What is a real life word problem for the equation
y=2x

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:

a) [tex]\bar X = \frac{\sum_{i=1}^{27} X_i }{27}= \frac{809}{27}=29.96[/tex]

[tex] Median = 25[/tex]

b) [tex] Mode = 25, 35[/tex]

Since 25 and 35 are repeated 4 times, so then the distribution would be bimodal.

c) [tex] Midrange = \frac{70+13}{3}=41.5[/tex]

d) [tex] Q_1 = \frac{20+21}{2} =20.5[/tex]

[tex] Q_3 =\frac{35+35}{2}=35[/tex]

e) Min = 13 , Q1 = 20.5, Median=25, Q3= 35, Max = 70

f) Figura attached.

g) When we use a quantile plot is because we want to show the percentage or the fraction of values below or equal to an specified value for the distribution of the data.

By the other hand the quantile-quantile plot shows the quantiles of the distribution values against other selected distribution (specified, for example the normal distribution). If the points are on a straight line we assume that the data values fit very well to the hypothetical distribution selected.

Step-by-step explanation:

For this case w ehave the following dataset given:

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

Part a

The mean is calculated with the following formula:

[tex]\bar X = \frac{\sum_{i=1}^{27} X_i }{27}= \frac{809}{27}=29.96[/tex]

The median on this case since we have 27 observations and that represent an even number would be the 14 position in the dataset ordered and we got:

[tex] Median = 25[/tex]

Part b

The mode is the most repeated value on the dataset on this case would be:

[tex] Mode = 25, 35[/tex]

Since 25 and 35 are repeated 4 times, so then the distribution would be bimodal.

Part c

The midrange is defined as:

[tex] Midrange = \frac{Max+Min}{2}[/tex]

And if we replace we got:

[tex] Midrange = \frac{70+13}{3}=41.5[/tex]

Part d

For the first quartile we need to work with the first 14 observations

13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25

And the Q1 would be the average between the position 7 and 8 from these values, and we got:

[tex] Q_1 = \frac{20+21}{2} =20.5[/tex]

And for the third quartile Q3 we need to use the last 14 observations:

25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70

And the Q3 would be the average between the position 7 and 8 from these values, and we got:

[tex] Q_3 =\frac{35+35}{2}=35[/tex]

Part e

The five number summary for this case are:

Min = 13 , Q1 = 20.5, Median=25, Q3= 35, Max = 70

Part f

For this case we can use the following R code:

> x<-c(13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30,33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70)

> boxplot(x,main="boxplot for the Data")

And the result is on the figure attached. We see that the dsitribution seems to be assymetric. Right skewed with the Median<Mean

Part g

When we use a quantile plot is because we want to show the percentage or the fraction of values below or equal to an specified value for the distribution of the data.

By the other hand the quantile-quantile plot shows the quantiles of the distribution values against other selected distribution (specified, for example the normal distribution). If the points are on a straight line we assume that the data values fit very well to the hypothetical distribution selected.

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:

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:

Mean ;

Median;

Step-by-step explanation:

The mean is used when the data under consideration is more of quantitative and in which the data is devoid of outliers as such the values are assumed to follow a normal distribution.

The median on the other hand is considered when the data are more of qualitative and usually contain outliers. Median on the other hand is best used when there is a skewed symmetry in the values given.

Mean ;

Median;

Whether mean or median provides a better description of a data set depends on the skewness and outliers in the data. Generally, the mean is more sensitive to outliers whereas the median can better represent the central tendency of skewed distributions.

In statistics, mean and median are two measures of central tendency. The mean is the average of the data points, while the median is the middle value. Whether the mean or median provides a better description depends on the distribution of the data.

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Answer:

4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.

Step-by-step explanation:

For each installation, there are only two possible outcomes. Either it reduces the utility bill, or it does not. The probabilities for each installation reducing the utility bill are independent. So we use the binomial probability distribution to solve this question.

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:

Solar-heat installations successfully reduce the utility bill 60% of the time, which means that [tex]p = 0.6[/tex]

What is the probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill?

This is [tex]P(X \geq 9)[/tex] when [tex]n = 10[/tex]. So

[tex]P(X \geq 9) = P(X = 9) + P(X = 10)[/tex]

In which

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

[tex]P(X = 9) = C_{10,9}.(0.6)^{9}.(0.4)^{1} = 0.0363[/tex]

[tex]P(X = 10) = C_{10,10}.(0.6)^{10}.(0.4)^{0} = 0.0060[/tex]

So

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) = 0.0363 + 0.0060 = 0.0423[/tex]

4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.

The probability of at least 90% success in solar-heat installations is found by using the binomial probability formula to calculate and add together the probabilities of exactly 9 and 10 successful installations out of 10.

The problem in question is a classic scenario of binomial probability. Here, each solar-heat installation attempt is independent and each attempt is a success (reduces the utility bill) 60% of the time. We are interested in the probability of having 90% or more success in ten attempts.

