A particle moves according to a law of motion s = f(t), 0 ≤ t ≤ 6, where t is measured in seconds and s in feet. f(t) = cos(πt/3) (a) Find the velocity at time t (in ft/s). v(t) = (b) What is the velocity after 2 s? (Round your answer to two decimal places.) v(2) = ft/s

Answer:

The player should be required to pay $5 to make this a fair game.

Step-by-step explanation:

U ~ Uniform(0, 10)

E[U] = (0 + 10)/2

         = 5

X | U ~ Poisson(U)

E[X | U] = U

By law of total probability for expectations,

E[X] = E[E[X|U]] = E[U] = $5

Therefore the player should be required to pay $5 to make this a fair game.

Answer:

  no solution

Step-by-step explanation:

First inequality:

  12x < -18 . . . . subtract 7

  x < -18/12 . . . divide by 12

  x < -1.5 . . . . . . write as decimal

__

Second inequality:

  5x -8 ≥ 40

  5x ≥ 48 . . . . . . add 8

  x ≥ 9.6 . . . . . . . divide by 5

__

There are no solutions to this pair of inequalities. No value of x can be both less than -1.5 and greater than 9.6.

Answer:

240°

Step-by-step explanation:

[tex] \frac{4\pi^{c} }{3} = \frac{4 \times 180 \degree}{3} = 4 \times 60 \degree = 240 \degree \\ [/tex]

Answer:

Among 4.1 and 4.009 The greater one is 4.1

Hope it will help.

Answer:

(3; - 4)

Step-by-step explanation:

Blue: y = -x - 1

Red: y = -2x + 2

The given system of linear equations have solution as  x = 3 and y = -4.

To represent a straight line on a graph consider two points namely x and y intercepts of the line. To find x-intercept put y = 0 and for y-intercept put x = 0. Then draw a line passing through these two points.

The system of equations are given as,

y =−2x + 2          (1)

y = −x − 1            (2)

The above equations are linear equation in two variables.

Their graph are straight lines which  shows their intersection at point (3, -4).

Hence, the solution of the given system of linear equations is x = 3 and y = -4.

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Given:

The given parent function is [tex]f(x)=x^2[/tex]

We need to determine the equation of the new translated (shifted) function g(x).

Vertical stretch:

The general rule to shift the graph f(x), to shift c units upward is [tex]g(x)=f(x)+c[/tex]

From the graph, it is obvious that the graph f(x) is shifted 1 unit upwards.

Thus, applying the above rule, we get;

[tex]g(x)=x^2+1[/tex]

Horizontal stretch:

The general rule to shift the graph f(x) to shift c units to the left is [tex]g(x)=f(x+c)[/tex]

From, the graph, it is obvious that the graph f(x) is shifted 2 units to the left.

Thus, applying the above rule, we have;

[tex]g(x)=(x+2)^2[/tex]

Equation of the new function g(x):

From the figure, it is obvious that the graph g(x) is shifted 1 unit upwards and 2 units to the left.

Thus, we have;

[tex]g(x)=(x+2)^2+1[/tex]

Therefore, the equation of the new function g(x) is [tex]g(x)=(x+2)^2+1[/tex]

Answer:

It is shown in the explanation

Step-by-step explanation:

p(s) = s² + bs + c

a = 1

b = b

c = c

We get Δ as follows

Δ = (b²-4*a*c) = b² - 4*1*c = b² - 4c > 0  ⇒   b² > 4c  ⇔ c > 0

s = (-b + √(b² - 4c))/2(1)

⇒   s₁ = (-b + √(b² - 4c))/2

s₂ = (-b - √(b² - 4c))/2(1)

⇒   s₂ = (-b - √(b² - 4c))/2

We have that -b < 0  ⇔ b > 0

then s₁ < 0 and s₂ < 0 ⇔ c > 0 and b > 0

Answer:

For roots to lie on the left half plane, b ⊃ 0 and c ⊃0

Step-by-step explanation:

From quadratic formula, we have;

x = -b±√(b²-4ac)/2a

From the given expression p(s) = s² + bs + c,

x = s

a = 1

b = b

c = c

The quadratic formula can then be written as;

s =  -b±√(b²-4*1*c)/2*1

   =  -b±√(b²-4c)/2

s₁ =  -b+√(b²-4c)/2

s₂ =  -b±√(b²-4c)/2

From the equation above,

Sum of root = -b

Product of root = c

If both the root lie on left side of the s-plane, then sum of roots will be negative. Hence, -b ∠0. That is, b ⊃0

Also, the product root will be positive, c ⊃ 0

Hence, for roots to lie on the left half plane, b ⊃ 0 and c ⊃0

Given:

Given that the table with values of x and y.

