Find the direction cosines and direction angles of the given vector. (Round the direction angles to two decimal places.) a = 5, 9, 3 cos(α) = cos(β) = cos(γ) = α = ° β = ° γ = °

Answer:

P ( P U T ) = 0.6481

Dependent events

P ( P / T ) = (359/756) = 0.4749

Step-by-step explanation:

Given:

- Pink : P

- Teenagers: T

Find:

- P( T u P )

- Are the events T and R dependent, independent, mutually exclusive, or complementary

- If the events are independent, then we can compare P ( P / T ).

Solution:

a)

For first part we will determine the total outcome for T and P:

                   Total outcomes P or T =  359 + 252 - 121 = 490

                                             (P U T) = (P) + (T) - (P n T)

The total number of possible outcomes are = 756

Hence, P ( P U T ) = 490 / 756 = 0.6481

b)

We are to investigate how the two events are related to one another:

- Check for dependent events:

              P ( T n P ) = P( T ) * P ( P / T )

              (121 / 756 ) = (252/756) * ( 121 / 252 )

              (121 / 756) = (121/756)  ....... Hence, events are dependent

- Check for independent events:

              P ( T n P ) = P( T ) * P ( P )

              (121 / 756 ) = (252/756) * ( 359 / 756 )

              (121 / 756) =/ (359/756)  ....... Hence, events are not independent

- Check for mutually exclusive events:

              P ( T U P ) = P( T )  + P ( P )

              (490 / 756 ) = (252/756) + ( 359 / 756 )

              (121 / 756) =/ (611/756)  ... Hence, events are not mutually exclusive

Hence, the two events are dependent on each other.

c)

If the events are said to be independent then the event:

               P ( P / T ) = P (P)

               P ( P / T ) = (359/756) = 0.4749

The probability that a randomly selected survey respondent is a Teenager OR selects Pink (Strawberry) as their favorite Starburst color (flavor) can be calculated by adding the individual probabilities of these two events occurring.

The probability of a respondent being a Teenager is 85/756, and the probability of selecting Pink (Strawberry) as their favorite flavor is 359/756. So, the probability of either event happening is (85/756) + (359/756) = 0.4749.

Based on these calculations, the events "a randomly selected person is a Teenager" and "a randomly selected person selects Pink (Strawberry) as their favorite Starburst color (flavor)" are NOT mutually exclusive events because they can both happen.

They are also NOT complementary events because they don't cover all possible outcomes.

Whether they are independent or dependent events can be determined by comparing P(Pink | Teenager) with the probability of Pink (Strawberry) regardless of age group, P(Pink).

If P(Pink | Teenager) is equal to P(Pink), they are independent events; otherwise, they are dependent.

In this case, P(Pink | Teenager) is 114/85, and P(Pink) is 359/756. Comparing these probabilities, we find that P(Pink | Teenager) is not equal to P(Pink), so the events are dependent.

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To lift a 1400-kg satellite to an altitude of 2×10⁶ meters above the Earth's surface, the work required is approximately 2.079×10¹¹ Joules. This is calculated based on changes in gravitational potential energy using given values for mass, radius, and the gravitational constant.

To calculate the work required to lift a 1400-kg satellite to an altitude of 2×10⁶ meters (2000 km) above the Earth's surface, we need to consider the gravitational potential energy change.

Given:

The total distance from the center of the Earth to the satellite is:

The work required is equal to the change in gravitational potential energy:

The gravitational potential energy at a distance r from the Earth’s center is given by:

The work done (W) to move the satellite from the Earth's surface to this altitude is the difference in potential energy:

Substitute the given values:

Calculate the values inside the brackets first:

Now, multiply:

The work required to lift the satellite to the desired altitude is approximately 2.079×10¹¹ Joules.

Answer:

Kamala had the best GPA compared to other students at his school, since his GPA is 2 standard deviations above his school's mean.

Step-by-step explanation:

The z-score measures how many standard deviation a score X is above or below the mean.

it is given by the following formula:

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

In which

[tex]\mu[/tex] is the mean, [tex]\sigma[/tex] is the standard deviation.

In this problem:

The student with the best GPA compared to other students at his school is the one with the higher z-score, that is, the one whose grade is the most standard deviations above the mean for his school

Thuy

Student GPA| School Average GPA| School Standard Deviation

Thuy 2.7| 3.2| 0.8

So [tex]X = 2.7, \mu = 3.2, \sigma = 0.8[/tex]

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

[tex]Z = \frac{2.7 - 3.2}{0.8}[/tex]

[tex]Z = -0.625[/tex]

Thuy's score is -0.625 standard deviations below his school mean.

Vichet

Student GPA| School Average GPA| School Standard Deviation

Vichet 88| 75| 20

So [tex]X = 88, \mu = 75, \sigma = 20[/tex]

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

[tex]Z = \frac{88 - 75}{20}[/tex]

[tex]Z = 0.65[/tex]

Vichet's score is 0.65 standard deviation above his school mean

Kamala

Student GPA| School Average GPA| School Standard Deviation

Kamala 8.8| 8| 0.4

So [tex]X = 8.8, \mu = 8, \sigma = 0.4[/tex]

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

[tex]Z = \frac{8.8 - 8}{0.4}[/tex]

[tex]Z = 2[/tex]

Kamala's score is 2 standard deviations above his school mean.

