In a sample of nequals16 lichen​ specimens, the researchers found the mean and standard deviation of the amount of the radioactive​ element, cesium-137, that was present to be 0.009 and 0.003 microcurie per​ milliliter, respectively. Suppose the researchers want to increase the sample size in order to estimate the mean mu to within 0.001 microcurie per milliliter of its true​ value, using a​ 95% confidence interval. Complete parts a through c. a. What is the confidence level desired by the​ researchers? The confidence level is 95. b. What is the sampling error desired by the​ researchers? The sampling error is 0.001. c. Compute the sample size necessary to obtain the desired estimate. The sample size is nothing. ​(Type a whole​ number.)

Given:

Given that the radius of the circle is 12 inches.

The angle of the shaded sector is 210°

We need to determine the area of the shaded sector.

Area of the shaded sector:

The area of the shaded sector can be determined using the formula,

[tex]Area =(\frac{\theta}{360}) \pi r^2[/tex]

Substituting [tex]\theta=210[/tex] and r = 12, we get;

[tex]Area =(\frac{210}{360}) (3.142)(12)^2[/tex]

Simplifying, we get;

[tex]Area =(\frac{210}{360}) (3.142)(144)[/tex]

[tex]Area =263.9 \ in^2[/tex]

Thus, the area of the shaded sector is 263.9 square inches.

Hence, Option c is the correct answer.

Answer:

-17

Step-by-step explanation:

-2+3(8+-13)

-2+3(8-13)

-2+3(-5)

-2-15

-17

[tex]\text{Solve:}\\\\-2+3(8+-13)\\\\\text{Solve the variables inside of the parenthesis}\\\\-2+3(-5)\\\\-2-15\\\\\boxed{-17}[/tex]

Answer:

x  = 16

Step-by-step explanation:

The sum of measures of these three angles of any triangle is equal to 180 °

54 + 90 + (2x + 4) = 180

144 + 2x + 4 = 180

148 + 2x = 180

2x = 180 - 148

2x = 32

[tex]x= \frac{32}{2} \\\\x=16[/tex]

Focus on triangle KLM. Since JKLM is a rectangle, angle KLM is right. Since the sum of the interior angles of every triangle is 180, we deduce that

[tex]\hat{K}+\hat{L}+\hat{M}=180[/tex]

Now plug the values for each angle and you have

[tex]54+90+(2x+4)=180[/tex]

Move 54 and 90 to the right side to get

[tex]2x+4=36 \iff 2x=32 \iff x=16[/tex]

Answer:

(a) [tex]X\sim N(0.10,\ 0.02^{2})[/tex].

(b) The probability that a randomly selected bag of marijuana sold on the street will have a THC content greater than 11% is 31%.

(c) The 60th percentile is 11%.

Step-by-step explanation:

The complete question is:

The average THC content of marijuana sold on the street is 10%. Suppose the THC content is normally distributed with standard deviation of 2%. Let X be the THC content for a randomly selected bag of marijuana that is sold on the street. Round all answers to two decimal places and give THC content in units of percent. For example, for a THC content of 11%, write "11" not 0.11.  

A. X ~ N( , )  

B. Find the probability that a randomly selected bag of marijuana sold on the street will have a THC content greater than 11%.

C.Find the 60th percentile for this distribution.

Solution:

The random variable X is defined as the THC content for a bag of marijuana that is sold on the street.

(a)

The random variable X is Normally distributed with mean, μ = 0.10 and standard deviation, σ = 0.02.

Thus, [tex]X\sim N(0.10,\ 0.02^{2})[/tex].

(b)

Compute the value of P (X > 0.11) as follows:

[tex]P(X>0.11)=P(\frac{X-\mu}{\sigma}>\frac{0.11-0.10}{0.02})\\=P(Z>0.50)\\=1-P(Z<0.50)\\=1-0.69146\\=0.30854\\\approx 0.31[/tex]

*Use a z-table.