We need to determine the relationship between x and y.

Slope:

The slope of the relation can be determined using the formula,

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Substituting the points (2,11) and (4,9), we get;

[tex]m=\frac{9-11}{4-2}[/tex]

[tex]m=\frac{-2}{2}[/tex]

[tex]m=-1[/tex]

Thus, the slope of the relation is m = -1.

y - intercept:

The y - intercept of the relation is the value of y when x = 0.

Hence, from the table, it is obvious that when x = 0, the value of y is 13.

Thus, the y - intercept of the relation is b = 13.

Relationship between x and y:

The relationship between x and y can be determined using the formula,

[tex]y=mx+b[/tex]

Substituting m = -1 and y =13, we get;

[tex]y=-x+13[/tex]

Thus, the relationship between x and y is [tex]y=-x+13[/tex]

Use Hooke's law... (just kidding)

Break down each force vector into horizontal and vertical components.

[tex]\vec F_1=(1000\,\mathrm N)(\cos30^\circ\,\vec x+\sin30^\circ\,\vec y)\approx(866.025\,\mathrm N)\,\vec x+(500\,\mathrm N)\,\vec y[/tex]

[tex]\vec F_2=(1500\,\mathrm N)(\cos160^\circ\,\vec x+\sin160^\circ\,\vec y)\approx(-1409.54\,\mathrm N)\,\vec x+(513.03\,\mathrm N)\,\vec y[/tex]

[tex]\vec F_3=(750\,\mathrm N)(\cos195^\circ\,\vec x+\sin195^\circ\,\vec y)\approx(-724.444\,\mathrm N)\,\vec x+(-194.114\,\mathrm N)\,\vec y[/tex]

The resultant force is the sum of these vectors,

[tex]\vec F=\displaystyle\sum_{i=1}^3\vec F_i\approx(-1267.96\,\mathrm N)\,\vec x+(818.916\,\mathrm N)\,\vec y[/tex]

and has magnitude

[tex]|\vec F|\approx\sqrt{(-1267.96\,\mathrm N)^2+(818.916\,\mathrm N)^2}\approx1509.42\,\mathrm N[/tex]

The closest answer is D.

To determine the magnitude of the resultant force acting on a hook when three forces are applied, you can use vector addition. If you have the information of the forces and the angles between them, you can calculate the resultant force using trigonometric functions.

To determine the magnitude of the resultant force when three forces act on a hook, you must realize that forces are vector quantities. This means that they have both a magnitude (how much force is being applied) and a direction (the direction the force is being applied in).

If the forces are concurrent (i.e., they act at the same point), one usually uses the parallelogram law or the triangle rule to find the resultant force. You can add two forces to create a resultant, then add the third force to that resultant to find the total resultant. If the forces and the angles between them are known, you can use trigonometric functions to calculate the resultant force.

For instance, if the three forces are F1, F2, and F3, and the angles between them are θ1, θ2, and θ3, the resultant force R can be found using the following equation:

R = √[ (F1 + F2cosθ2 + F3cosθ3)^2 + (F2sinθ2 + F3sinθ3)^2 ]

This equation will give the magnitude of the resultant force. Please note that to use this equation, you must have enough information about the forces and the angles between them.

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It's a parallelogram, opposite sides congruent.

6x - 7 = 2x + 9

4x = 16

x = 4

12 = y + 3

9 = y

Answer: x=4, y=9

Answer:

x = 4 and y = 9

Step-by-step explanation:

This is a parallelogram, which we can tell because of the arrows. Basically, opposite sides are parallel. By definition, then, opposite sides of this polygon are equal: LM = ON and LO = MN. That means we can set the various expressions equal to each other:

LM = ON  ⇒  6x - 7 = 2x + 9  ⇒  4x = 16  ⇒  x = 4

LO = MN  ⇒  12 = y + 3  ⇒  y = 9

Thus, x = 4 and y = 9.

Hope this helps!

