Industrial engineers periodically conduct "work measurement" analyses to determine the time required to produce a single unit of output. At a large processing plant, the number of total worker-hours required per day to perform a certain task was recorded for 50 days. a. Compute the mean, median, and mode of the data set. b. Find the range, variance, and standard deviation of the data set. c. Construct the intervals using the Empirical rule. d. Find the 70th percentile for the data on total daily worker-hours.

Answer:

a) 23%

Step-by-step explanation:

To find the price change as percentage :

Multiply 100 by 160 then divide it by 130

100 × 160 ÷ 130 = 123 approximately which means there's an additional 23% to the 100% price

Answer:

Answers may vary but will most likely be close to 2.

Step-by-step explanation

                 first test:38%

                 second test:76%

    SIMULATION FIRST TEST

Randomly select a 2-digit number.

If the digit is between 00 and 35 then you passed the test,else you did not pass the test.

SIMULATION SECOND TEST

Randomly select a 2-digit number.

if the digit is between 00 and 75 then you passed the test,else you did not pass the test.

SIMULATION TRIAL

Perform the simulation of the first test.if you did not pass the first test then perform the simulation of the second test.

Record the number of trials needed to pass the first or second test.

Repeat 20 times and take the average of the 20 recorded number of trials

(what is the sum of recorded values divided by 20).

Note:you will most likely obtain a result of about two trials needed.

The estimated average number of tests drivers take to get a license is 1.9

To estimate the average number of tests drivers take to get a license, we can simulate the process using a simple random experiment. We will assume that each test attempt is independent of the others and that the probabilities of passing are 38% on the first attempt and 76% on subsequent attempts. Here's how we can perform the simulation:

1. For each driver, simulate the number of tests they take by generating a random number between 0 and 1. If the number is less than or equal to 0.38 (38%), the driver passes on the first attempt. Otherwise, they proceed to a second attempt.

2. For the second attempt and any subsequent attempts, generate another random number. If the number is less than or equal to 0.76 (76%), the driver passes.

3. Repeat this process until the driver passes the test. Record the number of attempts it took.

4. Perform this simulation for 20 drivers to get a sample size that will help estimate the average number of tests taken.

5. Calculate the average number of tests by summing the number of tests taken by all drivers and dividing by the number of drivers (20).

Let's perform the simulation:

1. For the first attempt, generate a random number for each of the 20 drivers and check if it's less than or equal to 0.38.

2. For any subsequent attempts, generate a random number for each driver who did not pass on the first attempt and check if it's less than or equal to 0.76.

3. Record the number of attempts for each driver.

4. Sum the total number of attempts and divide by 20 to find the average.

Now, let's calculate the average number of tests based on the simulation: Assuming we have performed the simulation and recorded the number of attempts for each driver, we would have a list of numbers. For example, it might look something like this:

 Driver 1: 1 attempt

Driver 2: 2 attempts

Driver 3: 1 attempt

Driver 20: 3 attempts

Let's say after performing the simulation, we have the following counts for each number of attempts:

1 attempt: 8 drivers

2 attempts: 7 drivers

3 attempts: 4 drivers

4 attempts: 1 driver

Now, we calculate the average number of tests taken by these 20 drivers:

Average = (8 * 1 + 7 * 2 + 4 * 3 + 1 * 4) / 20

Average = (8 + 14 + 12 + 4) / 20

Average = 38 / 20

Average = 1.9

Answer:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Step-by-step explanation:

The equation of the distance traveled by Plane A is

[tex]d(t) = 290t[/tex]

The plane B traveled 540 miles in 2 hours.

So in 1 hour, plane B traveled 540/2 = 270 miles:

How does the distance Plane A traveled in 1 hour compare to the distance Plane B traveled in 1 hour?

Plane A:

d(1) = 290*1 = 290

Plane A traveled 290 miles in 1 hour.

Plane B travaled 270 miles in 1 hour.

So the correct answer is:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Answer:

The distance Plane A traveled in 1 h is greater than the distance Plane B traveled in 1 h.

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

The type I error arises when we wrongfully reject the null hypothesis. The probability of  occurrence of type I error is denoted as α. Thus, α=0.05 means that there is 5% probability that we reject the null hypothesis when it is true. So, we can say that the α=0.05, means that 5% of the time, we reject the null hypothesis when it is correct.

Answer:

A

Step-by-step explanation:

Answer:

[tex]z=-1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 -1.28*0.15=2.058[/tex]

So the value of height that separates the bottom 10% of data from the top 90% is 2.06.

