Would you use a sample or a census to measure each of the following? (a) The number of cans of Campbell’s soup on your local supermarket's shelf today at 6:00 p.m. (b) The proportion of soup sales last week in Boston that was sold under the Campbell's brand. (c) The proportion of Campbell’s brand soup cans in your family's pantry.

Answer:

d = 13 / 5 units          

Step-by-step explanation:

Given:

- Point P = (-2 , 3 )

- line : 3x - 4y + 5 = 0

Find:

- Use projection to show the given formula

- use the formula to find the distance from given point and line.

Solution:

- Let line L be defined as:

                                     a*x + b*y + c = 0

- Choose A as fixed point on the line L whose coordinates are ( m , k ).

- Now the coordinates of any point (x , y ) can be given by the line L:

                                    a*( x - m ) + b*( y - k ) + c = 0

- We did that by subtracting the coordinates (x,y) from (m,k).

- The above line can represented by dot product of constants (a , b ) to vector coordinates between points (x,y) to (m,k):

                                   ( a , b ) . ( x - m , y - k) = 0

- For above expression to be true i.e the dot product of two vectors is only zero when the vectors are orthogonal. So vector (a , b ) is always perpendicular to every vector in direction of line L

- The vector:

                                         v =  ( x - m , y - k)

- It may represent the line connecting the points (x_1 , y_1) and (m, k).

Then the distance d from the point P_1: (x_1, y_1) to the line L is given by the magnitude of the scalar projection of v onto a, because the latter is the length of the side of a right  triangle with two of the vertices being P_1 and

( m , k), and the other vertex lying on L So:

                                      d = | component v along x |

                                      d =  | v . ( a , b ) | / | (a , b ) |

                                      d = | (x_1 - m , y_1 - k ) . ( a , b)| / sqrt(a^2 + b^2)

                                      d = | ax_1 + by_1 - (a*m + b*k) | / sqrt(a^2 + b^2)

Where point A (m,k) lies on line hence;

                                      a*m + b*k + c = 0

                                      c = - (a*m + b*k)

Hence,

                                      d = | ax_1 + by_1 + c | / sqrt(a^2 + b^2)

- Given point P (-2,3) and line L : 3x - 4y + 5 = 0

where,

                                    (x_1 , y_1) = (-2 , 3 )

                                     (a , b) = ( 3 , -4 )

                                            c = 5

-Evaluate:

                                    d = | 3*(-2) + -4*3 + 5 | / sqrt(3^2 + 4^2)

                                    d = | - 13 | / 5

                                    d = 13 / 5 units                                    

The distance from the point (-2,3) to the line [tex]\(3x - 4y + 5 = 0\)[/tex] is [tex]\(\frac{|3(-2) - 4(3) + 5|}{\sqrt{3^2 + (-4)^2}}\)[/tex].

To find the distance from a point [tex]\(P_1(x_1, y_1)\)[/tex] to a line [tex]\(ax + by + c = 0\)[/tex], one can use the formula derived from scalar projection:

[tex]\[ \text{Distance} = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \][/tex]

Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the line, and [tex]\(x_1\)[/tex] and [tex]\(y_1\)[/tex] are the coordinates of the point. The numerator represents the absolute value of the expression [tex]\(ax_1 + by_1 + c\)[/tex], which is the scalar projection of the vector from the origin to the point onto the normal vector of the line. The denominator is the magnitude of the normal vector of the line.

Given the point [tex]\(P_1(-2, 3)\)[/tex] and the line [tex]\(3x - 4y + 5 = 0\)[/tex], we substitute [tex]\(x_1 = -2\)[/tex] and [tex]\(y_1 = 3\)[/tex] into the formula:

[tex]\[ \text{Distance} = \frac{|3(-2) - 4(3) + 5|}{\sqrt{3^2 + (-4)^2}} \][/tex]

Now, we calculate the numerator:

[tex]\[ 3(-2) - 4(3) + 5 = -6 - 12 + 5 = -13 + 5 = -8 \][/tex]

The absolute value of [tex]\(-8\)[/tex] is [tex]\(8\)[/tex], so the numerator becomes [tex]\(8\)[/tex].

Next, we calculate the denominator, which is the magnitude of the normal vector of the line:

[tex]\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]

Finally, we divide the absolute value we found in the numerator by the denominator to get the distance:

[tex]\[ \text{Distance} = \frac{8}{5} \][/tex]

Therefore, the distance from the point (-2,3) to the line [tex]\(3x - 4y + 5 = 0\)[/tex] is [tex]\(\frac{8}{5}\)[/tex].

Answer:

B. The number of girls is significantly low.

Step-by-step explanation:

Given that 200 births are randomly selected and 6 of the births are girls.

Normally we expect 50% of births to be boys and 50% girls.  If it is not exactly 50% atleast near to 50% we expect

But here out of 200 births, only 6 births are girls.

This means percentage of girls birth= [tex]\frac{6}{200} =0.03[/tex]=3%

while that of boys is 97%

This is a very unbalanced figure posing danger to the future generations.

B. The number of girls is significantly low.

The number of girls out of 200 randomly selected births is significantly low.

The number of girls out of a randomly selected group of 200 births is 6. To determine if this number is significantly high or low, we can use statistical methods. To do this, we can calculate the expected number of girls using probability. Assuming a 50% probability of having a boy or a girl, the expected number of girls would be 200 * 0.5 = 100. Comparing the observed number of girls (6) to the expected number (100), we can see that the number of girls is significantly low.

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Answer:

[tex]z=\frac{0.4 -0.48}{\sqrt{\frac{0.48(1-0.48)}{115}}}=-1.717[/tex]  

[tex]p_v =P(z<-1.717)=0.0429[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.1[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of people that have traded in their old car is lower than 0.48 or 48%.  

Step-by-step explanation:

Data given and notation

n=115 represent the random sample taken

X=46 represent the number of people that have traded in their old car.