Kamala has the higher z-score, so he had the best GPA when compared to other students at his school.

The correct answer is:

Kamala had the best GPA compared to other students at his school, since his GPA is 2 standard deviations above his school's mean.

Final answer:

Kamala had the best GPA relative to her school's average, with a z-score of 2.00, indicating her GPA is 2 standard deviations above the average.

Explanation:

To determine which student had the best GPA compared to other students at their school, we need to calculate how many standard deviations each student's GPA is away from their school's average GPA. This can be done using the formula:

z = (x - μ) / σ

Where z is the number of standard deviations, x is the student's GPA, μ (mu) is the school's average GPA, and σ (sigma) is the school's standard deviation.

Kamala has the highest z-score, which means her GPA is the furthest above her school's average when compared to Thuy and Vichet. Therefore, Kamala had the best GPA relative to her peers at her school.

The variance of the number of children in a randomly chosen family is approximately 9.4148.

To determine the variance of the number of children in a randomly chosen family, we can use the properties of the Poisson distribution, assuming that the number of children in a family follows a Poisson distribution with a mean of 2.3.

The Poisson distribution has a variance equal to its mean, so if the mean number of children per family is 2.3, then the variance is also 2.3.

However, we can also calculate the variance directly from the given information.

Let ( X ) be the random variable representing the number of children in a family.

Given:

- The mean number of children in a family, [tex]\( \mu = 2.3 \)[/tex]

- Each child has, on average, 1.6 siblings.

To find the variance, we use the formula:

\[ \text{Variance} = \text{E}(X^2) - [\text{E}(X)]^2 \]

[tex]First, we find \( \text{E}(X^2) \), the expected value of \( X^2 \):\[ \text{E}(X^2) = \text{Variance} + [\text{E}(X)]^2 \]\[ \text{E}(X^2) = 1.6^2 \times 2.3 + (2.3)^2 \]\[ \text{E}(X^2) = 4.096 \times 2.3 + 5.29 \]\[ \text{E}(X^2) = 9.4148 + 5.29 \]\[ \text{E}(X^2) = 14.7048 \][/tex]

Then, using the formula for variance:

[tex]\[ \text{Variance} = \text{E}(X^2) - [\text{E}(X)]^2 \]\[ \text{Variance} = 14.7048 - (2.3)^2 \]\[ \text{Variance} = 14.7048 - 5.29 \]\[ \text{Variance} = 9.4148 \][/tex]

So, the variance of the number of children in a randomly chosen family is approximately 9.4148.

Question:

In a town there are on average 2.3 children in a family and a randomly chosen child has on average 1.6 siblings. Determine the variance of the number of children in a randomly chosen family.

Answer:

a) 8 units

b) 3 units

c) 4 units

d) [tex]4\sqrt{5}\text{ units}[/tex]

e) [tex]\sqrt{73}\text{ units}[/tex]

f) 5 units

Step-by-step explanation:

We are given the following:

Point (3, −4, 8)

We have to find the distance of the point from the following:

Distance formula:

[tex](x_1,y_1,z_1),(x_2,y_2,z_2)\\\\d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}[/tex]

(a) the xy-plane

We have to find the distance from (3, −4, 8) to (3, −4, 0)

[tex]d = \sqrt{(3-3)^2 + (-4+4)^2 + (0-8)^2} = 8\text{ units}[/tex]

(b) the yz-plane

We have to find the distance from (3, −4, 8) to (0, −4, 8)

[tex]d = \sqrt{(0-3)^2 + (-4+4)^2 + (8-8)^2} = 3\text{ units}[/tex]

(c) the xz-plane

We have to find the distance from (3, −4, 8) to (3, 0, 8)

[tex]d = \sqrt{(3-3)^2 + (0+4)^2 + (8-8)^2} = 4\text{ units}[/tex]

(d) the x-axis

We have to find the distance from (3, −4, 8) to (3, 0, 0)

[tex]d = \sqrt{(3-3)^2 + (0+4)^2 + (0-8)^2} = \sqrt{80} = 4\sqrt{5}\text{ units}[/tex]

(e) the y-axis (0,−4,0)

We have to find the distance from (3, −4, 8) to (0, -4, 0)

[tex]d = \sqrt{(0-3)^2 + (-4+4)^2 + (0-8)^2} = \sqrt{73}\text{ units}[/tex]

(f) the z-axis (0,0,8)

We have to find the distance from (3, −4, 8) to (0, 0, 8)

[tex]d = \sqrt{(0-3)^2 + (0+4)^2 + (8-8)^2} = \sqrt{25} = 5\text{ units}[/tex]

At least 96.00% of the data in any data set lies within three standard deviations of the mean.

The Empirical Rule states that in a bell-shaped and symmetric distribution, approximately 68 percent of the data lies within one standard deviation of the mean, 95 percent lies within two standard deviations of the mean, and more than 99 percent lies within three standard deviations of the mean. Therefore, at least 96.00​% of the data lies within three standard deviations of the mean.

