Prove that among 502 positive integers, there are always two integers so that either their sum or their difference is divisible by 1000.

Answer:

[tex]P(57<X<59)=P(\frac{57-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{59-\mu}{\sigma})=P(\frac{57-61}{6}<Z<\frac{59-61}{6})=P(-0.67<Z<-0.33)[/tex]

[tex]P(-0.67<z<-0.33)=P(z<-0.33)-P(z<-0.67)[/tex]

[tex]P(-0.67<z<-0.33)=P(z<-0.33)-P(z<-0.67)=0.371-0.251=0.119[/tex]

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 lifetime for a TV of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(61,6)[/tex]  

Where [tex]\mu=61[/tex] and [tex]\sigma=6[/tex]

We are interested on this probability

[tex]P(57<X<59)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(57<X<59)=P(\frac{57-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{59-\mu}{\sigma})=P(\frac{57-61}{6}<Z<\frac{59-61}{6})=P(-0.67<Z<-0.33)[/tex]

And we can find this probability like this:

[tex]P(-0.67<z<-0.33)=P(z<-0.33)-P(z<-0.67)[/tex]

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

[tex]P(-0.67<z<-0.33)=P(z<-0.33)-P(z<-0.67)=0.371-0.251=0.119[/tex]

Answer:

a) There is a 59.87% probability that none of the LED light bulbs are defective.

b) There is a 31.51% probability that exactly one of the light bulbs is defective.

c) There is a 98.84% probability that two or fewer of the LED light bulbs are defective.

d) There is a 100% probability that three or more of the LED light bulbs are not defective.

Step-by-step explanation:

For each light bulb, there are only two possible outcomes. Either it fails, or it does not. This means that 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 combinatios 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]n = 10, p = 0.05[/tex]

a) None of the LED light bulbs are defective?

This is P(X = 0).

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

[tex]P(X = 0) = C_{10,0}*(0.05)^{0}*(0.95)^{10} = 0.5987[/tex]

There is a 59.87% probability that none of the LED light bulbs are defective.

b) Exactly one of the LED light bulbs is defective?

This is P(X = 1).

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

[tex]P(X = 1) = C_{10,1}*(0.05)^{1}*(0.95)^{9} = 0.3151[/tex]

There is a 31.51% probability that exactly one of the light bulbs is defective.

c) Two or fewer of the LED light bulbs are defective?

This is

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

[tex]P(X = 2) = C_{10,2}*(0.05)^{2}*(0.95)^{8} = 0.0746[/tex]

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.5987 + 0.3151 + 0.0746 0.9884[/tex]

There is a 98.84% probability that two or fewer of the LED light bulbs are defective.

d) Three or more of the LED light bulbs are not defective?

Now we use p = 0.95.

Either two or fewer are not defective, or three or more are not defective. The sum of these probabilities is decimal 1.

So

[tex]P(X \leq 2) + P(X \geq 3) = 1[/tex]

[tex]P(X \geq 3) = 1 - P(X \leq 2)[/tex]

In which

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

[tex]P(X = 0) = C_{10,0}*(0.95)^{0}*(0.05)^{10}\cong 0[/tex]

[tex]P(X = 1) = C_{10,1}*(0.95)^{1}*(0.05)^{9} \cong 0[/tex]

[tex]P(X = 2) = C_{10,1}*(0.95)^{2}*(0.05)^{8} \cong 0[/tex]

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0[/tex]

[tex]P(X \geq 3) = 1 - P(X \leq 2) = 1[/tex]

There is a 100% probability that three or more of the LED light bulbs are not defective.

The question relates to binomial distribution in probability theory. The probabilities calculated include those of none, one, two or less, and three or more LED bulbs being defective out of a random sample of 10.

This question relates to the binomial probability distribution. A binomial distribution is applicable because there are exactly two outcomes in each trial (either the LED bulb is defective or it's not) and the probability of a success remains consistent.

a) In this scenario, 'none of the bulbs being defective' means 10 successes. The formula for probability in a binomial distribution is p(x) = C(n, x) * [p^x] * [(1-p)^(n-x)]. Plugging in the values, we find p(10) = C(10, 10) * [0.95^10] * [0.05^0] = 0.5987 or 59.87%.

b) 'Exactly one of the bulbs being defective' implies 9 successes and 1 failure. Following the same formula, we get p(9) = C(10, 9) * [0.95^9] * [0.05^1] = 0.3151 or 31.51%.

c) 'Two or less bulbs being defective' means 8, 9 or 10 successes. We add the probabilities calculated in (a) and (b) with that of 8 successes to get this probability. Therefore, p(8 or 9 or 10) = p(8) + p(9) + p(10) = 0.95.

d) 'Three or more bulbs are not defective' means anywhere from 3 to 10 successes. As the failure rate is low, it's easier to calculate the case for 0, 1 and 2 successes and subtract it from 1 to find this probability. This gives us p(>=3) = 1 - p(2) - p(1) - p(0) = 0.98.