For the upper limit since the distribution is symmetrical we can do this:

[tex]z=1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 +1.28*0.15=2.442[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 2.44.

And the best answer for this case would be:

C. [2.06, 2.44]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the amount of juice of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(2.25,0.15)[/tex]  

Where [tex]\mu=2.25[/tex] and [tex]\sigma=0.15[/tex]

For this case we want the limits for the middle 80% values  of the distribution. so then we need 100-80= 20% of the area in the tails and 10% on each tail since the distribution is symmetrical.

We can use this condition for the lower limits

[tex]P(X>a)=0.9[/tex]   (a)

[tex]P(X<a)=0.1[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.1 of the area on the left and 0.9 of the area on the right it's z=-1.28. On this case P(Z<-1.28)=0.1 and P(z>-1.28)=0.9

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.1[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.1[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 -1.28*0.15=2.058[/tex]

So the value of height that separates the bottom 10% of data from the top 90% is 2.06.

For the upper limit since the distribution is symmetrical we can do this:

[tex]z=1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 +1.28*0.15=2.442[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 2.44.

And the best answer for this case would be:

C. [2.06, 2.44]

Answer:

$7826.78

Step-by-step explanation:

Total income = 7950

With a service fee charge of 1.55%, the service fee charge = [tex]1.55\%\times7950 = \dfrac{1.55}{100}\times7950 = 123.225[/tex]

Net income = 7950 - 123.225 = 7826.775 = $7826.78

An alternative solution:

Service charge = 1.55%

Net income = (100 - 1.55)% × 7950

= 98.45% × 7950 = [tex]\dfrac{98.45}{100}\times7950 = 7826.775=7826.78[/tex]

Answer:

Step-by-step explanation:

Accepted credit= $ 7950

Charges for services =1.55% of the total  

7950 - 1.55% of 7950

7950 - (1.55/100)*7950

7950 - 123.225

7826.775

7826,78  two decimal places

This is the net income earned during the month of July  

Answer:

a. 74.63%

b. 33.63%

c. 87.67%

Step-by-step explanation:

If 59% (0.59) of the workers are married, then It means (100-59 = 41%) of the workers are not married.

If 43% (0.43) of the workers are College graduates, then it means (100-43= 57%) of the workers are not college graduates.

If 1/3 of college graduates are married, it means portion of graduate that are married = 1/3 * 43% = 1/3 * 0.43 = 0.1433.

For question a, Probability that the worker is neither married nor a college graduate becomes:

= (probability of not married) + (probability of not a graduate) - (probability of not married * not a graduate)

= 0.41 + 0.57 - (0.41*0.57) = 0.98 - 0.2337

= 0.7463 = 74.63%

For question b, probability that the worker is married but not a college graduate becomes:

=(probability of married) * (probability of not a graduate.)

= 0.59 * 0.57

= 0.3363 = 33.63%

For question c, probability that the worker is either married or a college graduate becomes:

=probability of marriage + probability of graduate - (probability of married and graduate)

= 0.59 + 0.43 - (0.1433)

= 0.8767. = 87.67%

The probability that a randomly chosen worker is neither married nor a college graduate is 0%. The probability that a worker is married but not a college graduate is approximately 34%. The probability that a worker is married or a college graduate is approximately 75%.

To calculate the probabilities, we need to use the formulas for conditional probability. Let's solve each part:

a) To find the probability that a randomly chosen worker is neither married nor a college graduate, we can subtract the probability that the worker is married and the probability that the worker is a college graduate from 1. The probability of being married is 59%, and the probability of being a college graduate is 43%. Using the formula, 1 - 0.59 - 0.43 = 0.

b) To find the probability that a worker is married but not a college graduate, we can multiply the probability of being married and the probability of not being a college graduate. The probability of being married is 59%, and the probability of not being a college graduate is 57% (100% - 43%). Using the formula, 0.59 * 0.57 = 0.3363 (approximately 34%).

c) To find the probability that a worker is married or a college graduate, we can add the probabilities of being married and being a college graduate, and then subtract the probability of being both married and a college graduate to avoid double counting. The probability of being married is 59%, the probability of being a college graduate is 43%, and the probability of being both married and a college graduate is 1/3 of the college graduates (1/3 * 43%). Using the formula, 0.59 + 0.43 - (1/3 * 0.43) = 0.745 (approximately 75%).

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Answer:

The parenthesis need to be kept intact while applying the DeMorgan's theorem on the original equation to find the compliment because otherwise it will introduce an error in the answer.