Answer:

a) quantitative discrete data

b)  categorical variable

c)  categorical variable

d) quantitative continuous variable

e) quantitative discrete variable

f)  categorical variable

Step-by-step explanation:

a) The number of days (to the nearest day) the dog has been at Huron Valley Humane Society

Since days will always have whole number numerical values it is a quantitative discrete data.

b) Whether or not the dog has a microchip

This will be answered with a yes or a no. Thus, it is a categorical variable. It does not have any numerical value.

c) The breed of the dog

This is a categorical variable. It does not have any numerical value. It will have non-parametric values.

d) How much the dog weighs (in pounds)

Since weight is always measured and not counted. Also weight can take decimal values, thus it is quantitative continuous variable.

e) The amount of food (in cups) the dog eats

Since this will have whole number values. Also, number of cups will be counted. Thus, it is a quantitative discrete variable.

f) The number of people who have taken the dog out for a walk

Since this will have whole number values. Also, number of people will be counted. Thus, it is a quantitative discrete variable.

g) Whether you decide to adopt the dog

This will be answered with a yes or a no. Thus, it is a categorical variable. It does not have any numerical value.

a,f are quantitave discrete b,c,g are categorical d,e,are quantitative continous.

a.he number of days (to the nearest day) the dog has been at Huron Valley Humane Society: This is quantitative discrete data because the number of days is counted in whole numbers.
b.Whether or not the dog has a microchip: This is categorical data because it categorizes the dog as either having or not having a microchip.
c.The breed of the dog: This is categorical data because breeds represent different categories.
d.How much the dog weighs (in pounds): This is quantitative continuous data because weight can be measured to a very fine degree.
e.The amount of food (in cups) the dog eats: This is quantitative continuous data because the amount can be measured to any level of precision.
f.The number of people who have taken the dog out for a walk: This is quantitative discrete data because it involves counting distinct individuals.
g.Whether you decide to adopt the dog: This is categorical data because it categorizes your decision into adopt or not adopt.

Answer:

The answer to your question is below

Step-by-step explanation:

Inequality 1

                       -4x + 1 > 13

                       -4x > 13 - 1

                      -4x > 12

                         x < 12/-4

                         x < -3

Inequality 2

                      4(x + 2) ≤ 4

                      4x + 8 ≤ 4

                      4x ≤ 4 - 8

                       4x ≤ - 4

                         x ≤ -4/4

                         x ≤ -1

Interval notation (-∞, -3)

See the graph below

The solution to these inequalities in where both intervals crosses

Answer:

Subjective probability

Step-by-step explanation:

At first we should know the following:

Classical probability ⇒ when there are n equally likely outcomes.

Subjective probability ⇒ is based on whatever information is available.

Empirical probability ⇒ when the number of times the event happens is divided by the number of observations.

So, according to the previous definitions:

This person has a 75% chance of a full recovery

There is no equally likely outcomes, and the percentage of full recovery is based on the information available about the person and also it is based on educated guess.

So, this is Subjective probability

Answer:

P1: An ant releases a chemical when it dies, and its fellows then carry it away to the compost heap.

P2: A healthy ant painted with the death chemical will be dragged to the funeral heap again and again.

C: Apparently the communication is highly effective

Step-by-step explanation:

Premises are statements that are always assumed to be truth or statements that have been given and made to be true.

Premises will be labelled as P1, P2, P3..........

Conclusions are derived statements from premises. Its truthfulness is derived from the truthfulness of premises. Meaning that, when the premises is true, the conclusion is also true.

Conclusions will be labelled as C1, C2, C3........

In the question above, it's noted that the conclusion starts at "Apparently the" while the premises are statements before "Apparently the"

The argument's premises are as follows:

1. An ant releases a chemical when it dies.

2. Its fellow ants then carry the dead ant to the compost heap.

3. A healthy ant painted with the death chemical will be dragged to the funeral heap again and again.

The conclusion is that the communication among ants when one of them dies is highly effective.

In the given passage, there is an implicit argument about the effectiveness of communication among ants when one of them dies. Let's identify the premises and the conclusion:

Premises:

1. An ant releases a chemical when it dies.

2. Its fellow ants then carry the dead ant to the compost heap.

3. A healthy ant painted with the death chemical will be dragged to the funeral heap again and again.

Conclusion:

4. Apparently, the communication among ants when one of them dies is highly effective.

These premises and conclusion can be organized as follows:

Premise 1: An ant releases a chemical when it dies.

Premise 2: Its fellow ants then carry the dead ant to the compost heap.

Premise 3: A healthy ant painted with the death chemical will be dragged to the funeral heap again and again.

Conclusion: Apparently, the communication among ants when one of them dies is highly effective.

In this argument, premises 1, 2, and 3 provide evidence or observations related to ant behavior when a member of their colony dies. The conclusion, stated in premise 4, is drawn from these observations and suggests that the communication system among ants regarding the death of a fellow ant is effective.

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Answer:

a) Magnitude = 10.72

b) = 0.32 and 0.400

c) Drug B is closest to having the desired effect.

Step-by-step explanation:

The knowledge of vector projection and dot product is applied as shown in the step by step explanation from the attached file.

The probability that a randomly chosen household has at least two televisions can be calculated by dividing the number of households with at least two televisions by the total number of households.

To find the probability that a randomly chosen household has at least two televisions, we need to use the data from the marketing survey. Let's assume there are 'n' households in total. The probability of a household having at least two televisions can be calculated by dividing the number of households with at least two televisions by the total number of households, which is 'n'.

