Answer:
Please see the Step-by-step explanation for the answers
Step-by-step explanation:
1)
∑[tex]\left \ {{5} \atop {j=1}} \right.[/tex] 2j
The sum of series from j=1 to j=5 is:
∑ = 2(1) + 2(2) + 2(3) + 2(4) + 2(5)
= 2 + 4 + 6 + 8 + 10
∑ = 30
2)
This question is not given clearly so i assume the following series that will give you an idea how to solve this:
∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] 2k²
The sum of series from k=1 to j=4 is:
∑ = 2(1)² + 2(2)² + 2(3)² + 2(4)²
= 2(1) + 2(4) + 2(9) + 2(16)
= 2 + 8 + 18 + 32
∑ = 60
∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] (2k)²
∑ = (2*1)² + (2*2)² + (2*3)² + (2*4)²
= (2)² + (4)² + (6)² + (8)²
= 4 + 16 + 36 + 64
∑ = 120
∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] (2k)²- 4
∑ = (2*1)²-4 + (2*2)²-4 + (2*3)²-4 + (2*4)²-4
= (2)²-4 + (4)²-4 + (6)²-4 + (8)²-4
= (4-4) + (16-4) + (36-4) + (64-4)
= 0 + 12 + 32 + 60
∑ = 104
∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] 2k²- 4
∑ = 2(1)²-4 + 2(2)²-4 + 2(3)²-4 + 2(4)²-4
= 2(1)-4 + 2(4)-4 + 2(9)-4 + 2(16)-4
= (2-4) + (8-4) + (18-4) + (32-4)
= -2 + 4 + 14 + 28
∑ = 44
3)
∑[tex]\left \ {{6} \atop {k=3}} \right.[/tex] (2k-10)
∑ = (2×3−10) + (2×4−10) + (2×5−10) + (2×6−10)
= (6-10) + (8-10) + (10-10) + (12-10)
= -4 + -2 + 0 + 2
∑ = -4
4)
1+1/2+1/4+1/8+1/16+1/32+1/64
This is a geometric sequence where first term is 1 and the common ratio is 1/2 So
a = 1
This can be derived as
1/2/1 = 1/2 * 1 = 1/2
1/4/1/2 = 1/4 * 2/1 = 1/2
1/8/1/4 = 1/8 * 4/1 = 1/2
1/16/1/8 = 1/16 * 8/1 = 1/2
1/32/1/16 = 1/32 * 16/1 = 1/2
1/64/1/32 = 1/64 * 32/1 = 1/2
Hence the common ratio is r = 1/2
So n-th term is:
[tex]ar^{n-1}[/tex] = [tex]1(\frac{1}{2})^{n-1}[/tex]
So the answer that represents the series in sigma notation is:
∑[tex]\left \ {{7} \atop {j=1}} \right.[/tex] [tex](\frac{1}{2})^{j-1}[/tex]
5)
−3+(−1)+1+3+5
This is an arithmetic sequence where the first term is -3 and the common difference is 2. So
a = 1
This can be derived as
-1 - (-3) = -1 + 3 = 2
1 - (-1) = 1 + 1 = 2
3 - 1 = 2
5 - 3 = 2
Hence the common difference d = 2
The nth term is:
a + (n - 1) d
= -3 + (n−1)2
= -3 + 2(n−1)
= -3 + 2n - 2
= 2n - 5
So the answer that represents the series in sigma notation is:
∑[tex]\left \ {{5} \atop {j=1}} \right.[/tex] (2j−5)
For (1) the sum is 30, for (2) the sum is 90, for (3) the sum is -4, for(4) the sigma notation is [tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex] where j = 1 to j = 7, and for (5) the sigma notation is [tex]\rm\sum j = (2j-5)[/tex] where j = 1 to j = 5.
We have different series in the question.
It is required to find the sum of all series.
What is a series?In mathematics, a series can be defined as a group of data that followed certain rules of arithmetic.
1) We have:
[tex]\rm \sum j=2j[/tex] where j = 1 to j = 5
After expanding the series, we get:
= 2(1)+2(2)+2(3)+2(4)+2(5)
=2(1+2+3+4+5)
= 2(15)
=30
2) We have:
[tex]\rm \sum k=(2k^2-4)[/tex] where k = 1 to k = 4
After expanding the series, we get:
[tex]\rm = (2(1)^2-4)+(2(2)^2-4)+(2(3)^2-4)+(2(4)^2-4)+(2(5)^2-4)\\[/tex]
[tex]\rm = 2[1^2+2^2+3^2+4^2+5^2]-4\times5\\\\\rm=2[55]-20\\\\\rm = 90[/tex]
3) We have:
[tex]\rm \sum k= (2k-10)[/tex] where k = 3 to k = 6
After expanding the series, we get:
[tex]= (2(3)-10)+(2(4)-10)+(2(5)-10)+(2(6)-10)\\\\=2[3+4+5+6] - 10\times4\\\\=2[18] - 40\\\\= -4[/tex]
4) The series given below:
[tex]1, \frac{1}{2} ,\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\frac{1}{64}[/tex]
It is a geometric progression:
[tex]\rm n^t^h[/tex] for the geometric progression is given by:
[tex]\rm a_n = ar^{n-1}[/tex]
[tex]\rm a_n = 1(\frac{1}{2})^{n-1}\\\\\rm a_n = (\frac{1}{2})^{n-1}\\[/tex]
In sigma notation we can write:
[tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex] where j = 1 to j = 7
5) The given series:
−3+(−1)+1+3+5, it is arithmetic series.