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Answer:

a)   (y+ Oz)

b) (ab + ac)(abc + cd + ce)

Step-by-step explanation:

a) xy + xOz  (First find the common factor between xy and xOz, and it is x as it appears in both xy and xOz. Factoring out x will result in

 x(y+ Oz)

There are no more common factors thus the answer is x(y + Oz)

Since the question was asking after removing redundant terms it will on be y + Oz

b) (ab0 f + a 0 cf)(a 0 bcf + c 0 f(d + e))  simplfying the expression

(ab0 f + a 0 cf)(aObcf + cOfd + cOfe) common factor Of is factored out as below

Of[ (ab + ac)(abc + cd + ce) ] therefore after removing redundant terms the answer is (ab + ac)(abc + cd + ce)

Answer:

Step-by-step explanation:

To prove sin(a+b)*sin(a-b)=cos^2b-cos^2a

we simplify the left side  sin(a+b)*sin(a-b) first

sin(a+b) = sin a cos b + cos a sin b

sin(a-b) = sin a cos b - cos a sin b

sin(a+b)*sin(a-b) = (sin a cos b + cos a sin b) x (sin a cos b -cos a sin b)

sin a cos b((sin a cos b + cos a sin b) - cos a sin b (sin a cos b + cos a sin b)

open the bracket

sin a cos b(sin a cos b) + sin a cos b(cos a sin b) -cos a sin b (sin a cos b)+ cos a sin b ( cos a sin b)

sin²a cos²b + sin a cos b cos a sin b - cos a sin b sin a cos b + cos²a sin²b

sin²a cos²b  + 0 + cos²a sin²b

sin²a cos²b + cos²a sin²b

(1-cos² a)cos² b - cos² a(1-cos² b)

= cos² b - cos² a cos² b - cos² a +cos² a cos² b

cos² b - cos² a - cos² a cos² b + cos² a cos² b = cos² b - cos² a  + 0

cos² b - cos² a

left hand side equals right hand side

Answer:

The value of x increased by 3.25.

Step-by-step explanation:

It is given that the value of x varies from x=2 to x=5.25.

We need to find the change is x.

If value of a variable x varies from a to b, then the change in te value of variable is

Change in x = b - a

For the given problem the variable is x, a=2 and b=5.25.

Change in x = 5.25 - 2

                  = 3.25

If the change is positive, then the value of variable increased and if the change is negative, then the value of variable decreased.

Therefore, the value of x increased by 3.25.

The value of x changed by 3.25 as it varies from x=2 to x=5.25.

The change in the value of x from x=2 to x=5.25 can be calculated by subtracting the initial value from the final value. In this case, the change in x is 5.25 - 2 = 3.25. Thus, the value of x has changed by 3.25.

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Answer:

Each packet of orange can make 2 cups of juice and each packets of Apple can make 3 cups of juice.

Step-by-step explanation:

Let number of cups of juice in each packet of orange be 'x'.

Let number of cups of juice in each pack of apples be 'y'.

Given:

On Tuesday;

Number of packets of oranges = 8

Number of packets of apples = 6 packets.

Amount of fruit juice = 34 cups

So we can say that;

Amount of fruit juice is equal to sum of Number of packets of oranges multiplied by number of cups of juice in each packet of orange and Number of packets of apples multiplied by number of cups of juice in each pack of apples.

framing in equation form we get;

[tex]8x+6y=34 \ \ \ \ equation \ 1[/tex]

On Wednesday;

Number of packets of oranges = 14

Number of packets of apples = 6

Amount of fruit juice = 46 cups

So we can say that;

Amount of fruit juice is equal to sum of Number of packets of oranges multiplied by number of cups of juice in each packet of orange and Number of packets of apples multiplied by number of cups of juice in each pack of apples.

framing in equation form we get;

[tex]14x+6y=46 \ \ \ \ equation \ 2[/tex]

Now Subtracting equation 1 from equation we get;

[tex]14x+6y-(8x+6y)=46-34\\\\14x+6y-8x-6y=12\\\\6x=12[/tex]

Now dividing both side by 6 we get;

[tex]\frac{6x}{6}=\frac{12}{6}\\\\x=2\ cups[/tex]

Now we will substitute the value of 'x' in equation 1 we get;

[tex]8x+6y=34\\\\8\times2+6y=34\\\\16+6y=34[/tex]

Subtracting both side by 16 we get;

[tex]16+6y-16=34-16\\\\6y= 18[/tex]

Now dividing both side by 6 we get;

[tex]\frac{6y}{6}=\frac{18}{6}\\\\y=3[/tex]

Hence Each packet of orange can make 2 cups of juice and each packets of Apple can make 3 cups of juice.

By organizing the data and determining the minimum, maximum, median, and quartiles for the volumes of the largest dams in the United States and South America, we can construct boxplots to compare the distribution of the data in each region   Create a text based boxplot

United States:

 +---------+---------+---------+---------+---------+

 | Minimum |   Q1    | Median  |   Q3    | Maximum |

 +---------+---------+---------+---------+---------+

         |-----|-----|-----|-----|-----|

         50,000        77,700  77,854      108,814       125,628

South America:

 +---------+---------+---------+---------+---------+

 | Minimum |   Q1    | Median  |   Q3    | Maximum |

 +---------+---------+---------+---------+---------+

         |-----|-----|-----|-----|-----|

         46,563        51,403  103,979     292,783       311,539

To construct a boxplot for the volumes of the largest dams in the United States and South America, we will first need to organize the data in ascending order. Then we can determine the minimum, maximum, median, and quartiles. The boxplot will allow us to compare the distribution of the data for each region.