[tex]\hat p=\frac{46}{115}=0.4[/tex] estimated proportion of people that have traded in their old car

[tex]p_o=0.48[/tex] is the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level

Confidence=90% or 0.9

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 proportion is less than 0.48.:  

Null hypothesis:[tex]p\geq 0.48[/tex]  

Alternative hypothesis:[tex]p < 0.48[/tex]  

When we conduct a proportion test we need to use the z statistic, 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{0.4 -0.48}{\sqrt{\frac{0.48(1-0.48)}{115}}}=-1.717[/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=0.1[/tex]. The next step would be calculate the p value for this test.  

Since is a left tailed test the p value would be:  

[tex]p_v =P(z<-1.717)=0.0429[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.1[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of people that have traded in their old car is lower than 0.48 or 48%.  

Answer:

z_{calc}=2.57[/tex]

Since is a two-sided test the p value would be:  

[tex]p_v =2*P(z>2.57)=0.0101[/tex]  

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is different from [tex]\mu_o[/tex] at 5% of signficance.  

Step-by-step explanation:

Data given and notation  

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

[tex]\sigma[/tex] represent the population standard deviation

[tex]n[/tex] sample size  

[tex]\mu_o [/tex] represent the value that we want to test  

Confidence = 95% or 0.95

[tex]\alpha=1-0.95=0.05[/tex] represent the significance level for the hypothesis test.  

z would represent the statistic (variable of interest)  

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

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is equal to an specified value [tex]\mu_o [/tex], the system of hypothesis would be:  

Null hypothesis:[tex]\mu =\mu_o[/tex]  

Alternative hypothesis:[tex]\mu \neq \mu_o[/tex]  

Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)  

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

We can replace in formula (1) and we got a value calculated let's say [tex] z_{calc}=2.57[/tex]

P-value  

Since is a two-sided test the p value would be:  

[tex]p_v =2*P(z>2.57)=0.0101[/tex]  

Conclusion  

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is different from [tex]\mu_o[/tex] at 5% of signficance.  

For a 95% confidence level in a two-tail hypothesis test, if the computed Z-value is 2.57, you would reject the null hypothesis, as the value lies outside the critical z-scores of ±1.96, indicating statistical significance.

If you use a 95% confidence level in a two-tail hypothesis test, the critical z-scores are ±1.96.

This means that if the computed test statistic falls beyond these values (either less than -1.96 or greater than +1.96), we reject the null hypothesis.

In the scenario where the computed value of the test statistic is Z = 2.57, it is greater than +1.96. Therefore, under these conditions, you will reject the null hypothesis.

The reason for this decision stems from the test statistic lying outside the range of values considered to be consistent with the null hypothesis at the 95% confidence level.

The absolute value of 2.57 is greater than 1.96, indicating that the observed effect is statistically significant, and the likelihood of observing such a result if the null hypothesis were true is less than 5% (which is the alpha level of 0.05 associated with a 95% confidence level).

This indicates that there is sufficient evidence to support the alternative hypothesis over the null hypothesis in a two-tailed test.

Answer:

Correct option: c. Double blind.

Step-by-step explanation:

A double blind experiment is an experiment where the participants are divided into two groups: one is the experimental group and the other is a control group. The participants in the experimental group are provided with a treatment and those in the control group are not provided with the treatment but are given a placebo.

In this experiment neither the researcher nor the participants know to which group a participant is placed.

After the experiment the results for the two groups are compared and the conclusion is hence drawn.

Here the researcher randomly assigned babies to either usual care or to a whole-body cooling group. The experimental group is the cooling group and the control group is the group provided with usual treatment.

Thus, the researchers are conducting a double blind experiment to determine whether reducing body temperature for three days after birth increased the rate of survival without brain damage or not.

Thus, the correct option is (c).

Final answer:

The possible locations of the third vertex of the triangle with a perimeter of 12 units, given two vertices at (4, 0) and (8, 0), are (12, 0) and (0, 0). The vertex at (0, 0) will produce a triangle with the largest area equals to 8 square units.

Explanation:

Possible Locations of Third Vertex:

To find the possible locations of the third vertex of the triangle, we need to consider the perimeter of the triangle. Given that the perimeter is 12 units and two vertices are located at (4, 0) and (8, 0), we can deduce that the third vertex must lie on the line segment between these two points.

Calculating the Third Vertex:

Since the distance between the two given vertices is 8 - 4 = 4 units, the remaining distance to achieve a perimeter of 12 units would be 12 - 4 = 8 units.

Therefore, the possible locations of the third vertex are (4 + 8, 0) = (12, 0) or (8 - 8, 0) = (0, 0).

Determining the Largest Area:

To determine which vertex would produce a triangle with the largest area, we need to consider the base and the height of the triangle. Since both the possible vertices lie on the x-axis, the base would be the same for both triangles, which is 4 units.

The height of the triangle can be determined by finding the distance between the base and the y-coordinate of the third vertex. If the third vertex is (12, 0), the height would be 0 units. If the third vertex is (0, 0), the height would be 4 units.

Therefore, the triangle with the third vertex at (0, 0) would produce the largest area, which is equal to 1/2 * base * height = 1/2 * 4 * 4 = 8 square units.

The possible locations of the third vertex of the triangle form a line segment parallel to the x-axis and symmetric about the y-axis, extending from (4, 3) to (8, -3) or from (4, -3) to (8, 3). The triangle with the largest area will be the one where the third vertex is directly above or below the midpoint of the line segment joining the first two vertices, at either (6, 3) or (6, -3).

Given two vertices of a triangle at (4, 0) and (8, 0), the distance between them is the length of the line segment joining them, which is 8 - 4 = 4 units. Since the perimeter of the triangle is 12 units, the sum of the lengths of the other two sides must be 12 - 4 = 8 units.