Thus, the probability that a randomly selected bag of marijuana sold on the street will have a THC content greater than 11% is 31%.

(c)

The pth percentile is a data value such that at least p% of the data-set is less than or equal to this data value and at least (100 - p)% of the data-set are more than or equal to this data value.

Let x represent the 60th percentile value.

The, P (X < x) = 0.60.

⇒ P (Z < z) = 0.60

The value of z for this probability is,

z = 0.26.

Compute the value of x as follows:

[tex]z=\farc{x-\mu}{\sigma}\\0.26=\farc{x-0.10}{0.02}\\x=0.10+(0.26\times 0.02)\\x=0.1052\\x\approx 0.11[/tex]

Thus, the 60th percentile is 11%.

Answer:

A

Step-by-step explanation:

[tex]81[/tex] is [tex]9^2[/tex] which happens to be [tex](3^2)^2[/tex] or [tex]3^4[/tex].

So [tex]\sqrt[4]{81} =3[/tex]

this eliminates options C and D

It can't be option B as both b and c aren't 4th rooted.

Answer:

Ok. so the Place value is 70000 and the face value is 7. Face value of a digit in a number is the digit itself.

More clearly, face value of a digit always remains same irrespective of the position where it is located

Place value of a digit in a number is the digit multiplied by thousand or hundred or whatever place it is situated.

Hope this helped.

Step-by-step explanation:

Answer: A. $1300

Step-by-step explanation:

The initial value is $7000

The Salvage value is $500

The car will depreciate under a straight-line method, meaning it will lose $6500 value in 5 years.

6500 / 5 = 1300

Answer:

a) We need to conduct a hypothesis in order to check if the true average number of days in the past month that adults walked for the purpose of health or recreation is lower than 5.5 days (a;ternative hypothesis) , the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 5.5[/tex]  

Alternative hypothesis:[tex]\mu < 5.5[/tex]  

b) For this case since we want to determine if the true mean is lower than an specified value is a one tailed test

c) For this case the significance level is given [tex]\alpha=0.01[/tex] and we want to find the critical value, so we need to find a critical value on the normal standard distribution who accumulate 0.01 of the area on the right tail and if we use excel or a table we got:

[tex]z_{cric}= -2.326[/tex]

And the rejection zone would be: [tex] (\infty ,-2.326)[/tex]

And the region is on the figure attached

Step-by-step explanation:

Part a: State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true average number of days in the past month that adults walked for the purpose of health or recreation is lower than 5.5 days (a;ternative hypothesis) , the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 5.5[/tex]  

Alternative hypothesis:[tex]\mu < 5.5[/tex]  

Part b

For this case since we want to determine if the true mean is lower than an specified value is a one tailed test

Part c

For this case the significance level is given [tex]\alpha=0.01[/tex]and we want to find the critical value, so we need to find a critical value on the normal standard distribution who accumulate 0.01 of the area on the right tail and if we use excel or a table we got:

[tex]z_{cric}= -2.326[/tex]

And the rejection zone would be: [tex] (\infty ,-2.326)[/tex]

And the region is on the figure attached

Answer:

Correct option:

The 95% confidence interval is wider than the 90% confidence interval.

Step-by-step explanation:

The (1 - α)% confidence interval for the population proportion is:

[tex]CI=\hat p\pm z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

The width of this interval is:

[tex]W=UL-LL\\=2\times z_{\alpha/2}\times \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

The width of the interval is directly proportional to the critical value.

The critical value of a distribution is based on the confidence level.

Higher the confidence level, higher will be critical value.

The z-critical value for 95% and 90% confidence levels are:

[tex]90\%: z_{\alpha /2}=z_{0.05}=1.645\\95\%: z_{\alpha /2}=z_{0.025}=1.96[/tex]

*Use a z-table.

The critical value of z for 95% confidence level is higher than that of 90% confidence level.

So the width of the 95% confidence interval will be more than the 90% confidence interval.