Let's say there are 'x' households with at least two televisions. The probability can be expressed as:

P(at least 2 TVs) = x/n

For example, if there are 100 households in total and 30 of them have at least two televisions, then the probability would be P(at least 2 TVs) = 30/100 = 0.3, which is 30%.

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Answer:

1 1/4 cups

Step-by-step explanation:

Their are 8 ounces in one cup and their are 2 ounces in 1/4's of a cup

CUP        FL. OZ

1                8        

3/4            6

2/3            5

1/2             4

1/3             3

1/4            2

1/8            1

1/16         0.5

Answer:

you go up by .75 each time per pound so it would be 5 pounds.

hope this helps!!

Answer:

5

Step-by-step explanation:

Or (A)

Answer:

P(A∪B)=17/20 or 0.85

P(A∪B')=2/5 or 0.4

P(A'∪B')=4/5 or 0.8

Step-by-step explanation:

There are four font colors so each color had equal chance and thus,

P(A)=1/4

There are 5 font sizes and so not the smallest fonts are 4.Thus,

P(B)=4/5

P(A∪B)=P(A)+P(B)-P(A∩B)

The design is generated randomly so event A and event B are independent.

P(A∩B)=P(A)*P(B)

P(A∩B)=1/4(4/5)=1/5

P(A∪B)=P(A)+P(B)-P(A∩B)

P(A∪B)=1/4+4/5-1/5=1/4+3/5

P(A∪B)=17/20 or 0.85

P(A∪B')=P(A)+P(B')-P(A∩B')

P(B')=1-P(B)=1-4/5=1/5

P(A∩B')=P(A)*P(B')=1/4*1/5=1/20

P(A∪B')=P(A)+P(B')-P(A∩B')

P(A∪B')=1/4+1/5-1/20=9/20-1/20=8/20

P(A∪B')=2/5 or 0.4

P(A'∪B')=P(A∩B)'

P(A'∪B')=1-P(A∩B)

P(A'∪B')=1-1/5=4/5

P(A'∪B')=4/5 or 0.8

The probability of event A or B occurring can be calculated using the addition rule. The probability of A occurring is 1/4, the probability of B occurring is 4/5, and the probability of both A and B occurring is 1/5. Using the addition rule, the probability of A or B occurring is 13/20.

To calculate the probability of events A or B occurring, we can use the addition rule. The addition rule states that the probability of A or B occurring is equal to the sum of their individual probabilities minus the probability that both A and B occur together.

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Answer:

S={1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T}

Step-by-step explanation:

The sample space for a six sided die is {1,2,3,4,5,6} as there are 6 possible outcomes and for flipping a coin is {H,T} as there are two possible outcomes.

If a person rolls a six sided die and then flips a coin then the sample space for this event will be written as

S={1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T}.

The number of elements in the sample space are

n(S)=12.

Answer:

Step-by-step explanation:If a person draws a playing card and checks its color and then spins a six dash space spinner​, describe the sample space of possible outcomes using Upper B comma Upper R for the card outcomes and 1 comma 2 comma 3 comma 4 comma 5 comma 6 for the spinner outcomes.​ (Make sure your answers reflect the order​ stated.)

The sample space is Sequals​{

012

12​}.

​(Use a comma to separate answers as​ needed.)

Answer and Step-by-step explanation:

From the question statement we get know that it is Binomial distribution because there are only two possible outcomes so we need to use Binomial Probability Distribution for this question.

Formula for the Binomial Probability Distribution:

P(X)=   p^x q^(n-x)

Where,

Answer and explanation for each part of the question are as follow:

a.What is the probability that among 10 randomly observed individuals exactly 4 do not cover their mouth when sneezing?

Solution:

Given that

n=10  

p=0.267 (because p is the probability of success which is “number of individuals not covering their mouths when sneezing” in the question)

q=1-0.267=0.733  

x=4 (number of successes i.e. individuals not covering their mouths)

C_x^n=n!/(n-x)!x!=10!/(10-4)!4!=210

P(X)=C_x^n   p^x q^(n-x)=210×〖(0.267)〗^4×〖0.733〗^(10-4)

P(X)=210×0.00508×0.155  

P(X)=0.165465  

b. What is the probability that among 10 randomly observed individuals fewer than 3 do not cover their mouth when sneezing?

Solution:

Given that

n=10  

p=0.267 (because p is the probability of success which is “number of individuals not covering their mouths when sneezing” in the question)

q=1-0.267=0.733  

x=3 (number of successes i.e. individuals not covering their mouths)

C_x^n=n!/(n-x)!x!=10!/(10-3)!3!=120

P(X)=C_x^n   p^x q^(n-x)=120×(0.267)^3×〖0.733〗^(10-3)

P(X)=120×0.01903×0.1136  

P(X)=0.25962  

c. Would you be surprised if, after observing 18 individuals, fewer than half covered their mouth when sneezing? why?