Let's denote the third vertex as (x, y). The distance from (4, 0) to (x, y) is one side of the triangle, and the distance from (8, 0) to (x, y) is the other side. Using the distance formula, we have:

[tex]\[ \sqrt{(x - 4)^2 + y^2} + \sqrt{(x - 8)^2 + y^2} = 8 \][/tex]

To simplify the problem, we can look for points where the sum of the distances to the two fixed points is 8 units. Since the two given vertices are on the x-axis, the third vertex must be such that its x-coordinate is between 4 and 8 (inclusive) to form a triangle.

For the y-coordinate, we know that the sum of the distances from the third vertex to the two fixed points is 8 units. If we consider the case where y = 0, the third vertex would be on the x-axis, which would not form a triangle. Therefore, y must be non-zero.

Let's consider the two sides of the triangle formed by the third vertex and the two fixed points. The lengths of these sides can be thought of as the radii of two circles, one centered at (4, 0) and the other at (8, 0), both with a radius of 4 units (since each circle's radius is half the sum of the lengths of the two sides not on the x-axis).

The third vertex must lie on the intersection of these two circles. Since the circles are of equal radius and their centers are 4 units apart, they will intersect in two points, which will be equidistant from the line segment joining the first two vertices.

To find these points, we can set up the following system of equations based on the distance formula:

[tex]\[ (x - 4)^2 + y^2 = 16 \][/tex]

[tex]\[ (x - 8)^2 + y^2 = 16 \][/tex]

The line segment joining these two points, (6, 3) and (6, -3), represents all possible locations of the third vertex. This line segment is parallel to the x-axis and symmetric about the y-axis, and it extends from (4, 3) to (8, -3) or from (4, -3) to (8, 3), depending on which side of the x-axis the third vertex is located.

To find the vertex that produces the largest area, we note that for a given base length (which is fixed at 4 units in this case), the area of a triangle is maximized when the height is maximized. The height is maximized when the third vertex is directly above or below the midpoint of the base, which is at (6, 0). Therefore, the third vertex should be at either (6, 3) or (6, -3) to produce the triangle with the largest area. The area of this triangle is 1/2 * base * height = 1/2 * 4 * 3 = 6 square units.

We have found all possible locations of the third vertex because any point outside the line segment joining (6, 3) and (6, -3) would result in a perimeter greater than 12 units, and any point inside would result in a perimeter less than 12 units. Thus, the line segment from (4, 3) to (8, -3) or from (4, -3) to (8, 3) represents the complete set of solutions for the third vertex that satisfy the given perimeter constraint.

Answer:

a) [tex] \hat p = \frac{X}{n}= \frac{31}{72}= 0.431[/tex]

b) [tex]0.431 - 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.317[/tex]

[tex]0.431 + 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.545[/tex]

And the 95% confidence interval would be given (0.317;0.545).

Step-by-step explanation:

Part a

The best estimator for the population proportion is the sample proportion given by:

[tex] \hat p = \frac{X}{n}= \frac{31}{72}= 0.431[/tex]

Where X represent the adults in the sample that support the death penalty and n the sample size selected

Part b

The confidence interval would be given by this formula

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

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.96[/tex]

And replacing into the confidence interval formula we got:

[tex]0.431 - 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.317[/tex]

[tex]0.431 + 1.96 \sqrt{\frac{0.431(1-0.431)}{72}}=0.545[/tex]

And the 95% confidence interval would be given (0.317;0.545).

The sample proportion is approximately 0.431, and the 95% confidence interval for the population proportion is (0.317, 0.545), providing a range for the likely proportion of adults in Santa Clarita who support the death penalty based on the sample data.

In a random sample of 72 adults in Santa Clarita, California, the sample proportion ([tex]\hat{p}[/tex]) of adults who support the death penalty is calculated using the formula [tex]\hat{p}[/tex] = X/n, where X is the number of adults in the sample supporting the death penalty (31), and n is the sample size (72).

In this case, [tex]\hat{p}[/tex] = 31/72 ≈ 0.431, representing the estimated proportion of adults in Santa Clarita supporting the death penalty.

To construct a 95% confidence interval for the population proportion, the formula for the confidence interval is employed:

Confidence Interval = [tex]\left( \hat{p} - Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right)[/tex]

Here, Z is the z-score associated with a 95% confidence interval, which is approximately 1.96. Substituting the given values into the formula, the confidence interval is calculated as (0.317, 0.545), indicating that we can be 95% confident that the true proportion of adults in Santa Clarita supporting the death penalty lies within this range.

To learn more about sample proportion

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Answer:

[tex] 1.05-2.65 \frac{0.23}{\sqrt{65}} \approx 0.98[/tex]

[tex] 1.05+2.65 \frac{0.23}{\sqrt{65}} \approx 1.12[/tex]

So on this case the 99% confidence interval for the mean would be given by (0.98;1.12)

And the best option is:

C..98 to 1.12

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=1.05[/tex] represent the sample mean for the sample  

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

s=0.23 represent the sample standard deviation

n=65 represent the sample size  

Solution to the problem

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=65-1=64[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,64)".And we see that [tex]t_{\alpha/2}=2.65[/tex]

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

[tex] 1.05-2.65 \frac{0.23}{\sqrt{65}} \approx 0.98[/tex]

[tex] 1.05+2.65 \frac{0.23}{\sqrt{65}} \approx 1.12[/tex]

So on this case the 99% confidence interval for the mean would be given by (0.98;1.12)

And the best option is:

C..98 to 1.12

Answer:

-5n + 6

Step-by-step explanation:

It goes up in the - 5 times tables and you have to add 6 to get the real answer.

Find the difference between numbers to find the n. (-5n bit)

Answer:

L(x,y) = (2,-8,0) + (0,-1,1)*t

Step-by-step explanation:

for the planes

x + y + z = -6  and y + z = -8

the intersection can be found subtracting the equation of the planes

x + y + z - ( y + z ) = -6 - (-8)

x= 2

therefore

x=2

z=z

y= -8 - z

using z as parameter t and the point (2,-8,0) as reference point , then

x= 2

y= -8 - t

z= 0 + t

another way of writing it is

L(x,y) = (2,-8,0) + (0,-1,1)*t

Final answer:

The parametrization of the line where the planes x + y + z = -6 and y + z = -8 intersect is found by solving the equations together and using a point with z = 0. This leads to parametric equations x(t) = 2, y(t) = -8 - t, and z(t) = t.