Step-by-step explanation:

According to DeMorgan's Theorem:

(W.X + Y.Z)'

(W.X)' . (Y.Z)'

(W'+X') . (Y' + Z')

Note that it is important to keep the parenthesis intact while applying the DeMorgan's theorem.

For the original function:

(W . X + Y . Z)'

= (1 . 1 + 1 . 0)

= (1 + 0) = 1

For the compliment:

(W' + X') . (Y' + Z')

=(1' + 1') . (1' + 0')

=(0 + 0) . (0 + 1)

=0 . 1 = 0

Both functions are not 1 for the same input if we solve while keeping the parenthesis intact because that allows us to solve the operation inside the parenthesis first and then move on to the operator outside it.

Without the parenthesis the compliment equation looks like this:

W' + X' . Y' + Z'

1' + 1' . 1' + 0'

0 + 0 . 0 + 1

Here, the 'AND' operation will be considered first before the 'OR', resulting in 1 as the final answer.

Therefore, it is important to keep the parenthesis intact while applying DeMorgan's Theorem on the original equation or else it would produce an erroneous result.

The error originates from an incorrect application of DeMorgan's theorem. The correct complement of (W · X) + (Y · Z) is (W' + X') · (Y' + Z'). This corrects the discrepancy seen for WXYZ = 1110.

The confusion here likely originates from a mistake in the DeMorgan's theorem application. According to DeMorgan's Theorem, the complement of (W · X) + (Y · Z) is given by (W' + X') · (Y' + Z'), not W' + X' · Y' + Z'.

So, if we have WXYZ = 1110, the given function (W · X) + (Y · Z) equals 1 because we have (1 · 1) + (1 · 0) = 1 + 0 = 1. Whereas, the correct complement function, (W' + X') · (Y' + Z') equals 0 because we have (0 + 0) · (0 + 1) = 0 · 1 = 0.

This explains why we were seeing both original function and the incorrectly applied complement function evaluating to 1 for the same input combination.

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Answer: The question is incomplete ad some details are missing.

it says Suppose the relationship is given by T(w)=0.1w2+2.155w+20

where T is the temperature in degrees Celsius and w is the power input in watts.

a) How many watts of power are needed to maintain the temperature at exactly 200degree celsius

= 33watts of power are needed

Step-by-step explanation:

The detailed steps and appropriate substitution is as shown in the attachment

Answer:

49.34% probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

Step-by-step explanation:

For each television, there are only two possible outcomes. Either they have defective speakers, or they do not. The probabilities of each television having defective speakers are independent from each other. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

[tex]p = 0.01, n = 500[/tex]

Find an approximate probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

This is

[tex]P(5 \leq X \leq 8) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)[/tex]

So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 5) = C_{500,5}.(0.01)^{5}.(0.99)^{495} = 0.1764[/tex]

[tex]P(X = 6) = C_{500,6}.(0.01)^{6}.(0.99)^{494} = 0.1470[/tex]

[tex]P(X = 7) = C_{500,7}.(0.01)^{7}.(0.99)^{493} = 0.1048[/tex]

[tex]P(X = 8) = C_{500,8}.(0.01)^{8}.(0.99)^{492} = 0.0652[/tex]

[tex]P(5 \leq X \leq 8) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.1764 + 0.1470 + 0.1048 + 0.0652 = 0.4934[/tex]

49.34% probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

The approximate probability that 5 to 8 out of 500 televisions have defective speakers is 0.49. This was calculated using the binomial distribution with n = 500 and p = 0.01. We summed P(X = 5), P(X = 6), P(X = 7), and P(X = 8) to find the result.

To find the approximate probability that 5 to 8 televisions out of 500 have defective speakers, we can use the binomial distribution.

The probability mass function for a binomial distribution is given by:

[tex]P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)[/tex]

where C(n, k) is the binomial coefficient.

We need to calculate P(5 ≤ X ≤ 8), which is P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8).

Using a binomial calculator or statistical software, we get:

Summing these probabilities gives:

P(5 ≤ X ≤ 8) ≈ 0.1755 + 0.1447 + 0.1040 + 0.0681

≈ 0.4923

Therefore, the approximate probability that 5 to 8 televisions have defective speakers is 0.49 (rounded to two decimal places)

Answer: the selling price of the boat is $57.5

Step-by-step explanation:

The original price of a toy boat was $50. The boat is marked up 15% before it’s sold. This means that the amount by which the original price of the toy boat was increased would be

15/100 × 50 = 0.15 × 50 = $7.5

The selling price of the toy boat would be the sum of its original price and the amount by which it was marked up. It becomes.