Solution:

Given that

n=18  

p=0.267 (because p is the probability of success which is “number of individuals not covering their mouths when sneezing” in the question)

q=1-0.267=0.733  

x=9 (x is the number of successes “number of individuals not covering their mouths when sneezing”, if less than half cover their mouth then more than half will not cover), so let x=9

C_x^n=n!/(n-x)!x!=18!/(18-9)!9!=48620

P(X)=48620×(0.267)^9×〖0.733〗^(18-9)  

P(X)=48620×0.00000689×0.0610  

P(X)=0.020  

Yes, I am surprised that probability of less than 9 individuals covering their mouth when sneezing is 0.020. Which is extremely is small.

Answer:

Interquartile range: Difference between the third and the first quartile.

Range: Difference between the highest and the lowest value of a dataset.

Since the range is not in the middle of a set(like the quartiles), it is more sensitive to extreme values.

Step-by-step explanation:

The interquartile range is the difference between the third quartile and the first quartile.

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 range is the difference between the highest and the lowest value of a dataset.

Lets see two different sets of cardinality 8

Set 1: 5,5,6,7,8,9,13,13

Range = 13-5 = 8

Q1 = Element at the position 0.25*8 = 2 = 5

Q3 = Element at the position 0.75*8 = 6 = 9

IQR = Q3 - Q1 = 9 - 5 = 4

Now, replacing the first and the last element by extreme values

Set 1: 0,5,6,7,8,9,13,20

Range = 20-0 = 20

Q1 = Element at the position 0.25*8 = 2 = 5

Q3 = Element at the position 0.75*8 = 6 = 9

IQR = Q3 - Q1 = 9 - 5 = 4

The range changes and the IQR does not. Since extreme values are not in the middle of the distribution(in which the quartiles are), the range is more sensitive to extreme values, as this example showed.

Final answer:

The interquartile range measures the spread of the middle 50% of data and is less sensitive to extreme values compared to the range, which measures the spread of all data points by looking at the difference between the maximum and minimum values.

Explanation:

Difference Between Interquartile Range and Range in Data Analysis:

The difference between the interquartile range (IQR) and the range is that while the range measures the spread of all data points by subtracting the smallest value from the largest, the interquartile range measures the spread of the middle 50% of data, between the 25th and 75th percentiles (Q1 and Q3 respectively). To calculate the range in a dataset, if the highest score is 100 and the lowest is 70, the range is 100 - 70 = 30. On the other hand, if Q1 is 72.25 and Q3 is 86.75, the interquartile range would be 86.75 - 72.25 = 14.50.

The range is more sensitive to extreme values or outliers because it takes into account the most extreme scores in the dataset. In contrast, the IQR is less sensitive to extreme values since it focuses on the central portion of the data set and ignores the extremes. This is why IQR is often used to identify potential outliers, which are suspected if they lie more than 1.5 times the IQR below Q1 or above Q3.

Therefore, the IQR provides a more robust measure of data variability, especially when dealing with skewed distributions or datasets with outliers.

Answer:

a) [tex]183-1.984\frac{20}{\sqrt{100}}=179.032[/tex]    

[tex]183+1.984\frac{20}{\sqrt{100}}=186.968[/tex]    

So on this case the 95% confidence interval would be given by (179.032;186.968)    

b) [tex]190-1.984\frac{23}{\sqrt{100}}=185.437[/tex]    

[tex]190+1.984\frac{23}{\sqrt{100}}=194.563[/tex]    

So on this case the 95% confidence interval would be given by (185.437;194.563)    

c) For Summer the confidence interval was (179.032;186.968) and as we can see our upper limit is <190 so then we can conclude that they are below the specification of 190 at 5% of significance

For Winter the confidence interval was (185.437;194.563) and again the upper limit is <190 so then we can conclude that they are below the specification of 195 at 5% of significance

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

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".

[tex]\bar X[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Part a : Summer

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=100-1=99[/tex]

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,99)".And we see that [tex]t_{\alpha/2}=1.984[/tex]

Now we have everything in order to replace into formula (1):

[tex]183-1.984\frac{20}{\sqrt{100}}=179.032[/tex]    

[tex]183+1.984\frac{20}{\sqrt{100}}=186.968[/tex]    

So on this case the 95% confidence interval would be given by (179.032;186.968)    

Part b: Winter

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=100-1=99[/tex]

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,99)".And we see that [tex]t_{\alpha/2}=1.984[/tex]

Now we have everything in order to replace into formula (1):

[tex]190-1.984\frac{23}{\sqrt{100}}=185.437[/tex]    

[tex]190+1.984\frac{23}{\sqrt{100}}=194.563[/tex]    

So on this case the 95% confidence interval would be given by (185.437;194.563)    

Part c

For Summer the confidence interval was (179.032;186.968) and as we can see our upper limit is <190 so then we can conclude that they are below the specification of 190 at 5% of significance

For Winter the confidence interval was (185.437;194.563) and again the upper limit is <190 so then we can conclude that they are below the specification of 195 at 5% of significance

Final answer:

Using statistical methods, we have constructed 95% confidence intervals for the mean summer and winter weights of Frontier Airlines passengers, finding that the new FAA recommendations align with our calculated intervals but are positioned conservatively towards the upper limits.

Explanation:

To answer the student's question about constructing and interpreting 95% confidence intervals for the mean summer and winter weights of Frontier Airlines passengers, and then commenting on the new FAA recommendations, we'll follow statistical methods.