Thus, the correct option is:

"The 95% confidence interval is wider than the 90% confidence interval."

Answer:

P(yellow, then orange WITHOUT replacement) = 0.0158

Step-by-step explanation:

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

This problem asks:

The probability of first getting a yellow marble, then an orange marble, without replacemtn.

Yellow marble first:

3 yellow marbles out of 3+6+1+8+2 = 20 marbles.

So P(a) = 3/20.

Then the orange marble:

Now the yellow marble is retired.

2 orange marbles out of 19.

So P(b) = 2/19

Probability:

P(yellow, then orange WITHOUT replacement) = P(a)P(b) = (3/20)*(2/19) = 0.0158

Answer:

Y=1/2x +5

Step-by-step explanation:

(0,5) (-10,0)

0-5/-10-0=1/2

(0,5) (x, y)

Y-5/x-10=1/2(crossmultiply)

2y-10=x

2y= x +10(divide all sides by 2)

Y=1/2x+5

Answer:

0.64 ± 0.1117 or

[tex]0.64\pm 1.645*\sqrt{\frac{0.64*(0.36)}{50}}[/tex]

Step-by-step explanation:

Sample size (n) = 50

Z-score for a 90% confidence interval (z) = 1.645

Proportion of students that read at least one book (p):

[tex]p=\frac{32}{50}=0.64[/tex]

The confidence interval is given by:

[tex]p\pm z*\sqrt{\frac{p*(1-p)}{n} }[/tex]

Applying the given data:

[tex]0.64\pm 1.645*\sqrt{\frac{0.64*(1-0.64)}{50}}\\ 0.64\pm 0.1117[/tex]

The confidence interval is 0.64 ± 0.1117

The 90 percent confidence interval for the proportion of all students in her district who read at least 1 book last month is [tex]0.64\pm 0.11178[/tex].

Given information:

From a random sample of 50 (n) students, Alma found that 32 students read at least 1 book last month.

Alma is estimating the proportion of students in her school district who, in the past month, read at least 1 book.

It is required to find the 90 percentage confidence interval for the proportion of all students in her district who read at least 1 book last month.

Now, from tha table, the z-score of 90 percent confidence level is,

[tex]z=1.645[/tex]

The probability p for the proportion of students who read atleast one book is,

[tex]p=\dfrac{32}{50}\\p=0.64[/tex]

So, the 90 percent confidence interval will be calculated as,

[tex]p\pm z\sqrt{\dfrac{p(1-p)}{n}}=0.64\pm 1.645\times \sqrt{\dfrac{0.64(1-0.64)}{50}}\\=0.64\pm 0.11178[/tex]

Therefore, the 90 percent confidence interval for the proportion of all students in her district who read at least 1 book last month is [tex]0.64\pm 0.11178[/tex].

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Answer:

Using either method, we obtain:  [tex]t^\frac{3}{8}[/tex]

Step-by-step explanation:

a) By evaluating the integral:

 [tex]\frac{d}{dt} \int\limits^t_0 {\sqrt[8]{u^3} } \, du[/tex]

The integral itself can be evaluated by writing the root and exponent of the variable u as:   [tex]\sqrt[8]{u^3} =u^{\frac{3}{8}[/tex]

Then, an antiderivative of this is: [tex]\frac{8}{11} u^\frac{3+8}{8} =\frac{8}{11} u^\frac{11}{8}[/tex]

which evaluated between the limits of integration gives:

[tex]\frac{8}{11} t^\frac{11}{8}-\frac{8}{11} 0^\frac{11}{8}=\frac{8}{11} t^\frac{11}{8}[/tex]

and now the derivative of this expression with respect to "t" is:

[tex]\frac{d}{dt} (\frac{8}{11} t^\frac{11}{8})=\frac{8}{11}\,*\,\frac{11}{8}\,t^\frac{3}{8}=t^\frac{3}{8}[/tex]

b) by differentiating the integral directly: We use Part 1 of the Fundamental Theorem of Calculus which states:

