Final answer:
Using the given probabilities, we find that the probability is 0.965, or 96.5%.
Explanation:
To find the probability that a random person does not pass through the system and is without any security problems, we need to calculate the complement of two events: a person being a security hazard and the security system denying a person without security problems.
First, let's calculate the probability of a person being a security hazard:
Probability of a person being a security hazard = 0.02
Next, let's calculate the probability of the security system denying a person without security problems:
Probability of the security system denying a person without security problems = 1.5% = 0.015
To find the probability that a person does not pass through the system and is without any security problems, we can use the formula:
Probability = (1 - probability of being a security hazard) * (1 - probability of the security system denying a person without security problems)
Probability = (1 - 0.02) * (1 - 0.015)
Probability = 0.98 * 0.985
Probability = 0.9653
Therefore, the probability that a random person does not pass through the system and is without any security problems is 0.965, or 96.5% (rounded to 3 decimal places).
For the following sequence, determine if it is an arithmeticsequence, a geometric sequence, or neither. If it is either arithmetic or geometric, give the next term in the sequence.
5, 8,13, 21, 34, 55,...
Answer:
Neither
Step-by-step explanation:
We are given that a sequence
5,8,13,21,34,55,....
We have to determine the sequence is an arithmetic , geometric or neither.
[tex]a_1=5,a_2=8,a_3=13,...a_6=55[/tex]
We know that if a sequence is an arithmetic then the difference between consecutive terms are equal.
If a sequence is geometric then the ratio of consecutive terms is constant.
[tex]d_1=a_2-a_1=8-5=3[/tex]
[tex]d_2=a_3-a_2=13-8=5[/tex]
[tex]d_1\neq d_2[/tex]
Hence, the difference between the consecutive terms is not equal .Therefore, sequence is not an arithmetic sequence.
[tex]r_1=\frac{a_2}{a_1}=\frac{8}{5}[/tex]
[tex]r_2=\frac{a_3}{a_2}=\frac{13}{8}[/tex]
[tex]r_1\neq r_2[/tex]
Hence, the ratio of consecutive terms is not equal .Therefore, given sequence is not geometric sequence.
Therefore, given sequence is not an arithmetic nor geometric sequence.
Prove that H c G is a normal subgroup if and only if every left coset is a right coset, i.e., aH = Ha for all a e G
[tex]\Rightarrow[/tex]
Suppose first that [tex]H\subset G[/tex] is a normal subgroup. Then by definition we must have for all [tex]a\in H[/tex], [tex]xax^{-1} \in H[/tex] for every [tex]x\in G[/tex]. Let [tex]a\in G[/tex] and choose [tex](ab)\in aH[/tex] ([tex]b\in H[/tex]). By hypothesis we have [tex]aba^{-1} =abbb^{-1}a^{-1}=(ab)b(ab)^{-1} \in H[/tex], i.e. [tex]aba^{-1}=c[/tex] for some [tex]c\in H[/tex], thus [tex]ab=ca \in Ha[/tex]. So we have [tex]aH\subset Ha[/tex]. You can prove [tex]Ha\subset aH[/tex] in the same way.
[tex]\Leftarrow[/tex]
Suppose [tex]aH=Ha[/tex] for all [tex]a\in G[/tex]. Let [tex]h\in H[/tex], we have to prove [tex]aha^{-1} \in H[/tex] for every [tex]a\in G[/tex]. So, let [tex]a\in G[/tex]. We have that [tex]ha^{-1} =a^{-1}h'[/tex] for some [tex]h'\in H[/tex] (by the hypothesis). hence we have [tex]aha^{-1}=h' \in H[/tex]. Because [tex]a[/tex] was chosen arbitrarily we have the desired .
How many sets of two or more consecutive positive integers can be added to obtain a sum of 1800?
Answer:
n = 60
Step-by-step explanation:
GIVEN DATA:
Total sum of consecutive number is 1800
sum of n number is given as
[tex] sum = \frac{ n(n+1)}{2}[/tex]
where n is positive number and belong to natural number i.e 1,2,3,4,...
from the data given we have[tex]1800 = \frac{n(n+1)}{2}[/tex]
solving for n we get
[/tex]n^2 + n -3600 = 0[/tex]
n = 59.5, -60.5
therefore n = 60
Yesterday's World Cup final had viewing figures of 138,695,157.
What is the value of the 3?
Answer:
The value of the 3 is 30,000,000.
Step-by-step explanation:
From the digit at the right, you go multiplying each element by 10 powered to a counter that starts at zero and increases at every digit. So:
Our counter is i
i = 0;
v(7) is the value of the 7
[tex]v(7) = 7*10^{0} = 7[/tex]
i = 1;
v(5) is the value of the 5
[tex]v(5) = 5*10^{1} = 50[/tex]
i = 2;
v(1) is the value of the 1
[tex]v(1) = 1*10^{2} = 100[/tex]
i = 3;
v(5) is the value of the 5
[tex]v(5) = 5*10^{3} = 5,000[/tex]
i = 4;
v(9) is the value of the 9
[tex]v(9) = 9*10^{4} = 90,000[/tex]
i = 5;
v(6) is the value of the 6
[tex]v(6) = 6*10^{5} = 600,000[/tex]
i = 6;
v(8) is the value of the 8
[tex]v(8) = 8*10^{6} = 8,000,000[/tex]
i = 7;
v(3) is the value of the 3
[tex]v(3) = 3*10^{7} = 30,000,000[/tex]
The value of the 3 is 30,000,000.
The digit 3 in the number 138,695,157 represents a value of 300 million, indicating its substantial contribution to the overall magnitude of the figure in the context of place value and powers of 10.
The value of the digit 3 in the number 138,695,157 is 3 hundred million. In this number, each place value represents a power of 10, with the rightmost digit being ones, the next one being tens, the next hundreds, and so on. The digit 3 in the hundred million's place means that it represents 3 multiplied by 100,000,000.