United States volumes (in cubic yards) in ascending order:

50,000

52,435

62,850

66,500

77,700

78,008

92,000

125,628

South America volumes (in cubic yards) in ascending order:

46,563

56,242

102,014

105,944

274,026

311,539

Now, we can find the minimum, maximum, median, and quartiles for each dataset:

United States:

- Minimum: 50,000

- Maximum: 125,628

- Median: The middle value is the average of the two middle values, (77,700 + 78,008) / 2 = 77,854

- First Quartile (Q1): The median of the lower half of the dataset, (52,435 + 62,850) / 2 = 57,642.5

- Third Quartile (Q3): The median of the upper half of the dataset, (92,000 + 125,628) / 2 = 108,814

South America:

- Minimum: 46,563

- Maximum: 311,539

- Median: The middle value is the average of the two middle values, (102,014 + 105,944) / 2 = 103,979

- First Quartile (Q1): The median of the lower half of the dataset, (46,563 + 56,242) / 2 = 51,402.5

- Third Quartile (Q3): The median of the upper half of the dataset, (274,026 + 311,539) / 2 = 292,782.5

With this information, we can construct boxplots for each region. The boxplots will visually represent the minimum, first quartile, median, third quartile, and maximum values for each dataset.

Answer: The equation of the sphere is:

(x-2)^2 + (y-2)^2 + (z-5/2)^2 = sqrt(245)/2

Step-by-step explanation:

The centre of the sphere is the midpoint of the diameter, which is

1/2

[(-4,7,6) + (8,-3,5)] =(2,2,5/2)

The length of the diameter is

=sqrt |(8,-3,5) - (-4,7,6)|^2

=sqrt (12^2 + (-10)^2 + (-1)^2)

=sqrt (144+100+1)

=sqrt(245)

so the radius of

the sphere is:

1/2(sqrt(245)) = sqrt(245)/2.

The equation of the sphere is:

(x-2)^2 + (y-2)^2 + (z-5/2)^2 = sqrt(245)/2

Answer:

1) This random variable should be modelled using a binomial distribution since we have independence between the events and a bernoulli trial each time when the experiment is conducted, a fixd value for the sample size n and for the probability of success.

Let X the random variable of interest, on this case th distribution would be given by:

[tex]X \sim Binom(n=10, p=0.02)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x} = (10Cx) (0.02)^x (1-0.02)^{10-x}[/tex]

2) For this case we don't have a sample size provided and we just have an average rate for a given period, so then we can assume that the best distribution for this case is the Poisson distribution.

Let X the random variable that represent the number of claims per car. We know that [tex]X \sim Poisson(\lambda)[/tex]

The probability mass function for the random variable is given by:

[tex]f(x)=\frac{e^{-\lambda} \lambda^x}{x!} , x=0,1,2,3,4,...[/tex]

Where [tex]\lambda=0.2[/tex] represent the mean of occurrences in the interval of 2 years provided.

And f(x)=0 for other case.

Step-by-step explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Random variable 1

This random variable should be modelled using a binomial distribution since we have independence between the events and a bernoulli trial each time when the experiment is conducted, a fixd value for the sample size n and for the probability of success.

Let X the random variable "number of failed drawers", on this case th distribution would be given by:

[tex]X \sim Binom(n=10, p=0.02)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x} = (10Cx) (0.02)^x (1-0.02)^{10-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

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

Random variable 2

For this case we don't have a sample size provided and we just have an average rate for a given period, so then we can assume that the best distribution for this case is the Poisson distribution.

Let X the random variable that represent the number of claims per car. We know that [tex]X \sim Poisson(\lambda)[/tex]

The probability mass function for the random variable is given by:

[tex]f(x)=\frac{e^{-\lambda} \lambda^x}{x!} , x=0,1,2,3,4,...[/tex]

Where [tex]\lambda=0.2[/tex] represent the mean of occurrences in the interval of 2 years provided.

And f(x)=0 for other case.

Answer:

The percentage of inter-rater reliability is 80%.

Step-by-step explanation:

The rate of inter-rater reliability is the division of the number of intervals in which they agreed number of total intervals.

In this problem, we have that:

There are 10 intervals.

The observers agreed on 8 of them.

So the rate of agreement is 8/10 = 0.8.

As a percentage, we multiply the rate by 100, so 0.8*100 = 80%.

The percentage of inter-rater reliability is 80%.

Answer:

a)0.08  , b)0.4  , C) i)0.84  , ii)0.56

Step-by-step explanation:

Given data

P(A) =  professor arrives on time

P(A) = 0.8

P(B) =  Student aarive on time

P(B) = 0.6

According to the question A & B are Independent  

P(A∩B) = P(A) . P(B)

Therefore  

[tex]{A}'[/tex] & [tex]{B}'[/tex] is also independent

[tex]{A}'[/tex] = 1-0.8 = 0.2

[tex]{B}'[/tex] = 1-0.6 = 0.4

part a)

Probability of both student and the professor are late

P(A'∩B') = P(A') . P(B')  (only for independent cases)

= 0.2 x 0.4

= 0.08

Part b)

The probability that the student is late given that the professor is on time

[tex]P(\frac{B'}{A})[/tex] = [tex]\frac{P(B'\cap A)}{P(A)}[/tex] = [tex]\frac{0.4\times 0.8}{0.8}[/tex] = 0.4

Part c)

Assume the events are not independent

Given Data

P[tex](\frac{{A}'}{{B}'})[/tex] = 0.4

=[tex]\frac{P({A}'\cap {B}')}{P({B}')}[/tex] = 0.4

[tex]P[/tex][tex]({A}'\cap {B}')[/tex] = 0.4 x P[tex]({B}')[/tex]

