In estimating the mean score on a fitness exam, we use an original sample of size n 3 and a bootstrap distribution containing 5000 bootstrap samples to obtain a 95% confidence interval of 67 to 73. A change in this process is described below. If all else stays the same, which of the following confidence intervals (A, B, or C) is the most likely result after the change: Using an original sample of size n 16

O A. 66 to 74
O B. 67 to 73
c. 67.5 to 72.5

Answer:

Step-by-step explanation:

Final answer:

The value of sin(3π/2) is -1, and the value of cos(3π/2) is 0, as determined by their positions on the unit circle at an angle of 3π/2 radians.

Explanation:

To find the values of sin(3π/2) and cos(3π/2) using the unit circle, we need to understand the positions on the circle.

At 3π/2 radians, which is the same as 270°, the point on the unit circle is directly below the center of the circle, at coordinates (0, -1).

This position corresponds to a cosine value of 0 (since the x-coordinate is 0) and a sine value of -1 (since the y-coordinate is -1).

Answer:

it’s everything multiplied by 10

so

200

280

500

Step-by-step explanation:

Answer: x = -6

Step-by-step explanation: answer on edge

Answer: don’t use k for the other numbers

Step-by-step explanation:

Answer:

6k -5

Step-by-step explanation:

19-6(-k+4)

Distribute

19 -6*-k -6*4

19 +6k -24

Combine like terms

6k +19-24

6k -5

Answer:

the probability that the wait time is between 6 and 7.3 minutes = 0.725

Step-by-step explanation:

Given -

Average wait time  [tex](\nu)[/tex]  =  7.6 minutes

Standard deviation [tex](\sigma)[/tex] = 30 second = .5 minute

Let X be the wait times in line at a grocery store

the probability that the wait time is between 6 and 7.3 minutes

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

[tex]P (7.3> X>6 )[/tex] = [tex]P (\frac{7.3 - 7.6}{.5}> Z>\frac{6 - 7.6 }{.5} )[/tex]

                         = [tex](.6> Z> -3.2 )[/tex]

[Using z table]

                         = Area to the left of z = .6 - area to the left of z = -.32

                         = .7257 - .0007 = 0.725

To find the probability of a wait time between 6 and 7.3 minutes at the grocery store, convert the times to z-scores and use the standard normal distribution to find the probability between these scores.

To calculate the probability that the wait time is between 6 and 7.3 minutes for a normally distributed data set with a mean of 7.6 minutes and a standard deviation of 30 seconds, we first need to convert the wait times into z-scores. Using the formula z = (X - \\(mu\\)) / \\(sigma\\), where X is the value in the data set, \\(mu\\) is the mean, and \\(sigma\\) is the standard deviation, we find the z-scores for 6 minutes and 7.3 minutes.

The z-score for 6 minutes is z1 = (6 - 7.6) / 0.5 = -3.2, and the z-score for 7.3 minutes is z₂ = (7.3 - 7.6) / 0.5 = -0.6. Using a standard normal distribution table or calculator, we find the probabilities corresponding to these z-scores, P(z₁) and P(z₂), and subtract them to get the probability of a wait time between 6 and 7.3 minutes, which is P(z₂) - P(z₁).

The answer is 12 because 3 x 4 is 12

Answer:

x=0

Step-by-step explanation:

To solve, we need to get all the variables on one side of the equation, and all the numbers on the other

6(x + 1) – 5x = 8 + 2(x - 1)

First, distribute the 6 on the left

6*x+6*1 -5x=8 +2(x-1)

6x+6-5x=8+2(x=1)

Combine like terms on the left

(6x-5x)+6=8+2(x-1)

x+6=8+2(x-1)

Distribute the 2 on the right

x+6=8+2*x+2*-1

x+6=8+2x-2

Combine like terms on the right

x+6=2x+(8-2)

x+6=2x+6

Subtract x from both sides

6=x+6

Subtract 6 from both sides

x=0

Hope this helps! :)

Answer:

The dimensions of the box that minimizes the amount of material of construction is

Square base = (5.98 × 5.98) in²

Height of the box = 2.99 in.

Step-by-step explanation:

Let the length, breadth and height of the box be x, z and y respectively.