"If f is continuous on [a,b] then

[tex]g(x)=\int\limits^x_a {f(t)} \, dt[/tex]

is continuous on [a,b], differentiable on (a,b) and  [tex]g'(x)=f(x)[/tex]

Since this this function [tex]u^{\frac{3}{8}[/tex] is continuous starting at zero, and differentiable on values larger than zero, then we can apply the theorem. That means:

[tex]\frac{d}{dt} \int\limits^t_0 {u^\frac{3}{8} } } \, du=t^\frac{3}{8}[/tex]

Answer:

w= -0.163265306122449

Step-by-step explanation:

49w = -8

divide both sides by 49

w= -8/49 = -0.163265306122449

check answer by multiplying -0.163265306122449  by 49

49 * -0.163265306122449 = -8

Answer:

w = -8/49

Step-by-step explanation:

49w = −8

Divide each side by 49

49w/49 = -8/49

w = -8/49

Answer:

: The percentage of newborn infants who are exclusively breastfed at the time of hospital discharge.

Data Source: Rhode Island Department of Health, Center for Health and Data Analysis, Newborn Developmental Risk Screening Program Database and Maternal and Child Health Database.

Footnotes: The year indicated is the mid-year of a five-year period of data.

Step-by-step explanation:

Cramér's V value of .22 in the study indicates a small to medium association between whether the mother ate breakfast and the gender of her baby.

In the study, Cramér's V measuress the strength of association between two nominal variables, in this case, the mother's breakfast-eating habits and the gender of her baby. The value of Cramer’s V ranges from 0 (indicating no association) to 1 (indicating a perfect association)). Given the value of Cramér's V at .22 for this study, there is a small to medium association between whether a mother ate breakfast and the gender of her baby. It does not mean that 22% of the variation in baby gender can be explained by the mother's breakfast-eating habits. Rather, it quantifies the extent of the association between these two factors.

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Answer:

With n = 50, both conditions(np > 5, n(1-p) > 5) are satisfied, so a random sample of 50 cars would have been large enough.

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to the normal distribution if:

np > 5, n(1-p) > 5

70% of the drivers traveling on a major interstate highway exceed the speed limit.

So p = 0.7.

Would a random sample of 50 cars have been large enough and why?

With n = 50

np = 50*0.7 = 35 > 5

n(1-p) = 50*0.3 = 15 > 5

With n = 50, both conditions(np > 5, n(1-p) > 5) are satisfied, so a random sample of 50 cars would have been large enough.

Answer:

5040$

Step-by-step explanation:

28000 x .18 = 5040

We are given that if Tiffany closes any car on a car she gets 18% as commission. For a car that cost $28,000 her commission will be  $5,040

Given data

Commission = 18%

Cost of car  =  $ 28,000.00

Let us find 18% of  $28,000.00

Commission = 18/100* 28,000.00

Commission = 0.18* 28,000.00

Commission = $5,040

Hence her commission will be: $5,040

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Answer:

9 cups

Step-by-step explanation:

3/4 cups of sugar for 14 cookies

There are 168÷14 = 12 set of 14 cookies in 168

So we need 12 times 3/4 sugar

12 × 3/4 = 9 cups of sugar needed

To find the probability that the archer gets exactly 4 bull's-eyes out of 10 shots, use the binomial probability formula. The probability is approximately 0.221.

To find the probability that the archer gets exactly 4 bull's-eyes out of 10 shots, we can use the binomial probability formula. The formula is:

P(x) = nCx * p^x * (1-p)^(n-x)

Where:

In this case, n = 10, x = 4, and p = 0.53 (since the archer hits the bull's-eye 53% of the time).

Plugging these values into the formula:

P(4) = 10C4 * (0.53)^4 * (1-0.53)^(10-4)

Calculating this gives us P(4) ≈ 0.221

Final answer:

To find the probability of exactly 4 bull's-eyes out of 10 shots for an archer with a 53% success rate, use the binomial probability formula, resulting in approximately a 20.6% chance.