= 0.4 x 0.4 = 0.16

[tex]P({A}'\cap {B}')[/tex] = 0.16

i)

The probability that at least one of them is on time

[tex]P(A\cup B)[/tex] = 1- [tex]P({A}'\cap {B}')[/tex]  

=  1 - 0.16 = 0.84

ii)The probability that they are both on time

P[tex](A\cap B)[/tex] = 1 - [tex]P({A}'\cup {B}')[/tex] = 1 - [tex][P({A}')+P({B}') - P({A}'\cap {B}')][/tex]

= 1 - [0.2+0.4-0.16] = 1-0.44 = 0.56

Answer:

0.71%

Step-by-step explanation:

Given that a normally distributed set of data has a mean of 102 and a standard deviation of 20.

Let X be the random variable

Then X is N(102, 20)

We can convert this into standard Z score by

[tex]z=\frac{x-102}{20}[/tex]

We are to find the probability and after wards percentage of scores in the data set expected to be below a score of 151.

First let us find out probability using std normal table

P(X<151) = [tex]P(Z<\frac{151-102}{20} \\=P(Z<2.45)\\=0.5-0.4929\\\\=0.0071[/tex]

We can convert this into percent as muliplying by 100

percent of scores in the data set expected to be below a score of 151.

=0.71%

The a chi-square of the experimental data is 4.32.

Observed data(o)

Purple tall=206

White tall=65

Purple short=83

White short=30

Total=384

Expected (e)

9 Purple tall=216

3 White tall=72

3 Purple short=72

1 White short=24

Chi-square

Purple tall=(206-216)²/216=0.46

White tall=(65-72)²/72=0.68

Purple short=(83-72)²/72=1.68

White short=(30-24)²/24=1.5

X²=0.46+0.68+1.68+1.5

X²=4.32

Inconclusion  a chi-square of the experimental data is 4.32.

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Final answer:

To analyze the dihybrid cross data, we calculate the expected frequencies using the Mendelian 9:3:3:1 ratio, perform a chi-square analysis, and then conclude whether the observed data significantly deviates from the expected phenotypic ratio.

Explanation:

To analyze the given data, we must first establish our expected ratio. For a dihybrid cross involving two independently assorting traits with complete dominance, Mendel's laws predict a 9:3:3:1 phenotypic ratio among the F2 progeny. These numbers represent the ratio of the offspring displaying both dominant traits, one dominant and one recessive trait, one recessive and one dominant trait, and both recessive traits, respectively.

With the given data:

206 purple tall (dominant traits)

65 white tall (one recessive trait)

83 purple short (one recessive trait)

30 white short (recessive traits)

Adding these together, we have a total of 384 plants. To get the expected counts based on the 9:3:3:1 ratio: Expected purple tall = 9/16 * 384 = 216, Expected white tall = 3/16 * 384 = 72, Expected purple short = 3/16 * 384 = 72, and Expected white short = 1/16 * 384 = 24. Now, we can perform a chi-square analysis using these expected counts.

The chi-square formula is: X^2 = ∑ (observed - expected)^2 / expected. Plugging in our values:

X^2 for purple tall = (206 - 216)^2 / 216

X^2 for white tall = (65 - 72)^2 / 72

X^2 for purple short = (83 - 72)^2 / 72

X^2 for white short = (30 - 24)^2 / 24

After calculating these values, they need to be added to get the total chi-square value. We then compare this value to a critical value from a chi-square distribution table (typically at a 0.05 significance level) with the degrees of freedom equal to the number of classes - 1, here df = 3.

Based on this comparison, we determine if the observed ratios significantly deviate from the expected 9:3:3:1 ratio. If the calculated chi-square value is less than or equal to the critical value, our null hypothesis that purple color and tall height are dominant and assort independently holds true. However, if it's greater, we may have to reject the null hypothesis.

Answer:

Option A is the correct answer.

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis

change in the value of y = y2 - y1

Change in value of x = x2 -x1

y2 = final value of y

y 1 = initial value of y

x2 = final value of x

x1 = initial value of x

From the graph,

y2 = 6

y1 = 2

x2 = 3

x1 = 1

Slope,m = (6 - 2)/(3 - 1) = 4/2 = 2

To determine the intercept, we would substitute x = 3, y = 6 and

m= 2 into y = mx + c. It becomes

6 = 2 × 3 + c = 6 + c

c = 6 - 6 = 0

The equation becomes

y = 2x

Answer:

a) [tex] E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000[/tex]

b) [tex] Sd(T) = \sqrt{0.2^2 Var(Y)}=\sqrt{0.2^2 8000^2}= 1600[/tex]

c) Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:

[tex] E(T) = 20000000*20000= 40000 millions[/tex]

And if we replace into the formula of T we have:

[tex] T = 0.2*400000x10^6 -1000= 790000 millions[/tex]

Approximately.  

Step-by-step explanation:

For this case we knwo that Y represenet the random variable "Income" and we have the following properties:

[tex] E(Y) = 20000, Sd(Y) = 8000[/tex]

We define a new random variable T "who represent the taxes"

[tex] T = 0.2(Y-5000) = 0.2Y -1000[/tex]

Part a

For this case we need to apply properties of expected value and we have this:

[tex] E(T) = E(0.2 Y -1000)[/tex]

We can distribute the expected value like this:

[tex] E(T) = E(0.2 Y) -E(1000)[/tex]

We can take the 0.2 as a factor since is a constant and the expected value of a constant is the same constant.