Answer:

[tex]z=\frac{0.253 -0.24}{\sqrt{\frac{0.24(1-0.24)}{75}}}=0.264[/tex]  

[tex]p_v =2*P(z>0.264)=0.792[/tex]  

So the p value obtained was a very high value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of students said that blue is their favorite color is not different from 0.24

Step-by-step explanation:

Data given and notation

n=75 represent the random sample taken

X=19 represent the students said that blue is their favorite color

[tex]\hat p=\frac{19}{75}=0.253[/tex] estimated proportion of  students said that blue is their favorite color

[tex]p_o=0.24[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true proportion is different from 0.24.:  

Null hypothesis:[tex]p=0.24[/tex]  

Alternative hypothesis:[tex]p \neq 0.24[/tex]  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.253 -0.24}{\sqrt{\frac{0.24(1-0.24)}{75}}}=0.264[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z>0.264)=0.792[/tex]  

So the p value obtained was a very high value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of students said that blue is their favorite color is not different from 0.24

Answer:

0.792

Step-by-step explanation:

Answer:

a) 30.50% probability that  at least 150 stay on the line for more than one minute.

b) 0% probability that more than 200 stay on the line for more than one minute.

Step-by-step explanation:

Binomial probability distribution

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

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

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.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].

In this problem, we have that:

[tex]n = 800, p = 0.18[/tex]

So

[tex]\mu = E(X) = np = 800*0.18 = 144[/tex]

[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{800*0.18*0.82} = 10.87[/tex]

a. at least 150 stay on the line for more than one minute.

Using continuity correction, [tex]P(X \geq 150 - 0.5) = P(X \geq 149.5)[/tex], which is 1 subtracted by the pvalue of Z when X = 149.5. So

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

[tex]Z = \frac{149.4 - 144}{10.87}[/tex]

[tex]Z = 0.51[/tex]

[tex]Z = 0.51[/tex] has a pvalue of 0.6950

1 - 0.6950 = 0.3050

30.50% probability that  at least 150 stay on the line for more than one minute.

b. more than 200 stay on the line.

Using continuity correction, [tex]P(X \geq 200 + 0.5) = P(X \geq 200.5)[/tex], which is 1 subtracted by the pvalue of Z when X = 200.5. So

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

[tex]Z = \frac{200.5 - 144}{10.87}[/tex]

[tex]Z = 5.2[/tex]

[tex]Z = 5.2[/tex] has a pvalue of 1

1 - 1 = 0

0% probability that more than 200 stay on the line for more than one minute.

Step-by-step explanation:

Given, the face of a clock is divided into 12 equal parts.

Angle of each part = [tex]\frac{360}{12}[/tex] = 30°

(i) When one hand points at 2 and the other points at 4, this is can be divided into two parts, 2 to 3 and 3 to 4.

The angle formed = 2 (30) = 60°

Option (i) is correct

(ii) The circumference of the clock is ,

Circumference of circle = 2πr,

where r is the radius = 6 and π = 3.14.

Substituting the values in the formula, we get

Circumference of circle = 37.68.

Option (ii) is wrong.

(iii) With one hand at 5 and the other at 10, this is 5 parts

The angle formed= 30(5) = 150°.  

The arc length =[tex]\frac{150}{360}[/tex](37.68) = 15.7

Option (iii) is correct

(iv) When one hand points at 1 and the other points at 9, this is 4 parts,

30(4) = 120°.  T

Option (iv) is wrong

(v) The length of the minor arc from 11 to 2, this is 3 parts

3(30) = 90°  

minor arc from 7 to 10 is 3(30) = 90°  

Option (v) is correct

Answer: options 1,3,5

Answer:

i think it's -0.0608

Step-by-step explanation:

The Answer is : ABC = 40

Answer:

it is 61.3

Step-by-step explanation:

Answer:

Approximately 20,579 units.

Answer:

8b+7p=20

p=2b

Step-by-step explanation:

You're welcome. Thou shall complete thou work without any trouble.

Answer:

a. length  = 0.7211 ft

b. width  = 0.7211 ft

c. height = 140.3846 ft

Step-by-step explanation:

This is an optimiztion with restriction problem.

We have to minimize the cost, with the restriction of the volume being equal to 72 ft3.

As the cost for the sides is constant, we know that length and width are equal.