Explanation:

The probability that an archer who hits the bull's-eye 53% of the time getting exactly 4 bull's-eyes out of 10 shots can be calculated using the binomial probability formula: P(X = k) = nCk × p^k × (1 - p)^(n - k), where:

n is the number of trials (10 arrows)

k is the number of successes (4 bull's-eyes)

p is the probability of success on a single trial (0.53)

nCk is the number of combinations of n things taken k at a time

Applying these values:

Calculate the number of combinations for getting 4 successes out of 10 trials:

10C4 = 210.

Calculate the probability of getting exactly 4 bull's-eyes:

P(X = 4) = 210 × (0.53)^4 × (0.47)^6.

Compute the result:

P(X = 4) ≈ 0.206 (rounded to three decimal places).

The archer has approximately a 20.6% chance of hitting exactly 4 bull's-eyes out of 10 shots.

Answer:

a) P(Job offer) = 0.65

b) P(S|J) = 0.862

P(M|J) = 0.143

P(W|J) = 0.0179

c) P(S|J') = 0.400

P(M|J') = 0.343

P(W|J') = 0.257

Step-by-step explanation:

- Let the event that she gets a job be J

- Let the event that she does not get the job be J'

- Let the event that she receives a strong recommendation be S

- Let the event that she receives a moderate recommendation be M

- Let the event that she receives a weak recommendation be W.

Given in the question,

P(J|S) = 80% = 0.8

P(J|M) = 40% = 0.4

P(J|W) = 10% = 0.1

P(S) = 0.70

P(M) = 0.20

P(W) = 0.10

a) How certain is she that she will receive the new job offer?

P(J) = P(J n S) + P(J n M) + P(J n W) (since S, M and W are all of the possible outcomes that lead to a job)

But note that the conditional probability, P(A|B) is given mathematically as,

P(A|B) = P(A n B) ÷ P(B)

P(A n B) is then given as

P(A n B) = P(A|B) × P(B)

So,

P(J n S) = P(J|S) × P(S) = 0.80 × 0.70 = 0.56

P(J n M) = P(J|M) × P(M) = 0.40 × 0.20 = 0.08

P(J n W) = P(J|W) × P(W) = 0.10 × 0.10 = 0.01

P(J) = P(J n S) + P(J n M) + P(J n W)

P(J) = 0.56 + 0.08 + 0.01 = 0.65

b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation?

This probability = P(S|J)

P(S|J) = P(J n S) ÷ P(J) = 0.56 ÷ 0.65 = 0.862

i. a moderate recommendation?

P(M|J) = P(J n M) ÷ P(J) = 0.08 ÷ 0.65 = 0.143

ii. a weak recommendation?

P(W|J) = P(J n W) ÷ P(J) = 0.01 ÷ 0.65 = 0.0179

c) Probability that she doesn't get job offer, given she got a strong recommendation = P(J'|S)

P(J'|S) = 1 - P(J|S) = 1 - 0.80 = 0.20

Probability that she doesn't get job offer, given she got a moderate recommendation = P(J'|M)

P(J'|M) = 1 - P(J|M) = 1 - 0.40 = 0.60

Probability that she doesn't get job offer, given she got a weak recommendation = P(J'|S)

P(J'|W) = 1 - P(J|W) = 1 - 0.10 = 0.90

Total probability that she doesn't get job offer

P(J') = P(J' n S) + P(J' n M) + P(J' n W)

P(J' n S) = P(J'|S) × P(S) = 0.20 × 0.70 = 0.14

P(J' n M) = P(J'|M) × P(M) = 0.60 × 0.20 = 0.12

P(J' n W) = P(J'|W) × P(W) = 0.90 × 0.10 = 0.09

Total probability that she doesn't get job offer

P(J') = P(J' n S) + P(J' n M) + P(J' n W)

= 0.14 + 0.12 + 0.09 = 0.35

Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation?