[tex] E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000[/tex]

Part b

For this case we need to first find the variance of T we need to remember that if a is a constant and X a random variable [tex] Var(aX) = a^2 Var(X)[/tex]

[tex] Var(T) = Var (0.2Y -1000)[/tex]

[tex] Var(T)= Var(0.2Y) -Var(1000) + 2 Cov(0.2Y, -1000)[/tex]

The covariance between a random variable and a constant is 0 and a constant not have variance so then we have this:

[tex] Var(T) =0.2^2 Var(Y)[/tex]

And the deviation would be:

[tex] Sd(T) = \sqrt{0.2^2 Var(Y)}=\sqrt{0.2^2 8000^2}= 1600[/tex]

Part c

Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:

[tex] E(T) = 20000000*20000= 40000 millions[/tex]

And if we replace into the formula of T we have:

[tex] T = 0.2*400000x10^6 -1000= 790000 millions[/tex]

Approximately.  

Answer:

2^c

Step-by-step explanation:

Final answer:

To solve -30 ÷ (6a) when a = -5, substitute the value of a into the expression. The answer in simplest form is 1.

Explanation:

To solve the expression -30 ÷ (6a), we need to substitute the value of a, which is -5.

Substituting the value of a, -30 ÷ (6a) becomes -30 ÷ (6*(-5)).

Using the order of operations, first, we solve the multiplication inside the parentheses: -30 ÷ (-30).

Dividing a negative number by a negative number results in a positive number, so -30 ÷ (-30) = 1.

Therefore, the answer in simplest form is 1.

Answer:

For each n get its posting then AND them

Step-by-step explanation:

1. Lets presume we have multiple n terms

2. We get the posting from each n term

3. We use the function AND and apply it to each n term

4. We start with the tiniest set and we continue from there

5. We have a pair of n terms

6. We get the posting from 1st n

7. We get the posting from 2nd n

8. We apply n1 AND n2

Answer:

Therefore, we conclude that the center of sphere at point

(27/3, -8/3, -8/3) with a radius 6.89.

Step-by-step explanation:

We have the formula for distance, we get

\sqrt{(x+3)^2+(y-4)^2+(z-4)^2} =2· \sqrt{(x-6)^2+(y-3)^2+(z+1)^2}

(x+3)^2+(y-4)^2+(z-4)^2=4·[(x-6)^2+(y-3)^2+(z+1)^2]

x²+6x+9+y²-8y+16+z²-8z+16=4x²-48x+4y²-24y+4z²+8z+184

3x²+3y²+3z²-54x+16y+16z=-143

(x²-54x/3)+(y²+16y/3)+(z²+16z/3)=-143/3

(x²-54x/3+729/9)+(y²+16y/3+64/9)+(z²+16z/3+64/9)=-143/3+729/9+2·64/9

(x-27/3)²+(y+8/3)²+(z+8/3)²=428/9

We calculate a radius \sqrt{428/9} =6.89

Therefore, we conclude that the center of sphere at point

(27/3, -8/3, -8/3) with a radius 6.89.

To find the equation of the sphere in question, use the property that the distance from P to A is twice the distance from P to B, and solve the resulting equation to find the sphere's center and radius.

To find an equation of a sphere where the distance from a generic point P to the point A(−3, 4, 4) is twice the distance from P to the point B(6, 3, −1), we use the geometric properties of spheres and distances in three-dimensional space. The distance from a point P to another point Q in 3D space, with coordinates P(x1, y1, z1) and Q(x2, y2, z2), can be found using the distance formula d = √((x2 − x1)² + (y2 − y1)² + (z2 − z1)²).

Let's denote the coordinates of P as (x, y, z). To satisfy the given condition, we have to solve the equation (x + 3)² + (y − 4)² + (z − 4)² = 4[(x − 6)² + (y − 3)² + (z + 1)²]. This equation is derived from setting the distance from P to A as twice the distance from P to B and then squaring both sides to eliminate the square root.

Solving this equation will give us the center and radius of the sphere.

Answer:

Step-by-step explanation:

Let

                               [tex]f(x,y,z) = z-x^2-y^4-e^{xy}[/tex].

The partial derivatives of this function are

                                    [tex]\frac{\partial f}{\partial x} (x,y,z) = ye^{xy} - 2x \\ \frac{\partial f}{\partial y} (x,y,z) = xe^{xy} - 4y^3 \\\frac{\partial f}{\partial z} (x,y,z) = 1[/tex]

The tangent plane equation through a point [tex]A(x_1,y_1,z_1)[/tex] is given by                     [tex]f'_x (x_1,y_1,z_1)(x-x_1) + f'_y(x_1,y_1,z_1)(y-y_1) + f'_z(x_1,y_1,z_1)(z-z_1) = 0[/tex]

In this case, we have

                                  [tex]x_1 = 1, y_1 = 0, z_1 = 2.[/tex]

The values of the partial derivatives in this point are

                                 [tex]\frac{\partial f}{\partial x} (1,0,2) = 0 \cdot e^{0} - 2\cdot 1 = -2 \\ \frac{\partial f}{\partial y} (1,0,2) = 1 \cdot e^{0} - 0 = 1 \\ \frac{\partial f}{\partial y} (1,0,2) = 1[/tex]

So, the equation is

                        [tex]-2(x-1) + 1 \cdot (y-0) + 1\cdot (z-2) = 0[/tex]

Therefore, the equation for the plane tangent to the surface at the point [tex](1,0,2)[/tex] is given by

                                      [tex]2x-y-z = 0[/tex]

The points A(1, 1, 3), B(2, -5, 6), C(4, -2, -1), and D(3, 4, -4) form a parallelogram with an area of [tex]\sqrt{(1483)[/tex] square units.