In a binomial distribution, the formula for calculating the probability of k successes in n attempts is:

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

where C(n, k) is the binomial coefficient ('n choose k'), p is the probability of success on an individual trial, n is the number of trials, and k is the number of successes.

To calculate the probability that at least 9 out of 10 solar-heat installations are successful, we need to calculate P(X=9) and P(X=10) and add these probabilities together.

Calculations like these help inform decisions in a range of fields - from individual choices about energy saving at homes to policy and planning decisions at the level of energy utilization for entire nations.

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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.

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:

2.2 times

Step-by-step explanation:

Given that,

Sales = $4,400,000

Average total assets (excluding long-term investments) = 2,000,000

Therefore, it is as follows;

Asset turnover ratio:

= Sales ÷ Average total assets (excluding long-term investments)

= $4,400,000 ÷ 2,000,000

= 2.2 times

Hence, the asset turnover ratio of this company is 2.2 times.

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

Focus on the sub-triangle on the left. It is a right triangle with legs 9 and 6, so its hypothenuse is

[tex]\sqrt{9^2+6^2}=\sqrt{81+36}=\sqrt{117}[/tex]

Now focus on the sub-triangle on the right. It is a right triangle with legs 6 and x, so its hypothenuse is

[tex]\sqrt{6^2+x^2}=\sqrt{x^2+36}[/tex]

Now, the entire triangle has legs [tex]\sqrt{117}[/tex] and [tex]\sqrt{x^2+36}[/tex], and its hypothenuse is [tex]9+x[/tex]. Write the Pytagorean theorem one last time to get

[tex]117+(x^2+36)=(9+x)^2\iff x^2+153=81+18x+x^2 \iff 18x+81=153[/tex]

Subtract 81 from both sides to get

[tex]18x=72 \iff x=\dfrac{72}{18}=4[/tex]

Answer: x = 4

Step-by-step explanation:

The attached photo shows a clearer illustration of the given triangle.

Looking at the photo, assuming ∆BCD is a right angle triangle. To determine BC, we would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side² + adjacent side²

BC² = 9² + 6²

BC² = 81 + 36 = 117

BC = √117

To determine θ, we would apply the tangent trigonometric ratio.

Tan θ opposite side/adjacent side

Tan θ = 6/9 = 0.6667

θ = 33.6914

Considering ∆ABC,

Hypotenuse = x + 9

Adjacent = √117

Cos θ = adjacent side/ hypotenuse

Cos 33.6914 = √117/(x + 9)

Cross multiplying, it becomes

0.8320 = √117/(x + 9)

x + 9 = √117/0.8320

x + 9 = 13

x = 13 - 9 = 4

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:

Step-by-step explanation:

This is a circle with radius 2 and z = y

All points on or within the circle x2 + y2 +4 and in the plane z = y

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: 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:

24, 204

Step-by-step explanation:

to find the height

we would use the heron formula  when all the sides are given

S which is equal to half of the perimeter of the triangle which is (a+b+c)/2.

S = (17+ 25+ 26)/2 = 68/2 = 34

Area = √S(S-A)(S-B)(S-C)

Area= (1/2)bh

we equate it back to the formula Area = √S(S-A)(S-B)(S-C)

it becomes

(1/2)bh  = √S(S-A)(S-B)(S-C)

A = IABI= 25

B= IBCIM = 26

C= IACI = 17

b = base = IACI = 17

S = 34

(1/2)bh  = √S(S-A)(S-B)(S-C)

(1/2)17h  = √34(34-25)(34-26)(34-17)

(17/2)h = √34(9)(8)(17) = √34 x 9 x 8 x 17

(17/2)h = √41616 = 204

17h/2 = 204

17h = 204 x 2 = 408

h = 408/17 = 24 inch

height = h = IBDI = 24 in

Area = (1/2)bh

      = (1/2) x 17 x 24

    =  12 x 17 = 204  or we use the heron formula just like the above which we get 204 before multiplication by 2.

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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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:

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:

[tex]\frac{3}{14}[/tex]

Step-by-step explanation:

There are 252 (=31+54+42+20+47+58) employees in total. 54 of those are in research. So the chances that one check that gets lost belongs to a research employee can be calculated as follows:

[tex]P=\frac{54}{252}= \frac{3}{14}[/tex]

Answer:

...the size tolerance controls both the measurements or dimensions of a piece, and the size.

Step-by-step explanation:

However, dimensional tolerance controls neither the shape, nor the position, nor the orientation of the elements to which said tolerance applies. In manufacturing, geometric irregularities occur that can affect the shape, position or orientation of the different elements of the pieces. An applied dimensional tolerance, for example, has an effect on the parallelism and flatness of that piece.