In other words, the digit 3 in this context signifies 300 million. This is because when you see a digit in a number, its place value determines its weight in terms of powers of 10. So, the digit 3 in the hundred million's place is equivalent to 3 x 100,000,000, which is indeed 300 million.
So, in the number 138,695,157, the digit 3 holds the value of 300 million, contributing significantly to the overall magnitude of the figure.
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A concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The following data represent the strength of nine randomly selected casts (in psi). 3970, 4100, 3100, 3200, 2950, 3830, 4100, 4050, 3460 Compute the mean, median and mode strength of the concrete (in psi). Compute the mean strength of the concrete. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The mean strength of the concrete is nothing psi of pressure. (Round to the nearest tenth as needed.) B. The mean does not exist.
Answer:
Mean = 3640
Mode = 4100
Median = 3830.
Step-by-step explanation:
We are given the following data in the question:
Strength of casts (in psi):
3970,4100,3100,3200,2950,3830,4100,4050,3460
Formula:
[tex]Mean = \displaystyle\frac{\text{Sum of all observation}}{\text{Total number of observations}}[/tex]
[tex]\displaystyle\frac{3970+4100+3100+ 3200+ 2950+ 3830+4100+ 4050+ 3460}{9} = \displaystyle\frac{32760} {9} = 3640[/tex]
Mode is the entry with most frequency. Thus, for the given sample mode = 4100.
Median
Since n = 9 is odd,
Formula:
[tex]Median = \displaystyle\frac{n+1}{2}th~term[/tex]
Data in ascending order:
2950,3100,3200,3460,3830,3970,4050,4100,4100
Median = 5th term = 3830.
The mean strength of the concrete is 3640.
The mode strength of the concrete is 4100.
The median strength of the concrete is 3830.
Given
A concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure.
The following data represent the strength of nine randomly selected casts (in psi).
3970, 4100, 3100, 3200, 2950, 3830, 4100, 4050, 3460
What formula is used to calculate the mean value?The mean value of the data is given by;
[tex]\rm Mean = \dfrac{Sum \ of \ all \ observation}{Total \ number \ of \ observation}[/tex]
The mean of the strength of the concrete is;
[tex]\rm Mean = \dfrac{Sum \ of \ all \ observation}{Total \ number \ of \ observation}\\\\Mean =\dfrac{3970+4100+3100+3200+2950+3830+ 4100+4050+3460 }{9}\\\\Mean =\dfrac{32760}{9}\\\\Mean=3640[/tex]
The mean strength of the concrete is 3640.
Mode is the entry with the most frequency.
Thus, for the given sample mode = 4100.
The mode strength of the concrete is 4100.
The mid-value of the data is called the median.
The median strength of the concrete is 3830.
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Both Alex and Chris left their homes at 7:00 a.m. and walked to school. The graph shows the distances that each boy was from school as they walked. Which statement is best supported by the graph?
Alex lives farther from the school than Chris lives, and Alex walked to school at a faster rate than Chris walked.
Alex lives closer to the school than Chris lives, and Alex walked to school at a faster rate than Chris walked.
Chris lives farther from the school than Alex lives, and Chris walked to school at a faster rate than Alex walked.
Chris lives closer to the school than Alex lives, and Chris walked to school at a faster rate than Alex walked
Answer:
Chris lives farther from school than Alex lives, and Chris walks to school at a faster rate than Alex
Step-by-step explanation:
Chris walked 4 miles.
Alex walked 2.
Alex takes 25 minutes to walk one mile. (50÷2) and Chris takes 15 minutes to walk one mile (60÷4), meaning Chris walks at a faster rate although he lives farther away.
Answer:
Chris lives farther from the school than Alex lives, and Chris walked to school at a faster rate than Alex walked.
The gross domestic product (GDP) of a certain country, which measures the overall size of the economy in billions of dollars, can be approximated by the function g(y) = 557y +8883, where y = 10 corresponds to the year 2010. Estimate the GDP (to the nearest billion dollars) in the given years. (a) 2007 (b) 2011 (c) 2012 (a) What value of y corresponds to the year 2007? y=O (Type a whole number.)
Answer:
The estimated GDP in 2007 is 12782 billion dollars.
The estimated GDP in 2011 is 15010 billion dollars.
The estimated GDP in 2012 is 15567 billion dollars.
Step-by-step explanation:
Consider the provided function.
[tex]g(y) = 557y +8883[/tex]
Where y = 10 corresponds to the year 2010.
Part (b) Estimate the GDP (to the nearest billion dollars) in 2007.
If y = 10 corresponds to the year 2010 then y = 7 corresponds to the year 2007.
Substitute the value of y = 7 in the provided function.
[tex]g(y) = 557(7) +8883[/tex]
[tex]g(y) = 3899+8883[/tex]
[tex]g(y) = 12782[/tex]
Hence, the estimated GDP in 2007 is 12782 billion dollars.
Part (b) Estimate the GDP (to the nearest billion dollars) in 2011.
If y = 10 corresponds to the year 2010 then y = 11 corresponds to the year 2011.
Substitute the value of y = 11 in the provided function.
[tex]g(y) = 557(11) +8883[/tex]
[tex]g(y) = 6127+8883[/tex]
[tex]g(y) = 15010[/tex]
Hence, the estimated GDP in 2011 is 15010 billion dollars.
Part (C) Estimate the GDP (to the nearest billion dollars) in 2012.
If y = 10 corresponds to the year 2010 then y = 12 corresponds to the year 2012.
Substitute the value of y = 12 in the provided function.
[tex]g(y) = 557(12) +8883[/tex]
[tex]g(y) = 6684+8883[/tex]
[tex]g(y) = 15567[/tex]
Hence, the estimated GDP in 2012 is 15567 billion dollars.