Volume of the box = xyz = 107 in³

The box has a square base and an open top.

x = z

V = x²y = 107 in³

The task is to minimize the amount of material used in its construction, that is, minimize the surface area of the box.

Surface area of the box (open at the top) = xz + 2xy + 2yz

But x = z

S = x² + 2xy + 2xy = x² + 4xy

We're to minimize this function subject to the constraint that

x²y = 107

The constraint can be rewritten as

x²y - 107 = constraint

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x and y

L(x,y,z) = x² + 4xy - λ(x²y - 107)

We then take the partial derivatives of the Lagrange function with respect to x, y and λ. Because these are turning points, at the turning points each of the partial derivatives is equal to 0.

(∂L/∂x) = 2x + 4y - 2λxy = 0

λ = (2x + 4y)/2xy = (1/y) + (2/x)

(∂L/∂y) = 4x - λx² = 0

λ = (4x)/x² = (4/x)

(∂L/∂λ) = x²y - 107 = 0

We can then equate the values of λ from the first 2 partial derivatives and solve for the values of x and y

(1/y) + (2/x) = (4/x)

(1/y) = (2/x)

x = 2y

Hence, at the point where the box has minimal area,

x = 2y

Putting these into the constraint equation or the solution of the third partial derivative,

x²y - 107 = 0

(2y)²y = 107

4y³ = 107

y³ = (107/4) = 26.75

y = ∛(26.75) = 2.99 in.

x = 2y = 2 × 2.99 = 5.98 in.

Hence, the dimensions of the box that minimizes the amount of material of construction is

Square base = (5.98 × 5.98) in²

Height of the box = 2.99 in.

Hope this Helps!!!

To minimize packaging costs for a box with a square base and a fixed volume, we differentiate the surface area function with respect to the side length of the base and find the optimal dimensions, resulting in an approximately 5.15 inches base and a height of approximately 4.03 inches.

To minimize the material used for the open box with a square base and a given volume, we must use calculus to find the dimensions that will give us a box with minimum surface area.

Let the side of the square base be x inches and the height be h inches. Since the volume of the box is fixed at [tex]107 in^3[/tex], we have [tex]V = x^2h[/tex], which implies [tex]h = V / x^2 = 107 / x^2[/tex].

To minimize the surface area, we need to minimize the function [tex]S(x) = x^2 + 4xh[/tex]. Substituting in the equation for h, we get [tex]S(x) = x^2 + 4x(107) / x^2[/tex]. To find the minimum, take the derivative of S with respect to x and set it equal to zero. Solve for x to find the optimal dimension for the base.

After solving, we find that the dimension for the sides of the square base, x, is approximately x = 5.15 in (rounded to two decimal places). The height h would then be [tex]h = 107 / x^2[/tex] ≈ 4.03 in (rounded to two decimal places).

Answer:

x = 8, x = -8

Step-by-step explanation:

Hello!

We can use the Difference of Squares Formula: [tex](a + b)(a - b) = a^2 - b^2[/tex]

We need the two terms to be perfect squares for this formula to work, and subtraction has to be the operation. As you can see x² is a square of x, and 64 is the square of 8.

Solve:

Set each factor to zero. (Zero Product Property)

And

The solutions are x = 8, and x = -8.

Answer:

The current monthly cost is 1$,388.625

Step-by-step explanation:

We are given the following in the question:

[tex]C(x) = 0.001x^3 + 0.07x^2 + 16x + 700[/tex]

where C(x) is the monthly cost in dollars and x are the number of chairs produced.

Monthly chair production = 35

We have to evaluate monthly cost.

Putting x= 35 in the equation, we get,

[tex]C(35) = 0.001(35)^3 + 0.07(35)^2 + 16(35) + 700\\C(35) =1388.625[/tex]

Thus, the current monthly cost is 1$,388.625

Answer:

232 votes

Step-by-step explanation:

This question is just asking what 80% of 290 is. In order to find this, you convert 80% into a decimal, then multiply by 290.

Converting percentages into decimals is just moving the decimal left twice.

0.8 x 290 = 232

You can check this by doing 232/290, and making sure it's 80%.

I hope this helps!