[tex]\rm n^t^h[/tex] for the arithmetic progression is given by:
[tex]\rm a_n = a+(n-1)d[/tex]
[tex]\rm a_n = -3+(n-1)(2)\\\\\rm a_n = 2n-5[/tex]
In sigma notation we can write:
[tex]\rm\sum j = (2j-5)[/tex] where j = 1 to j = 5
Thus, for (1) the sum is 30, for (2) the sum is 90, for (3) the sum is -4, for(4) the sigma notation is [tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex] where j = 1 to j = 7, and for (5) the sigma notation is [tex]\rm\sum j = (2j-5)[/tex] where j = 1 to j = 5.
Learn more about the series here:
https://brainly.com/question/10813422
What number should replace the question mark?
Answer:
1 Cruz it makes sense I hope this helps tho
Find the cost of fencing a square plot area 11025 sq m. at the rate of rupees 85 per sq m.
Answer:
Rate per meter=85 Rs. hence , the cost of fencing the square plot at rate of 85 rs. per meter is 937125 Rs.
Given below are descriptions of two lines. Line 1: Goes through (-2,10) and (1,1) Line 2: Goes through (-2,8) and (2,-4)
Answer:
Option (2)
Step-by-step explanation:
1). If two lines have the same slope, lines are defined as parallel.
m₁ = m₂
2). If the multiplication of the slopes of two lines is (-1), lines will be perpendicular.
m₁ × m₂ = (-1)
Line 1 : It passes through two points (-2, 10) and (1, 1).
Slope of the line 1 = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
= [tex]\frac{1+2}{10-1}[/tex]
= [tex]\frac{3}{9}[/tex]
m₁ = [tex]\frac{1}{3}[/tex]
Line 2 : It passes through two points (-2, 8) and (2, -4).
Slope of the line 2 = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
= [tex]\frac{8+4}{-2-2}[/tex]
= [tex]-\frac{12}{4}[/tex]
m₂ = -3
Since, m₁ × m₂ = [tex]\frac{1}{3}\times (-3)[/tex]
= (-1)
Therefore, given lines are perpendicular to each other.
Option (2) is the correct option.
In a study of academic procrastination, researchers reported that for a random sample of 41 undergraduate students preparing for a psychology exam, the mean time spent studying was 11.9 hours with a standard deviation of 4.5 hours. Compute a 95% confidence interval for μ, the mean time spent studying for the exam among all students taking this course.
Answer:
The 95% confidence interval is [tex]10.5 < \mu <13.3[/tex]
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 41[/tex]
The sample mean is [tex]\= x = 11.9 \ hr[/tex]
The standard deviation is [tex]\sigma = 4.5[/tex]
For a 95% confidence interval the confidence level is 95%
Given that the confidence level is 95% then the level of significance can be mathematically represented as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical values of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table
The values is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{ \sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{ 4.5 }{ \sqrt{41} }[/tex]
[tex]E = 1.377[/tex]
The 95% confidence interval is mathematically represented as
[tex]\= x - E < \mu < \= x - E[/tex]
substituting values
[tex]11.9 - 1.377 < \mu <11.9 + 1.377[/tex]
[tex]10.5 < \mu <13.3[/tex]
The expression −50x+100 represents the balance, in dollars, of a bank account after x months. What is the rate of change, in dollars per month, of the bank account balance?
Answer:
-50
Step-by-step explanation:
Basically get two slopes -50(1)+100 will get you 1,50 (1 is x and 50 is y since its the answer)
-50(0)+100 (0,100) Y₂-Y₁/X₂-X₁ 50-100/1-0
Rate of change per month = -$50
A professor graded the final exams and found that the mean score was 70 points. Which of the following can you conclude?
A- All of the above.
B- The median score was 70 points.
C- 50% of the students scored below 70 points.
D- This would be a normal distribution.
Answer: C) 50% of the students scored below 70%
Step-by-step explanation:
Mean is the average. To find the mean (aka average) you add up all of the scores and divide by the number of tests.
B) The mean can be 70 without any test scoring 70% so B is not true.
A) Since B is not true, then A is not a valid option.
D) We don't know any of the other data so don't know if it is skewed left, skewed right, or normal. Therefore, option D is not true.
C) If the average is 70%, then half received grades above that score and half received grades below that score. So, option C is TRUE!
Suppose that a box contains 6 cameras and that 3 of them are defective. A sample of 2 cameras is selected at random. Define the random variable X as the number of defective cameras in the sample. Write the binomial probability distribution for X . Round to two decimal places.
Answer:
X ~ Binom (n = 6, p = 0.50)
Step-by-step explanation:
We are given that a box contains 6 cameras and that 3 of them are defective.
A sample of 2 cameras is selected at random.
Let X = Number of defective cameras in the sample.