Final answer:
To estimate the GDP of a certain country for different years using the function g(y) = 557y + 8883, the value of y is adjusted according to the respective year, and the GDP is calculated by substituting this value into the formula. For 2007, 2011, and 2012, respective GDPs estimated are 12019, 14311, and 14868 billion dollars.
Explanation:
The question asks us to estimate the GDP of a certain country for different years using the function g(y) = 557y + 8883, where y=10 corresponds to the year 2010. To find the GDP for the given years, we first need to calculate the correct value of y for each year and then substitute these values into the function.
For the year 2007, y would be 7 (as y=0 corresponds to 2000, so 2007-2000=7). Substituting y=7 into the function gives us a GDP of g(7) = 557*7 + 8883 = 12019 (to the nearest billion dollars).For 2011, y would be 11 (2011-2000), and substituting it yields g(11) = 557*11 + 8883 = 14311.For 2012, with y=12, the GDP would be g(12) = 557*12 + 8883 = 14868.This approach allows us to quantify economic growth over different years, comparing these figures to gain insight into the economic health and trends of the country.
Find the optimal solution for the following problem. (Round your answers to 3 decimal places.)
Maximize C = 9x + 7y
subject to 8x + 10y ≤ 17
11x + 12y ≤ 25
and x ≥ 0, y ≥ 0.
1. what is the optimal value of x?
2. What is the optimal value of y?
3. What is the maximum value of the objective function?
Answer:
Maximize C =[tex]9x + 7y[/tex]
[tex]8x + 10y \leq 17[/tex]
[tex]11x + 12y\leq 25[/tex]
and x ≥ 0, y ≥ 0
Plot the lines on graph
[tex]8x + 10y \leq 17[/tex]
[tex]11x + 12y\leq 25[/tex]
[tex]x\geq 0[/tex]
[tex]y\geq 0[/tex]
So, boundary points of feasible region are (0,1.7) , (2.125,0) and (0,0)
Substitute the points in Maximize C
At (0,1.7)
Maximize C =[tex]9(0) + 7(1.7)[/tex]
Maximize C =[tex]11.9[/tex]
At (2.125,0)
Maximize C =[tex]9(2.125) + 7(0)[/tex]
Maximize C =[tex]19.125[/tex]
At (0,0)
Maximize C =[tex]9(0) + 7(0)[/tex]
Maximize C =[tex]0[/tex]
So, Maximum value is attained at (2.125,0)
So, the optimal value of x is 2.125
The optimal value of y is 0
The maximum value of the objective function is 19.125
There are several ways to calculate optimal solution; one of them, is by using graphs.
The optimal value of x is 2.125The optimal value of y is: 0The maximum value of the objective function is: 19.125The given parameters are:
[tex]\mathbf{Max\ C = 9x + 7y}[/tex]
[tex]\mathbf{8x + 10y \le 17}[/tex]
[tex]\mathbf{11x + 12y \le 25}[/tex]
[tex]\mathbf{x,y\ge 0}[/tex]
See attachment for the graphs of [tex]\mathbf{8x + 10y \le 17}[/tex] and [tex]\mathbf{11x + 12y \le 25}[/tex]
From the graph, the feasible regions are:
[tex]\mathbf{(x,y) = (0,1.7), (2.125,0)}[/tex]
Test these values in the objective function
[tex]\mathbf{Max\ C = 9x + 7y}[/tex]
[tex]\mathbf{C = 9 \times 0 + 7 \times 1.7 = 11.9}[/tex]
[tex]\mathbf{C = 9 \times 2.125 + 7 \times 0 = 19.125}[/tex]
So, the value of C is maximum at: [tex]\mathbf{(x,y) = (2.125,0)}[/tex]
So, we have:
The optimal value of x is 2.125The optimal value of y is: 0The maximum value of the objective function is: 19.125Read more about maximizing functions at:
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In an arithmetic sequence, the nth term an is given by the formula an=a1+(n−1)d, where a1 is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1,
where r is the common ratio. Use these formulas to determine
the indicated term in the given sequence.
The 105th term of 1/2, 1, 3/2, 2,..
Answer:
The 105th term of given sequence is [tex]\frac{105}{2}[/tex].
Step-by-step explanation:
The given sequence is
[tex]\frac{1}{2},1,\frac{3}{2},2[/tex]
It can be rewritten as
[tex]0.5,1,1.5,2[/tex]
Here the first term is 0.5.
It is an arithmetic sequence because it has common difference.
[tex]d=a_2-a_1=1-0.5=0.5[/tex]
[tex]d=a_3-a_2=1.5-1=0.5[/tex]
[tex]d=a_4-a_3=2-1.5=0.5[/tex]
The nth term of an AP is
[tex]a_n=a_1+(n-1)d[/tex]
where, [tex]a_1[/tex] is first term and d is common difference.
Substitute [tex]a_1=0.5[/tex] and [tex]d=0.5[/tex] in the above formula.
[tex]a_n=0.5+(n-1)0.5[/tex]
[tex]a_n=0.5+0.5n-0.5[/tex]
[tex]a_n=0.5n[/tex]
We need to find the 105th term of given sequence.
Substitute n=105 in the above equation.
[tex]a_n=0.5(105)[/tex]
[tex]a_n=52.5[/tex]
[tex]a_n=\frac{105}{2}[/tex]
Therefore the 105th term of given sequence is [tex]\frac{105}{2}[/tex].
Define fn : [0,1] --> R by the
equationfn(x) = xn. Show that the
sequence(fn(x)) converges for each x belongs to [0,1],
but thatthe sequence (fn) does not converge uniformly.
Answer:
The sequence of functions [tex]\{x^{n}\}_{n\in \mathbb{N}}[/tex] converges to the function
[tex]f(x)=\begin{cases}0&0\leq x<1\\1&x=1\end{cases}[/tex].