Answer:100 people

Step-by-step explanation:

140 people x 5 ladle/1 person = 700 ladles full

700 ladles x 1 person/7 ladles = 100 people fed

Step-by-step explanation:

(2575÷25)(1000÷25) = 10340 when reduced to the simplest form. As the numerator is greater than the denominator, we have an IMPROPER fraction, so we can also express it as a MIXED NUMBER, thus 25751000 is also equal to 22340 when expressed as a mixed number.

Answer: B The width of the interval would increase

Step-by-step explanation:

i just took this and go the answers back

Decreasing the sample size from 100 to 50 would widen the 95 percent confidence interval for the difference in mean seat widths between two airlines. This is because smaller samples have more variability, requiring a larger interval to capture the population mean with the same level of certainty.

If the consumer group used a sample size of 50 instead of 100 for each airline, that would increase the width of the 95 percent confidence interval for the difference in population mean widths of seats between the two airlines. This is because smaller sample sizes result in more variability, requiring a wider interval to capture the true population mean with the same level of certainty.

As with the unoccupied seats example, where the sample mean of 11.6 and standard deviation of 4.1 were used to form a confidence interval, the size of the interval is dependent on the variability within the sample, and smaller samples generally have higher variability. Similarly in the case of exam scores, with a lower confidence level of 90 percent, a narrower interval is needed compared to a higher confidence level of 95 percent.

Therefore, in effect, decreasing the sample size from 100 to 50 would make the confidence interval wider, as more variability is expected and a larger interval is needed to capture the true population mean with a 95 percent confidence level.

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Answer:

Find the difference between the two scores for a number of sample distributions. Make a plot of the differences and check for outliers.

Step-by-step explanation:

Checking for Normality means basically checking if one's data distribution approximates a normal distribution.

A normal distribution is represented by a bell-shaped curve, peaking around the mean, indicating that all of the data spreads out from the mean.

Th original aim of the teacher is to check the effects of the particular added unit on the performance of students in the subject.

The teacher goes about this by testing the students before and after learning the unit.

The best way to compare of course, is to take a difference of the test scores for different samples. This first gives the idea of whether the newly introduced unit affects performance.

This set of differences is then checked for normality.

So, the best manner to make a plot of these differences. Like we mentioned earlier, a normal distribution is bell shaped. So, the plot of these differences would be a bell shaped curve if the distribution was normal and we wouldn't get a bell shaped curve if the distribution wasn't normal.

Checking for outliers help to eliminate part of data that can totally scatter the regular behaviour of the data distribution.

So, the best way for the teacher to check for normality is to find the difference between the two scores for a number of sample distributions. Make a plot of the differences and check for outliers.

Hope this Helps!!!

Since the data include the pretest and post test scores of each student or subject, then, it is approached as a paired t test. Hence, checking for normality would require obtaining the difference between the two scores for each subject, then make a plot of the differences and check for outliers.

Hence, the most appropriate check for normality in this scenario is to check for outliers in the plot made from the score difference.

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The difference in wire required per wall hanging, rounded to the nearest hundredth, is roughly 10.99 inches.

Given that Poppy is using a metal wire wall hanging with radius of 6.25 inches, his sister is making same with radius 1 3/4 inches longer than the Poppy.

We need to find how much more wire did she need.

Calculating the circumference of each wall hanging will help us determine how much wire Poppy's sister used in comparison to Poppy.

The radius of Poppy's wall hanging is 6.25 inches, hence the following formula can be used to determine its circumference:

C = 2πr, where r is the radius.

Putting the values,

C = 2 x 3.14 x 6.25

C ≈ 39.25 inches

The diameter of Poppy's sister's wall hanging is 1 3/4 inches larger than Poppy's. Her wall hanging's radius would be 6.25 + 1 3/4 inches.

Converting 1 3/4 to an improper fraction: 1 3/4 = 7/4

Radius of Poppy's sister's wall hanging = 6.25 + 7/4

Radius ≈ 6.25 + 1.75

Radius ≈ 8 inches

Now we can calculate the circumference of Poppy's sister's wall hanging:

C = 2 x 3.14 x 8

C ≈ 50.24 inches

By deducting Poppy's circumference from Poppy's sister's circumference, we can calculate the difference in wire utilized per wall hanging:

Difference = Poppy's sister's circumference - Poppy's circumference

Difference ≈ 50.24 - 39.25

Difference ≈ 10.99 inches

Hence the difference in wire used per wall hanging is approximately 10.99 inches.

