If z= 0.65, then the raw score is 0.65 standard deviations above the mean. True or false? explain

Answer:

(a) The average value of the given function is 12/π

(b) c = 1.238 or 2.808

Step-by-step explanation:

The average value of a function on a given interval [a, b] is given as

f(c) = (1/(b - a))∫f(x)dx;

from x = b to a

Now, given the function

f(x) = 6sin(x) - 3sin(2x), on [0, π]

The average value of the function is

1/(π-0) ∫(6sinx - 3sin2x)dx

from x = 0 to π

= (1/π) [-6cosx + (3/2)cos2x]

from 0 to π

= (1/π) [-6cosπ + (3/2)cos 2π - (-6cos0 + (3/2)cos0)]

= (1/π)(6 + (3/2) - (-6 + 3/2) )

= (1/π)(12) = 12/π

f(c) = 12/π

b) if f_(ave) = f(c), then

6sinx - 3sin2x = 12/π

2sinx - sin2x = 4/π

But sin2x = 2sinxcosx, so

2sinx - 2sinxcosx = 4/π

sinx - sinxcosx = 2/π

sinx(1 - cosx) = 2/π

This equation can only be estimated to be x = 1.238 or 2.808

Solving sin(x)(1 - cos(x)) = 2/π on [0, π] yields x = π/2 as the only solution. This corresponds to approximately 1.5708, satisfying the given equation.

To solve the equation sin(x)(1 - cos(x)) = 2/π, we can use the double-angle identity for sine, which states that sin(2x) = 2sin(x)cos(x). Rewrite the equation in terms of sin(2x):

sin(x)(1 - cos(x)) = 2/π

sin(x) - sin(x)cos(x) = 2/π

Now, substitute sin(x) = 2/π into the equation:

(2/π) - (2/π)cos(x) = 2/π

Multiply both sides by π to simplify:

2 - 2cos(x) = 2

Subtract 2 from both sides:

-2cos(x) = 0

Divide by -2:

cos(x) = 0

Now, find the values of x where cos(x) = 0, which occurs at x = π/2 and 3π/2. Since we are looking for solutions in the interval [0, π], x = π/2 is the only valid solution.

So, the solution to the equation sin(x)(1 - cos(x)) = 2/π on the interval [0, π] is x = π/2, which is approximately 1.5708.

To learn more about equation

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Answer:

there is no stagnation point

Step-by-step explanation:

for the velocity field V→(u,v)= (0.46 + 2.1x)i→ + (−2.8 - 2.1y)j , the stagnation point is found when the velocity vectors converge in one point ( thus also stays in that place when the point is reached). Thus the stagnation point can be found when the divergence of the velocity field is <0 ( thus it does not diverge , but converges)

div(V) = ∇*V= d/dx (0.46 + 2.1x) + d/dy (−2.8 - 2.1y) = 0

2.1 - 2.1 = 0

since div(V) can never be  <0 , there is no stagnation point

Answer:

B. 0.4542 to 0.5105

Step-by-step explanation:

A 90% confidence interval for p is calculated as:

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

This apply if n*p≥5 and n*(1-p)≥5

Where p is the proportion of sample, n is the size of the sample and [tex]z_{\alpha /2}[/tex] is equal to 1.645 for a 90% confidence.

Then, in this case p, n*p and n*(1-p) are calculated as:

[tex]p=\frac{410}{850} =0.4824[/tex]

n*p = (850)(0.4824) = 410

n*(1-p) = (850)(1-0.4824) = 440

So, replacing values we get:

[tex]0.4824-1.645\sqrt{\frac{0.4824(1-0.4824)}{850} }\leq p\leq 0.4824+1.645\sqrt{\frac{0.4824(1-0.4824)}{850} }[/tex]

[tex]0.4824-0.0282\leq p\leq 0.4824+0.0282[/tex]

[tex]0.4542\leq p\leq 0.5105[/tex]