Step-by-step explanation:
The limit [tex]\lim_{n\to \infty }c^{n}[/tex] exists and converges to zero whenever [tex]\lvert c \rvert <1 [/tex]. But, if [tex]c=1[/tex] the sequence [tex]\{c^{n}\}[/tex] is constant and all its terms are equal to [tex]1[/tex], then converges to [tex]1[/tex]. Using this result, consider the sequence of functions [tex]\{f_{n}\}[/tex] defined on the interval [tex][0,1][/tex] by [tex]f_{n}(x)=x^{n}[/tex]. Then, for all [tex]0\leq x<1[/tex] we have that [tex]\lim_{n\to \infty}x^{n}=0[/tex]. Now, if [tex]x=1[/tex], then [tex]\lim_{n\to \infty }x^{n}=1[/tex]. Therefore, the limit function of the sequence of functions is
[tex]f(x)=\begin{cases}0&0\leq x<1\\1&x=1\end{cases}[/tex].
To show that the convergence is not uniform consider [tex]0<\varepsilon<1[/tex]. For any [tex]n>1[/tex] choose [tex]x\in (0,1)[/tex] such that [tex]\varepsilon^{1/n}<x<1[/tex]. Then
[tex]\varepsilon <x^{n}=\lvert f(x)-f_{n}(x)\rvert[/tex]
This implies that the convergence is not uniform.
convert 1 cal/(m^2 * sec * °C) into BTU/(ft^2 * hr * °F)
Its easy enough to convert the energy, time, and area units, but how am I suppose to convert the temp units?
Answer:
[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
Step-by-step explanation:
To find : Convert [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}[/tex] into [tex]\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
Solution :
We convert units one by one,
[tex]1\text{ m}^2=10.7639\text{ ft}^2[/tex]
[tex]1\text{ sec}=\frac{1}{3600}\text{ hour}[/tex]
[tex]1\text{ cal}=0.003968\text{ BTU}[/tex]
Converting temperature unit,
[tex]^\circ C\times \frac{9}{5}+32=^\circ F[/tex]
[tex]1^\circ C\times \frac{9}{5}+32=33.8^\circ F[/tex]
So, [tex]1^\circ C=33.8^\circ F[/tex]
Substitute all the values in the unit conversion,
[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{10.7639\times \frac{1}{3600}\times 33.8}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{0.101061}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
Therefore, The conversion of unit is [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]
Compare the values of the underline digits (2 is underline in both numbers in problem 1)
1.) 2,783 and 7,283 The value of 2 in_____ is ___times the value of 2 in____
(7 is underline in both numbers for problem 2)
2.) 503,497 and 26, 475 The value of 7 in____is___times the value of 7 in____
(4 is underline in both numbers for problem 3)
3.) 34,258 and 47,163 The value of 4 in____is___times the value of 4 in____
Answer with Step-by-step explanation:
1.We are given
2783 and 7283
We have to compare the values of the underline digits
Place value of 2 in 2783 =2000 because 2 is at thousand place
2 in 7283 is at hundred place
Therefore, the place value of 2 in 7283=200
Therefore, the value of 2 in 2783 is 10 times the value of 2 in 7283.
2.We have to compare the values of 7 in both numbers
Place of 7 in 503497=one's
Therefore, place value of 7 =7
7 in 26475 is at tens place
Therefore, the place value of 7 in 26475=70
Hence, the value of 7 in 26475 is 10 times the value of 7 in 503497.
3.We have to compare the values of 4 in both given numbers
Place of 4 in 34258=Thousand
Place value of 4 in 34258=4000
Place of 4 in 47163=Ten thousand
Place value of 4 in 47163=40000
Hence, the value of 4 in 47163 is 10 times the value of 4 in 34258.
Among users of automated teller machines (ATMs), 94% use ATMs to withdraw cash and 28% use them to check their account balance. Suppose that 95% use ATMs to either withdraw cash or check their account balance (or both). Given a woman who uses an ATM to check her account balance, what the probability that she also uses an ATM to get cash?
Answer:
96%
Step-by-step explanation:
Conditional probability is defined as:
P(A|B) = P(A∩B) / P(B)
Or, in English:
Probability that A occurs, given that B has occurred = Probability that both A and B occur / Probability that B occurs
We want to find the probability that a woman uses an ATM to get cash, given that she uses an ATM to check her balance.
P(withdraws cash | checks account)
Using the definition of condition probability, this equals:
P = P(withdraws cash AND checks account) / P(checks account)
We know that P(checks account) = 0.28.
But we don't know what P(withdraws cash AND checks account) is. To find that, we need to use the definition of P(A∪B):
P(A∪B) = P(A) + P(B) − P(A∩B)
This says that the probability of A or B occurring (or both) is the probability of A occurring plus the probability of B occurring minus the probability of both A and B occurring.
P(withdraws cash OR checks account) = P(withdraws cash) + P(checks account) − P(withdraws cash AND checks account)
0.95 = 0.94 + 0.28 − P(withdraws cash AND checks account)
P(withdraws cash AND checks account) = 0.27
Therefore:
P = 0.27 / 0.28
P ≈ 0.96
Final answer:
The probability that a woman who checks her account balance at an ATM also withdraws cash is approximately 96.43%.
Explanation:
To solve the problem, we can apply the probability rule for conditional probability. We are provided with the following probabilities:
The probability that ATM users withdraw cash (P(Cash)) is 94%, or 0.94.The probability that ATM users check their account balance (P(Balance)) is 28%, or 0.28.The probability that ATM users either withdraw cash or check their account balance (or both) (P(Cash ∪ Balance)) is 95%, or 0.95.Using this information, we're interested in finding the probability that a user who checks their account balance also withdraws cash, represented as P(Cash|Balance).
