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andrezito [222]

3 years ago

11

With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the en

tire batch is accepted if every item in the sample is okay. The ABC Electronics Company has just manufactured 1950 write-rewrite CDs, and 120 are defective. If 4 of these CDs are randomly selected for testing, what is the probability that the entire batch will be accepted?

1 answer:

zysi [14]3 years ago

4 0

Answer:

0.7755

Step-by-step explanation:

If 120 CDs are defective, the number of non-defective CDs in the batch is:

$n=1950-120\\n=1830$

In order for the batch to be accepted, all of the 4 CDs in the sample must not be defective, the probability of that happening is:

$P=\frac{1830}{1950}*\frac{1829}{1949} *\frac{1828}{1948} *\frac{1827}{1947}\\P=0.7755$

The probability that the entire batch will be accepted is 0.7755.

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Particle P moves along the y-axis so that its position at time t is given by y(t)=4t−23 for all times t. A second particle, part

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a) The limit of the position of particle $Q$ when time approaches 2 is $-\pi$ .

b) The velocity of particle $Q$ is $v_{Q}(t) = \frac{2\pi\cdot \cos \pi t-\pi\cdot t \cdot \cos \pi t -\sin \pi t}{(2-t)^{2}}$ for all $t \ne 2$ .

c) The rate of change of the distance between particle $P$ and particle $Q$ at time $t = \frac{1}{2}$ is $\frac{4\sqrt{82}}{9}$ .

<h3>How to apply limits and derivatives to the study of particle motion</h3>

a) To determine the limit for $t = 2$ , we need to apply the following two <em>algebraic</em> substitutions:

$u = \pi t$ (1)

$k = 2\pi - u$ (2)

Then, the limit is written as follows:

$x(t) = \lim_{t \to 2} \frac{\sin \pi t}{2-t}$

$x(t) = \lim_{t \to 2} \frac{\pi\cdot \sin \pi t}{2\pi - \pi t}$

$x(u) = \lim_{u \to 2\pi} \frac{\pi\cdot \sin u}{2\pi - u}$

$x(k) = \lim_{k \to 0} \frac{\pi\cdot \sin (2\pi-k)}{k}$

$x(k) = -\pi\cdot \lim_{k \to 0} \frac{\sin k}{k}$

$x(k) = -\pi$

The limit of the position of particle $Q$ when time approaches 2 is $-\pi$ . $\blacksquare$

b) The function velocity of particle $Q$ is determined by the <em>derivative</em> formula for the division between two functions, that is:

$v_{Q}(t) = \frac{f'(t)\cdot g(t)-f(t)\cdot g'(t)}{g(t)^{2}}$ (3)

Where:

$f(t)$ - Function numerator.
$g(t)$ - Function denominator.
$f'(t)$ - First derivative of the function numerator.
$g'(x)$ - First derivative of the function denominator.

If we know that $f(t) = \sin \pi t$ , $g(t) = 2 - t$ , $f'(t) = \pi \cdot \cos \pi t$ and $g'(x) = -1$ , then the function velocity of the particle is:

$v_{Q}(t) = \frac{\pi \cdot \cos \pi t \cdot (2-t)-\sin \pi t}{(2-t)^{2}}$

$v_{Q}(t) = \frac{2\pi\cdot \cos \pi t-\pi\cdot t \cdot \cos \pi t -\sin \pi t}{(2-t)^{2}}$

The velocity of particle $Q$ is $v_{Q}(t) = \frac{2\pi\cdot \cos \pi t-\pi\cdot t \cdot \cos \pi t -\sin \pi t}{(2-t)^{2}}$ for all $t \ne 2$ . $\blacksquare$

c) The vector <em>rate of change</em> of the distance between particle P and particle Q ( $\dot r_{Q/P} (t)$ ) is equal to the <em>vectorial</em> difference between respective vectors <em>velocity</em>:

$\dot r_{Q/P}(t) = \vec v_{Q}(t) - \vec v_{P}(t)$ (4)

Where $\vec v_{P}(t)$ is the vector <em>velocity</em> of particle P.