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Answer:

g(-1) = -3

Step-by-step explanation:

g(-1) is the y value of the graph when x = -1

when x=-1 y=-3

g(-1) = -3

Answer:

a. 20

b. -5

c. 5

Step-by-step explanation:

Please see attachment

Answer:

[tex]x+3y=-27[/tex]

Step-by-step explanation:

We are given that

[tex]y=(-\frac{1}{3})x-9[/tex]

We have to find the  standard form of given equation

[tex]y=\frac{-x-27}{3}[/tex]

[tex]3y=-x-27[/tex]

By using multiplication property of equality

[tex]x+3y=-27[/tex]

We know that

Standard form of equation

[tex]ax+by=c[/tex]

Therefore, the standard form of given equation  is given by

[tex]x+3y=-27[/tex]

The equation y = -1/3x - 9 can be converted to standard form by eliminating fractions and rearranging. The final equation in standard form is x - 3y = 27.

The equation you provided is already in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. However, you're asked to convert this equation into standard form, which is Ax + By = C, with A, B, and C being integers, and A and B not both equal to zero.

To convert the given equation y = -1/3x - 9 into standard form, we first need to eliminate the fractions. We can achieve this by multiplying every term by -3, giving us 3y = x + 27. To make it fit the standard form, we can rearrange as -x + 3y = -27.

Remember, standard form shouldn't have any negatives in front of the x term, so we multiply everything by -1. The final equation in standard form is x - 3y = 27.

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Answer:

The mean is zero and the standard deviation is one.

Correct, since the sample mean is not always defined by a distribution with mean 0 and deviation 1. Since the normal standard distribution is not always representative of the sample mean dsitribution

Step-by-step explanation:

For this case we want to analyze the distribution of the sample mean given by:

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

Let's analyze one by one the possible options:

The distribution of the values is obtained by repeated sampling.

False, for this case we assume that the data of each observation [tex] X_i[/tex] , [tex] i =1,2,...,n[/tex] is obtained from a repated sampling method

The samples are all of size n.

False, for this case we assume that the sample mean is obtained from n data values

The samples are all drawn from the same population.

False, for this case we assume that the data comes from the same distribution

The mean is zero and the standard deviation is one.

Correct, since the sample mean is not always defined by a distribution with mean 0 and deviation 1. Since the normal standard distribution is not always representative of the sample mean dsitribution

The mean is zero and the standard deviation is one.

It is not characteristic of the sampling distribution of a sample mean.

In the Sampling Distribution of the Sample Mean , If repeated random samples of a given size n are taken from a population .where the population mean is μ  and the population standard deviation is σ .then the mean of all sample means  is population mean μ .

Characteristic of Sampling Distribution ,

So, The mean is zero and the standard deviation is one in sampling distribution is not any mandatory condition.

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Answer:

The relation between [tex]W_{1} \ and \ W_{2}[/tex] is [tex]W_{2} = 3 \ W_{1}[/tex]

Step-by-step explanation:

Natural length = 0.23 m

Spring stretches from 23 cm to 33 cm. now

Work done [tex]W_{1}[/tex] in stretching the spring

[tex]W_{1} = \int\limits^a_b {kx} \, dx[/tex]

where b = 0 & a = 0.1 m

[tex]W_{1} = k [\frac{x^{2} }{2} ][/tex]

With limits b = 0 & a = 0.1 m

Put the values of limits we get

[tex]W_{1} = k [\frac{0.1^{2} }{2} ][/tex]

[tex]W_{1} = 0.005 k[/tex] ------- (1)

Now the work done in stretching the spring from 33 cm to 43 cm.

[tex]W_{1} = \int\limits^a_b {kx} \, dx[/tex]

With limits b = 0.1 m to a = 0.2 m

[tex]W_{2} = k [\frac{x^{2} }{2} ][/tex]

With limits b = 0.1 m to a = 0.2 m

[tex]W_{2} = k [\frac{0.2^{2} - 0.1^{2} }{2} ][/tex]

[tex]W_{2} =0.015[/tex]

[tex]\frac{W_{2} }{W_{1} } = \frac{0.015}{0.005}[/tex]

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

Thus

[tex]W_{2} = 3 \ W_{1}[/tex]

This is the relation between [tex]W_{1} \ and \ W_{2}[/tex].