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:

Step-by-step explanation:

Since each trial is independent of the other

no of mistakes he does is binomial with p = 1/3

a) the probability that he makes no mistakes on his first 10 orders but the 11th order is a mistake

= [tex](\frac{2}{3}) ^{10} *\frac{1}{3}\\=\frac{2^{10} }{3^{11} }[/tex]

b) Prob that shanker quits = P(Shankar does I one mistake and Fran does not do the first one)+Prob (Shanker does mistake in the II one while Fran does both right)

= [tex]\frac{1*5}{3*6} +\frac{2}{3} \frac{1}{3}(\frac{5}{6})^2 =\frac{5}{18} +\frac{50}{216} \\=\frac{55}{108}[/tex]

Final answer:

Calculating probabilities in scenarios involving making mistakes in orders in a coffee shop setting. he probability that Shankar quits and goes to grad school before Fran is approximately 0.651.

Explanation:

a. For Shankar to make no mistakes on his first 10 orders but the 11th order is a mistake, the probability of making no mistakes on the first 10 orders and then making a mistake on the 11th order can be calculated as follows:

[tex]\[ P(\text{No mistakes on first 10 orders}) \times P(\text{Mistake on 11th order}) \][/tex]

[tex]\[ = \left(\frac{2}{3}\right)^{10} \times \frac{1}{3} \][/tex]

[tex]\[ = \left(\frac{1024}{59049}\right) \times \frac{1}{3} \][/tex]

[tex]\[ \approx 0.0173 \][/tex]

b. For Shankar to quit and go to grad school before Fran, Shankar must make a mistake before Fran does. The probability of Shankar quitting and going to grad school can be calculated as follows:

[tex]\[ P(\text{Shankar quits}) = 1 - P(\text{Fran quits first}) \][/tex]

Since Fran's probability of making a mistake on an order is [tex]\( \frac{1}{6} \)[/tex], the probability of Fran making no mistakes on an order is [tex]\( \frac{5}{6} \).[/tex] Thus, the probability of Fran not making a mistake before Shankar is:

[tex]\[ P(\text{Fran makes no mistakes before Shankar}) = \left(\frac{5}{6}\right)^{10} \][/tex]

Therefore,

[tex]\[ P(\text{Shankar quits}) = 1 - \left(\frac{5}{6}\right)^{10} \][/tex]

[tex]\[ \approx 0.651 \][/tex]

So, the probability that Shankar quits and goes to grad school before Fran is approximately \(0.651\).

Answer:

[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]

The correct answer would be:

NO

Step-by-step explanation:

For this case we have the following dataset given

Hours    Number of students (f)

_______________________________

   4                     15

   5                     11

    6                     19

   7                      6

   8                      9

    9                     16

    10                     2

______________________________

Total                  78

For this case we have defined the following events:

A = event the student took at most 9 hours

B = event the student took at least 9 hours

And we can find the empirical probability for both elements like this:

[tex] P(A) = \frac{78-2}{78}= \frac{76}{78}[/tex]

[tex] P(B) = \frac{16+2}{78}= \frac{18}{78}[/tex]

And for this case we want to see if A and B are disjoint

From definition two events X and Y are disjoint if the two sets not have a common elements, and we satisfy that:

[tex] P(X \cap Y) =0[/tex]

So this case the intersection for the events A and B is X=9, because at most 9 means [tex] X \leq 9[/tex] and at least 9 means [tex] X \geq 9[/tex] and the intersection between [tex] X \leq 9[/tex]  and [tex] X \geq 9[/tex]  is X=9

So then the probability:

[tex] P(A \cap B) = P(X=9) =\frac{16}{78} \neq 0[/tex]

So then we can conclude that the two events not are disjoint

The correct answer would be:

NO

No, the events A and B are not disjoint.

If two events have no outcomes in common, then they are called disjoint.

We have data of the number of hours sixth grade students took to complete a research project as:

For this case we have the following dataset given

Hours    Number of students (f)

 4                     15

5                     11

6                     19

7                      6

8                      9

9                     16

10                     2

Total                  78

Two events are:

A = event the student took at most 9 hours

B = event the student took at least 9 hours

Now, the number of students who took at most 9 hours

= 78 - 2

= 76

So, [tex]P(A)=\frac{76}{78}[/tex]

The number of students who took at least 9 hours

=16 +2

=18

So, [tex]P(B)=\frac{16}{78}[/tex]

Number of students who read exactly 9 hours

P(A n B)[tex]=\frac{16}{78}[/tex][tex]\neq 0[/tex]

Therefore the events A and B disjoint are not disjoint.

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Answer:

The probability is 0.27

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that:

Odds of a win are 27 to 73.

This means that for each 27 games that you are expcted to win, you are also expected to lose 73.

So

Desired outcomes:

27 wins

Total outcomes:

27 + 73 = 100 games

Probability

[tex]P = \frac{27}{100} = 0.27[/tex]

Answer:

The probability is 0.27

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) 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:

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

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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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]