The above situation can be represented through binomial distribution;
[tex]P(X=r)= \binom{n}{r}\times p^{r} \times (1-p)^{n-r} ;x = 0,1,2,3,......[/tex]
where, n = number of trials (samples) taken = 2 cameras
x = number of success
p = probabilitiy of success which in our question is probability that
cameras are defective, i.e. p = [tex]\frac{3}{6}[/tex] = 0.50
So, X ~ Binom (n = 2, p = 0.50)
Now, the binomial probability distribution for X is given by;
[tex]P(X=r)= \binom{6}{r}\times 0.5^{r} \times (1-0.5)^{6-r} ;r = 0,1,2[/tex]
Here, the number of success can be 0, 1, or 2 defective cameras.
According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $1,999. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $574. (Round your z-score computation to 2 decimal places and final answers to 2 decimal places.) What percent of the adults spend more than $2,550 per year on reading and entertainment?
Answer:
The probability is [tex]P(X > x ) = 0.19215[/tex]
Step-by-step explanation:
From the question we are told that
Th The population mean [tex]\mu = \$ 1,999[/tex]
The standard deviation is [tex]\sigma = \$ 574[/tex]
The values considered is [tex]x = \$ 2,500[/tex]
Given that the distribution of the amounts spent follows the normal distribution then the percent of the adults spend more than $2,550 per year on reading and entertainment is mathematically represented as
[tex]P(X > x ) = P(\frac{ X - \mu}{\sigma } > \frac{ x - \mu}{\sigma } )[/tex]
Generally
[tex]X - \mu}{\sigma } = Z (The \ standardized \ value \ of \ X )[/tex]
So
[tex]P(X > x ) = P(Z > \frac{ x - \mu}{\sigma } )[/tex]
substituting values
[tex]P(X > 2500 ) = P(Z > \frac{ 2500 - 1999}{574 } )[/tex]
[tex]P(X > 2500 ) = P(Z >0.87 )[/tex]
From the normal distribution table the value of [tex]P(Z >0.87 )[/tex] is
[tex]P(Z >0.87 ) = 0.19215[/tex]
Thus
[tex]P(X > x ) = 0.19215[/tex]
pls help:Find all the missing elements:
Answer:
B = 48.7° , C = 61.3° , b = 12Step-by-step explanation:
In order to find B we must first angle C
To find angle C we use the sine rule
That's
[tex] \frac{ |a| }{ \sin(A) } = \frac{ |c| }{ \sin(C) } [/tex]
From the question
a = 15
A = 70°
c = 14
So we have
[tex] \frac{15}{ \sin(70) } = \frac{14}{ \sin(C) } [/tex]
[tex] \sin(C) = \frac{14 \sin(7 0 ) }{15} [/tex]
[tex]C = \sin^{ - 1} ( \frac{14 \sin(70) }{15} ) [/tex]
C = 61.288
C = 61.3° to the nearest tenthSince we've found C we can use it to find B.
Angles in a triangle add up to 180°
To find B add A and C and subtract it from 180°
That's
A + B + C = 180
B = 180 - A - C
B = 180 - 70 - 61.3
B = 48.7° to the nearest tenthTo find b we can use the sine rule
That's
[tex] \frac{ |a| }{ \sin(A) } = \frac{ |a| }{ \sin(B) } [/tex]
[tex] \frac{15}{ \sin(70) } = \frac{ |b| }{ \sin(48.7) } [/tex]
[tex] |b| = \frac{15 \sin(48.7) }{ \sin(70) } [/tex]
b = 11.9921
b = 12.0 to the nearest tenthHope this helps you
I need to know what’s 500+0+12+44+55+500+0+12+44+55+ 500+0+12+44+55+500+0+12+44+55+
Answer:
Step-by-step explanation:
2,444
Answer:
2,444.
Step-by-step explanation:
500+0+12+44+55+500+0+12+44+55+ 500+0+12+44+55+500+0+12+44+55 = 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 = 4(500 + 12 + 44 + 55) = 4(512 + 99) = 4(611) = 2,444.
Hope this helps!
find the sum 7+7(2)+7(2^2)+...+7(2^9)
Answer:
7161
Step-by-step explanation:
7 + 7(2) + 7(2)² + ... + 7(2)⁹
= ∑₁¹⁰ 7(2)ⁿ⁻¹
= 7 (1 − 2¹⁰) / (1 − 2)
= 7161
I will rate you brainliest/ / / Tonicha is creating two triangular gardens in her backyard. The total area of each triangle varies jointly with the length of the triangle's base and the height. The area of the smaller triangle is 16 square feet. It has a base that is 8 feet in length and the height is 4 feet. What is the area of the larger triangle when its base is 12 feet in length and its height is 8 feet?
Answer:
The area of larger triangle is 48 ft².