This probability = P(S|J')

P(S|J') = P(J' n S) ÷ P(J') = 0.14 ÷ 0.35 = 0.400

i. a moderate recommendation?

P(M|J') = P(J' n M) ÷ P(J') = 0.12 ÷ 0.35 = 0.343

ii. a weak recommendation?

P(W|J') = P(J' n W) ÷ P(J') = 0.09 ÷ 0.35 = 0.257

Hope this Helps!!!

She has 62% certainty of getting the job. If she gets it, there are high chances (~90%) that the recommendation was strong. In case of rejection, she's slightly more likely to have received a strong recommendation than a moderate or weak one.

The subject of this problem involves the understanding of conditional probabilities and the principle of total probability. Let's denote the events as follows:

a. The total probability that she will receive the job offer is calculated as follows:

P(B)=P(A1)P(B|A1)+P(A2)P(B|A2)+P(A3)P(B|A3) = 0.7*0.8+0.2*0.4+0.1*0.1=0.62, so she is 62% certain that she will receive the job offer.

b. If she does receive the offer, the probabilities that she got a strong, moderate and weak recommendation are calculated using Bayes' Theorem:

c. If she doesn't get the job, the probabilities are calculated similarly:

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Answer:

95 percent confidence interval for μ is determined by

(60.688 ,  69.312)

a) be the same

Step-by-step explanation:

Step (i):-

Given data a random sample of 121 automobiles traveling on an interstate showed an average speed of 65 mph.

The sample size is n= 121

The mean of the sample x⁻ = 65mph.

The standard deviation of the population is 22 mph

σ = 22mph

Step (ii) :-

Given  The 95 percent confidence interval for μ is determined as

(61.08, 68.92)

we are to reduce the sample size to 100 (other factors remain unchanged) so

given The sample size is n= 100

The mean of the sample x⁻ = 65mph.

The standard deviation of the population is 22 mph

σ = 22mph

Step (iii):-

95 percent confidence interval for μ is determined by

[tex](x^{-} - 1.96\frac{S.D}{\sqrt{n} } , x^{-} + 1.96 \frac{S.D}{\sqrt{n} } )[/tex]

[tex](65 - 1.96\frac{22}{\sqrt{100} } , 65 + 1.96 \frac{22}{\sqrt{100} } )[/tex]

(65 - 4.312 ,65 +4.312)

(60.688 ,  69.312) ≅(61 ,69)

This is similar to (61.08, 68.92) ≅(61 ,69)

Conclusion:-

it become same as (61.08, 68.92)

In the given scenario, the 95 percent confidence interval for μ would become wider if the sample size is reduced from 121 to 100. This is due to the increased uncertainty in estimation as a result of a smaller sample size.

In statistics, a confidence interval is a range in which a population parameter is estimated to lie. The calculation of a confidence interval involves several factors, one of which is the sample size. The width of the confidence interval decreases as the sample size increases, assuming that all other factors remain the same.

In this case, the sample size is proposed to decrease from 121 to 100. As a result, the confidence interval for μ would become wider. This is because as the sample size decreases, the standard error increases, which in turn widens the confidence interval, allowing for more uncertainty in estimations.

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Answer:

C: The area of ABC is 1 square unit greater than the

area of ADEF

Step-by-step explanation:

Due to the area of triangle ABC and triangle DEF, the ABC triangle has dimensions that are 1 square unit greater that the area of ADEF. This is because the lines have different lengths. Good Luck on the Rest of Your Questions!!

Answer: C

Step-by-step explanation:

Answer:

2/16

Step-by-step explanation:

1/8 x 2 = 2/16

Both 2 and 16 divide by 8, so one can know this is correct.

Also, both 8 and 16 are divisible by 8, so one can see this is the common denominator.

Answer:

The monthly cost is $690.17.