The work done on a spring is calculated using Hooke's Law, and it depends on the change in length of the spring (Δx) and the spring constant (k). Since the change in length is the same (10 cm) when stretching the spring from 23 cm to 33 cm (W1) and from 33 cm to 43 cm (W2), the work done during both stretches, W1 and W2, are equal.

The work done on a spring, using Hooke's Law, is calculated with the formula W = 0.5 * k * (Δx)², where k is the spring constant and Δx is the change in length of the spring.

To find the work W1 done in stretching the spring from 23 cm to 33 cm, Δx = 33-23 = 10 cm. Thus, W1 = 0.5 * k * (10)².

The work W2 done in stretching it from 33 cm to 43 cm would be calculated similarly with Δx = 43-33 = 10 cm. Thus, W2 = 0.5 * k * (10)².

As you can see, since the stretch (Δx) is same in both cases (10 cm), W1 and W2 are equal.

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Answer:

[tex]s^2=1.32[/tex]

Step-by-step explanation:

-This is a binomial probability distribution problem.

-Given that p=0.86 and n=11, the sample variance can be calculated using the formula:

[tex]\sigma^2=np(1-p)[/tex]

#We substitute the given parameters in the formula to solve for variance:

[tex]\sigma^2=np(1-p)\\\\\\=11\times 0.86(1-0.86)\\\\=1.324\approx1.32[/tex]

Hence, the sample variance is 1.32

Final answer:

The variance in a sample of 11 flights that have arrived on time at Denver International Airport, where 86% of flights are on time, is calculated using a binomial distribution with the formula variance = np(1-p), resulting in a variance of 1.33 after rounding to two decimal places.

Explanation:

The subject of this question is about finding the variance in a sample of flights that have arrived on time at Denver International Airport, given that 86% of recent flights have arrived on time and that a sample of 11 flights is studied. To calculate the variance for a binomial distribution, which is applicable in this context because each flight can either be on time or not, we use the formula:

Variance = np(1-p).

Where 'n' is the number of trials (or flights, in this case), which is 11, and 'p' is the probability of success on each trial (a flight arriving on time), which is 0.86. Thus, the variance for the 11 flights can be calculated as follows:

Variance = 11 × 0.86 × (1 - 0.86) = 11 × 0.86 × 0.14 = 1.3284.

After rounding to two decimal places, the variance for a sample of 11 flights is 1.33.

Answer:

  (1000 m)/(1 km) and (60 min)/(1 h)

Step-by-step explanation:

Put the unit you don't want in a position to cancel the unit in the given number. Write the fraction so that the equivalent amount of the unit you do want is on the other side of the fraction bar.

Here we have km/min with km in the numerator. To cancel that, we need a fraction with km in the denominator. We want meters (m) in the numerator, so we need a fraction that has a number of meters equivalent to 1 km. That will be ...

  (1000 m)/(1 km)

This is one of the conversion factors we will need to multiply by.

__

We also have "min" in the denominator. To cancel that, we need a conversion factor with min in the numerator. The unit we want in the denominator is h (hours), so we need an equivalent for hours and minutes. That would be 60 min = 1 h, so we write the conversion factor as ...

  (60 min)/(1 h)

So, our conversion factors are ...

  (1000 m)/(1 km) and (60 min)/(1 h)

_____

The converted number is ...

  (4 km/min)(1000 m/km)(60 min/h) = 240,000 m/h

__

Comment on conversion factors

As you can see, we write the fraction so equal amounts are in numerator and denominator. Since the amounts are equal, the value of the fraction is 1. Multiplying by 1 in this form doesn't change the original value, it only changes the units.

Answer:

  5√2 units ≈ 7.07 units

Step-by-step explanation:

The ratio of leg to hypotenuse in an isosceles right triangle is ...

  leg/hypotenuse = 1/√2

Multiplying by the length of the hypotenuse, we have ...

  leg = hypotenuse/√2 = 10/√2

  leg = 5√2 . . . . rationalize the denominator

The length of one leg is 5√2 units, about 7.07107 units.

Final answer:

In a 45-45-90 triangle with a hypotenuse of 10 units, each leg would be of length 5√2 units.