Step-by-step explanation:
Assuming that the area of triangle formula is A = 1/2 × base × height. Then, you have to substitute the following values :
[tex]area = \frac{1}{2} \times b \times h[/tex]
[tex]let \: b = 12,h = 8[/tex]
[tex]area = \frac{1}{2} \times 12 \times 8[/tex]
[tex]area = \frac{1}{2} \times 96[/tex]
[tex]area = 48 \: {feet}^{2} [/tex]
Given that;
A∞bh
A=kbh. where k is a constant
For smaller triangle
A=kbh
16=8(4)k
16=32k
k=½
For bigger triangle
A=kbh
Where k=½ , b= 12 and h=8
A=½(12)8
A=48 square feet
Activity 12-4: A large monohybrid crossa corn ear with purple and yellow kernels The total number of purple and yellow kernels on 8 different corn ears were counted: Purple kernels 3593 Yellow kernels 1102 What is the ratio of purple kernels to yellow kernels
Complete Question
The complete question is shown on the first uploaded image
Answer:
The correct option is C
Step-by-step explanation:
From the question we are told that
The number of purple kernel is [tex]n_k = 3593[/tex]
The number of yellow kernel is [tex]n_y = 1102[/tex]
Generally the ration of the purple to the yellow kernels is mathematically evaluated as
[tex]r = \frac{n_k}{n_y}[/tex]
substituting values
[tex]r = \frac{3593}{1102}[/tex]
[tex]r = 3.3[/tex]
[tex]r \approx 3[/tex]
Therefore the ratio is
[tex]1 \ Yellow : 3 \ Purple[/tex]
Find the perimeter of the rectangle with the following vertices. (−6, −2), (0, −10), (5, 2), (−1, 10) 23 52 46 40
Answer:
46
Step-by-step explanation:
See attached for reference
The points given:
(−6, −2), (0, −10), (5, 2), (−1, 10)They form a rectangle as seen in the picture.
We can notice that this is a parallelogram, as respective lines have same difference of coordinates.
So calculating only the two of the sides will be sufficient to get its perimeter:
a = √(-1+6)² + (10+2)² = √25+144 = √169= 13b = √(0+6)² + (-10+2)² = √36+64 = √100 = 10So, the perimeter:
P = 2(13+10) = 46
Find the length of the leg of a right triangle with leg length b= 21.5 inches and the hypotenuse c= 31.9 inches. Use a calculator to estimate the square root to one decimal place.
Answer:
23.6 (approximate value)
Step-by-step explanation:
a*a+b*b=c*c Pythagorean thereom(21.5)^2 + b*b = (31.9)^2b*b= 1017.61 - 462.25√b*b= √555.36b=23.6(approximate value)Answer:
Step-by-step explanation:
23.6
Following is a portion of the regression output for an application relating maintenance expense (dollars per month) to usage (hours per week) for a particular brand of computer terminal.
ANOVA
df SS MS F Significance F
Regression 1 1575.76
Residual 8 349.14
Total 9 1924.90
Coefficient Standard Error t Stat P-value
Intercept 6.1092 0.9361
Usage 0.8931 0.149
A) Write the estimated regression equation (to 4 decimals).
B) Use a t test to determine whether monthly maintenance expense is related to usage at the .05 level of significance (to 2 decimals, if necessary).
1. Reject the null hypothesis
2. Do not reject the null hypothesis
C) Monthly maintenance expense______to usage.
1. Is related
2. Is not related
D) Did the estimated regression equation provide a good fit?
1. yes
2. no
E) Explain.
Answer:
Explained below.
Step-by-step explanation:
The ANOVA and Regression output for an application relating maintenance expense (dollars per month) to usage (hours per week) for a particular brand of computer terminal is provided.
(A)
The estimated regression equation equation is:
[tex]y=6.1092+0.8931x[/tex]
Here,
y = maintenance expense (dollars per month)
x = usage (hours per week) for a particular brand of computer terminal
(B)
Consider the Regression output.
The hypothesis to test whether monthly maintenance expense is related to usage is:
H₀: The monthly maintenance expense is not related to usage, i.e. β = 0.
Hₐ: The monthly maintenance expense is related to usage, i.e. β ≠ 0.
Compute the test statistic as follows:
[tex]t=\frac{b}{S.E._{b}}=\frac{0.8931}{0.149}=5.99[/tex]
Compute the p-value as follows:
[tex]p-value=2\times P (t_{8}<5.99}=0.00033[/tex]
The null hypothesis will be rejected if the p-value is less than the significance level.
p-value = 0.00033 < α = 0.05
Reject the null hypothesis.
(C)
Monthly maintenance expense is related to usage.
(D)
Yes, the estimated regression equation provide a good fit.
Since the regression coefficient is significant it can be concluded that the regression equation estimated is a good fit.
From the regression output given, the solution to the questions given are outlined thus ;
[tex] Null \: hypothesis : H_{0} : β = 0 [/tex] [tex] Alternative \: hypothesis : H_{1} : β ≠ 0 [/tex]1.)
Regression equation :
y = bx + c b = slope ; c = interceptHence, the estimated regression equation is;
y = 0.8931x + 6.10922.)
We can calculate the T-statistic value thus ;
[tex] T-statistic = \frac{b}{SE_{b}}[/tex] [tex]SE_{b} = Standard \: error \: of \: slope[/tex] df = 8Hence, the T-statistic is given as ;
[tex] T-statistic = \frac{0.8931}{0.149} = 5.99[/tex]
Pvalue (2 tailed) = 0.00033
Decison Region :
[tex] Reject \: H_{0} \: if \: Pvalue \: < \: α [/tex]Since 0.00033 < 0.05 ; we reject the Null hypothesis.