50 + 7.5 = $57.5

Answer: John must make at least 44 tips to reach his goal.

Step-by-step explanation:

Let x represent the number of tips that John must make this week to reach his goal.

John works at a restaurant and gets paid $80 per week plus $5 per tip. This means that the total amount that John would earn in a week if he makes x tips would be

80 + 5x

This week, John wants to earn at least $300. This is expressed as

80 + 5x ≥ 300

5x ≥ 300 - 80

5x ≥ 220

x ≥ 220/5

x ≥ 44

Answer:

a)

[tex]H_0: \pi\geq0.78\\\\H_a: \pi<0.78[/tex]

b) The Type I error occurs when we reject a null hypothesis that is actually true. In this case, it means we conclude that the arrival time have improved, when it didn't.

The Type II error occurs when we accept a null hypothesis that is actually false. In this case, although the arrival times have really improved, the evidence from the sample was not enough to show that improvement.

c) In this case, the Type I error is more serious, because it gives the wrong impression of improvement and no further actions will be taken to reduce the times.

Step-by-step explanation:

a) If you want to determine if the responders are arriving within 8 minutes of the call more often, you have to evaluate the proportion of accidents in which the arrival time is less than 8 minutes and compare it with the known proportion of π=0.78.

The sample parameter "p: proportion of accidents with arrival time of 8 minutes or less" will be used to test the hypothesis.

The null and alternative hypothesis will be:

[tex]H_0: \pi\geq0.78\\\\H_a: \pi<0.78[/tex]

Final answer:

The hypotheses for a significance test to determine whether first responders are arriving more often within 8 minutes are H0: p = 0.78 and HA: p > 0.78, with p representing the proportion of calls responded to within this timeframe. A Type I error involves mistakenly concluding an improvement, while a Type II error occurs by overlooking an actual improvement. The potential demotivating effect of a Type II error may render it more serious in this context.

Explanation:

The question seems to revolve around the concept of hypothesis testing in statistics and how it applies to the emergency response times in a city. The parameter of interest would be the proportion of emergency calls responded to within 8 minutes. So for part (a), the hypotheses could be stated as follows:

H0: p = 0.78 (The proportion of calls responded to within 8 minutes is 78% as it was last year.)

HA: p > 0.78 (The proportion of calls responded to within 8 minutes has increased from last year.)

In part (b), a Type I error would occur if the city concludes that the proportion of calls responded to within 8 minutes has increased when in reality, it has not. The consequence of a Type I error would be misallocating resources based on false success. A Type II error would occur if the city fails to recognize an actual improvement in response times. The consequence of this could lead to a lack of recognition and continued encouragement for first responders who have actually improved.

Part (c) asks which error is more serious. A Type II error may be considered more serious in this setting, as failing to acknowledge and react to an actual improvement could demotivate emergency personnel and affect future performances, possibly leading to life-threatening delays for accident victims.

Answer: 4.85%

Step-by-step explanation: productivity = parcel delivered / number of drivers.

At when parcel delivered was 103, 000 the average number of drivers was 84 thus making the productivity to be

103,000/ 84 = 1226.19

At when the parcel delivered was 112, 800 the average number of drivers was 96 thus making the productivity to be

112, 000/ 96 = 1166.67

Thus the change in productivity = 1226.19 - 1166.67 = 59.52.

Initial productivity = 1226.19

final productivity = 1166.67

% change in productivity = change in productivity / initial productivity * 100

% change in productivity = 59.52 /1226 * 100

% change in productivity = 4.85%

Answer:

The probability that more than 8 and less than 12 customers pay in cash is 0.0931.

Step-by-step explanation:

Let X = a customer at David's gasoline station pay in cash.

The probability of a customer paying in cash is, P (X) = p = 0.40

The number of customers at the gasoline station during a 1-hour period is,

n = 15.

Then the random variable X follows a binomial distribution, Bin (15, 0.40).

The probability function for a Binomial distribution is:

[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x}[/tex]

Compute the probability that more than 8 and less than 12 customers pay in cash as follows:

[tex]P(8< X< 12)=P(X<12)-P(X<8)\\=P(X=9)+P(X=10)+P(X=11)\\=[{15\choose 9}(0.40)^{9}(1-0.40)^{15-9}]+[{15\choose 10}(0.40)^{10}(1-0.40)^{15-10}]\\+[{15\choose 11}(0.40)^{11}(1-0.40)^{15-11}]\\=0.0612+0.0245+0.0074\\=0.0931[/tex]

Thus, the probability that more than 8 and less than 12 customers pay in cash is 0.0931.