The formula for conditional probability is:
P(A|B) = P(A ∩ B) / P(B)
Where A and B are two events, and P(A|B) is the conditional probability of A given B.
Using the inclusion-exclusion principle, we can express P(Cash ∩ Balance) as:
P(Cash ∩ Balance) = P(Cash) + P(Balance) - P(Cash ∪ Balance)
Substitute the given probabilities:
P(Cash ∩ Balance) = 0.94 + 0.28 - 0.95 = 0.27
The probability that a woman who checks her balance also gets cash (P(Cash|Balance)) is:
P(Cash|Balance) = P(Cash ∩ Balance) / P(Balance)
P(Cash|Balance) = 0.27 / 0.28 ≈ 0.9643
Therefore, the probability is approximately 96.43%.
The accompanying observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added: 2767 2924 3042 2844 2895 (a) What are the values of the sample mean x and sample median x tilde?
Answer:
Step-by-step explanation:
Given are the observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added:
2767 2924 3042 2844 2895
No of items = 5
If written in ascending order the order would be
2767 2844 2895 2924 3042
Hence median is the middle value in the ordered row = 2895
Mean = sum/5
=[tex]\frac{14472}{5} =2894.4[/tex]
Way back in the olden days, Blockbuster tallied all their US movie rental data and found that on average, individuals rent 10 movies a year with a standard deviation of 3. Treat these as population statistics. They wanted to see if movie rental rates in Yuma, Arizona, were different from those of the country as a whole (why Yuma? Who knows ). A random sample of 25 blockbuster members in Yuma yielded a mean rental rate of 11.3 movies per year. Use alpha = .05
By considering the given information we have ,
[tex]H_0: \mu = 10\\\\ H_a: \mu\neq10[/tex]
Since, the alternative hypothesis is two tailed so the test is a two-tailed test.
Given : Population mean : [tex]\mu=10[/tex]
Standard deviation: [tex]\sigma= 3[/tex]
Sample size : n=25 , whihc is less than 30 so the sample is small and we use t-test.
Sample mean : [tex]\overline{x}=11.3[/tex]
Significance level : [tex]\alpha= 0.5[/tex]
Formula to find t-test statistic is given by :-
[tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
i.e. [tex]t=\dfrac{11.3-10}{\dfrac{3}{\sqrt{25}}}\approx2.17[/tex]
By using the standard normal distribution table,
The p-value corresponds 2.17 (two-tailed)=0.0300068
Since , the p-value is less than the significance level, so we reject the null hypothesis.
Hence, we conclude that there are enough evidence to to support the claim that movie rental rates in Yuma, Arizona, were different from those of the country as a whole .
A hypothesis test is conducted to see if the average movie rental rate in Yuma, Arizona, is statistically different from the national average. This problem is resolved in several steps including stating hypotheses, formulating an analysis plan, analyzing sample data, and interpreting the results. The rental rate is then compared to a critical value determined by the significance level (α = .05).
Explanation:The subject here is a hypothesis testing problem related to the mean rental rate of DVDs in Yuma, Arizona. Blockbuster found that the average nation-wide movie rental rate was 10 movies per year with a standard deviation of 3. In Yuma, a sample of 25 people resulted in a mean rental rate of 11.3 movies per year. The company wanted to check whether this difference was significant or not. So, they used an alpha level of .05 to conduct this hypothesis test.
Here are the steps of the hypothesis test:
State the hypotheses. The null hypothesis H0 would be that the mean rental rate in Yuma is the same as the average across the US (μ = 10). The alternative hypothesis Ha would be that the mean rental rate in Yuma is not equal to the average across the US (μ ≠ 10).Formulate an analysis plan. For this analysis, the significance level is defined as alpha (α) = .05. As per the conditions, the population standard deviation (σ) is known and equals 3.Analyze sample data. Using the sample data and the information provided, we can calculate the test statistic (z).Interpret the results. If the test statistic is beyond the critical value, we reject the null hypothesis. Otherwise, we do not have enough evidence to reject it.Learn more about Hypothesis Testing here:https://brainly.com/question/34171008
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The price of a calculator is currently $23, which is a 532% decrease from the price thirty years ago. What was the price of the calculator thirty years ago?
Answer:
The price of calculator before 30 years = $30+$122.36 = $145.36
Step-by-step explanation:
We have given price current price of calculator = $23
It is given that current price of calculator is after decrease of 532 %
We have to find the price of calculator before 30 years
The price of calculator will be more than 532 % from the current price
So 532% of 23 [tex]=\frac{23\times 532}{100}=$122.36[/tex]
So the price of calculator before 30 years = $30+$122.36 = $145.36
A store has clearance items that have been marked down by 25%. They are having a sale, advertising an additional 40% off clearance items. What percent of the original price do you end up paying? Give your answer accurate to at least one decimal place.
Final answer:
To find the percent of the original price you end up paying after a 25% discount and an additional 40% discount, first calculate the discounted prices and then determine the final price. In this case, you end up paying 45% of the original price.
Explanation:
To find the percent of the original price you end up paying, you need to calculate the final price after both discounts. Let's say the original price of the item is $100. First, apply the 25% discount by multiplying the original price by 0.75 (1 - 0.25 = 0.75). This gives you a price of $75. Next, apply the additional 40% discount by multiplying the discounted price by 0.60 (1 - 0.40 = 0.60). This gives you a final price of $45. Therefore, you end up paying 45% of the original price.
Consider all the whole numbers from 0 to 1500. What is the sum of all digits needed to write down these numbers? No calculators
Answer:757
Step-by-step explanation:
Mercoxit Energy Inc. has a retention ratio of 0.79, an ROA of 5.94%, and an EM of 1.11. This means that they will have a sustainable growth rate of ____%
Final answer:
To calculate the sustainable growth rate of Mercoxit Energy Inc., the retention ratio is multiplied by the calculated Return on Equity (ROE), which is derived from the ROA and EM figures given. The result yields a sustainable growth rate of 5.21% for Mercoxit Energy Inc.