3.)
Hence, we conclude that monthly expense is related to usage.
4.)
Since, the correlation Coefficient, β ≠ 0 ; Yes, the correlation provides a good fit as it is significant.
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PLZ answer quick i will give brainliest if right no explanation needed Joe is responsible for reserving hotel rooms for a company trip. His company changes plans and increases how many people are going on the trip, so they need at least 50 total rooms. Joe had already reserved and paid for 161616 rooms, so he needs to reserve additional rooms. He can only reserve rooms in blocks, and each block contains 8 rooms and costs $900. Let B represent the number of additional blocks that Joe reserves. 1) Which inequality describes this scenario? Choose 1 answer: a: 16+8B≤50 b: 16+8B≥50 c: 16+B≤50 d: 16+B≥50 2) What is the least amount of additional money Joe can spend to get the rooms they need?
Answer:
16 + 8b ≥ 50
4500
Step-by-step explanation:
He needs at least 50 rooms and has already reserved 16
They are in groups of 8
16 + 8b ≥ 50
Subtract 16 from each side
16+8b-16 ≥ 50 -16
8b≥ 34
Divide by 6
8b/8 ≥ 34/8
b≥ 4.25
We need to round up since we need at least 50
b = 5 since we want the least amount of rooms
Each block is 900
5*900 = 4500 more that he will have to spend
A potato chip company makes potato chips in two flavors, Regular and Salt & Vinegar. Riley is a production manager for the company who is trying to ensure that each bag contains about the same number of chips, regardless of flavor. He collects two random samples of 10 bags of chips of each flavor and counts the number of chips in each bag. Assume that the population variances of the number of chips per bag for both flavors are equal and that the number of chips per bag for both flavors are normally distributed. Let the Regular chips be the first sample, and let the Salt & Vinegar chips be the second sample. Riley conducts a two-mean hypothesis test at the 0.05 level of significance, to test if there is evidence that both flavors have the same number of chips in each bag. (a) H0:μ1=μ2; Ha:μ1≠μ2, which is a two-tailed test. (b) t≈1.44 , p-value is approximately 0.167 (c) Which of the following are appropriate conclusions for this hypothesis test?
A. There is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.B. There is sufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.C. Reject H0.D. Fail to reject H0.
Answer:
A. There is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.
D. Fail to reject H0.
Step-by-step explanation:
From the summary of the given test statistics.
The null and the alternative hypothesis are:
[tex]H_0:\mu_1=\mu_2 \\ \\ Ha:\mu_1 \neq \mu_2[/tex]
This test is also a two tailed test.
Similarly, the t value for the test statistics = 1.44
The p- value - 0.167
The level of significance ∝ = 0.05
The objective we are meant to achieve here is to determine which of the following from the given options are appropriate conclusions for this hypothesis test.
From what we have above:
Decision Rule: We fail to reject the null hypothesis since the p-value is greater than the level of significance at 0.05
CONCLUSION: Therefore, we can conclude that there is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag as we fail to reject H0.
Can somebody explain how these would be done? The selected answer is incorrect, and I was told "Nice try...express the product by first multiplying the coefficients...then adding your "like term" angles...for instance, cos (2pi/5) + cos (-pi/2) = cos (2pi/5 + -pi/2)...then use the calculator in RADIAN mode to evaluate." Doing those steps, I got the correct constant but a coefficient that was completely off. For the second one, I was told "Good effort...express the quotient by first dividing the coefficients...then subtract your "like term" angles...for instance, cos (2pi/5) - cos (-pi/2) = cos (pi/6 - pi/3)...Finally, use the calculator (in radian MODE) to evaluate."
Answer:
Solution ( Second Attachment ) : - 2.017 + 0.656i
Solution ( First Attachment ) : 16.140 - 5.244i
Step-by-step explanation:
Second Attachment : The quotient of the two expressions would be the following,
[tex]6\left[\cos \left(\frac{2\pi }{5}\right)+i\sin \left(\frac{2\pi \:}{5}\right)\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
So if we want to determine this expression in standard complex form, we can first convert it into trigonometric form, then apply trivial identities. Either that, or we can straight away apply the following identities and substitute,
( 1 ) cos(x) = sin(π / 2 - x)
( 2 ) sin(x) = cos(π / 2 - x)
If cos(x) = sin(π / 2 - x), then cos(2π / 5) = sin(π / 2 - 2π / 5) = sin(π / 10). Respectively sin(2π / 5) = cos(π / 2 - 2π / 5) = cos(π / 10). Let's simplify sin(π / 10) and cos(π / 10) with two more identities,
( 1 ) [tex]\cos \left(\frac{x}{2}\right)=\sqrt{\frac{1+\cos \left(x\right)}{2}}[/tex]
( 2 ) [tex]\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos \left(x\right)}{2}}[/tex]
These two identities makes sin(π / 10) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and cos(π / 10) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex].