Then, we can express the volume as:

[tex]V=x\cdot y\cdot z=x^2z=73[/tex]

being x: length and z: height

We can express the height in function of the length as:

[tex]x^2z=73\\\\z=73x^{-2}[/tex]

Then, the cost of the box can be expressed as:

[tex]C=0.62(x^2)+4*0.15(xz)+0.18(x^2)=(0.62+0.18)x^2+0.60xz\\\\C=0.8x^2+0.60x*x^{-2}=0.8x^2+0.6x^{-1}[/tex]

To optimize C, we derive and equal to zero

[tex]\dfrac{dC}{dx}=\dfrac{d}{dx}[0.8x^2+0.6x^{-1}]=1.6x-0.6x^{-2}=0\\\\\\1.6x=0.6x^{-2}\\\\x^{1+2}=0.6/1.6=0.375\\\\x=\sqrt[3]{0.375} =0.7211[/tex]

The height z is then

[tex]z=73x^{-2}=\dfrac{73}{0.7211^2}=\dfrac{73}{0.52}=140.3846[/tex]

Answer:

41.25$

Step-by-step explanation:

Answer:

41.25

Step-by-step explanation:

55/4=13.75

So 13.75 is 25% of 55

So then you would do 55-13.75

Because that is 25% off

The final answer would be $41.25 for the sale price.

Answer:

43.62% probability that it will take more than an additional 34 minutes

Step-by-step explanation:

To solve this question, we need to understand the exponential distribution and the conditional probability formula.

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

[tex]f(x) = \mu e^{-\mu x}[/tex]

In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.

The probability that x is lower or equal to a is given by:

[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]

Which has the following solution:

[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]

The probability of finding a value higher than x is:

[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]

Conditional probability formula:

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

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Taking more than 24 minutes.

Event B: Taking ore than 24+34 = 58 minutes.

P(A)

More than 24, use the exponential distribution.

Mean of 41, so [tex]m = 41, \mu = \frac{1}{41} = 0.0244[/tex]

[tex]P(A) = P(X > 24) = e^{-0.0244*24} = 0.5568[/tex]

Intersection:

More than 24 and more than 58, the intersection is more than 58. So

[tex]P(A \cap B) = P(X > 58) = e^{-0.0244*58} = 0.2429[/tex]

Then:

[tex]P(B|A) = \frac{0.2429}{0.5568} = 0.4362[/tex]

43.62% probability that it will take more than an additional 34 minutes

Answer:

Step-by-step explanation:

if you have 1/4 of a rope and you need to give 7/16 to your friend how much rope did you give to your friend?

Final answer:

Divide 1/4 by the reciprocal of 7/16 to get 4/7. A real-world example is when needing ¼ cup of sugar with only a 7/16 cup measure, fill it approximately 4/7 full to obtain the needed amount of sugar.

Explanation:

To calculate 1/4 divided by 7/16, you would multiply 1/4 by the reciprocal of 7/16, which is 16/7. This would give you (1/4) * (16/7) = 16/28, which can be simplified to 4/7 after dividing both numerator and denominator by 4. A real-world situation involving this expression could be as follows: Imagine you have a recipe that requires 1/4 of a cup of sugar, but you only have a measuring cup that measures 7/16 of a cup.

To find out how many times you need to fill the 7/16 cup to get the 1/4 cup needed, you would calculate 1/4 divided by 7/16, which will give you 4/7. So, you would fill the 7/16 measuring cup approximately 4/7 of the way full to have 1/4 cup of sugar for your recipe.

Answer:

[tex]3,125[/tex]

Step-by-step explanation:

If you want to fill a pool, you will use the formula for finding the volume:

[tex]v=l*w*h[/tex]

In this case, height being depth:

[tex]v=l*w*d[/tex]

Insert values

[tex]v=50*25*2.5[/tex]

Simplify

[tex]v=1,250*2.5\\v=3,125[/tex]

You would need a lot of water.

Answer:

3125000 liter

Step-by-step explanation:

hope i helped

if i can be brainliest that would be great

Answer:

61 degrees

Step-by-step explanation:

87+32=119

180 (total degrees for triangle)-119=67 degrees

Answer:

a) The null hypothesis is represented as

H₀: p ≥ 0.08

The alternative hypothesis is represented as

Hₐ: p < 0.08

b) A type I error for this question would be that

we conclude that the proportion of defective products is less than 8% when in reality, the proportion of defective products, is more than or equal to 8%.

c) At most, the number of defective products in the sample for you to agree to use the new product = 7

d) If minimum of 5000 pieces are purchased, 90% confidence interval for minimum number of flawed pieces will be (103, 497)

Step-by-step explanation:

For hypothesis testing, the first thing to define is the null and alternative hypothesis.