The problem comes to calculating binomial probabilities for when 9, 10, and 11 customers pay in cash and adding them together. This scenario applies to binomial distribution, where we have a success (paying in cash) happening with a probability of 40%.

This question is a problem of the binomial distribution. The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial (often referred to as success and failure). In this case, the success is the customer paying in cash, which happens 40% of the time according to past evidence.

The formula for the binomial distribution is:

P(X = k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where

Carry out this calculation for k=9, 10, 11, and then add these probabilities together to get the probability that more than 8 and less than 12 customers pay in cash.

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Answer:

a)

Step-by-step explanation:

Given problem:

Which of the following best describes galena’s cleavage (see animation in placemark)?

Options:

a)three directions at ~90

b) three directions not at ~90°

c) two directions at ~90°

d) two directions not at ~90°

Solution:

- As per the animation where the place-mark is observed as a point. We can see that corner point is a point of intersection of three planes, left, right and bottom. We can also see that these three planes are at right angles @ 90 degrees to each other. Hence, option A.

Answer:

0.5882 or 58.82%

Step-by-step explanation:

The probability that both balls were red (A) is:

[tex]P(A)=\frac{6}{9}*\frac{5}{9}=0.3704[/tex]

The probability that the first ball was blue and the second ball was red (B) is:

[tex]P(B) = \frac{3}{9}*\frac{7}{9}=0.2593[/tex]

The conditional probability that the first ball selected is red, given that the second ball was red is:

[tex]P = \frac{P(A)}{P(A)+P(B)}=\frac{0.3704}{0.3704+0.2593} =0.5882[/tex]

Answer: QINGHAOSU

Step-by-step explanation:Artemisinin is an antimalarial (Drug used to cure Malaria) drug whose active ingredient QINGHAOSU was isolated in the 1970s from the Plant called Artemisia annua by a Chinese Physician.

Artemisinin has been in use in ancient Chinese communities since the 4th century to cure Diseases. Artemisinin is potent in killing several forms of the Plasmodium specie and has been used to derive other antimalarial Drugs in use today like Artemeter and Artesunate.

Answer:

Our answer is 0.8172

Step-by-step explanation:

P(doubles on a single roll of pair of dice) =(6/36) =1/6

therefore P(in 3 rolls of pair of dice at least one doubles)=1-P(none of roll shows a double)

=1-(1-1/6)3 =91/216

for 12 players this follows binomial distribution with parameter n=12 and p=91/216

probability that at least 4 of the players will get “doubles” at least once =P(X>=4)

=1-(P(X<=3)

=1-((₁₂ C0)×(91/216)⁰(125/216)¹²+(₁₂ C1)×(91/216)¹(125/216)¹¹+(₁₂ C2)×(91/216)²(125/216)¹⁰+(₁₂ C3)×(91/216)³(125/216)⁹)

=1-0.1828

=0.8172

Answer:

a) W'(t) = 1.51 -0.0138t + 0.000762t² pounds/day

Step-by-step explanation:

The median weight of a boy with age between 0 and 36 months is given by:

[tex]W(t) = 8.38+1.51t-0.0069t^2 + 0.000254t^3[/tex]

To find the rate of change of weight with respect to time, that is, the change in weight measured in pounds caused by a unit change in time, measured in days, simply derivate the weight function over time:

[tex]\frac{d(W(t)}{dt}= W'(t) = 1.51-0.0138t + 0.000762t^2[/tex]

The rate of change is 1.51 -0.0138t + 0.000762t² pounds/day.

Answer:

m arc DC=48°

Step-by-step explanation:

Angles in a Circle

We know [tex]\overline{BE}[/tex] and [tex]\overline{AC}[/tex] are diameters, so

[tex]m\angle BFA+m\angle AFE=180^o[/tex].