Therefore cos(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and sin(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex]. Substitute,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
Remember that cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting those values,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex]
And now simplify this expression to receive our answer,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex] = [tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}+\frac{3\sqrt{3-\sqrt{5}}}{4}i[/tex],
[tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}[/tex] = [tex]-2.01749\dots[/tex] and [tex]\:\frac{3\sqrt{3-\sqrt{5}}}{4}[/tex] = [tex]0.65552\dots[/tex]
= [tex]-2.01749+0.65552i[/tex]
As you can see our solution is option c. - 2.01749 was rounded to - 2.017, and 0.65552 was rounded to 0.656.
________________________________________
First Attachment : We know from the previous problem that cos(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], sin(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex], cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting we receive a simplified expression,
[tex]6\sqrt{5+\sqrt{5}}-6i\sqrt{3-\sqrt{5}}[/tex]
We know that [tex]6\sqrt{5+\sqrt{5}} = 16.13996\dots[/tex] and [tex]-\:6\sqrt{3-\sqrt{5}} = -5.24419\dots[/tex] . Therefore,
Solution : [tex]16.13996 - 5.24419i[/tex]
Which rounds to about option b.
A sprinkler system is being installed in a newly renovated building on campus. The average activation time is supposed to be at most 20 seconds. A series of 12 fire alarm/sprinkler system tests results in an average activation time of 21.5 seconds. Do these data indicate that the design specifications have not been met? The hypotheses to be tested are H0: m = 20 versus Ha: m > 20, where m = the true average activation time of the sprinkler system. Assume that activation times for this system are Normally distributed with s = 3 seconds.
(a) What is the value of the observed test statistic?
(b) What is the value of the P-value?
(c) Are the data statistically significant at the 5% significance level? Explain briefly.
(d) What does the decision you made mean with respect to the question "Do these data indicate that the design specifications have not been met?"
(e) If the true average activation time of the sprinkler system is, in fact, equal to 20 seconds, what type of error would you have made?
Answer:
A) t = 1.73
B) p-value = 0.0558
C) Data is not statistically significant because the p-value of 0.0558 is more than the significance value of 0.05
D) The decision means that the design specifications are not met.
E) Type II error
Step-by-step explanation:
The hypotheses are:
H₀: μ = 20
H₁: μ > 20
A) Formula for the test statistic is;
t = (x' - μ)/(s/√n)
Now, we are given;
x' = 21.5
μ = 20
s = 3
n = 12
Thus;
t = (21.5 - 20)/(3/√12)
t = 1.73
B) we have our t-value as 1.73
Now, Degree of freedom(DF) = n - 1
So,DF = 12 - 1 = 11
Using significance level of α = 0.05, t-value = 1.73 and DF = 11, one tailed hypothesis, from online P-value calculator attached, we have;
p-value = 0.0558
C) Data is not statistically significant because the p-value of 0.0558 is more than the significance value of 0.05
D) We will not reject the null hypothesis. The decision means that the design specifications are not met.
E) If the true average activation time of the sprinkler system is, in fact, equal to 20 seconds, then the null hypothesis is false.
Since we did not reject the null hypothesis even though it is false, the error that was committed was therefore a type II error.
heeeeeeeeelpppppppppppp
Answer:
1). x = 2.67 units
2). x = 4.80 units
3). x = 6.00 units
Step-by-step explanation:
1). By applying Pythagoras theorem,
Hypotenuse² = [Leg(1)]² + [leg(2)]²
12² = x² + b² [Let the base of both the triangles = b units]
144 = x² + b² ------(1)
Similarly, 13² = (x + 3)² + b²
169 = x² + 6x + 9 + b²
169 - 9 - 6x = x² + b²
160 - 6x = x² + b² ------(2)
From equation (1) and (2)
144 = 160 - 6x
6x = 160 - 144
x = [tex]\frac{16}{6}[/tex]
x = 2.67 units
2). By applying Pythagoras theorem,
10² = x² + h² [Let the height of the triangle = h]
100 = x² + h² ------(1)
13² = (2x)² + h²
169 = 4x² + h² -----(2)
By substituting equation (1) from equation (2),
169 - 100 = (4x² + h²) - (x² + h²)
69 = 3x²
x² = 23
x = √23
x = 4.795
x ≈ 4.80 units
3). By applying Pythagoras theorem,
9² = x² + h² [Let the height of the triangle = h units]
81 = x² + h² ------(1)
7² = (x - 4)² + h²
49 = x² + 16 - 8x + h²
49 - 16 = x² + h² - 8x
33 + 8x = x² + h² -------(2)
From equation (1) and (2)
81 = 33 + 8x
8x = 48
x = 6.00 units
What is an equation of the line that passes through the points (2, -7) and (8, -4)?
Answer:
The answer is
[tex]y = \frac{1}{2} x - 8[/tex]Step-by-step explanation:
To find the equation of the line that passes through two points , first find the slope and then use the formula
y - y1 = m(x - x1)
where m is the slope
(x1 , y1) are any of the points
To find the slope of the line using two points we use the formula
[tex]m = \frac{y2 - y1}{x2 - x1} [/tex]
Slope of the line using points
(2, -7) and (8, -4) is
[tex] \frac{ - 4 + 7}{8 - 2} = \frac{3}{6} = \frac{1}{2} [/tex]Now the equation of the line using point (2 , - 7) and slope 1/2 is
[tex] y + 7 = \frac{1}{2} (x - 2)[/tex][tex]y + 7 = \frac{1}{2} x - 1[/tex][tex]y = \frac{1}{2} x - 1 - 7[/tex]We have the final answer as
[tex]y = \frac{1}{2} x - 8[/tex]Hope this helps you
How to graph the line y=4/3x
Answer:
make a table of values
Step-by-step explanation:
then plot using those values
The required graph has been attached which represents the line y = 4/3x
What is a graph?A graph can be defined as a pictorial representation or a diagram that represents data or values.