Explanation:

To find a parametrization of the line in which the planes x + y + z = -6 and y + z = -8 intersect, we first solve these two equations together to find the relationship between x, y, and z. Since both equations involve y and z, we can set them equal to isolate x.

1. Subtract the second equation from the first to isolate x: x = 2.

2. Using the second equation y + z = -8, we express y in terms of z: y = -8 - z.

Now, to use a point with z = 0 to determine the parametrization, we plug z = 0 into our equations. This gives us x = 2 and y = -8 for the point (2, -8, 0).

With z as our parameter t, the parametrization of the line can be given as x = 2, y = -8 - t, and z = t. Therefore, the parametric equations describing the intersection line are x(t) = 2, y(t) = -8 - t, and z(t) = t.

Answer:

t = 45 cubic light years to find a star with this certainty.

Step-by-step explanation:

The Poisson random probability equation is given by:

[tex]P(k events in interval t)=\frac{(\lambda t)^{k}e^{-\lambda t}}{k!}[/tex]

We can use the next equation to quantify how many cubic light years of space must be studied so that the probability of one or more stars exceeds 0.94.

[tex]P(k\ge 1) \ge 0.94[/tex]

[tex]1-f(0)=1-\frac{(\frac{1}{16}*t)^{0}e^{-\frac{1}{16}*t}}{0!}=1-e^{-\frac{1}{16}*t}} \ge 0.94[/tex]

So, here we just need to solve it for t:

[tex]1-e^{-\frac{1}{16}*t}} \ge 0.94[/tex]

[tex]e^{-\frac{1}{16}*t}} \ge 0.06[/tex]

[tex]ln(e^{-\frac{1}{16}*t}}) \ge ln(0.06)[/tex]

[tex]-\frac{1}{16}*t \ge -2.8[/tex]

[tex]t \ge 44.8[/tex]

Therefore t = 45 cubic light years to find a star with this certainty.

I hope it helps you!

To find the volume of space where the probability of encountering one or more stars exceeds 0.94 in the Milky Way, use the inverse of the cumulative distribution function of the Poisson distribution, with the given star density of one per 16 cubic light years, and solve for the volume V such that the probability of zero stars is below 0.06.

In order to solve the problem of how many cubic light years of space must be studied so that the probability of one or more stars exceeds 0.94, we need to use the properties of the Poisson distribution. The Poisson distribution is often used for modeling the number of events in fixed intervals of time or space under certain conditions. If we let λ denote the average number of stars in a given volume, then for our Milky Way Galaxy in the vicinity of our solar system where the density is one star per 16 cubic light years, λ is 1/16 per cubic light year. The probability of finding no stars in a volume V is given by e^{-λV}. To find when the probability of one or more stars exceeds 0.94, we need to find V such that the probability of finding no stars is less than 0.06 (which is 1 - 0.94).

Let's calculate the volume V:

We first solve the inequality e^{-λV} < 0.06 for V.

Taking the natural logarithm of both sides gives us -λV < ln(0.06).

We then solve for V: V > ln(0.06) / -λ.

Substituting λ (1/16) gives us V > ln(0.06) / -(1/16).

Finally, calculate the numeric value and round to the nearest integer to find the minimum volume V that meets the requirement.

By performing these calculations, you can find how many cubic light years of space must be studied so that the probability of one or more stars exceeds 0.94.

Answer:

The required system of equations to the given parametric equations are:

5x1 - x2 = 6

x1 + x2 = -6

Step-by-step explanation:

Given the parametric equations:

x1 = t

x2 = -6 + 5t

Eliminating the parameter t, we obtain one of the equations of a system in two variables, x1 and x2 that has the solution set given by the parametric equations.

Doing that, we have:

5x1 - x2 = 6

Again a second equation can be a linear combination of x1 and x2

x1 + x2 = -6 + 6t

x1 + x2 = -6 (putting t=0)

And they are the required equations.

Answer:

  see attached

Step-by-step explanation:

The attached list shows base-10 numbers in the left column, followed by their equivalents in base 8, base 16, and base 12.

Counting is done in the usual way: when you come to the last digit of the particular base, you increment the next digit to the left, and start over.

The Octal numbers from 16 to 32 and the Hexadecimal numbers from 16 to 32 are listed. Additionally, numbers from 8 to 28 are provided in base 12 system, where A and B are two last digits representing 10 and 11 respectively.

The Octal numbers from 16 to 32 are as follows: 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 40.

The Hexadecimal numbers from 16 to 32 are: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20.

For base 12 numbers from 8 to 28, where A stands for 10 and B for 11, they are: 8, 9, A, B, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 20, 21, 22, 23.

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Answer:

The answer to the question is

Inferential statistics

Step-by-step explanation:

Inferential statistics is used to make informed conclusions about a population that cannot be completely sampled due to the population size.

With Inferential statistics, it is possible to make predictions or inferences from available data. It involves collecting data from a random sample of individuals within the population concerned and make generalizations about the entire population from those samples.

Answer: The answer to the question is

Inferential statistics

Step-by-step explanation:

Inferential statistics is used to make informed conclusions about a population that cannot be completely sampled due to the population size.

With Inferential statistics, it is possible to make predictions or inferences from available data. It involves collecting data from a random sample of individuals within the population concerned and make generalizations about the entire population from those samples.

Answer:

2√10 / 7

Step-by-step explanation:

to solve cos(sin^-1(3/7))

we break it into simpler terms

sin^-1(3/7) ------ these will be taken as an angle when dealing with cos

sin Ф = opposite / hypothenus = 3/7

Using Pythagoras Theorem

Hypothenus ² = opposite² + adjacent ²

7² = 3² + a²

49 = 9 + a²

a² = 49 - 9 = 40

a = √40 = √2x2x10 = 2√10

cos Ф = adjacent / hypothenus = 2√10 / 7  

cos(sin^-1(3/7)) = 2√10 / 7  

The value of the given expression is 2√10 / 7.