Since [tex]m\angle AFE=3x+12[/tex] and [tex]m\angle BFA=4x[/tex]

We set the equation

[tex]3x+12+4x=180^o[/tex]

Solving

[tex]7x=180-12=168[/tex]

[tex]x=24^o[/tex]

Thus

[tex]m\angle BFA=4(24)=96^o[/tex]

We also know

[tex]m\angle CFD\cong m\angle DFE[/tex]

Being [tex]\angle CFE[/tex] opposite to [tex]\angle BFA[/tex]

Then [tex]m\angle CFE=96^o[/tex]

It's divided in two equal angles [tex]\angle CFD\ and\ \angle DFE[/tex], thus m arc DC is half of 96°:

m arc DC=48°

First option

Answer:

The Matrix of T is

[tex]\left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\-sin(\pi/6)&-cos(\pi/6)\end{array}\right][/tex]

Step-by-step explanation:

Rotate -pi/6 Clockwise is the same as rotating pi/6 anticlockwise. The matrix of that rotation is

[tex]\left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\sin(\pi/6)&cos(\pi/6)\end{array}\right][/tex]

The matrix of the reflection through the x1-axis is

[tex]\left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]

Therefore, the composition is the product of both matrices is the matrix of T

[tex]MT = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] * \left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\sin(\pi/6)&cos(\pi/6)\end{array}\right] = \left[\begin{array}{cc}cos(\pi/6)&-sin(\pi/6)\\-sin(\pi/6)&-cos(\pi/6)\end{array}\right][/tex]

I hope that works for you!

To find the standard matrix of the linear transformation [tex]\( T \)[/tex], we need to perform two operations in sequence: a rotation through [tex]\( -\frac{\pi}{6} \)[/tex] radians (clockwise) and a reflection through the horizontal [tex]\( x_1 \)-axis[/tex]. We will find the matrices for each of these transformations and then multiply them to get the standard matrix for [tex]\( T \)[/tex].

1. Rotation Matrix [tex]\( R \)[/tex]:

The matrix that represents a rotation through an angle [tex]\( \theta \)[/tex] in the counterclockwise direction is given by:

[tex]\[ R(\theta) = \begin{bmatrix} \cos(\theta) -\sin(\theta) \\ \sin(\theta) \cos(\theta) \end{bmatrix} \][/tex]

For a clockwise rotation, we use a negative angle, so for [tex]\( -\frac{\pi}{6} \)[/tex] radians, the rotation matrix is:

[tex]\[ R\left(-\frac{\pi}{6}\right) = \begin{bmatrix} \cos\left(-\frac{\pi}{6}\right) -\sin\left(-\frac{\pi}{6}\right) \\ \sin\left(-\frac{\pi}{6}\right) \cos\left(-\frac{\pi}{6}\right) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ -\frac{1}{2} \frac{\sqrt{3}}{2} \end{bmatrix} \][/tex]

2. Reflection Matrix [tex]\( M \)[/tex]:

The matrix that represents a reflection through the horizontal[tex]\( x_1 \)[/tex]-axis is given by:

[tex]\[ M = \begin{bmatrix} 1 0 \\ 0 -1 \end{bmatrix} \][/tex]

3. Standard Matrix of [tex]\( T \)[/tex]

To find the standard matrix of[tex]\( T \)[/tex], we multiply the rotation matrix [tex]\( R \)[/tex] by the reflection matrix [tex]\( M \)[/tex]:

[tex]1 0 \\ 0 -1[/tex]

[tex]\end{bmatrix} \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ -\frac{1}{2} \frac{\sqrt{3}}{2} \end{bmatrix} \] \[ T = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ \frac{1}{2} -\frac{\sqrt{3}}{2} \end{bmatrix} \][/tex]

Therefore, the standard matrix of the linear transformation [tex]\( T \)[/tex] is:

[tex]\[ \boxed{T = \begin{bmatrix} \frac{\sqrt{3}}{2} \frac{1}{2} \\ \frac{1}{2} -\frac{\sqrt{3}}{2} \end{bmatrix}} \][/tex]

This matrix represents the transformation that first rotates points through [tex]\( -\frac{\pi}{6} \)[/tex] radians (clockwise) and then reflects them through the horizontal [tex]\( x_1 \)-axis.[/tex]

Answer: The strength of an electric field is E = - 0,05.10⁹ N/C.

Step-by-step explanation: According to the question, the plastic sphere is in equilibrium in an electric field. This sugests that the forces acting on the sphere, which are Gravitational Force (Fg) and Electric Force (Fe) are also in equilibrium, denotating Fg=Fe.

As Fg = m . g, with m = 0,009kg and g= 9,8m/s², we have Fg = 0,0882N.

Knowing the value of Fe, the strength of the electric field can be calculated as

E = Fe/Q, in which Q is the electric charge.

E = (0,0882) / (-1,6·10⁻⁹)

E = - 0,05·10⁹N/C

Answer:

How many measurements must be made = 9

Step-by-step explanation:

The steps are as shown in the attachment.

To reduce the uncertainty in the diameter of a disk from 1.5mm to 0.5mm, three times more measurements of the circumference would need to be made. This is due to the relationship between the circumference and diameter, and how the uncertainty propagates through this relationship.