The null hypothesis plays the devil's advocate and is usually stating the opposite of the theory is being tested. It usually maintains that random chance is responsible for the outcome or results of any experimental study/hypothesis testing. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis takes the other side of the hypothesis; that there is indeed a significant difference between two proportions being compared. It usually confirms the the theory being tested by the experimental setup. It usually maintains that other than random chance, there are significant factors affecting the outcome or results of the experimental study/hypothesis testing. It usually contains the signs ≠, < and > depending on the directions of the test

For this question, we want to prove that less than 8% of the products we subsequently purchase will be defective.

So, the null hypothesis will be that there is not enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective. That is, the proportion of the sample that are defective is more than or equal to 8%.

And the alternative hypothesis is that there is enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective.

Mathematically,

The null hypothesis is represented as

H₀: p ≥ 0.08

The alternative hypothesis is represented as

Hₐ: p < 0.08

b) A type I error involves rejecting the null hypothesis and accepting the alternative hypothesis when in reality, the null hypothesis is true. It involves saying that there is enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective when in reality, there isn't enough evidence to arrive at this conclusion.

That is, the proportion of defective products in reality, is more than or equal to 8% and we have concluded that the proportion is less than 8%.

c) Out of the 100 samples provided by the manufacturer, at most how many can be defective for you to agree to use the new product?

The engineer agrees to use the new product when less than 8% of the products we subsequently purchase will be defective.

8% of the product = 0.08 × 100 = 8.

Meaning that the engineer agrees to subsequently purchase the product if less than 8 out of 100 are defective.

So, the maximum number of defective product in the sample that will still let the engineer purchase the products will be 7.

(d) For better or worse, your boss convinces you to go through with the deal. Turns out the minimum order is 5000 pieces. Assuming you purchase that many pieces of the new product, and that you found 6 defective pieces out of the 100, generate a 90% two-sided confidence interval for the number of pieces that will be flawed.

Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample proportion) ± (Margin of error)

Sample proportion = 0.495

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error)

Critical value at 90% confidence interval for sample size of 100 using the t-tables since information on the population standard deviation.

Degree of freedom = n - 1 = 100 - 1 = 99

Significance level = (100-90)/2 = 5% = 0.05

Critical value = t(0.05, 99) = 1.660

Standard error of the mean = σₓ = √[p(1-p)/n]

p = 0.06

n = sample size = 100

σₓ = (0.06/√100) = 0.006

σₓ = √[0.06(0.94)/100] = 0.0237486842 = 0.02375

90% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]

CI = 0.06 ± (1.660 × 0.02375)

CI = 0.06 ± 0.039425

90% CI = (0.020575, 0.099425)

90% Confidence interval = (0.0206, 0.0994)

If minimum of 5000 pieces are purchased, 90% confidence interval for minimum number of flawed pieces will be

5000 × (0.0206, 0.0994) = (103, 497)

Hope this Helps!!!

(a) The null hypothesis [tex]\(H_0\)[/tex] is that the proportion of defective products is 8% or less (b)  A Type I error occurs when the null hypothesis is true (the actual proportion of defective products is 8% or less), but we incorrectly reject it (c) at most 12 defective products can be found in the sample for the deal to proceed. (d) the 90% two-sided confidence interval for the number of defective pieces in the order of 5000 is from 105 to 496.

(a) The null hypothesis [tex]\(H_0\)[/tex] is that the proportion of defective products is 8% or less. The alternative hypothesis [tex]\(H_1\)[/tex] is that the proportion of defective products is greater than 8%. Mathematically, this can be expressed as:

[tex]\(H_0: p \leq 0.08\) \(H_1: p > 0.08\)[/tex]

(b) A Type I error occurs when the null hypothesis is true (the actual proportion of defective products is 8% or less), but we incorrectly reject it, concluding that the proportion of defective products is greater than 8%. This would mean unnecessarily turning down a good deal and potentially incurring additional costs to find another supplier.