Trigonometry is the branch of mathematics that set up a relationship between the sides and angle of the right-angle triangles.

Trigonometric identities are equality conditions in trigonometry that hold for all values of the variables that appear and are defined on both sides of the equivalence. These are identities that, geometrically speaking, involve certain functions of one or more angles.

Solve cos(sin⁻¹(3/7)). Break it into simpler terms sin⁻¹(3/7) these will be taken as an angle when dealing with cos.

sin Ф = opposite / hypothenus = 3/7

Using Pythagoras Theorem

Hypothesis ² = opposite² + adjacent ²

7² = 3² + a²

49 = 9 + a²

a² = 49 - 9 = 40

a = √40 = √2x2x10 = 2√10

cos Ф = adjacent / hypothenus = 2√10 / 7  

cos(sin⁻¹(3/7)) = 2√10 / 7  

Hence, the value of cos(sin⁻¹(3/7)) will be 2√10 / 7.

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Answer: 0.0136

Step-by-step explanation:

Let events are:

A = alarm A will fail

B=  alarm B will fail

We have given ,

P(A) =  0.004  ,   P(B)=0.01

Since both events are independent , so

P(A and B) = P(A) x P(B)

= 0.04 x 0.01 =0.0004

i.e. P(A and B) =0.0004

Now , P(A or B) = P(A)+P(B)-P(A and B)

= 0.004+ 0.01-0.0004=0.0136

Hence, the probability that at least one alarm fails in the event of a fire is 0.0136 .

The probability that at least one alarm fails in the event of a fire is approximately 0.014 or 1.4%.

The probability that at least one alarm fails in the event of a fire is given by:

[tex]\[ P(\text{at least one fails}) = 1 - P(\text{both work}) \][/tex]

Since the failures are independent, the probability that both alarms work is the product of their individual probabilities of working:

[tex]\[ P(\text{both work}) = P(\text{A works}) \times P(\text{B works}) \][/tex]

The probability that alarm A works is the complement of the probability that it fails, which is [tex]\( 1 - 0.004 \)[/tex]. Similarly, the probability that alarm B works is  1 - 0.01 .

Thus, we have:

[tex]\[ P(\text{A works}) = 1 - 0.004 = 0.996 \] \[ P(\text{B works}) = 1 - 0.01 = 0.99 \][/tex]

Now we can calculate [tex]\( P(\text{both work}) \)[/tex]:

[tex]\[ P(\text{both work}) = 0.996 \times 0.99 \] \[ P(\text{both work}) = 0.98604 \][/tex]

Finally, we find the probability that at least one alarm fails:

[tex]\[ P(\text{at least one fails}) = 1 - 0.98604 \] \[ P(\text{at least one fails}) = 0.01396 \][/tex]

Answer:

Step-by-step explanation:

When data points are plotted between two variables, seeing the scatter plot we can form idea about the curve which is close to all points.

In other words, using least squares method, the curve of best fit would be the curve which has squares of deviations form the observed points a minimum

Thus conditions are

i) The scatter plot suggests a linear relationship

ii) Pearson correlation coefficient has value near to 1 or -1 suggesting linearlity

iii) There seems to be constant increase of y for unit increase in x

Under the above linear line fits

In all the other cases, quadratic or cubic function fits well.

Answer:

64bags of ice were sold when the temperature is 82 degree F

Step-by-step explanation:

From the equation ; Y =-192.20 + 3.13x

where Y = predicted number of bags of ice sold

x = temperature, in degrees Fahrenheit (degree)

To find Y when x = 82

substitute the value of x in the equation given

= Y = -192.20 + 3.13(82)

Y = 64.46 and approximately, Y = 64

To Compute the residual at this temperature;

residual = 67 - 64.46= 2.53 which is approximately 3

Answer:

Step-by-step explanation:

Prime numbers are numbers that are divisible by itself and '1'. This means prime numbers are numbers with just two factors while composite numbers are numbers that are divisible by more than two numbers, that is they have more than 2 factors.

Example of prime numbers are 5,7,11 etc. We see that these numbers are only divisible by themselves and by '1' while examples of composite numbers are 10, 15, 20, etc. We clearly see that these numbers have more than 2factors.

Prime numbers are divisible by only 2 numbers: 1 and themselves.

Composite numbers have 3 or more factors.

Hope this helps!

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[tex]GraceRosalia[/tex]

~Just a joyful teen

Answer:

The t-score is -1.8432    

Step-by-step explanation:

We are given the following in the question:  

98, 99.6, 97.8, 97.6, 98.7, 98.4, 98.9, 97.1, 99.2, 97.4, 99.1, 96.9, 98.8, 99.9, 96.8, 97, 98.7, 97.6, 98.7, 98.2

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{1964.4}{20} = 98.22[/tex]

Sum of squares of differences = 16.152

[tex]s = \sqrt{\dfrac{16.152}{49}} = 0.922[/tex]

Population mean, μ = 98.6

Sample mean, [tex]\bar{x}[/tex] = 98.22

Sample size, n = 20

Sample standard deviation, s = 0.922

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 98.6\\H_A: \mu < 98.6[/tex]

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{98.22 - 98.6}{\frac{0.922}{\sqrt{20}} } = -1.8432[/tex]

The t-score is -1.8432    

Final answer:

The t-score is calculated using the sample mean, the population mean under the null hypothesis, the sample's standard deviation, and the sample size. For the provided data, these values lead to a computed t-score of approximately -0.4594.