To determine if the points A(1, 1, 3), B(2, -5, 6), C(4, -2, -1), and D(3, 4, -4) form the vertices of a parallelogram, we need to check two conditions:

Opposite sides are parallel.

Opposite sides are of equal length.

First, let's calculate the vectors representing the sides of the quadrilateral:

Vector AB = (2-1, -5-1, 6-3) = (1, -6, 3)

Vector BC = (4-2, -2-(-5), -1-6) = (2, 3, -7)

Vector CD = (3-4, 4-(-2), -4-(-1)) = (-1, 6, -3)

Vector DA = (1-3, 1-4, 3-(-4)) = (-2, -3, 7)

Next, we check if opposite sides are parallel. AB is parallel to CD, and BC is parallel to DA, as the direction ratios are proportional.

Now, we need to verify that opposite sides have the same length. Using the distance formula, we find:

[tex]|AB| = \sqrt{(1^2 + (-6)^2 + 3^2)} = \sqrt{(46)[/tex]

[tex]|CD| = \sqrt{((-1)^2 + 6^2 + (-3)^2)} = \sqrt{(46)[/tex]

[tex]|BC| = \sqrt{(2^2 + 3^2 + (-7)^2)} = \sqrt{(62)[/tex]

[tex]|DA| = \sqrt{((-2)^2 + (-3)^2 + 7^2)} = \sqrt{(62)[/tex]

Since opposite sides are both parallel and have equal lengths, the given points form the vertices of a parallelogram.

To find the area of the parallelogram, we can use the magnitude of the cross product of vectors AB and BC (or vectors BC and CD):

Area = |AB x BC| = |(1, -6, 3) x (2, 3, -7)| = |(-33, 17, 15)| = [tex]\sqrt{(33^2 + 17^2 + 15^2)} = \sqrt{(1483).[/tex]

So, the area of the parallelogram formed by the given points is sqrt(1483) square units.

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To find out if the given points A(1, 1, 3), B(2, −5, 6), C(4, −2, −1), and D(3, 4, −4) are the vertices of a parallelogram, we need to prove that the opposite sides are equal and parallel. In a parallelogram, both pairs of opposite sides are parallel and have the same length.

Step 1: Find the vectors representing the sides of the potential parallelogram.

Vectors AB, BC, CD, and DA can be calculated using the coordinates of points A, B, C, and D.

AB = B - A = (2 - 1, -5 - 1, 6 - 3) = (1, -6, 3)
BC = C - B = (4 - 2, -2 + 5, -1 - 6) = (2, 3, -7)
CD = D - C = (3 - 4, 4 + 2, -4 - (-1)) = (-1, 6, -3)
DA = A - D = (1 - 3, 1 - 4, 3 - (-4)) = (-2, -3, 7)

Step 2: Check if opposite sides are equal and parallel.

For sides AB and CD to be parallel and equal in length, vector AB should be equal to vector CD or AB should be a scalar multiple of CD.

Looking at vectors AB and CD:
AB = (1, -6, 3)
CD = (-1, 6, -3)

Vector CD is indeed the negative of vector AB, which means these vectors have the same length but opposite directions, so AB and CD are parallel and equal in length.

Now we need to check if vector BC is equal and parallel to vector DA.

BC = (2, 3, -7)
DA = (-2, -3, 7)

Vector DA is also the negative of vector BC. This means that they too have the same length but opposite directions, so BC and DA are parallel and equal in length.
Since both pairs of opposite sides are parallel and equal in length, points A, B, C, and D form a parallelogram.
Step 3: Calculate the area of the parallelogram.
The area of a parallelogram can be found using the cross product of two adjacent sides. The magnitude of the cross product vector gives us the area.
Let's calculate the cross product of vectors AB and AD:
AB = (1, -6, 3)
AD = (-2, -3, 7)  (Remember, AD is the opposite of DA)
The cross product AB x AD is:
| i   j  k  |
| 1  -6  3 |
| -2 -3  7 |
= i((-6)(7) - (3)(-3)) - j((1)(7) - (3)(-2)) + k((1)(-3) - (-6)(-2))
= i(-42 + 9) - j(7 + 6) + k(-3 - 12)
= i(-33) - j(13) + k(-15)
So the cross product of AB and AD is (-33, -13, -15).
The magnitude of this vector is √((-33)^2 + (-13)^2 + (-15)^2) = √(1089 + 169 + 225) = √(1483) ≈ 38.50 units squared.
Therefore, the area of the parallelogram formed by points A, B, C, and D is approximately 38.50 square units.