Explanation:
The sustainable growth rate (SGR) is a company's maximum growth rate in sales using internal financial resources while maintaining a constant debt to equity ratio. To calculate the SGR for Mercoxit Energy Inc., one needs to use the following formula:
SGR = Retention Ratio x Return on Equity (ROE)
While we are given the retention ratio and the Return on Assets (ROA), we are not directly given the ROE. However, we can calculate ROE using the provided Equity Multiplier (EM), since ROE can be written as:
ROE = ROA x EM
Considering the given figures:
Retention Ratio = 0.79
ROA = 5.94%
EM = 1.11
Now calculate ROE:
ROE = 0.0594 x 1.11 = 0.065934 (or 6.5934%)
Finally, we calculate the SGR:
SGR = 0.79 x 0.065934 = 0.05208786 (or 5.208786%)
If we round to two decimal places, Mercoxit Energy Inc. has a sustainable growth rate of 5.21%.
FOCUS ON HIGHER ORDER THINKING Work Area Katie said, "Negative numbers are integers." What was Explain the Error her error?
Step-by-step explanation:
Consider the provided information.
Integers: Integers are the set of whole number and the negatives of the natural numbers, i.e, … ,-2, -1, 0, 1, 2,...
Rational number: A number is said to be rational, if it is in the form of p/q. Where p and q are integer and denominator is not equal to 0.The decimal expansion of rational numbers may terminate or become periodic.
Irrational number: A number is irrational if it cannot be expressed be expressed by dividing two integers. The decimal expansion of Irrational numbers are neither terminate nor periodic.
Now consider the provided statement "Negative numbers are integers."
The statement is incorrect as all negative numbers are not integers they might be rational number or irrational numbers.
For example: -2.5 is a negative number but it is not an integer. The number is a rational number.
Hence, her statement is incorrect.
She can say that all negative whole numbers are integers.
Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees 0° and standard deviation of 1.00 degrees °C. Assume 3 3% of the thermometers are rejected because they have readings that are too high and another 3 3% are rejected because they have readings that are too low. Draw a sketch and find the two readings that are cutoff values separating the rejected thermometers from the others.
Answer:
The two readings that are cutoff values are T=1.84 deg C and T=-1.84 deg C.
Step-by-step explanation:
Thermometers rejected by measurements above normal represent 3.3% of the total, which indicates, by normal probability distribution data, that accepted thermometers are 96.7% likely to measure less than the maximum allowable temperature.
This value (P(X>x)=0.967) corresponds to z = 1.8388. Since the mean and standard deviation values are the same as the standard normal probability distribution (mean = 0, sd = 0), the z value is equivalent to the measured value (temperature).
Given the symmetry of the probability distribution, we can affirm that the thermometers rejected by measurements below the permissible measured a temperature lower than -1.8388.
Final answer:
The cutoff values separating the rejected thermometers are -1.88°C for the low end and 1.88°C for the high end, based on a normal distribution with a mean of 0°C and a standard deviation of 1.00°C.
Explanation:
To find the two readings that are cutoff values separating the rejected thermometers, we look at the normal distribution curve with a mean of 0°C and a standard deviation of 1.00°C. Since 3% of thermometers are rejected for high readings and another 3% for being too low, these correspond to the tail ends of the distribution.
Using the Z-score table, find the Z-score that has 3% in the tail. For the lower end, we seek the Z-score where the left tail (the area to the left of the Z-score) is 0.03, and for the higher end, we look for a Z-score where the right tail is 0.03. These Z-scores are approximately -1.88 for the low end and +1.88 for the high end (using the 97th percentile since we want the upper 3%).
To find the actual thermometer readings:
For the low cutoff: cutoff low = mean + (Z-score * standard deviation) = 0 + (-1.88 * 1) = -1.88°C.
For the high cutoff: cutoff high = mean + (Z-score * standard deviation) = 0 + (1.88 * 1) = 1.88°C.
Therefore, thermometers reading lower than -1.88°C or higher than 1.88°C will be rejected.
There are 14 juniors and 16 seniors in a chess club. a) From the 30 members, how many ways are there to arrange 5 members of the club in a line? b) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line? 0 c) If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there? d) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?
Answer:
No. of juniors = 14
No. of seniors = 16
Total students = 30
A) From the 30 members, how many ways are there to arrange 5 members of the club in a line?
Since we are asked about arrangement so we will use permutation
Formula : [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]
n = 30
r = 5
[tex]^{30}P_5=\frac{30!}{(30-5)!}[/tex]
[tex]^{30}P_5=17100720[/tex]
So, From the 30 members, there are 17100720 ways to arrange 5 members of the club in a line?
B) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line?
Out of 16 seniors 2 will be selected
So, 3 places are vacant
Remaining students = 30-2 = 28
So, out of 28 students 3 students will be selected
No. of ways = [tex]^{16}P_2 \times ^{28}P_3[/tex]
No. of ways = [tex]\frac{16!}{(16-2)!}\times\frac{28!}{(28-3)!}[/tex]
= [tex]4717440[/tex]
There are 4717440 ways to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line.
C)If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there?
Since we are not asked about arrangement so we will use combination
Out of 16 seniors 2 will be selected
Out of 14 juniors 2 will be selected
Formula : [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
So, No. of possible groupings = [tex]^{16}C_2 \times ^{14}C_2[/tex]
= [tex]\frac{16!}{2!(16-2)!} \times \frac{14!}{2!(14-2)!}[/tex]
= [tex]10920[/tex]
If the club sends 2 juniors and 2 seniors to the tournament, there are 10920 possible groupings
D) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?
