- 7,14,........,98,105,.............,994,1001,1008.....
Calculating the sum of smallest 2 digit multiple of 9, 8 and 7
The smallest positive 2 digit multiple of:
- 9 is 18
- 8 is 16
- 7 is 14
Sum = 18 + 16 + 14
Sum = 48
Calculating the sum of smallest 3 digit multiple of 9, 8 and 7
The smallest positive 3 digit multiple of:
- 9 is 108
- 8 is 104
- 7 is 105
Sum = 108 + 104 + 105
Sum = 317
Calculating the sum of smallest 4 digit multiple of 9, 8 and 7
The smallest positive 4 digit multiple of:
- 9 is 1008
- 8 is 1000
- 7 is 1001
Sum = 1008 + 1000 + 1001
Sum = 3009
Sum of All = Sum of 2 digit + Sum of 3 digit + Sum of 4 digit
Sum of All = 48 + 317 + 3009
Sum of All = 3374
Anyone can help me with my math homework please?
Answer:
Step-by-step explanation:
hello,
so we know y in terms of t and x in terms of t and we need to find y in terms of x
[tex]x=21t^2<=>\sqrt{x}=\sqrt{21}*t \ \ as \ \ t>=0 \ \ So\\t=\sqrt{\dfrac{x}{21}}[/tex]
and then
[tex]y=f(x)=3\sqrt{\dfrac{x}{21}}+5=\sqrt{\dfrac{9x}{21}}+5=\sqrt{\dfrac{3x}{7}}+5[/tex]
hope this helps
A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. Find the rate of growth after 3 hours.
Step-by-step explanation:
Rate of growth equals 420/100 = 4.2 times per hour
So after t=three hours,
size of culture = 100*(4.2)^t = 100*4.2^3=7408.8 bacteria,
round to nearest unit 7409 bacteria after three hours (after initial size of 100).
Answer:
a) 100•4.2^t
b) P(3)= about 7409 bacteria
c) P’(3)= about 10,632 bacteria per hour
d) t= about 3.2 hours
Step-by-step explanation:
Identify the axis of symmetry and vertex of f(x) = –x2 –2x–1.
Answer:
Vertex: (-1, 0)
Axis of Symmetry: x = -1
Step-by-step explanation:
Use a graphing calc.
solve for x enter the solution from least to greatest x^2+3x-28=0
Answer:
x+4=0,x=−4x−7=0,x=7
Find the length of a picture frame whose width is 3 inches and whose proportions are the same as a 9-inch wide by 12-inch long picture frame.
Answer:
4 inches
Step-by-step explanation:
We can set up a proportion to find out the length value (assuming x is the length of the frame)
[tex]\frac{3}{x} = \frac{9}{12}[/tex]
We multiply 12 and 3...
[tex]12\cdot3=36[/tex]
And divide by 9...
[tex]36\div9=4[/tex]
So, the length of the frame is 4 inches.
Hope this helped!
Answer:
Step-by-step explanation:
4 inches
If the area of a circular cookie is 28.26 square inches, what is the APPROXIMATE circumference of the cookie? Use 3.14 for π.
75.2 in.
56.4 in.
37.6 in.
18.8 in.
Answer:
Step-by-step explanation:
c= 2(pi)r
Area = (pi)r^2
28.26 = (pi) r^2
r =[tex]\sqrt{9}[/tex] = 3
circumference = 2 (3.14) (3)
= 18.8 in
Answer: approx 18.8 in
Step-by-step explanation:
The area of the circle is
S=π*R² (1) and the circumference of the circle is C= 2*π*R (2)
So using (1) R²=S/π=28.26/3.14=9
=> R= sqrt(9)
R=3 in
So using (2) calculate C=2*3.14*3=18.84 in or approx 18.8 in
consider the difference of squares identity a^2-2b^2=(a+b)(a-b)
Answer: a= 3x and b= 7
Step-by-step explanation:
^^
if karen was 27 and her oldest brother was 29 years older and and there dad was 22 when Karen was born how old is the dad?
Answer:
49
Step-by-step explanation:
When Karen was born, the dad is 22 so Karen is now 27 which means the dad is 22+27= 49
Answer:
Their father is 49.
Step-by-step explanation:
Her father had her when he was 22, meaning that he is 22 years older than Karen. Karen is 27 right now, so her fathers age is (27+22) 49 years old.
Hope this helps!
Which of the following functions is graphed below
Answer:
the answer is C. y=[x-4]-2
Answer:
Step-by-step explanation:
Y=(x+4)-2
what is the product?