Explanation:

The question is asking for the length of one of the legs of a 45-45-90 triangle with a hypotenuse length of 10 units. In a 45-45-90 triangle, also known as an isosceles right triangle, the lengths of the legs are equal, and each leg is √2 times smaller than the hypotenuse. To find the length of the leg (let's call it L), we set up the equation L = √2 / 2 × hypotenuse. Plugging in the hypotenuse length we get L = √2 / 2 × 10, which simplifies to L = 5√2. Thus, the length of each leg of the triangle is 5√2 units.

Final answer:

To find the most likely drawn coin from a bag of three biased coins after observing two heads and one tail from three flips, we create a Bayesian network and define the Conditional Probability Tables. Then, using Bayesian inference, the likelihood of the flip sequence for each coin is calculated, allowing us to determine which coin was most likely to have been drawn.

Explanation:

The task requires calculating the likelihood of which biased coin, a, b, or c, was drawn from the bag given that the observed flip outcomes are heads twice and tails once. To approach this problem, we apply Bayesian inference.

I. Bayesian Network and Conditional Probability Tables (CPTs)

Create a Bayesian network with node C representing the choice of coin and nodes X1, X2, X3 representing the flip outcomes. Calculate the CPTs considering the bias probabilities of each coin.

II. Most Likely Coin

Using Bayes' theorem:

Calculate the likelihood of the flip sequence (HH, T) for each coin.

Determine prior probabilities (1/3 for each coin).

Compute the posterior probabilities for each coin being drawn.

Identify the highest posterior probability to conclude the most likely drawn coin.

Generally, in a situation with unequal probabilities for different outcomes, the expected long-term results will align more closely with these probabilities, influencing the most likely outcomes.

Answer: I had a stroke trying to read and understand this sorry

Step-by-step explanation:

Answer:

$60.

Step-by-step explanation:

x / 2 - 10 = 0

x / 2 = 10

x = 20

Before she spent her money:

x / 2 - 10 = 20

x / 2 = 30

x = 60

She had $60 before she went shopping.

Feel free to let me know if you need more help. :)

Answer:

450 cubic meters

Step-by-step explanation:

Austin didn't add the 2 prisms, he multiplied the 1st prism and kept it as the answer.  

Austin is not correct as the combined volume of the prisms is 1100 cubic meter.

Volume is a three-dimensional quantity used to calculate a solid shape's capacity. That means that the volume of a closed form determines how much three-dimensional space it can fill.

From the figure the dimension of prism is 5 x 11 x 10.

Austin says the combined volume of the prisms is 225 cubic meters.

So, Volume of combined prism

= 2 x 5 x 11 x 10

= 2 x 55 x 10

= 110 x 10

= 1100 cubic meter.

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Answer:

(a) 95% confidence interval for the percentage of all car accidents that involve teenage drivers is [0.177 , 0.243].

(b) We are 95% confident that the percentage of all car accidents that involve teenage drivers will lie between 17.7% and 24.3%.

(c) We conclude that the the percentage of teenagers has not changed since you join the company.

(d) We conclude that the the percentage of teenagers has changed since you join the company.

Step-by-step explanation:

We are given that your manager asked you to check police records of car accidents and out of 576 accidents you selected randomly, teenagers were at the wheel in 120 of them.

(a) Firstly, the pivotal quantity for 95% confidence interval for the population proportion is given by;

                        P.Q. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex]  ~ N(0,1)

where, [tex]\hat p[/tex] = sample proportion teenage drivers = [tex]\frac{120}{576}[/tex] = 0.21

           n = sample of accidents = 576

           p = population percentage of all car accidents

Here for constructing 95% confidence interval we have used One-sample z proportion statistics.

So, 95% confidence interval for the population population, p is ;

P(-1.96 < N(0,1) < 1.96) = 0.95  {As the critical value of z at 2.5% level

                                                     of significance are -1.96 & 1.96}  

P(-1.96 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < 1.96) = 0.95

P( [tex]-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < [tex]{\hat p-p}[/tex] < [tex]1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ) = 0.95

P( [tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < p < [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ) = 0.95

95% confidence interval for p = [[tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] , [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex]]

  = [ [tex]0.21-1.96 \times {\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex] , [tex]0.21+1.96 \times {\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex] ]

  = [0.177 , 0.243]

Therefore, 95% confidence interval for the percentage of all car accidents that involve teenage drivers is [0.177 , 0.243].