Out of 16 seniors 4 will be selected
or
Out of 14 juniors 4 will be selected
So, No. of possible groupings = [tex]^{16}C_4 + ^{14}C_4[/tex]
= [tex]\frac{16!}{4!(16-4)!} + \frac{14!}{4!(14-4)!}[/tex]
= [tex]2821[/tex]
So,If the club sends either 4 juniors or 4 seniors, there are 2821 possible groupings .
How many kW-hr are used when one 100W light bulbs is used for 2 hours?
Answer:
0.2 kW-hr
Step-by-step explanation:
First, we are going to transform 100W in kW. We can use a rule of three in which we know that 1000W is equivalent to 1 kW, then how many kW are equivalent to 100W. This is:
1000W ------------- 1 kW
100W -------------- X
Where X is the the number of kW that are equivalent to 100W. Solving for X, we get:
[tex]X=\frac{100W * 1kW}{1000W} =0.1kW[/tex]
Then, for calculate the number of kW-hr we need to multiplicate the number of kW with the number of hours, This is:
0.1 kW * 2 hours = 0.2 kW-hr
Finally, when one 100W light bulbs is used for 2 hours, it used 0.2 kW-hr
Find the linear trend forecast for period 8 given the following data: Period Sales 1 19 2 18 3 15 4 20 5 18 6 22 7 20
Answer:
The equation of trend line is [tex]y = 0.500000x + 16.857143[/tex].
The linear trend forecast for period 8 is about 20.86.
Step-by-step explanation:
The given data table is
Period Sales
1 19
2 18
3 15
4 20
5 18
6 22
7 20
We need to find the linear trend forecast for period 8.
The general form of linear regression is
[tex]y=a+bx[/tex] .... (1)
where, a is y-intercept and b is slope.
[tex]b=\frac{\sum_{i=1}^nx_iy_i-n\overline{x}\overline{y}}{\sum_{i=1}^nx_i^2-n\overline{x}^2}[/tex]
[tex]a=\overline{y}-b\overline{x}[/tex]
Using the graphing calculator we get
[tex]a=16.857143[/tex]
[tex]b=0.500000[/tex]
Substitute these values in equation (1).
[tex]y = 0.500000x + 16.857143[/tex]
The equation of trend line is [tex]y = 0.500000x + 16.857143[/tex].
Substitute x=8 to find the linear trend forecast for period 8.
[tex]y = 0.500000(8) + 16.857143[/tex]
[tex]y =20.857143[/tex]
[tex]y \approx 20.86[/tex]
Therefore the linear trend forecast for period 8 is about 20.86.
A problem states: "There are 2 more horses than cows in a field. There are 16 animals in the field in all. How many horses are there in the field?"
Let h represent the number of horses.
Which equation represents the situation?
2h + 2 = 16
2(h+2)=16
h + 2 = 16
2h−2=16
Answer:
2h-2=16
Step-by-step explanation:
Let
h -----> the number of horses
c ----> the number of a cows
we know that
h+c=16 -----> equation A
h=c+2
c=(h-2) ----> equation B
Substitute equation B in equation A and solve for h
h+(h-2)=16
2h-2=16
2h=16+2
2h=18
h=9
therefore
The equation that represent the situation is
2h-2=16
Aminophylline is infusing at 30 mL/hr. The drug is mixed 250 mg in 500 ml D5W. How many mg/hr is the patient receiving?
Answer:
the patient receiving is 15 mg/hr
Step-by-step explanation:
given data
infusing = 30 mL/hr
drug mixed = 250 mg
D5W = 500ml
to find out
How many mg/hr is the patient receiving
solution
we know infusing rate is = 30 mL/hr ...................1
and we know mixed is = [tex]\frac{250mg}{500ml}[/tex] ..............2
so
now we multiply both equation 1 and 2
patient receiving = [tex]\frac{250mg}{500ml}[/tex] × [tex]\frac{30ml}{hr}[/tex]
solve we get here
patient receiving = [tex]\frac{15mg}{hr}[/tex]
so the patient receiving is 15 mg/hr
The patient is receiving 15 mg/hr of aminophylline.
Explanation:To calculate the mg/hr that the patient is receiving, we need to find the concentration of aminophylline in the infusion and multiply it by the infusion rate. The concentration of aminophylline is 250 mg in 500 ml of D5W. This means that there are 250 mg of aminophylline in 500 ml, or 0.5 mg/ml. The infusion rate is 30 ml/hr. So, to find the mg/hr, we multiply the concentration by the infusion rate: 0.5 mg/ml * 30 ml/hr = 15 mg/hr. Therefore, the patient is receiving 15 mg/hr of aminophylline.
Learn more about Calculating infusion dosage here:https://brainly.com/question/36528535
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Orders for a computer are summarized by the optional features that are requested. The proportion of orders with no optional features is 0.40. The proportion of orders with one optional feature is 0.34. The proportion of orders with more than one optional feature is 0.26. (a) What is the probability that an order requests at least one optional feature? Round your answer to two decimal places (e.g. 98.76). (b) What is the probability that an order does not request more than one optional feature? Round your answers to two decimal places (e.g. 98.76).
Answer:
a) The probability that an order requests at least one optional feature is 34%+26% = 60%.
b) The probability that an order does not request more than one optional feature is 40% + 34% = 74%.
Step-by-step explanation:
Probability:
What you want to happen is the desired outcome.
Everything that can happen iis the total outcomes.
The probability is the division of the number of possible outcomes by the number of total outcomes.
In our problem, the probabilities are:
-40% that no optional features are requested.
-34% that one optional feature is requested
-26% that more than one optional feature is requested.
(a) What is the probability that an order requests at least one optional feature?
There is a 34% probability that one optional feature is requested and a 26% probability that more than one optional feature is requested.
So the probability that an order requests at least one optional feature is 34%+26% = 60%.