(x-3)(2x²-5x+1)
C) 2x³-11x²+16x-3
Answer:
2x^3-11x^2+16x-3
Step-by-step explanation:
1) multiply each term inside the parentheses with all other terms:
(x*2x^2)=2x^3
x*-5x=-5x^2
x*1=x
-3*2x^2=-6x^2
-3*-5x=15x
and
-3*1=-3
so
2x^3-5x^2+x-6x^2+15x-3
is our equation
to simplify:
2x^3-11x^2+16x-3 is the answer
Can somebody help me i have to drag the functions on top onto the bottom ones to match their inverse functions.
Answer:
1. x/5
2. cubed root of 2x
3.x-10
4.(2x/3)-17
Step-by-step explanation:
Answer:
Step-by-step explanation:
1. Lets find the inverse function for function f(x)=2*x/3-17
To do that first express x through f(x):
2*x/3= f(x)+17
2*x=(f(x)+17)*3
x=(f(x)+17)*3/2 done !!! (1)
Next : to get the inverse function from (1) substitute x by f'(x) and f(x) by x.
So the required function is f'(x)=(x+17)*3/2 or f'(x)=3*(x+17)/2
This is function is No4 in our list. So f(x)=2*x/3-17 should be moved to the box No4 ( on the bottom) of the list.
2. Lets find the inverse function for function f(x)=x-10
To do that first express x through f(x):
x= f(x)+10
x=f(x)+10 done !!! (2)
Next : to get the inverse function from (2) substitute x by f'(x) and f(x) by x.
So the required function is f'(x)=x+10
This is function is No3 in our list. So f(x)=x-10 should be moved to the box No3 ( from the top) of the list.
3.Lets find the inverse function for function f(x)=sqrt 3 (2x)
To do that first express x through f(x):
2*x= f(x)^3
x=f(x)^3/2 done !!! (3)
Next : to get the inverse function from (3) substitute x by f'(x) and f(x) by x.
So the required function is f'(x)=x^3/2
This is function No2 in our list. So f(x)=sqrt 3 (2x) should be moved to the box No2 ( from the top) of the list.
4.Lets find the inverse function for function f(x)=x/5
To do that first express x through f(x):
x=f(x)*5 done !!! (4)
Next : to get the inverse function from (4) substitute x by f'(x) and f(x) by x.
So the required function is f'(x)=x*5 or f'(x)=5*x
This is function No1 in our list. So f(x)=x/5 should be moved to the box No1 ( on the top) of the list.
Use the data below, showing a summary of highway gas mileage for several observations, to decide if the average highway gas mileage is the same for midsize cars, SUV’s, and pickup trucks. Test the appropriate hypotheses at the α = 0.01 level.
n Mean Std. Dev.
Midsize 31 25.8 2.56
SUV’s 31 22.68 3.67
Pickups 14 21.29 2.76
Answer:
Step-by-step explanation:
Hello!
You need to test at 1% if the average highway gas mileage is the same for three types of vehicles (midsize cars, SUV's and pickup trucks) to compare the average values of the three groups altogether, you have to apply an ANOVA.
n | Mean | Std. Dev.
Midsize | 31 | 25.8 | 2.56
SUV’s | 31 | 22.68 | 3.67
Pickups | 14 | 21.29 | 2.76
Be the study variables :
X₁: highway gas mileage of a midsize car
X₂: highway gas mileage of an SUV
X₃: highway gas mileage of a pickup truck.
Assuming these variables have a normal distribution and are independent.
The hypotheses are:
H₀: μ₁ = μ₂ = μ₃
H₁: At least one of the population means is different.
α: 0.01
The statistic for this test is:
[tex]F= \frac{MS_{Treatment}}{MS_{Error}}[/tex]~[tex]F_{k-1;n-k}[/tex]
Attached you'll find an ANOVA table with all its components. As you see, to manually calculate the statistic you have to determine the Sum of Squares and the degrees of freedom for the treatments and the errors, next you calculate the means square for both and finally the test statistic.
For the treatments:
The degrees of freedom between treatments are k-1 (k represents the amount of treatments): [tex]Df_{Tr}= k - 1= 3 - 1 = 2[/tex]
The sum of squares is:
SSTr: ∑ni(Ÿi - Ÿ..)²
Ÿi= sample mean of sample i ∀ i= 1,2,3
Ÿ..= grand mean, is the mean that results of all the groups together.
So the Sum of squares pf treatments SStr is the sum of the square of difference between the sample mean of each group and the grand mean.