Explanation:

To calculate the t-score for a set of data when testing a hypothesis, you can use the following formula:

t = (X - \/u0) / (s/√n)

Where X is the sample mean, \/u0 is the population mean according to the null hypothesis, s is the sample standard deviation, and n is the sample size. To find the t-score, you need to know the sample mean (X), the standard deviation (s), and the sample size (n), all of which are usually provided in the problem or can be calculated from the data. In this case:

Plugging these values into the t-score formula gives us:

t = (98.59 - 98.6) / (0.0973/√20)

Perform the calculations:

t = -0.01 / (0.0973/4.4721)

t = -0.01 / 0.02176

t ≈ -0.4594

Thus, the calculated t-score is approximately -0.4594.

Answer:

OPTION B: 57x - 3

Step-by-step explanation:

Total tires produced is the sum of number of over-sized tires and the number of compact tires.

So, we have when x = 1, tires, y = 23 + 31 = 54

When x = 2, total tires y = 48 + 63  = 111

When x = 3, total tires y = 73 + 95  = 168

When x = 4, total tires y = 98 + 127 = 225

We can substitute the options to check which one will be the correct representation.

Option A: 55x - 1

When x = 1, y = 55(1) - 1    = 54

When x = 2, y = 55(2) - 1  = 109    [tex]$ \ne $[/tex]    111

So, this option is eliminated.

Option B: 57x - 1

When x = 1, y = 57(1) - 3    =   54

When x = 2, y = 57(2) - 3  =   111

When x = 3, y = 57(3) - 3  =   168

When x = 4, y = 57(4) - 3  =   225

These values exactly match with the values from the table. So, Option B is the right answer.

Option C: 110x - 2

When x = 1, y = 110(1) - 2 = 108  [tex]$ \ne $[/tex]  54

Hence, this option is incorrect.

Option D: 114x - 6

When x = 1, y = 114(1) - 6 [tex]$ \ne $[/tex]  54

Hence, this option is also eliminated.

Option B is the required answer.

Answer:

C. No, the probability of this outcome at 0.080, would be considered usual, so there is no problem

Step-by-step explanation:

To solve this problem, it is important to know the Central Limit Theorem and the Normal probability distribution.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/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 a probability is unusual?

A probability is said to be unusual if it is higher than 0.95 or lower than 0.05.

In this problem, we have that:

[tex]\mu = 20, \sigma = 0.11, n = 40, s = \frac{0.11}{\sqrt{15}} = 0.0284[/tex]

Should the machine be reset?

What is the probability of a mean of 20.04 or higher?

This is 1 subtracted by the pvalue of Z when X = 20.04. So

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

[tex]Z = \frac{20.04 - 20}{0.0284}[/tex]

[tex]Z = 1.41[/tex]

[tex]Z = 1.41[/tex] has a pvalue of 0.92.

So the probability of this outcome is 1-0.92 = 0.08 = 8%.

8% is still an usual probability.

So the correct answer is:

C. No, the probability of this outcome at 0.080, would be considered usual, so there is no problem

The sample mean of 20.04 ounces can be calculated using the z-score formula to determine if the machine should be reset. The probability of obtaining this sample mean is extremely close to 1, indicating it is likely to occur by random chance. Therefore, the machine does not need to be reset.

To determine if the machine should be reset, we can calculate the probability of obtaining a sample mean of 20.04 ounces or higher, assuming the machine is still set to the mean of 20 ounces. We can use the z-score formula to standardize the sample mean. The formula is z = (x - μ) / (σ / sqrt(n)), where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size. Plugging in the values, we have z = (20.04 - 20) / (0.11 / sqrt(15)) = 5.45.

Next, we can find the probability associated with this z-score using a z-table or a calculator. From the z-table, we find that the probability is approximately 0.99999999.

Since the probability is extremely close to 1, which means it is very likely to occur by random chance, we can conclude that the sample mean of 20.04 ounces is not significantly different from the mean of 20 ounces. Therefore, there is no need to reset the machine.

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Answer:

Volume of the solid generated = pi2/84 cubic unit

Step-by-step explanation:

The deatiled step and appropriate integration is donw as shown in the attached file.

To find the volume of the solid when the region bounded by the curves is revolved about the x-axis, use the disk method to integrate the cross-sectional areas of the disks formed. The limits of integration are determined by finding the intersection points of the curves. The formula for the cross-sectional area is A = πr^2, where r is the distance from the x-axis to the function y(x).

To find the volume of the solid when the region bounded by the curves is revolved about the x-axis, we can use the disk method. The disk method involves integrating the cross-sectional area of each disk formed by rotating a thin vertical strip of the region about the x-axis. The formula for the cross-sectional area of a disk is A = πr^2, where r is the distance from the x-axis to the function y(x).

First, we need to determine the limits of integration. The curves y = cos(21x) and y = 0 intersect at x = 0 and x = π/42. So we integrate from x = 0 to x = π/42.

The distance from the x-axis to the function y(x) is y(x) = cos(21x). Therefore, the cross-sectional area of each disk is A(x) = π[cos(21x)]^2. To find the volume, we integrate A(x) from x = 0 to x = π/42:

V = ∫0π/42π[cos(21x)]^2 dx

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Answer:

5

Step-by-step explanation:

The sample space is considered to be the total number of possibilities in a given sample or study. Here we are told that the bag contains two red marbles, two blue marbles, and one yellow marble. So the sample space is 5, the total number of marbles available and possible in a selection

Final answer:

The sample space for selecting two marbles without replacement from a bag with two red, two blue, and one yellow marble consists of the pairs RR, RB, RY, BR, BB, BY, YR, YB.

Explanation:

When we are picking two marbles without replacement from a bag containing two red marbles, two blue marbles, and one yellow marble, we are dealing with a probabilistic experiment. To find the sample space of this experiment, we need to list all possible pairs of marbles that could result from this process.

Here are the possible combinations without replacement:

Yellow and Blue (YB)

Note that combinations like Red and Blue (RB) and Blue and Red (BR) are distinct since the marbles are drawn one after the other. With this comprehensive listing, we have fulfilled the task of defining the sample space, which consists of the following pairs: RR, RB, RY, BR, BB, BY, YR, YB.

Answer:

The confidence interval is (0.444 ± 0.111).