We have been given the equation of a line below as:
y = 4/3x
Rewrite in slope-intercept form.
y = (4/3)x
Use the slope-intercept form to discover the slope and y-intercept.
Here the slope is 4/3 and y-intercept = (0, 0)
Any line can be graphed using two points. Select two x values, and plug them into the equation to find the corresponding y values.
When substitute the value of x = 0, then the value of y = 0, and When substitute the value of x = 3, then the value of y = -4,
Hence, the graph represents the line y = 4/3x
Therefore, the required graph of the line y=4/3x will be shown in the as attached file.
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If S is a compact subset of R and T is a closed subset of S, then T is compact. Prove this using the definition of compactness.
Answer:
It has been proved that T is compact
Step-by-step explanation:
To prove this using the definition of compactness, let's assume that T is
not compact. Now, if that be the case, an open cover of T will exist. Let's call this open cover "A". Now, this open cover will have no finite subcover.
Now, from the question, since T is closed, it’s complement R\T will be open.
Therefore, if we add the set R\T to the collection of sets A, then we'll have an open cover of R and also of S.
Due to the fact that S is compact, this
cover will have a finite sub - cover which we will call B.
Finally, either B itself or B\{R\T} would be a finite sub - cover of A. This is a contradiction.
Thus, it proves that T has to be compact if S is to be a compact subset of R and T is to be a closed subset of S.
The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. An article reported the accompanying observations on x = TOST time hr and y = RBOT time min for 12 oil specimens.TOST 4200 3575 3750 3700 4050 2770 4870 4500 3450 2675 3750 3300RBOT 370 340 375 310 350 200 400 380 285 220 345 280Required:Calculate the value of the sample correlation coefficient. Round your answer to four decimal places. r = _____
Answer:
0.9259
Step-by-step explanation:
Given the following data :
TOST(x) :4200 3575 3750 3700 4050 2770 4870 4500 3450 2675 3750 3300
RBOT(y) : 370 340 375 310 350 200 400 380 285 220 345 280
The correlation Coefficient tells about the strength of the statistical relationship which exists between two variables. The value of correlation Coefficient ranges from - 1 to +1.
The closer the value of correlation Coefficient is to ±1 , the stronger the correlation Coefficient with a negative and positive values signifying a negative and positive relationship respectively. Value of 1 depicts a perfect correlation while 0 means no relationship exists between them. Values close to zero denotes weak relationship.
Using the online Coefficient of correlation calculator ;
The r value of the data above is 0.9259 which signifies a very strong positive relationship between the variables.
In this exercise we have to calculate the value of the coefficient which can be descriptive statistics as:
0.9259
Given the following data :
[tex]TOST(x) :\\4200\\ 3575\\ 3750 \\3700\\ 4050\\ 2770\\ 4870\\ 4500\\ 3450\\ 2675\\ 3750\\ 3300[/tex][tex]RBOT(y) : \\370 \\340 \\375\\ 310\\ 350\\ 200\\ 400\\ 380\\ 285\\ 220\\ 345\\ 280[/tex]
The correlation Coefficient tells about the strength of the statistical relationship which exists between two variables. The value of correlation Coefficient ranges from - 1 to +1.
The closer the value of correlation Coefficient is to ±1 , the stronger the correlation Coefficient with a negative and positive values signifying a negative and positive relationship respectively. Value of 1 depicts a perfect correlation while 0 means no relationship exists between them. Values close to zero denotes weak relationship.
Using the online Coefficient of correlation calculator, the r value of the data above is 0.9259 which signifies a very strong positive relationship between the variables.
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A man saves 4% of his monthly
income of $19,540, the percentage
Savings is increased in the ratio
3:2 Calculate the savings from
the monthly
income.
Answer:
Although the question is not clear, It most likely looks like you were asking for the calculation of the savings for the month after increase.
savings for the month after increase = $1172.4
Step-by-step explanation:
First, let us calculate how much was saved before the increase in savings:
monthly income = $19,540
Percentage saved = 4% of monthly income
= 4/100 × 19,540 = 0.04 × 19,540 = $781.6
Next, we are given the ratio of increase in savings as 3:2
Let the new savings amount be x
3 : 2 = x : 781.6
[tex]\frac{3}{2} = \frac{x}{781.6} \\781.6\ \times 3\ =2x\\2344.8 = 2x\\x =\frac{2344.8}{2} \\x = \$1172.4[/tex]
therefore savings for the month after increase = $1172.4
Just incase you were looking for the savings before the increase, the answer is $781.6 (as calculated above)
Makayla wants to make 200 mL of a 18% saline solution but only has access to 8% and 24% saline mixtures.