Answer:

[tex] s= \sqrt{80.568}=8.97 [/tex]

[tex] Median =\frac{1+3}{2}=2[/tex]

Step-by-step explanation:

For this case we have the following data:

1.0 -5 3.0 -8 14 -11 12 0 16 -2 12 7

And ordering this data we have:

-11 -8 -5 -2 0 1 3 7 12 12 14 16  

And we are interested in find the standard deviation for the sample data. In order to do this the first step is find the mean given by this formula:

[tex] \bar X =\frac{\sum_{i=1}^n X_i}{n}=\frac{1-5+3-8+14-11+12+0+16-2+12+7}{12}=3.25[/tex]

Now with the sample mean we can calculate the sample variance with the following formula:

[tex] s^2 = \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]

And if we replace we got:

[tex] s^2 =80.568[/tex]

And for the sample deviation we just need to take the square root of the sample variance and we got:

[tex] s= \sqrt{80.568}=8.97 [/tex]

The median on this case would be given by:

[tex] Median =\frac{1+3}{2}=2[/tex]

Using the positions 5 and 6 for the average since the sample size is an even number.

Final answer:

The standard deviation of the weight changes for the 12 women is approximately 8.1 pounds, after computing the variance and taking the square root.

Explanation:

To calculate the sample standard deviation of the given weight changes, we will follow these steps:

Now let's apply these steps to the given data:

The data points are: 1.0, -5, 3.0, -8, 14, -11, 12, 0, 16, -2, 12, 7

The mean (step 1) is calculated as:

(1.0 - 5 + 3.0 - 8 + 14 - 11 + 12 + 0 + 16 - 2 + 12 + 7) / 12 = (49) / 12 ≈ 4.0833 pounds

Continuing with steps 2 to 5, the squared differences from the mean, their sum, and the variance are computed. Finally, the sample standard deviation is found to be approximately 8.1 pounds.

Therefore, the correct answer is option b, which is 8.1 pounds.

Answer:

So the parallelogram is in the first and second quadrants.

Step-by-step explanation:

From Exercise we have that the parallelogram have a vetrices:

A(negative 1​,4​) ​B(4​,0​) ​C(12​,3​) ​D(7​,7​). We use a site geogebra.org to plof a graph for the given parallelogram.

From the graphs we can see that the parallelogram is mostly in the first quadrant and smaller in the second quadrant. So the parallelogram is in the first and second quadrants.

Answer:

To produce 400 gallons of a chemical, it costs 700 dollars.

Step-by-step explanation:

We have that:

The cost, C (in dollars) to produce g gallons of a chemical can be expressed as C=f(g).

The interpretation is:

To produce g gallons of a chemical it costs C dollars.

So

f(400)=700

This means that to produce 400 gallons of a chemical, it costs 700 dollars.

Final answer:

The statement f(400)=700 means that the cost to produce 400 gallons of a chemical is 700 dollars. This cost function indicates the relationship between the number of gallons produced and the total cost, facilitating the calculation of expenses for chemical production.

Explanation:

The statement f(400)=700 means that to produce 400 gallons of a chemical, the cost is 700 dollars. The function f(g) represents the cost C, in dollars, to produce g gallons of the chemical. Therefore, when we input 400 into the function, it outputs the cost associated with producing that quantity, which in this case is 700 dollars.

Understanding the function as a rate of change, the cost to produce an additional gallon can be determined through ordinary algebra. If it were the case that this chemical firm is in a constant cost industry, where the cost per additional unit produced remains the same (e.g., 10 USD/oz), then this would simplify the exercise of calculating the cost for any number of gallons produced.

Conversion factors in chemistry are often used to relate amounts of substances in reactions, similarly in economics, conversion factors can tell us how much of a good can be bought or produced per unit of currency, like dollars per gallon, which is USD/unit.

Answer:

i need more information like what is the length width etc.

Step-by-step explanation:

Answer:

kilo is [tex]10^{3}[/tex]

nano is [tex]10^{-9}[/tex]

micro is [tex]10^{-6}[/tex]

centi is [tex]10^{-2}[/tex]

mega is [tex]10^{6}[/tex]

milli is [tex]10^{-3}[/tex]

Step-by-step explanation:

Examples of power of 10 are:

[tex]100 = 10^{2}[/tex]

[tex]10 = 10^{1}[/tex]

[tex]1 = 10^{0}[/tex]

[tex]0.1 = 10^{-1}[/tex]

[tex]0.01 = 10^{-2}[/tex]

kilo is 1000. So kilo is [tex]10^{3}[/tex]

Nano is 0.000000001. So nano is [tex]10^{-9}[/tex]

micro is 0.000001. So micro is [tex]10^{-6}[/tex]

centi is 0.01. So centi is [tex]10^{-2}[/tex]

Mega is 1000000. So mega is [tex]10^{6}[/tex]

milli is 0.001. So milli is [tex]10^{-3}[/tex]

The following are conversions.

[tex]\rm{kilo}=10^3\\\rm {nano}=10^{-9}\\\rm{micro}=10^{-6}\\\rm{centi}=10^{-2}\\\rm{Mega}=10^6\\\rm{milli}=10^{-3}[/tex]

On the basis of the power of 10 the following categories can be defined that are as follow:

[tex]\rm{kilo}=10^3\\\rm {nano}=10^{-9}\\\rm{micro}=10^{-6}\\\rm{centi}=10^{-2}\\\rm{Mega}=10^6\\\rm{milli}=10^{-3}[/tex]

The values of all are as follows.

kilo is 1000.

Nano is 0.000000001.

micro is 0.000001.

Centi is 0.01.

Mega is 1000000.

Milli is 0.001.

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Answer:

a) 120

b) 24

c) 18

Step-by-step explanation:

part a

The sample space is defined by the rank of the horses.