Part A: Summer Weight

For the summer weight, the mean is 183 lb, the standard deviation is 20 lb, and the sample size (n) is 100 passengers. The formula for a 95% confidence interval is: mean ± (z*standard error), where the standard error is (standard deviation/√n). Using z=1.96 for 95% confidence, we calculate the interval as 183 ± (1.96*(20/√100)), which simplifies to 183 ± 3.92. Thus, the 95% confidence interval for summer is ±183 lb ± 3.92 lb.

Part B: Winter Weight

Similar calculations for the winter weight (mean = 190 lb, standard deviation = 23 lb, n = 100) yield a 95% confidence interval of 190 ± (1.96*(23/√100)), or 190 ± 4.51. Therefore, the 95% confidence interval for winter is ±190 lb ± 4.51 lb.

Part C: Comment on FAA Recommendations

The new FAA recommendations are 190 lb for summer and 195 lb for winter. These fall within our confidence intervals, but lean towards the upper limit, suggesting a conservative approach by the FAA, possibly to ensure safety by accommodating potential underestimations of passenger weights.

Answer:

1) 0.375

2) 0.5

3) 0.875

4) 0.5                  

Step-by-step explanation:

We are given the following in the question:

Sample space, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

[tex]\text{Probability} = \displaystyle\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}[/tex]

1. The probability of getting exactly one tail

P(Exactly one tail)

Favorable outcomes ={HHT, HTH, THH}

[tex]\text{P(Exactly one tail)} = \dfrac{3}{8} = 0.375[/tex]

2. The probability of getting a head on the first toss

P(head on the first toss)

Favorable outcomes ={HHH, HHT, HTH, HTT}

[tex]\text{P(head on the first toss)} = \dfrac{4}{8} = \dfrac{1}{2} = 0.5[/tex]

3. The probability of getting at least one head

P(at least one head)

Favorable outcomes ={HHH, HHT, HTH, HTT, THH, THT, TTH}

[tex]\text{P(at least one head)} = \dfrac{7}{8} = 0.875[/tex]

4. The probability of getting at least two heads

P(Exactly one tail)

Favorable outcomes ={HHH, HHT, HTH,THH}

[tex]\text{P(Exactly one tail)} = \dfrac{4}{8} = \dfrac{1}{2} = 0.5[/tex]

Final answer:

The question involves calculating probabilities based on the outcomes of tossing a coin three times. The probabilities for getting exactly one tail, a head on the first toss, at least one head, and at least two heads are found to be 3/8, 1/2, 7/8, and 1/2 respectively, by counting the favorable outcomes within the complete sample space.

Explanation:

The question asks for the probability of various outcomes when tossing a coin three times, given the sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Let's solve each part step-by-step.

Probability of getting exactly one tail: There are three outcomes (HTT, THT, TTH) out of eight that satisfy this condition, so the probability is 3/8.

Probability of getting a head on the first toss: Four outcomes (HHH, HHT, HTH, HTT) satisfy this, so the probability is 4/8 or 1/2.

Probability of getting at least one head: Every outcome except TTT includes at least one head, so the probability is 7/8.

Probability of getting at least two heads: There are four outcomes (HHH, HHT, HTH, THH) that satisfy this condition, leading to a probability of 4/8 or 1/2.

Each probability is based on the fundamental principle of counting favorable outcomes over the total number of outcomes.

Answer:

The question is incomplete. Below is the complete question

"A fair die is tossed. A is the event that the outcome is odd. B is the event that the outcome is even. C is the event that the outcome is less than 4. Determine P(A), P(B), P(C), P(A∩B), P(B∩C), P(A|B), P(A|C), P(B|C).

answers:

a.1/2

b. 1/2

c. 1/2

d.0

e. 1/6

f. 0

g. 2/3

h. 1/3

Step-by-step explanation:

lets write out the probability of each event

a.A=outcome is odd

A={1,3,5}

P(A)=1/2

b.B=outcome is even

B={2,4,6}

P(B)=1/2

c. C=outcome is less than 4

C={1,2,3}

P(C)=1/2

d.P(AnB)={its odd and even i.e what is common between set A and B}

Therefore, P(AnB) is a null set i.e P(AnB)={ }=0

e. BnC= {2} i.e elements common between set B and C

Hence P(BnC)=1/6

f. P{A|B}= P(AnB)/P(B) since P(AnB)=0, P(B)=1/2. then, P(A|B)=0

g. P(A|C)= P(AnC)/P(C) since P(AnC)=1/3, P(C)=1/2. therefore, P(A|C)=2/3

h. P(B|C)= P(BnC)/P(C), since P(BnC)=1/6, P(C)=1/2. therefore, P(B|C)=1/3

Answer:

2.5 weeks

Step-by-step explanation:

Data provided in the question:

Number of container ships operated by the company = 10

Number of containers carried by each container = 5000

Number of container to be unloaded = 20,000

A shipping company operates 10 container ships

Now,

Capacity of 10 container ships = 10 × 5000 containers = 50,000 containers

Time required to load and upload 50,000 containers

= 50,000 containers  ÷ 20,000 containers per week

= 2.5 weeks

Answer:

0 < t < 5 is the required interval for the differential equation (t - 5)y' + (ln t)y = 6t to have a solution.