Which of the following system of equations correctly describes this situation if x represents the amount of the 8% solution used, and y represents the amount of the 24% solution used?
Answer:
x + y = 200
0.08x + 0.24y = 0.18(200)
Step-by-step explanation:
x + y = 200
0.08x + 0.24y = 0.18(200)
The equations which describes the amount of 8% solution used and the amount of 24% solution used are: x+ y=200 and x+3y=450.
What is equation?An equation is a relationship between two or more variables. They are mostly present in equal to form and are equated to find the value of variables present in them.
How to form equation?let the amount of 8% solution used be x and the amount of 24% solution used be y.
According to question the amount of total solution will be 200ml, So, the equation will be:
x +y=200
Now we have been said that the solution will be 18% saline and 8% saline mixture and 24% saline mixtures are used, So the next equation will be:
0.08x+ 0.24y=0.18*200
8x/100+24/100=18/100*200
8x+24y=3600
8(x+3y)=3600
x+3y=450
Hence the equations which shows the amount of solutions will be x+y=200 and x+3y=450.
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How to find probability from cumulative frequency graph
Answer:
find the difference of points on the graph
Step-by-step explanation:
The cumulative frequency graph (CDF) represents the integral of the probability distribution function (PDF). You find the probability that X is in some interval by subtracting the value of the CDF at the low end of the interval from the CDF value at the high end of the interval.
p(a < x < b) = cdf(b) -cdf(a)
Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 3x + 3, find x.
Answer:
x = 4
Step-by-step explanation:
2( 3x + 3) = 8x - 2
6x + 6 = 8x -2
6x + 8 = 8x
8 = 2x
4 = x
It is given that Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 3x + 3, So the value of x is x = 4.
What does a midpoint mean?Midpoint, as the word suggests, means the point which lies in the middle of something.
Midpoint of a line segment means a point which lies in the mid of the given line segment.
We have been given that Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 3x + 3, we need to find x.
We know that;
2(YZ) = XZ
Substitute in the values
2( 3x + 3) = 8x - 2
Use the Distributive Property
6x + 6 = 8x -2
6x + 8 = 8x
8 = 2x
4 = x
Switch the sides to make it easier to read
x = 4
It is given that Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 3x + 3, So the value of x is x = 4.
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Suppose that 1% of the employees of a certain company use illegal drugs. This company performs random drug tests that return positive results 99% of the time if the person is a drug user. However, it also has a 2% false positive rate. The results of the drug test are known to be independent from test to test for a given person.
a) Steve, an employee at the company, has a positive test. What is the probability that he is a drug user?
b) Knowing he failed his first test, what is the probability that Steve will fail his next drug test?
c) Steve just failed his second drug test. Now, what is the probability that he is a drug user?
Answer:
a) Pr(drug user| positive test) = 0.3333
b) The probability that he will failed his first test = 0.9703
c) the probability that he is a drug user since failed his second drug test
= 0.961165
Step-by-step explanation:
From the given information:
Suppose that 1% of the employees of a certain company use illegal drugs.
Probability of illegal drug user = 0.01
Probability of user that do not use drug = 1 - 0.01 = 0.99
From the person that is a illegal drug user, the company performs random drug tests that return positive results = 0.99
Therefore, the negative result for illegal drug user = 1 - 0.99 = 0.01
However, it also has a 2% false positive rate.
i.e the probability of the user that do not use drug has a positive result of 2% = 0.02
Thus, the probability of the user that do not use drug has a negative result of = 1 - 0.02
= 0.98
We are tasked to answer the following questions.
a) Steve, an employee at the company, has a positive test. What is the probability that he is a drug user?
i.e This employee we are taking about is a drug user and he has a positive test.
Thus;
Pr(drug user| positive test) = [tex]\dfrac{0.99 \times 0.01}{0.99 \times 0.01+ 0.02 \times 0.99}[/tex]
Pr(drug user| positive test) = [tex]\dfrac{0.0099}{0.0099+0.0198}[/tex]
Pr(drug user| positive test) = [tex]\dfrac{0.0099}{0.0297}[/tex]
Pr(drug user| positive test) = 0.3333
b) Knowing he failed his first test, what is the probability that Steve will fail his next drug test?
The probability that he will failed his first test = ((0.01 × 0.01) + (0.99×0.98))
The probability that he will failed his first test = ( 1 × 10⁻⁴ + 0.9702)
The probability that he will failed his first test = 0.9703
c) Steve just failed his second drug test. Now, what is the probability that he is a drug user?
the probability that he is a drug user since he failed his second drug test using Bayes theorem can be expressed as:
= [tex]\dfrac{0.01 \times(0.99\times 0.99)}{0.01 \times (0.99 \times0.99)+ 0.99(0.02 \times0.02)}[/tex]
the probability that he is a drug user since failed his second drug test
= [tex]\dfrac{0.01 \times(0.9801)}{0.01 \times (0.9801)+ 0.99(4 \times 10^{-4})}[/tex]
the probability that he is a drug user since failed his second drug test
= [tex]\dfrac{0.009801}{0.009801+ 3.96 \times 10^{-4}}[/tex]
the probability that he is a drug user since failed his second drug test
= 0.961165