Hence, 5 ranks would permute to = 5! = 120 outcomes

part b

Fixing the first position for horse A we are left with four horses and 4 positions, the position are permutated to 4! = 24 outcomes

part c

Fixing the first position for horse A we are left with four horses and 4 positions, and horse B can not finish second hence:

A _ B _ _  the rest can permute hence, 3! = 6 outcomes

A _ _ B _ the rest can permute hence, 3! = 6 outcomes

A _ _ _ B the rest can permute hence, 3! = 6 outcomes

Total outcomes is sum of 3 cases above = 18 outcomes

Answer:

a) Continuous since the weight can take decimals for example 2.3 pounds.

b) Discrete since the number of chapters in a book represent an integer, we can't say that abook have 4.3 chapters because not makes sense, we say the book have 4 or 5 chapters

c) Continuous since we can measure the width and we can get 8.3 inches so then the variable can takes decimals and for this reason is Continuous

d) Discrete since that's a binary variable and only take integers.

e) Discrete since the number of errors can't be 4.3 for example, and can't take decimals.

Step-by-step explanation:

Previous concepts

A continuous random variable by definition is a "random variable where the data can take infinitely many values" defined on a interval.

And a discrete random variable is a random variable that only can takes integers and is defined over a domain

Solution to the problem

a. Weight of the book (e.g., 2.3 pounds)

Continuous since the weight can take decimals for example 2.3 pounds.

b. Number of chapters in the book (e.g., 10 chapters)

Discrete since the number of chapters in a book represent an integer, we can't say that abook have 4.3 chapters because not makes sense, we say the book have 4 or 5 chapters

c. Width of the book (e.g., 8 inches)

Continuous since we can measure the width and we can get 8.3 inches so then the variable can takes decimals and for this reason is Continuous

d. Type of book (0=hardback or 1=paperback)

Discrete since that's a binary variable and only take integers.

e. Number of typographical errors in the book (e.g., 4 errors)

Discrete since the number of errors can't be 4.3 for example, and can't take decimals.

Answer:

In general, 50 % of the values in a data set lie at or below the median. 75 % of the values in a data set lie at or below the third quartile (Q3). If a sample consists of 1600 test scores, 0.5*1600 = 800 of them would be at or below the median. If a sample consists of 1600 test scores, 0.75*800 = 1200 of them would be at or above the first quartile (Q1)

Step-by-step explanation:

The median separates the upper half from the lower half of a set. So 50% of the values in a data set lie at or below the median, and 50% lie at or above the median.

The first quartile(Q1) separates the lower 25% from the upper 75% of a set. So 25% of the values in a data set lie at or below the first quartile, and 75% of the values in a data set lie at or above the first quartile.

The third quartile(Q3) separates the lower 75% from the upper 25% of a set. So 75% of the values in a data set lie at or below the third quartile, and 25% of the values in a data set lie at or the third quartile.

The answer is:

In general, 50 % of the values in a data set lie at or below the median. 75 % of the values in a data set lie at or below the third quartile (Q3). If a sample consists of 1600 test scores, 0.5*1600 = 800 of them would be at or below the median. If a sample consists of 1600 test scores, 0.75*800 = 1200 of them would be at or above the first quartile (Q1)

Using the principle of quartile distribution, the percentage amount of data at or below the median and third quartile are 50% and 75% respectively. Number of scores below the median and at or above the first quartile is 800 and 1200 respectively.

The median of 1600 can be calculated thus :

Above the first quartile :

Therefore, the score above the first quartile is 1200.

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Answer:

[tex]\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \boxed{\bold{4}}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

Derivative Property [Multiplied Constant]:
[tex]\displaystyle \bold{(cu)' = cu'}[/tex]
Derivative Rule [Basic Power Rule]:

Integration

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \bold{\int\limits^b_a {f(x)} \, dx = F(b) - F(a)}[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \bold{\int {cf(x)} \, dx = c \int {f(x)} \, dx}[/tex]

Multivariable Calculus

Partial Derivatives

Vector Calculus

Circulation Density:
[tex]\displaystyle \bold{F = M \hat{\i} + N \hat{\j} \rightarrow \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}[/tex]

Green's Theorem [Circulation Curl/Tangential Form]:
[tex]\displaystyle \bold{\oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy}[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy}[/tex]

[See Graph Attachment] Points (0, 0) → (1, 0) → (1, 1) → (0, 1)

↓

[tex]\displaystyle \text{Region:} \left \{ {{0 \leq x \leq 1} \atop {0 \leq y \leq 1}} \right.[/tex]

Step 2: Integrate Pt. 1

Step 3: Integrate Pt. 2

We can evaluate the Green's Theorem double integral we found using basic integration techniques listed above:
[tex]\displaystyle \begin{aligned}\oint_C {y^2x \, dx + 9x^2y \, dy} & = \int\limits^1_0 \int\limits^1_0 {16xy} \, dx \, dy \\& = \int\limits^1_0 {8x^2y \bigg| \limits^{x = 1}_{x = 0}} \, dy \\& = \int\limits^1_0 {8y} \, dy \\& = 4y^2 \bigg| \limits^{y = 1}_{y = 0} \\& = \boxed{\bold{4}}\end{aligned}[/tex]

∴ we have evaluated the line integral using Green's Theorem.

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Topic: Multivariable Calculus

Unit: Green's Theorem and Surfaces