(b) What is the probability that an order does not request more than one optional feature?
There is a 40% probability that no optional features are requested and a 34% probability that one optional feature is requested.
So the probability that an order does not request more than one optional feature is 40% + 34% = 74%.
Final answer:
The probability an order requests at least one optional feature is 0.60, and the probability an order does not request more than one optional feature is 0.74.
Explanation:
The question involves calculating probabilities based on provided proportions of orders with optional features.
Part (a): Probability of at least one optional feature
The proportion of orders with no optional features is 0.40. Therefore, the probability that an order requests at least one optional feature is 1 - 0.40 = 0.60. So, the probability is 0.60 when rounded to two decimal places.
Part (b): Probability of not more than one optional feature
We are given that orders with one optional feature make up 0.34 and those with no optional features constitute 0.40. Adding these together gives us a probability of 0.34 + 0.40 = 0.74 for orders not requesting more than one optional feature. Thus, this probability is 0.74, rounded to two decimal places.
On rainy days, Izzy goes from his house to the school by running 1.2 miles on West St, then makes a 90º turn and runs 0.5 miles on North Ave.
a. If Izzy runs 7.5 miles per hour, approximately how much time will it take her to run to school on rainy days?
b. On dry days, Izzy runs on the dashed path through the woods. How far is she traveling?
c. If Izzy runs 7.5 miles per hour, how much time will she save by cutting through the woods?
To calculate the time Izzy takes to run to school on rainy days, we use the distance and speed to find that she runs 1.7 miles in approximately 13.6 minutes. We are unable to calculate the distance through the woods or the time saved without further information.
Explanation:Calculation of Time and Distance
To calculate the time Izzy takes to run to school on rainy days, we use the following formula:
Time (in hours) = Distance (in miles) / Speed (in miles per hour)
Izzy runs a total distance of 1.2 miles on West St and then 0.5 miles on North Ave, summing up to 1.7 miles. Given Izzy's speed is 7.5 miles per hour, the time taken to run to school on rainy days can be calculated as:
Time = (1.2 + 0.5) miles / 7.5 mph = 1.7 / 7.5
To find the time in minutes, multiply the time in hours by 60:
Time in minutes = (1.7 / 7.5) imes 60
= 13.6 minutes (approximately)
As the dashed path through the woods on dry days is not described in the question, we cannot calculate the exact distance Izzy is traveling through the woods. Without this information, we also cannot calculate the time saved by cutting through the woods.
While completing a race, Edward spent 54 minutes walking. If his ratio of time walking to jogging was 6:5, how many minutes did he spend completing the race?
Answer: 99 minutes
Step-by-step explanation:
Given: While completing a race, Edward spent 54 minutes walking.
The ratio of time walking to jogging was 6:5 i.e. [tex]\dfrac{6}{5}[/tex] (1)
Let x be the time taken ( in minutes ) by him for jogging.
then, the ratio of time walking to jogging will be [tex]\dfrac{54}{x}[/tex] (2)
From (1) and (2), we have
[tex]\dfrac{6}{5}=\dfrac{54}{x}\\\\\Rightarrow\ 6x=54\times5\\\\\Rightarrow\ x=\dfrac{54\times5}{6}=45[/tex]
So, the number of minutes he took for jogging = 45 minutes
Now, the total time he spent on completing the race= 54+45=99 minutes
Find the general solution to each of the following ODEs. Then, decide whether or not the set of solutions form a vector space. Explain your reasoning. Compare your answers to the previous problem. Recall that the general solution has the form y(t) = yh(t) + yp(t).
(A) y' - 2y = 0
(B) y' - 2y = 1
(C) y" - 4y = 0
(D) y" - 4y = e^(3t)
Answer:
(A) [tex]y=ke^{2t}[/tex] with [tex]k\in\mathbb{R}[/tex].
(B) [tex]y=ke^{2t}/2-1/2[/tex] with [tex]k\in\mathbb{R}[/tex]
(C) [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex]
(D) [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex],
Step-by-step explanation
(A) We can see this as separation of variables or just a linear ODE of first grade, then [tex]0=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y \Rightarrow \frac{1}{2y}dy=dt \ \Rightarrow \int \frac{1}{2y}dy=\int dt \Rightarrow \ln |y|^{1/2}=t+C \Rightarrow |y|^{1/2}=e^{\ln |y|^{1/2}}=e^{t+C}=e^{C}e^t} \Rightarrow y=ke^{2t}[/tex]. With this answer we see that the set of solutions of the ODE form a vector space over, where vectors are of the form [tex]e^{2t}[/tex] with [tex]t[/tex] real.
(B) Proceeding and the previous item, we obtain [tex]1=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y+1 \Rightarrow \frac{1}{2y+1}dy=dt \ \Rightarrow \int \frac{1}{2y+1}dy=\int dt \Rightarrow 1/2\ln |2y+1|=t+C \Rightarrow |2y+1|^{1/2}=e^{\ln |2y+1|^{1/2}}=e^{t+C}=e^{C}e^t \Rightarrow y=ke^{2t}/2-1/2[/tex]. Which is not a vector space with the usual operations (this is because [tex]-1/2[/tex]), in other words, if you sum two solutions you don't obtain a solution.
(C) This is a linear ODE of second grade, then if we set [tex]y=e^{mt} \Rightarrow y''=m^2e^{mt}[/tex] and we obtain the characteristic equation [tex]0=y''-4y=m^2e^{mt}-4e^{mt}=(m^2-4)e^{mt}\Rightarrow m^{2}-4=0\Rightarrow m=\pm 2[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in the first items the set of solutions form a vector space.
(D) Using C, let be [tex]y=me^{3t} [/tex] we obtain that it must satisfies [tex]3^2m-4m=1\Rightarrow m=1/5[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in (B) the set of solutions does not form a vector space (same reason! as in (B)).