If we know that $\vec v_{P}(t) = (0, 4)$ , $\vec v_{Q}(t) = \left(\frac{2\pi\cdot \cos \pi t - \pi\cdot t \cdot \cos \pi t + \sin \pi t}{(2-t)^{2}}, 0 \right)$ and $t = \frac{1}{2}$ , then the vector rate of change of the distance between the two particles:

$\dot r_{P/Q}(t) = \left(\frac{2\pi \cdot \cos \pi t - \pi\cdot t \cdot \cos \pi t + \sin \pi t}{(2-t)^{2}}, -4 \right)$

$\dot r_{Q/P}\left(\frac{1}{2} \right) = \left(\frac{2\pi\cdot \cos \frac{\pi}{2}-\frac{\pi}{2}\cdot \cos \frac{\pi}{2} +\sin \frac{\pi}{2}}{\frac{3}{2} ^{2}}, -4 \right)$

$\dot r_{Q/P} \left(\frac{1}{2} \right) = \left(\frac{4}{9}, -4 \right)$

The magnitude of the vector <em>rate of change</em> is determined by Pythagorean theorem:

$|\dot r_{Q/P}| = \sqrt{\left(\frac{4}{9} \right)^{2}+(-4)^{2}}$

$|\dot r_{Q/P}| = \frac{4\sqrt{82}}{9}$

The rate of change of the distance between particle $P$ and particle $Q$ at time $t = \frac{1}{2}$ is $\frac{4\sqrt{82}}{9}$ . $\blacksquare$

<h3>Remark</h3>

The statement is incomplete and poorly formatted. Correct form is shown below:

<em>Particle </em> $P$ <em> moves along the y-axis so that its position at time </em> $t$ <em> is given by </em> $y(t) = 4\cdot t - 23$ <em> for all times </em> $t$ <em>. A second particle, </em> $Q$ <em>, moves along the x-axis so that its position at time </em> $t$ <em> is given by </em> $x(t) = \frac{\sin \pi t}{2-t}$ <em> for all times </em> $t \ne 2$ <em>. </em>

<em />

<em>a)</em><em> As times approaches 2, what is the limit of the position of particle </em> $Q?$ <em> Show the work that leads to your answer. </em>

<em />

<em>b) </em><em>Show that the velocity of particle </em> $Q$ <em> is given by </em> $v_{Q}(t) = \frac{2\pi\cdot \cos \pi t-\pi\cdot t \cdot \cos \pi t +\sin \pi t}{(2-t)^{2}}$ <em>.</em>

<em />

<em>c)</em><em> Find the rate of change of the distance between particle </em> $P$ <em> and particle </em> $Q$ <em> at time </em> $t = \frac{1}{2}$ <em>. Show the work that leads to your answer.</em>

To learn more on derivatives, we kindly invite to check this verified question: brainly.com/question/2788760

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2 years ago

Find the sum <br> 12x^2+9x^2

katen-ka-za [31]

Let's simplify step-by-step.
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</span></span></span></span><span>=<span>(<span><span>12<span>x^2</span></span>+<span>9<span>x^2</span></span></span>)
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3 years ago

Find the midpoint of a line<br> segment with endpoints of<br> (-5,9) and (7,-3).

Amiraneli [1.4K]

Answer:

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3 years ago

A poll of 1068 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At a

zubka84 [21]

Answer:

Z = -1.333

P-value = 0.09176

Decision Rule: Reject $H_o$ if ∝ is greater than the P-value

Conclusion: Since P-value is > the level of significance ∝, we fail to reject the null hypothesis, therefore there is insufficient evidence to conclude that at least half of all voters prefer the Democrat.

Step-by-step explanation:

Given that:

The sample size of the poll = 1068

The proportion of voters that preferred Democratic candidate is $\hat p$ = 0.48

To test the claim that at least half of all voters prefer the Democrat, i.e 1/2 = 0.5

The null hypothesis and the alternative hypothesis can be computed as:

$H_o : p \geq 0.50$

$H_1 : p < 0.50$

Using the Z test statistics which can be expressed by the formula:

$Z = \dfrac{\hat p - p}{\dfrac{\sqrt{p(1-p) }}{n}}}$

$Z = \dfrac{0.48 - 0.5}{\dfrac{\sqrt{0.5(1-0.5) }}{1068}}}$

$Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.5(0.5) }}{1068}}}$

$Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.25}}{1068}}}$

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P-value = P(Z< -1.33)

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P-value = 0.09176

The level of significance ∝ = 0.05

Decision Rule: Reject $H_o$ if ∝ is greater than the P-value

Conclusion: Since P-value is > the level of significance ∝, we fail to reject the null hypothesis, therefore there is insufficient evidence to conclude that at least half of all voters prefer the Democrat.

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$\bf \textit{arc's length}\\\\
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3 years ago

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