(c) To ensure a 90% confidence level with a maximum defective rate of 8%, we can use the binomial distribution to find the maximum number of defective products allowed in the sample of 100. The formula for a binomial confidence interval is given by:

[tex]\(n \cdot p \pm Z_{\alpha/2} \sqrt{n \cdot p \cdot (1 - p)}\)[/tex]

where [tex]\(n\)[/tex] is the sample size, [tex]\(p\)[/tex] is the defect rate, and [tex]\(Z_{\alpha/2}\)[/tex] is the Z-score corresponding to the desired confidence level. For a 90% confidence level, [tex]\(Z_{\alpha/2} = 1.645\)[/tex]. Plugging in the values:

[tex]\(100 \cdot 0.08 \pm 1.645 \sqrt{100 \cdot 0.08 \cdot (1 - 0.08)}\)[/tex]

[tex]\(8 \pm 1.645 \sqrt{100 \cdot 0.08 \cdot 0.92}\)[/tex]

[tex]\(8 \pm 1.645 \sqrt{7.36}\)[/tex]

[tex]\(8 \pm 1.645 \cdot 2.713\)[/tex]

[tex]\(8 \pm 4.46\)[/tex]

The interval is from [tex]\(8 - 4.46\) to \(8 + 4.46\)[/tex], which gives us a range from approximately 3.54 to 12.46. Since we cannot have a fraction of a defective product, we round down to 3. Therefore, at most 12 defective products can be found in the sample for the deal to proceed.

(d) To generate a 90% two-sided confidence interval for the number of defective pieces out of 5000, given that 6 defective pieces were found out of 100, we first calculate the sample proportion of defective products:

[tex]\(\hat{p} = \frac{6}{100} = 0.06\)[/tex]

The formula for the confidence interval is:

[tex]\(\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)[/tex]

where [tex]\(n\)[/tex] is the sample size (100 in this case), and [tex]\(Z_{\alpha/2}\)[/tex] is the Z-score for a 90% confidence level (1.645). Plugging in the values:

[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.06(1 - 0.06)}{100}}\)[/tex]

[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.06 \cdot 0.94}{100}}\)[/tex]

[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.0564}{100}}\)[/tex]

[tex]\(0.06 \pm 1.645 \cdot \sqrt{0.000564}\)[/tex]

[tex]\(0.06 \pm 1.645 \cdot 0.0237\)[/tex]

[tex]\(0.06 \pm 0.0391\)[/tex]

The interval is from [tex]\(0.06 - 0.0391\) to \(0.06 + 0.0391\)[/tex], which gives us a range from approximately 0.0209 to 0.0991. To find the number of defective pieces in the order of 5000, we multiply these proportions by 5000:

Lower bound: [tex]\(0.0209 \cdot 5000 = 104.5\)[/tex](round to 105)

Upper bound: [tex]\(0.0991 \cdot 5000 = 495.5\)[/tex] (round to 496)

Therefore, the 90% two-sided confidence interval for the number of defective pieces in the order of 5000 is from 105 to 496.

Answer:

The 95% confidence interval for the proportion of young adults with pierced tongues who have receding gums is (0.24, 0.506).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

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

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 51, \pi = \frac{19}{51} = 0.373[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.373 - 1.96\sqrt{\frac{0.373*0.627}{51}} = 0.24[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.373 + 1.96\sqrt{\frac{0.373*0.627}{51}} = 0.506[/tex]

The 95% confidence interval for the proportion of young adults with pierced tongues who have receding gums is (0.24, 0.506).

Answer:

The sample size is smaller than 30, so we need to assume that the underlying population is normally distributed.

The  sampling distribution of x overbar will be approximately normally distributed with mean 63 and standard deviation 5.69.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem

The sample size is smaller than 30, so we need to assume that the underlying population is normally distributed.

If it is:

[tex]\mu = 63, \sigma = 18, n = 10, s = \frac{18}{\sqrt{10}} = 5.69[/tex]

The  sampling distribution of x overbar will be approximately normally distributed with mean 63 and standard deviation 5.69.

Answer:

x = 4

Step-by-step explanation:

14x - 15 + 139 = 180

(Alternate & Supplementary angles)

14x = 56

x = 4

The day when they would meet first time after joining the gym together will be 12.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

Kirk goes to the gym every 3 days.

Deshawn goes to the gym every 4 days.

If they join the gym on the same day.

Then the day when they would meet first time after joining the gym together will be

LCM of 4, 3 will be 12.

Then the day will be 12.

More about the Algebra link is given below.

https://brainly.com/question/953809

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Final answer:

Kirk and Deshawn will be at the gym together on the 12th day since they joined.

Explanation:

Gym memberships for Kirk and Deshawn occur every 3 days and 4 days respectively. To find the first day they'll be at the gym together, we need to find the lowest common multiple of 3 and 4.

LCM(3, 4) = 12. Therefore, Kirk and Deshawn will be at the gym together on the 12th day since they joined the gym.