To calculate the grand mean you can sum the means of each group and dive it by the number of groups:
Ÿ..= (Ÿ₁ + Ÿ₂ + Ÿ₃)/ 3 = (25.8+22.68+21.29)/3 = 23.256≅ 23.26
[tex]SS_{Tr}[/tex]= (Ÿ₁ - Ÿ..)² + (Ÿ₂ - Ÿ..)² + (Ÿ₃ - Ÿ..)²= (25.8-23.26)² + (22.68-23.26)² + (21.29-23.26)²= 10.6689
[tex]MS_{Tr}= \frac{SS_{Tr}}{Df_{Tr}}= \frac{10.6689}{2}= 5.33[/tex]
For the errors:
The degrees of freedom for the errors are: [tex]Df_{Errors}= N-k= (31+31+14)-3= 76-3= 73[/tex]
The Mean square are equal to the estimation of the variance of errors, you can calculate them using the following formula:
[tex]MS_{Errors}= S^2_e= \frac{(n_1-1)S^2_1+(n_2-1)S^2_2+(n_3-1)S^2_3}{n_1+n_2+n_3-k}= \frac{(30*2.56^2)+(30*3.67^2)+(13*2.76^2)}{31+31+14-3} = \frac{695.3118}{73}= 9.52[/tex]
Now you can calculate the test statistic
[tex]F_{H_0}= \frac{MS_{Tr}}{MS_{Error}} = \frac{5.33}{9.52}= 0.559= 0.56[/tex]
The rejection region for this test is always one-tailed to the right, meaning that you'll reject the null hypothesis to big values of the statistic:
[tex]F_{k-1;N-k;1-\alpha }= F_{2; 73; 0.99}= 4.07[/tex]
If [tex]F_{H_0}[/tex] ≥ 4.07, reject the null hypothesis.
If [tex]F_{H_0}[/tex] < 4.07, do not reject the null hypothesis.
Since the calculated value is less than the critical value, the decision is to not reject the null hypothesis.
Then at a 1% significance level you can conclude that the average highway mileage is the same for the three types of vehicles (mid size, SUV and pickup trucks)
I hope this helps!
Suppose μ1 and μ2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. Use the two-sample t test at significance level 0.01 to test H0: μ1 − μ2 = −10 versus Ha: μ1 − μ2 < −10 for the following data: m = 8, x = 115.6, s1 = 5.04, n = 8, y = 129.3, and s2 = 5.32.
Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)
t = ________
P-value = _________
Answer:
Step-by-step explanation:
This is a test of 2 independent groups. Given that μ1 and μ2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems, the hypothesis are
For null,
H0: μ1 − μ2 = - 10
For alternative,
Ha: μ1 − μ2 < - 10
This is a left tailed test.
Since sample standard deviation is known, we would determine the test statistic by using the t test. The formula is
(x1 - x2)/√(s1²/n1 + s2²/n2)
From the information given,
x1 = 115.6
x2 = 129.3
s1 = 5.04
s2 = 5.32
n1 = 8
n2 = 8
t = (115.6 - 129.3)/√(5.04²/8 + 5.32²/8)
t = - 2.041
Test statistic = - 2.04
The formula for determining the degree of freedom is
df = [s1²/n1 + s2²/n2]²/(1/n1 - 1)(s1²/n1)² + (1/n2 - 1)(s2²/n2)²
df = [5.04²/8 + 5.32²/8]²/[(1/8 - 1)(5.04²/8)² + (1/8 - 1)(5.32²/8)²] = 45.064369/3.22827484
df = 14
We would determine the probability value from the t test calculator. It becomes
p value = 0.030
Since alpha, 0.01 < the p value, 0.03, then we would fail to reject the null hypothesis.
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Answer:
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What is the value of x in equation 1/3 (12x -24) = 16
Thank you
Answer:
The value of x is x = 6
Step-by-step explanation:
[tex]\frac{1}{3}(12x - 24) = 16\\ 12x - 24 = 48\\12x = 48+ 24\\12x = 72\\12/12 = x\\72/12 = 6\\x=6[/tex]
Hope this helped! :)
A publisher reports that 45% of their readers own a laptop. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 370 found that 40% of the readers owned a laptop. Is there sufficient evidence at the 0.02 level to support the executive's claim?
Answer:
At a significance level of 0.02, there is not enough evidence to support the claim that the percentage of readers that own a laptop is significantly different from 45%.
P-value = 0.06
Step-by-step explanation:
This is a hypothesis test for a proportion.
The claim is that the percentage of readers that own a laptop is significantly different from 45%.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi=0.45\\\\H_a:\pi\neq 0.45[/tex]
The significance level is 0.02.
The sample has a size n=370.
The sample proportion is p=0.4.
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.45*0.55}{370}}\\\\\\ \sigma_p=\sqrt{0.000669}=0.026[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.4-0.45+0.5/370}{0.026}=\dfrac{-0.049}{0.026}=-1.881[/tex]
This test is a two-tailed test, so the P-value for this test is calculated as:
[tex]\text{P-value}=2\cdot P(z<-1.881)=0.06[/tex]
As the P-value (0.06) is greater than the significance level (0.02), the effect is not significant.
The null hypothesis failed to be rejected.
At a significance level of 0.02, there is not enough evidence to support the claim that the percentage of readers that own a laptop is significantly different from 45%.