This question pertains to the areas of accuracy, precision, and uncertainty in measurements. Understanding these concepts is vital in the field of physics. The circumference of a disk and its diameter are related by the constant π (Pi): Diameter = Circumference / π.

The question states that there is an uncertainty of 1.5 mm in measuring the circumference. Given that the diameter and circumference are directly connected, when you reduce the uncertainty in the measurement of the circumference (e.g., by taking more measurements), you also reduce the uncertainty in the diameter. However, the relationship is not linear. Through propagation of uncertainty principles, the uncertainty in diameter would be the uncertainty in the circumference divided by π. To reduce this to 0.5 mm, you would require three times more measurements.

The precision of a measurement system is closely linked to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. Various factors can contribute to the uncertainty of a measurement, including the smallest division on a given tool, the ability of the person making the measurement, irregularities in the object being measured, and unforeseen circumstances that affect the outcome.

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Answer:

a) 99.24% chance HLI will find a sample mean between 5.5 and 7.1 hours.

b) 81.64% probability that the sample mean will be between 5.9 and 6.7 hours.

Step-by-step explanation:

To solve this question, it is important to know the Normal probability distribution and the Central Limit Theorem

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].

In this problem, we have that:

[tex]\mu = 6.3, \sigma = 2.1, n = 49, s = \frac{2.1}{\sqrt{49}} = 0.3[/tex]

A) What is the chance HLI will find a sample mean between 5.5 and 7.1 hours?

This is the pvalue of Z when X = 7.1 subtracted by the pvalue of Z when X = 5.5.

By the Central Limit Theorem, the formula for Z is:

[tex]Z = \frac{X - \mu}{s}[/tex]

X = 7.1

[tex]Z = \frac{7.1 - 6.3}{0.3}[/tex]

[tex]Z = 2.67[/tex]

[tex]Z = 2.67[/tex] has a pvalue of 0.9962

X = 5.5

[tex]Z = \frac{5.5 - 6.3}{0.3}[/tex]

[tex]Z = -2.67[/tex]

[tex]Z = -2.67[/tex] has a pvalue of 0.0038

So there is a 0.9962 - 0.0038 = 0.9924 = 99.24% chance HLI will find a sample mean between 5.5 and 7.1 hours.

B) Calculate the probability that the sample mean will be between 5.9 and 6.7 hours.

This is the pvalue of Z when X = 6.7 subtracted by the pvalue of Z when X = 5.9

X = 6.7

[tex]Z = \frac{6.7 - 6.3}{0.3}[/tex]

[tex]Z = 1.33[/tex]

[tex]Z = 1.33[/tex] has a pvalue of 0.9082

X = 5.9

[tex]Z = \frac{5.9 - 6.3}{0.3}[/tex]

[tex]Z = -1.33[/tex]

[tex]Z = -1.33[/tex] has a pvalue of 0.0918.

So there is a 0.9082 - 0.0918 = 0.8164 = 81.64% probability that the sample mean will be between 5.9 and 6.7 hours.

Answer:

(a) x1 = 11/13, x2 = 50/13, x3 = -17/13

(b) x = 54/235, y = 6/47, z = 24/235

(c) x1 = 22/9, x2 =164/9, x3 = 139/9, x4 = -37/3

Step-by-step explanation:

Gaussian Elimination Method was the matrix method used in solving the system of equations.

It is done by writing the equations given in an augmented form, this is shown in the attachment. The coefficients of each variable is taken to form a matrix.

Row operations are then performed on the augmented matrix. This operation can be addition, subtraction, multiplication, or division.

For convenience, Row is written as R1, Row 2 as R2, and so on

R2 - R3 means Subtract Row 3 from Row 2, and so on.

The step by step operations for each question are shown in the attachment.

The matrix method involves representing the systems of equations as matrices, and then using matrix operations or the inverse matrix method to solve for the variables. This method can only be used when the system has a unique solution.

Using the matrix method to solve systems of equations involves first representing the system as a matrix. For example, the first system of equations can be represented as a matrix: [[3,2,4], [2,5,3], [7,2,2]][[x1],[x2],[x3]] = [[5], [17], [11]]. Similarly, the second and third systems can be written in matrix form. Then you can use various matrix operations or the inverse matrix method to solve for the variables. Note that this method is used only when the system has a unique solution - that is, when the coefficient matrix is invertible.