Step-by-step explanation:

Each semester costs $4100.

In each semester, there are $41 spent on textbooks.

In a year:

2*4100 + 2*41 = $8282

Monthly cost

A year has 12 months. So

8282/12 = 690.17.

The monthly cost is $690.17.

Let [tex]f(x,y,z)=x+y+z-8e^{xyz}[/tex]. The tangent plane to the surface at (0, 0, 8) is

[tex]\nabla f(0,0,8)\cdot(x,y,z-8)=0[/tex]

The gradient is

[tex]\nabla f(x,y,z)=\left(1-8yze^{xyz},1-8xze^{xyz},1-8xye^{xyz}\right)[/tex]

so the tangent plane's equation is

[tex](1,1,1)\cdot(x,y,z-8)=0\implies x+y+(z-8)=0\implies x+y+z=8[/tex]

The normal vector to the plane at (0, 0, 8) is the same as the gradient of the surface at this point, (1, 1, 1). We can get all points along the line containing this vector by scaling the vector by [tex]t[/tex], then ensure it passes through (0, 0, 8) by translating the line so that it does. Then the line has parametric equation

[tex](1,1,1)t+(0,0,8)=(t,t,t+8)[/tex]

or [tex]x(t)=t[/tex], [tex]y(t)=t[/tex], and [tex]z(t)=t+8[/tex].

(See the attached plot; the given surface is orange, (0, 0, 8) is the black point, the tangent plane is blue, and the red line is the normal at this point)

Final answer:

This detailed answer explains how to find the equation of the tangent plane and the normal line to a given surface at a specified point. The parametric equations of the normal line are [tex]\( (x(t), y(t), z(t)) = (t, t, 8 + t) \).[/tex]

Explanation:

To find the equations of the tangent plane and the normal line to the given surface x + y + z = 8exyz at the specified point 0, 0, 8) we first need to find the partial derivatives of x + y + z - 8exyz with respect to x, y, and z.

Given the surface equation:

[tex]\[ x + y + z = 8exyz \][/tex]

Let's denote the function as [tex]\(F(x, y, z) = x + y + z - 8exyz\).[/tex]

(a) To find the equation of the tangent plane, we use the gradient vector of F at the point 0, 0, 8 as the normal vector to the tangent plane.

The gradient vector of F is given by:

[tex]\[ \nabla F = \left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right) \][/tex]

Let's calculate the partial derivatives:

[tex]\[ \frac{\partial F}{\partial x} = 1 - 8eyz \][/tex]

[tex]\[ \frac{\partial F}{\partial y} = 1 - 8exz \][/tex]

[tex]\[ \frac{\partial F}{\partial z} = 1 - 8exy \][/tex]

Now, evaluate these partial derivatives at the point (0, 0, 8):

[tex]\[ \frac{\partial F}{\partial x} = 1 - 8e(0)(0) = 1 \][/tex]

[tex]\[ \frac{\partial F}{\partial y} = 1 - 8e(0)(0) = 1 \][/tex]

[tex]\[ \frac{\partial F}{\partial z} = 1 - 8e(0)(0) = 1 \][/tex]

So, the gradient vector of [tex]\(F\) at \((0, 0, 8)\) is \( \nabla F = (1, 1, 1) \).[/tex]

Therefore, the equation of the tangent plane is:

[tex]\[ x + y + z = 8 \][/tex]

(b) To find the equation of the normal line, we use the same gradient vector [tex]\( \nabla F = (1, 1, 1) \)[/tex] as the direction vector for the line.

Given the point \((0, 0, 8)\) on the surface, the parametric equations of the normal line are:

[tex]\[ x(t) = 0 + t(1) = t \][/tex]

[tex]\[ y(t) = 0 + t(1) = t \][/tex]

[tex]\[ z(t) = 8 + t(1) = 8 + t \][/tex]

So, the parametric equations of the normal line are [tex]\( (x(t), y(t), z(t)) = (t, t, 8 + t) \).[/tex]

Answer:

13 cards

Step-by-step explanation:

One deck has 52 cards.