Step-by-step explanation:

Given the differential equation

(t - 5)y' + (ln t)y = 6t

and the condition y(1) = 6

We can rewrite the differential equation by dividing it by (t - 5) as

y' + [(ln t)/(t - 5)]y = 6t/(t - 5)

(ln t)/(t - 5) is continuous on the interval (0, 5) and (5, +infinity).

6t/(t - 5) is continuous on (-infinity, 5) and (5, +infinity)

We see that for these expressions, we have continuity at the intervals (0, 5) and (5, +infinity).

But the initial condition is y = 6, when t = 1.

The solution to differential equation is certain to exist at (0, 5)

Which implies that

0 < t < 5

is the required interval.

Answer:

Total distance is 8 meters.

Step-by-step explanation:

First, let's take the equation and analyse it:

v(t) = 6 + t²

The delta t = 2

To find the acceleration, we need to differentiate v with respect to t like this:

dv/dt = d/dt (6 + t²)

         = 2t

At t = 2, the change in velocity (dv/dt) = 2×2 = 4 m/s². This is the acceleration,a

The final velocity is given by v = 6 + (2)²

                                                   = 10 m/s

The distance is given by the formula:

s = ut + 1/2at²

initial velocity, u = 0

Therefore, the equation becomes s = 1/2at²

                                                            = 1/2× 4×4

                                                             = 8 m Ans

The upper estimate of the distance traveled is 150 meters, the lower estimate is 12 meters, and the average of these estimates is 81 meters.

This math problem involves the concept of approximating the distance traveled using velocity and small intervals of time. The distance traveled, in approximation, can be calculated by taking the velocity at a certain time, and multiplying it by the change in time, which is given as 2 seconds in this case.

To find the upper estimate, we calculate using the endpoint of the interval, t = 6. To find the lower estimate, we calculate using the beginning of the interval, t = 0.

For upper estimate: v(6) x 2 = [6 +  (6^2)] x 2 = 150 meters

For lower estimate: v(0) x 2 = [6 + (0^2)] x 2 = 12 meters

The average of the two estimates is then found by adding them together and dividing by 2, which results in 81 meters.

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Answer:

It does not appear ​that, as a​ group, the students are reasonably good at estimating one minute.

Step-by-step explanation:

We are given the following data in the question:

75, 88, 51, 73, 49, 31, 69, 74, 72, 59, 72, 81, 99, 101, 73

Formula:

[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]  

where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{1067}{15} = 71.13[/tex]

Sum of squares of differences = 4739.733

[tex]S.D = \sqrt{\frac{4739.733}{14}} = 18.39[/tex]

Population mean, μ = 60 minutes

Sample mean, [tex]\bar{x}[/tex] = 71.13 minutes

Sample size, n = 15

Alpha, α = 0.10

Sample standard deviation, s = 18.39 minutes

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 60\text{ minutes}\\H_A: \mu \neq 60\text{ minutes}[/tex]

We use Two-tailed t test to perform this hypothesis.

Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{71.13 - 60}{\frac{18.39}{\sqrt{15}} } = 2.34[/tex]

Calculating the p-value from the table, we have,

P-value = 0.034354

Since the p-value is lower than the significance level, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.

Thus, we conclude that it does not appear ​that, as a​ group, the students are reasonably good at estimating one minute.

Option A is the correct set of hypotheses for testing if students can estimate one minute. The test statistic and p-value are needed to make a decision, typically using a t-test. The conclusion depends on the p-value relative to the significance level.

The null and alternative hypotheses you are working with for testing whether students are good at estimating one minute are:

To determine if students are good at estimating one minute, we have option A as correct, which is H0: \\(\mu = 60\\) and Ha: \\(\mu \not= 60\\).

The test statistic and p-value should be calculated using appropriate statistical methods (typically a t-test assuming we do not know the population standard deviation, which would require the mean, standard deviation, and sample size). After calculating the test statistic and p-value, you will compare the p-value to the significance level (alpha). If the p-value is less than alpha, you reject H0; otherwise, you fail to reject H0.

Based on the decision to reject or fail to reject H0, you can make a conclusion regarding the original claim about whether students are reasonably good at estimating one minute.

Answer:

a. Mean = 4.5, Standard Deviation = 1.775

b. 0.0152

Step-by-step explanation:

Given

n = 15 purchasers

p = Success = 30%

p = 0.3

q =Failure = 70%

q = 0.7

a.

Mean = np

Mean = 15 * 0.3

Mean = 4.5

Standard Deviation = √Variance

Variance = npq

Variance = 15 * 0.3 * 0.7

Variance = 3.15

Standard Deviation = √3.15

Standard Deviation = 1.774823934929884

Standard Deviation = 1.775 ---------- Approximated

b.