Step-by-step explanation:

The general form of a confidence interval for single proportion is: [tex](\hat p- E<p<p\hat + E)=\hat p \pm E[/tex]

The interval provided is: (0.333 < p < 0.555)

Then

[tex]\hat p-E=0.333...(i)\\\hat p + E = 0.555...(ii)[/tex]

Solving the two equation simultaneously:

Add (i) and (ii),

[tex]2\hat p=0.888\\\hat p=\frac{0.888}{2}\\ =0.444[/tex]

Substitute the value of [tex]\hat p[/tex] in (i) to compute the value of E:

[tex]\hat p-E=0.333\\0.444-E=0.333\\E=0.444-0.333\\=0.111[/tex]

Thus, the confidence interval is,

[tex](0.444-0.111<p<0.444+0.111)=0.444 \pm 0.111[/tex]

Answer:

a. probability that exactly four sets detect the missile is 0.06561

probability that at least 1 set detect the missile is 0.99999

b. n = 3

Step-by-step explanation:

a. The probability that exactly 4 sets with probability of detection being 0.9 and 1 set fail with probability of 1 - 0.9 = 0.1 is

0.9*0.9*0.9*0.9*0.1 = 0.06561

The probability of having at least 1 set detect the missile is the inverse of the probability of having none of the set detecting the missile, which means all of the set fail to detect the missile, which is

0.1*0.1*0.1*0.1*0.1 = 0.00001

So the probability that at least 1 set detect the missile is

1 - 0.00001 = 0.99999

b. For the system to have a success rate of 0.999, this means at least 1 radar could detect the missile with probability of 0.999, which means all of them can fail with probability of 0.001. For this to happen:

[tex]0.1^n = 0.001[/tex]

[tex](10^{-1})^n = 10^{-3}[/tex]

[tex]10^{-1n} = 10^{-3}[/tex]

[tex]-n = -3[/tex]

[tex]n = 3[/tex]

You need 3 radars

The probability that exactly four out of five radar sets detect a missile is about 0.33, while the probability that at least one set detects it is almost 1 (0.99999). In order to achieve a detection probability of .999 or higher, there should be at least 11 radar sets.

The subject matter of this question is in the realm of probability and statistics, specifically binomial distributions. Probability is the measure of the likelihood that an event will occur in a random experiment.

a) To find the probability that exactly four sets detect the missile, we use the formula for a binomial probability, that is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the number of combinations of n items taken k at a time. So, the probability is C(5, 4) * (0.9^4) * ((1-0.9)^(5-4)) = 0.32805. The probability that at least one set detects the missile equals to 1 minus the probability that none of the sets detect the missile, which is 1-(0.1^5) = 0.99999.

b) In order to achieve a probability of at least .999 of detecting a missile, we'd need to solve the inequality 1-(1-p)^n >= .999 for n. This yields n as greater than or equal to log(.001)/log(.1), which rounded up to the nearest whole number is 11.

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Answer:

a) 90.1

b) $35.6

c) 3.0915 hours

Step-by-step explanation:

The z-score measures how many standard deviations 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 and [tex]\sigma[/tex] is the standard deviaition.

In all three cases, we have to find X

a. Final exam scores: Allison’s z-score = 2.30, μ = 74, σ = 7.

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

[tex]2.30 = \frac{X - 74}{7}[/tex]

[tex]X - 74 = 7*2.3[/tex]

[tex]X = 90.1[/tex]

b. Weekly grocery bill: James’ z-score = –1.45, μ = $53, σ = $12.

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

[tex]-1.45 = \frac{X - 53}{12}[/tex]

[tex]X - 53 = -1.45*12[/tex]

[tex]X = 35.6[/tex]

Mean and standard deviation in dollars, so the answer also in dollars.

c. Daily video game play time: Eric’s z-score = –0.79, μ = 4.00 hours, σ = 1.15 hours.

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

[tex]-0.79 = \frac{X - 4}{1.15}[/tex]

[tex]X - 4 = -0.79*1.15[/tex]

[tex]X = 3.0915[/tex]

Mean and standard deviation in hours, answer in hours.

Answer:

A = {2,6; 6,2; 3,4; 4,3}

B = {6,5; 6,4; 6,3; 6,2; 6,1; 5,4; 5,3; 5,2; 5,1; 4,3; 4,2; 4,1; 3,2; 3,1; 2,1}

C = {1,3; 3,1; 2,2; 3,5; 5,3; 4,4; 6,6}

D = {1,1; 1,2; 1,3; 1,4; 1,5; 1,6; 3,1; 3,2; 3,3; 3,4; 3,5; 3,6}

Step-by-step explanation:

Let the pair (O,B) be the number rolled on the orange and blue dice respectively.

For event A:  The product of the two numbers that show is 12, the sample space is:

A = {2,6; 6,2; 3,4; 4,3}

For event B: The number on the orange die is strictly larger than the number on the blue die, the sample space is:

B = {6,5; 6,4; 6,3; 6,2; 6,1; 5,4; 5,3; 5,2; 5,1; 4,3; 4,2; 4,1; 3,2; 3,1; 2,1}

For event C: The sum of the numbers is divisible by four, the sum must be 4, 8 or 12 and the sample space is:

C = {1,3; 3,1; 2,2; 3,5; 5,3; 4,4; 6,6}

For event D: The number on the orange die is either 1 or 3, the sample space is:

D = {1,1; 1,2; 1,3; 1,4; 1,5; 1,6; 3,1; 3,2; 3,3; 3,4; 3,5; 3,6}

Answer:

(a) Probability that at least one component needs service during the warranty period = 0.4044.

(b) Probability that exactly one of the components needs service during the warranty period = 0.3419.

Step-by-step explanation:

Given A1 be the event that the receiver functions properly throughout the warranty period.

A2  be the event that the speakers function properly throughout the warranty period.

A3 be the event that the CD player functions properly throughout the warranty period.

Also P(A1) = 0.91, P(A2) = 0.85, and P(A3) = 0.77.