2 decks have 2 * 52 cards, or 104 cards.

104 cards are divided among 8 players.

104/8 = 13

Answer: 13 cards

Answer:

a) 88.54% probability of a diameter between 3.8 in and 4.3 in

b) 25.14% probability of a diameter smaller than 3.9in

c) 90.82% probability of a diameter larger than 4.2 in

Step-by-step explanation:

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.

In this problem, we have that:

[tex]\mu = 4, \sigma = 0.15[/tex]

(a) a diameter between 3.8 in and 4.3 in,

This is the pvalue of Z when X = 4.3 subtracted by the pvalue of Z when X = 3.8.

X = 4.3

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

[tex]Z = \frac{4.3 - 4}{0.15}[/tex]

[tex]Z = 2[/tex]

[tex]Z = 2[/tex] has a pvalue of 0.9772

X = 3.8

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

[tex]Z = \frac{3.8 - 4}{0.15}[/tex]

[tex]Z = -1.33[/tex]

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

0.9772 - 0.0918 = 0.8854

88.54% probability of a diameter between 3.8 in and 4.3 in

(b) a diameter smaller than 3 9 in,

This is the pvalue of Z when X = 3.9. So

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

[tex]Z = \frac{3.9 - 4}{0.15}[/tex]

[tex]Z = -0.67[/tex]

[tex]Z = -0.67[/tex] has a pvalue of 0.2514

25.14% probability of a diameter smaller than 3.9in

(c) a diameter larger than 4.2 in

This is 1 subtracted by the pvalue of Z when X = 4.2. So

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

[tex]Z = \frac{4.2 - 4}{0.15}[/tex]

[tex]Z = 1.33[/tex]

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

90.82% probability of a diameter larger than 4.2 in

Answer: 0.8

Step-by-step explanation:

There are 5 possible equally sized sections meaning there is a 0.2 or 20% of landing on each section. We can subtract the probability of getting a squirrel from the whole thing (1 as probability always equals 1) to determine the probability of not squirrel.

P(squirrel) = 0.2

1 - P(squirrel) = P(not a squirrel)

1 - 0.2 = 0.8

Answer: the anserw is 0.8

Step-by-step explanation:

Answer:

0.0062 = 0.62% probability that a randomly selected truck from the fleet will have to be inspected

Step-by-step explanation:

Z-score

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.

What is the probability that a randomly selected truck from the fleet will have to be inspected?

Values of Z above 2.5. So this probability is 1 subtracted by the pvalue of Z = 2.5.

Z = 2.5 has a pvalue of 0.9938

1 - 0.9938 = 0.0062

0.0062 = 0.62% probability that a randomly selected truck from the fleet will have to be inspected

Answer:

For this case we want this probability:

[tex] P(X > \mu +2.5 \sigma) [/tex]

And we can use the z score formula given by:

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

And replacing we got:

[tex] z = \frac{\mu +2.5 \sigma -\mu}{\sigma}= 2.5[/tex]

We want this probability:

[tex] P(Z>2.5) = 1-P(Z<2.5)[/tex]

And using the normal standard table or excel we got:

[tex] P(Z>2.5) = 1-P(Z<2.5)= 1-0.9938= 0.0062[/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 mileage of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu,\sigma)[/tex]  

For this case we want this probability:

[tex] P(X > \mu +2.5 \sigma) [/tex]

And we can use the z score formula given by:

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

And replacing we got:

[tex] z = \frac{\mu +2.5 \sigma -\mu}{\sigma}= 2.5[/tex]

We want this probability:

[tex] P(Z>2.5) = 1-P(Z<2.5)[/tex]

And using the normal standard table or excel we got:

[tex] P(Z>2.5) = 1-P(Z<2.5)= 1-0.9938= 0.0062[/tex]