It means that a  90% confidence interval for p is 0.4542 to 0.5105

Answer:

The correct answer in the option is;

B. 0.4542 to 0.5105.

Step-by-step explanation:

To solve the question, we note that

Total number of residents, n = 850

Number supporting property tax levy = 410

Proportion supporting tax levy, p = [tex]\frac{410}{850}[/tex] = 0.48235

The formula for confidence interval is

[tex]p +/-z*\sqrt{\frac{p(1-p)}{n} }[/tex]

Where

z = z value

The z value from the tables at 90 % = 1.64

Therefore we have

The confidence interval given as

[tex]0.48235 +/-1.64*\sqrt{\frac{0.48235(1-0.48235)}{850} }[/tex] = 0.48235 ± 2.811 × 10⁻²

= 0.4542 to 0.5105

The confidence interval is 0.4542 to 0.5105.

Surface area of this rectangular prism is A=2(1x10+8x10+8x1) so the area for this one is 196 ft

Answer:

P-value of test statistics = 0.9773

Step-by-step explanation:

We are given that a publisher reports that 49% of their readers own a personal computer. A random sample of 200 found that 42% of the readers owned a personal computer.

And, a marketing executive wants to test the claim that the percentage is actually different from the reported percentage, i.e;

Null Hypothesis, [tex]H_0[/tex] : p = 0.49 {means that the percentage of readers who own a personal computer is same as reported 63%}

Alternate Hypothesis, [tex]H_1[/tex] : p [tex]\neq[/tex] 0.49 {means that the percentage of readers who own a personal computer is different from the reported 63%}

The test statistics we will use here is;

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

where, p = actual % of readers who own a personal computer = 0.49

            [tex]\hat p[/tex] = percentage of readers who own a personal computer in a

                  sample of 200 = 0.42

            n = sample size = 200

So, Test statistics = [tex]\frac{0.42 -0.49}{\sqrt{\frac{0.42(1- 0.42)}{200} } }[/tex]

                             = -2.00

Now, P-value of test statistics is given by = P(Z > -2.00) = P(Z < 2.00)

                                                                        = 0.9773 .

Answer:

22%

Step-by-step explanation:

In the event of changing a test answer there are three possible outcomes, which should add up to 100%: changing from incorrect to correct (51%),  changing from correct to incorrect (27%) and  changing from incorrect to incorrect (X). Therefore, the percent of changes from incorrect to​ incorrect was:

[tex]100\% = 51\%+27\%+X\\X= 22\%[/tex]

22% of the changes were  from incorrect to​ incorrect.

Answer:

Step-by-step explanation:

If two triangles are equal, it means that the ratio of the length of each side of one triangle to the length of the corresponding side of the other triangle is constant. Also, corresponding angles are congruent.

1) Triangle TUV is similar to triangle SQR because

Angle Q is congruent to angle U

TU/SQ = UV/QR = 2

2) Triangle ABC is not similar to triangle DEF because the ratio of AB to DF is not constant.

Answer:

So the probability is P=0.00256.

Step-by-step explanation:

We have 26 letters, so the probability that the first letter in the name is the same is 1/26.

The probability that the second letter in the name is the same is 1/26 and the probability that the third letter in the name is the same is 1/26.

Out of ten people we choose 2.

So the probability is:

[tex]P=C_2^{10}\left(\frac{1}{26}\right)^3\\\\P=\frac{10!}{2!(10-2)!}\cdot \left(\frac{1}{26}\right)^3\\\\P=\frac{45}{17576}\\\\P=0.00256\\[/tex]

So the probability is P=0.00256.

The total cost of the loan, including down payment, PMI, mortgage payments, and yearly taxes and insurance over 30 years, for purchasing a house at $138,000 with specific conditions rounds to approximately $333,400.00.

To calculate the total cost of the loan for purchasing a house at $138,000.00 with a 10% down payment, an interest rate of 4.875%, PMI payments of $25.88 for 77 months, yearly taxes of $2400.00, and insurance of $750.00 per year, the following steps are undertaken:

Rounding to the nearest hundred gives us a total cost of approximately $333,400.00.