(b) We are 95% confident that the percentage of all car accidents that involve teenage drivers will lie between 17.7% and 24.3%.

(c) We are also provided that before you were hired in the company, the percentage of teenagers who where involved in car accidents was 18%.

The manager wants to see if the percentage of teenagers has changed since you join the company.

Let p = percentage of teenagers who where involved in car accidents

So, Null Hypothesis, [tex]H_0[/tex] : p = 18%    {means that the percentage of teenagers has not changed since you join the company}

Alternate Hypothesis, [tex]H_A[/tex] : p [tex]\neq[/tex] 18%    {means that the percentage of teenagers has changed since you join the company}

The test statistics that will be used here is One-sample z proportion statistics;

                              T.S.  =  [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex]  ~ N(0,1)

where, [tex]\hat p[/tex] = sample proportion teenage drivers = [tex]\frac{120}{576}[/tex] = 0.21

           n = sample of accidents = 576

So, test statistics  =  [tex]\frac{0.21-0.18}{\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex]  

                              =  1.768

The value of the sample test statistics is 1.768.

Now at 0.05 significance level, the z table gives critical value of -1.96 and 1.96 for two-tailed test. Since our test statistics lies within the range of critical values of z, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.

Therefore, we conclude that the the percentage of teenagers has not changed since you join the company.

(d) Now at 0.1 significance level, the z table gives critical value of -1.6449 and 1.6449 for two-tailed test. Since our test statistics does not lie within the range of critical values of z, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the the percentage of teenagers has changed since you join the company.

Rounding to the nearest minute, your visit to the doctor will take approximately 36 minutes.

To determine how long your visit to the doctor will take, we need to find the time it takes for the remaining amount of radioactive dye in your system to be less than 2 milligrams.

We can use the exponential decay formula to model the amount of radioactive dye remaining in your system after a given time:

[tex]\[ A(t) = A_0 \times e^{-kt} \][/tex]

Where:

- [tex]\( A(t) \)[/tex] is the amount of radioactive dye remaining at time ( t )

- [tex]\( A_0 \)[/tex] is the initial amount of radioactive dye

- [tex]\( k \)[/tex] is the decay constant

- [tex]\( t \)[/tex] is the time (in minutes, in this case)

Given:

- [tex]\( A_0 = 11 \)[/tex] milligrams

- [tex]\( A(t) = 4.25 \)[/tex] milligrams

-[tex]\( t = 20 \)[/tex] minutes

We need to solve for ( k ) using the given data:

[tex]\[ 4.25 = 11 \times e^{-20k} \][/tex]

First, divide both sides by 11:

[tex]\[ \frac{4.25}{11} = e^{-20k} \][/tex]

Now, take the natural logarithm of both sides:

[tex]\[ \ln\left(\frac{4.25}{11}\right) = -20k \][/tex]

Now, solve for [tex]\( k \):[/tex]

[tex]\[ k = \frac{\ln\left(\frac{4.25}{11}\right)}{-20} \][/tex]

Now that we have the decay constant, we can use it to find the time it takes for the remaining amount of dye to be less than 2 milligrams:

[tex]\[ 2 = 11 \times e^{-kt} \][/tex]

Plug in the value of ( k ) and solve for ( t ):

[tex]\[ t = \frac{\ln\left(\frac{2}{11}\right)}{-k} \][/tex]

Let's calculate ( t ):

First, let's calculate ( k ):

[tex]\[ k = \frac{\ln\left(\frac{4.25}{11}\right)}{-20} \][/tex]

[tex]\[ k \approx \frac{\ln(0.3864)}{-20} \][/tex]

[tex]\[ k \approx \frac{-0.9515}{-20} \][/tex]

[tex]\[ k \approx 0.0476 \][/tex]

Now, let's use ( k ) to find ( t ):

[tex]\[ t = \frac{\ln\left(\frac{2}{11}\right)}{-0.0476} \][/tex]

[tex]\[ t \approx \frac{\ln(0.1818)}{-0.0476} \][/tex]

[tex]\[ t \approx \frac{-1.707}{-0.0476} \][/tex]

[tex]\[ t \approx 35.81 \][/tex]

Rounding to the nearest minute, your visit to the doctor will take approximately 36 minutes.