The probability that the number who want new copies is more than two standard deviations away from the mean value

Standard Deviation = 1.775

Mean = 4.5

2 Standard Deviation and Mean = 2 * 1.775 + 4.5

= 3.55 + 4.5

= 8.05

P(X>8.05) = P(9) + P(10) +........+ P(15)

Using the binomial distribution

(p + q) ^ n where p = 0.3, q = 0.7 , n = 15

Expanding (p+q)^n where n = 15 and r > 8

We have

15C9 p^9 q^6 + 15C10 p^10 q^5 + 15C11 p^11 q⁴ + 15C12 p^12 q³ + 15C13 p^13 q² + 15C14 p^14 q + p^15

= 5005 (0.3)^9 (0.7)^6 + 3003 (0.3)^10 (0.7)^5 + 1365 (0.3)^11 (0.7)⁴ + 455 (0.3)^12 (0.7)³ + 105 (0.3)^14 (0.7)² + 15 (0.3)^14 (0.7) + 0.3^15

=0.015234

Answer:

87.26 mm Hg is the reading that represents the 80th percentile of the distribution.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 78 mm Hg

Standard Deviation, σ = 11 mm Hg

We are given that the distribution of diastolic blood pressure is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

We have to find the value of x such that the probability is 0.80

P(X < x)  

[tex]P( X < x) = P( z < \displaystyle\frac{x - 78}{11})=0.8[/tex]  

Calculation the value from standard normal z table, we have,  

[tex]P(z < 0.842) = 0.8[/tex]

[tex]\displaystyle\dfrac{x - 78}{11} = 0.842\\\\x = 87.262 \approx 87.26[/tex]

87.26 mm Hg is the reading that represents the 80th percentile of the distribution.

To find the DBP reading representing the 80th percentile, we can use the z-score formula. The DBP reading is 86.8 mmHg.

To find the diastolic blood pressure (DBP) reading that represents the 80th percentile of the distribution, we can use the z-score formula. The z-score is calculated by subtracting the mean from the desired value and dividing it by the standard deviation. Then, we can use the z-score table or calculator to find the corresponding percentile. In this case, since we want the 80th percentile, we need to find the z-score that corresponds to an area of 0.8 to the left of it.

The formula for the z-score is: z = (x - mean) / standard deviation

Using the given values for Canada (mean = 78 mmHg, standard deviation = 11 mmHg), we can substitute them into the formula and solve for x:

0.8 = (x - 78) / 11

Multiplying both sides by 11, we get:

8.8 = x - 78

Adding 78 to both sides, we get:

x = 86.8 mmHg

Therefore, the DBP reading that represents the 80th percentile of the distribution is 86.8 mmHg.

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Answer:

546.94 meters is the height of the mountain.

Step-by-step explanation:

Pressure at the top of mountain = [tex]P_1=93.8 kPa[/tex]

Pressure at the bottom of the mountain = [tex]P_2=100.5 kPa[/tex]

Pressure difference  =[tex]P_2-P_1=100.5kPa-93.8kPa=6.7 kPa[/tex]

6.7 kPa = 6.7 × 1000 Pa (1 kPa= 1000 pa)

Density of the air = d = [tex]1.25 kg/m^3[/tex]

Acceleration due to gravity = g = [tex]9.8 m/s^2[/tex]

Height of the mountain = h

[tex]P=h\times d\times g[/tex]

[tex]h=\frac{P}{d\times g}=\frac{6.7\times 1000 Pa}{1.25 kg/m^3\times 9.8 m/s^2}[/tex]

[tex]h=546.94 m[/tex]

546.94 meters is the height of the mountain.

Answer:

[tex] P(X=6)[/tex]

If we use the probability mass function we got:

[tex] P(X=6) = \frac{e^{-3.3} 3.3^6}{6!}= 0.0662[/tex]

Step-by-step explanation:

Previous concepts

The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:

Solution to the problem

Let X the random variable that represent the number of students arrive at the office hour. We know that [tex]X \sim Poisson(\lambda=3.3)[/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]

And f(x)=0 for other case.

For this distribution the expected value is the same parameter [tex]\lambda[/tex]

[tex]E(X)=\mu =\lambda=3.3[/tex]

And we want this probability:

[tex] P(X=6)[/tex]

If we use the probability mass function we got:

[tex] P(X=6) = \frac{e^{-3.3} 3.3^6}{6!}= 0.0662[/tex]

Answer:

0.0662

Step-by-step explanation:

The given problem indicates the Poisson experiment.

The pdf of Poisson experiment is as follow

P(X=x)=μ^x(e^-μ)/x!

Here, the average student visits in office hour=3.3=μ.

We have to find the probability that the number of student arrivals is 6 in a randomly selected office hour.

So, x=6 and μ=3.3

The probability that the number of student arrivals is 6 in a randomly selected office hour

P(X=6)=3.3^6(e^-3.3)/6!

P(X=6)=1291.468(.0369)/720

P(X=6)=47.6552/720

P(X=6)=0.0662.

Thus, the probability that the number of student arrivals is 6 in a randomly selected office hour is 6.62 or 0.0662.

Answer: P(AUB) = 0.93 + 0.89 - 0.87 = 0.95

Therefore, the probability that at least one train is on time is 0.95.

Step-by-step explanation:

The probability that at least one train is on time is the probability that either train A, B or both are on time.

P(A) = P(A only) + P(A∩B)

P(B) = P(B only) + P(A∩B)

P(AUB) = P(A only) + P(B only) + P(A∩B)

P(AUB) = P(A) + P(B) - P(A∩B) ......1

P(A) = 0.93

P(B) = 0.89

P(A∩B) = 0.87

Then we can substitute the given values into equation 1;

P(AUB) = 0.93 + 0.89 - 0.87 = 0.95

Therefore, the probability that at least one train is on time is 0.95.