Now P(A1)' means that the receiver need service during the warranty period which is 1 - P(A1) = 1 - 0.91 = 0.09.

Similarly,P(A2)' =1 - P(A2) =1 - 0.85 =0.15  and  P(A3)' =1 - P(A3)=1 - 0.77 = 0.23

Note: The ' sign on the P(A1) represent compliment of A1 or not A1.

(a) The probability that at least one component needs service during the warranty period = 1 - none of the component needs service during the warranty period

And none of the component needs service during the warranty period means that all the three components functions properly during the warranty period .

So, Probability that at least one component needs service during the warranty period = 1 - P(A1) x P(A2) x P(A3) = 1 - (0.91 x 0.85 x 0.77) = 0.4044.

(b) Now to find the Probability that exactly one of the components needs service during the warranty period, there would be three cases for this:

And we have to add these three cases to calculate above probability.

Probability that exactly one of the components needs service during the warranty period = P(A1)' x P(A2) x P(A3) + P(A1) x P(A2)' x P(A3) + P(A1) x  P(A2) x P(A3)'

                         = 0.09 x 0.85 x 0.77 + 0.91 x 0.15 x 0.77 + 0.91 x 0.85 x 0.23

                         = 0.3419.

Answer:

(a) P (At least one component needs service) = 0.404

(b) P (Either component A₁ or A₂ or A₃) = 0.997

Step-by-step explanation:

Given:

[tex]A_{1}=[/tex] Event that the receiver functions properly throughout the warranty period.

[tex]A_{2}=[/tex] Event that the speakers function properly throughout the warranty period.

[tex]A_{3}=[/tex] Event that the CD player functions properly throughout the warranty period.

[tex]P(A_{1})=0.91,\ P(A_{2})=0.85\ and\ P(A_{3})=0.77[/tex]

(a)

Compute the probability that at least one component needs service during the warranty period:

P (At least one component needs service) = 1 - P (None of the one component needs service)

= 1 - {P ([tex]A_{1}[/tex]) × P ([tex]A_{2}[/tex]) × P ([tex]A_{3}[/tex])}

[tex]=1 - (0.91\times0.85\times0.77)\\=1-0.595595\\=0.404405\\\approx0.404[/tex]

Thus, the probability that at least one component needs service during the warranty period is 0.404.

(b)

Compute the probability that exactly one of the components needs service during the warranty period, i.e. P (Either A₁ or A₂ or A₃):

[tex]P(A_{1}\cup A_{2}\cup A_{3})=P(A_{1})+P(A_{2})+P(A_{3})-P(A_{1}\cap A_{2})-P(A_{2}\cap A_{3})-P(A_{3}\cap A_{1})+P(A_{1}\cap A_{2}\cap A_{3})\\=P(A_{1})+P(A_{2})+P(A_{3})-[P(A_{1})\tmes P(A_{2})]-[P(A_{2})\tmes P(A_{3})]-[P(A_{3})\tmes P(A_{1})] +[P(A_{1})\tmes P(A_{2})\times P(A_{3})]\\=0.91+0.85+0.77-(0.91\times0.85)-(0.85\times0.77)-(0.77\times0.91)+(0.91\times0.85\times0.77)\\=0.996895\\\approx0.997[/tex]

Thus, the probability that exactly one of the components needs service during the warranty period is 0.997

Answer:

a) We can use the following excel code to find it:"=NORM.DIST(-2.34,0,1,TRUE)"

[tex] P(Z \leq -2.34)=0.0096[/tex]

b) [tex] P(Z \geq -2.34)= 1-P(X \leq -2.34)= 1-0.0096=0.9904[/tex]

c) We can use the following excel code: "=1-NORM.DIST(1.74,0,1,TRUE)"

[tex] P(Z > 1.74)= 1-P(X \leq 1.74)=0.0409[/tex]

d) We can use the following excel code: "=NORM.DIST(1.74,0,1,TRUE)-NORM.DIST(-2.34,0,1,TRUE)"

[tex] P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)=0.959-0.0096=0.949[/tex]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

We want to find this probability:

[tex] P(Z \leq -2.34)[/tex]

And we can use the following excel code to find it:"=NORM.DIST(-2.34,0,1,TRUE)"

[tex] P(Z \leq -2.34)=0.0096[/tex]

Part b

We want to find this probability:

[tex] P(Z \geq -2.34)[/tex]

And for this case we can use the complement rule:

[tex] P(Z \geq -2.34)= 1-P(X \leq -2.34)[/tex]

And using the result from part a we got:

[tex] P(Z \geq -2.34)= 1-P(X \leq -2.34)= 1-0.0096=0.9904[/tex]

Part c

We want to find this probability:

[tex] P(Z > 1.74)[/tex]

And for this case we can use the complement rule:

[tex] P(Z > 1.74)= 1-P(X \leq 1.74)[/tex]

And we can use the following excel code: "=1-NORM.DIST(1.74,0,1,TRUE)"

[tex] P(Z > 1.74)= 1-P(X \leq 1.74)=0.0409[/tex]

Part d

We want to find this probability:

[tex] P(-2.34<Z <1.74)[/tex]

And for this case we can find this probability with this difference:

[tex] P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)[/tex]

And we can use the following excel code: "=NORM.DIST(1.74,0,1,TRUE)-NORM.DIST(-2.34,0,1,TRUE)"

[tex] P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)=0.959-0.0096=0.949[/tex]

The given z-scores correspond to various portions of a standard Normal distribution. Using a standard Normal distribution table, the proportions for the respective regions are derived as: z <= -2.34 is approximately 0.0094, z >= -2.34 is approximately 0.9906, z > 1.74 is approximately 0.0409, -2.34 < z < 1.74 is approximately 0.9497.

To answer your question, let's use the standard Normal distribution table which provides the proportion of observations less or equal to a particular z-score (for the values of z <= 0 and z >= 0).

Please remember that these proportions represent the areas under the curve (or the total probability) that the z-score will fall within the outlined regions.

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