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Answer:

14

Step-by-step explanation:

The inter-quartile range (IQR) is the difference between the third and first quartile. The data gathered is:

57, 59, 60, 62, 62, 63, 64, 68, 70, 70, 71, 71, 73, 79, 87, 89, 90

The data set has 17 values, the first quartile is the average between the 4th and 5th values, while the third quartile is the average between the 13th and 14th values:

[tex]Q_1 = \frac{62+62}{2}=62\\Q_3 = \frac{73+79}{2}=76[/tex]

The IQR is:

[tex]IQR = Q_3 - Q_1 = 76 -62\\IQR = 14[/tex]

The Interquartile Range (IQR) of the given data for resting heart rate among college students is 11, calculated by finding the difference between the upper (third) and lower (first) quartiles.

Given the following data for resting heartrate among college students: 57, 59, 60, 62, 62, 63, 64, 68, 70, 70, 71, 71, 73, 79, 87, 89, 90, we are to find the Interquartile Range (IQR). The IQR is a statistical term that measures the statistical spread, or variability, of data points. It is calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1).

First, we need to arrange the data points in ascending order, which has already been done for us in this case. Next, let's divide the data into two halves: the lower half and the upper half. For our data, the lower half is 57 to 68 and the upper half is 70 to 90. Find the median (middle value) of each half. The lower half median (Q1) is 62, and upper half median (Q3) is 73.

Now, subtract Q1 from Q3 to find the IQR: 73 - 62 = 11. So, the IQR of the resting heart rates of the given college students data is 11.

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Answer:

All options( option A, option B and option C)

Step-by-step explanation:

Mean, median and mode are used to summarize the data. Mode can be calculated for both quantitative and qualitative data but mean and median cannot be calculated for qualitative data.

Here outstanding balance on each account represents quantitative data and mode, median and mean all can summarize the quantitative data. So, mean, median and mode each can be used to summarize the data of outstanding balance on each account.

The value of B is determined by the equation 2π / 12.7, which corresponds to the tide's period. The variable C represents the time delay from midnight to the first high tide, which is a phase shift in the function.

The Bay of Fundy tidal pattern can be modeled using a cosine function. Since tides go through a complete cycle (360 degrees or 2π radians) every 12.7 hours, the value of B, the frequency, can be determined by dividing 2π by the period of the tide in hours.

Therefore, B = 2π / 12.7.

The variable C in the equation represents a phase shift. In this context, a phase shift refers to a horizontal shift of the cosine function, which corresponds to a time delay or advance of the tides. The meaning of C is the time delay between midnight and the first high tide of the day.

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The real and complex solutions of the cubic equation  [tex]x^3-64=0[/tex] are x=4 (real solution) and x= -2+2i√3, x= -2-2i√3 (complex solutions). This was found using the difference of cubes formula.

The polynomial equation asked in the question is [tex]x^3-64=0,[/tex] which is a cubic equation rather than a quadratic equation. Hence we need to use a different method to solve it rather than the quadratic formula. Here we can use the difference of cubes formula, which indicates  [tex]a^3-b^3[/tex] can be factored as [tex](a-b)(a^2+ab+b^2).[/tex] For this equation, the 'a' term is x (because  [tex]x^3 = a^3[/tex]) and the 'b' term is 4 (because 4^3 = 64 which is b^3).

Following this formula, we factor the equation as  [tex](x-4)(x^2+4x+16)=0.[/tex] Since this equation is set to equal zero, either the first factor equals zero (which gives us a solution x=4) or the second factor equals zero. After using the quadratic equation for the second factor, it has no real roots since its discriminant  [tex](b^2-4ac = 4^2 - 4*1*16 = 16 - 64 = -48)[/tex]is negative. However, it has complex roots, which are -2+2i√3 and -2-2i√3.

So, the real and complex solutions of the polynomial equation [tex]x^3-64=0[/tex]are x=4, x= -2+2i√3, x= -2-2i√3.

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The real solution for the equation x^3-64=0 is 4. The complex solutions are -2 + 2i√3 and -2 - 2i√3. Therefore, the complete solutions are {4, -2 + 2i√3, -2 - 2i√3}.

The given equation is x3-64=0. First, we can rewrite this equation as x3=64. This can be solved by taking the cube root of both sides, which gives us x = 4. Thus, 4 is the real solution.

To find the complex solutions, we need to use the fact that every non-zero number has three cube roots. The other two solutions can be found using the formula:
x = -2 + 2i√3
x= -2 - 2i√3

Therefore, the complete solution set of the equation x3-64=0 is {4, -2 + 2i√3, -2 - 2i√3}

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