Answer:

0.344 = 34.4% probability that at least 1 of them have blue eyes.

Step-by-step explanation:

For each person, there are only two possible outcomes. Either they have blue eyes, or they have not. The probability of a person having blue eyes is independent of any other person. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

A survey reports that the probability a person has blue eyes is 0.10.

This means that [tex]p = 0.1[/tex]

4 people are randomly selected at Miramar College

This means that [tex]n = 4[/tex]

Find the probability that at least 1 of them have blue eyes.

Either none of them have blue eyes, or at least one do. The sum of the probabilities of these events is decimal 1. So

[tex]P(X = 0) + P(X \geq 1) = 1[/tex]

We want [tex]P(X \geq 1)[/tex]. So

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{4,0}.(0.1)^{0}.(0.9)^{4} = 0.656[/tex]

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.656 = 0.344[/tex]

0.344 = 34.4% probability that at least 1 of them have blue eyes.

To find the probability that at least one out of four people has blue eyes at Miramar College, we use the complement rule and calculate that the probability is 0.344 after rounding to three decimal places.

The question asks us to calculate the probability that at least one person out of four randomly selected people at Miramar College will have blue eyes, given that the probability a person has blue eyes is 0.10. To find the probability of at least one person having blue eyes, we can use the complement rule. The complement of at least one person having blue eyes is that no person has blue eyes.

The probability that a single person does not have blue eyes is 1 - 0.10 = 0.90. For four independent selections, the probability that none of them have blue eyes is 0.90 raised to the fourth power. Therefore, the probability that at least one out of four people has blue eyes is 1 minus this result.

Calculation:

P(no one has blue eyes) = 0.90 ^ 4

P(no one has blue eyes) = 0.6561

P(at least one has blue eyes) = 1 - P(no one has blue eyes)

P(at least one has blue eyes) = 1 - 0.6561

P(at least one has blue eyes) = 0.3439

Thus, the probability that at least one person out of the four has blue eyes, rounded to three decimal places, is 0.344.

Answer:

a) P=0.2861

b) P=0.0954

c) P=0.3815

d) P=0.6185

Step-by-step explanation:

The question is incomplete:

Among the N=16 students taking a graduate statistics class, A=10 are master students and the other N-A=6 are doctorial students. A random sample of n=5 students is going to be selected to work on a class project. Use X to denote the number of master students in the sample. Keep at least 4 decimal digits if the result has more decimal digits.

a) The probability that exactly 4 master students are in the sample is closest to?

b) The probability that all 5 students in the sample are master students is closest to?

c) The probability that at least 4 students in the sample are master students is closest to?

d) The probability that at most 3 students in the sample are master students is closest to?

We use a binomial distribution with n=5, with p=10/16=0.625 (proportion of master students).

a)

[tex]P(k=4)=\binom{5}{4}p^4q^1=5*0.625^4*0.375=5*0.1526*0.3750\\\\P(k=4)=0.2861[/tex]

b)

[tex]P(k=5)=\binom{5}{5}p^5q^0=1*0.625^5*1=\\\\P(k=4)=0.0954[/tex]

c)

[tex]P(k\geq4)=P(k=4)+P(k=5)=0.2861+0.0954=0.3815[/tex]

d)

[tex]P(k\leq3)=1-P(x\geq4)=1-0.3815=0.6185[/tex]

Answer:

The column height of air after the inversion is 7 cm.

Step-by-step explanation:

The initial pressure balance is given as

P_1+20 cm=76 cm of Hg

P_1=76-20 cm of Hg

P_1=56 cm of Hg

The initial volume of the air with cross sectional area as 12 cm2 and the length of air column as 12 is given as V_1=12 cm *1 =12 cm3

After the inversion

P_2=20 cm+76 cm of Hg

P_2=96 cm of Hg

The volume of the air after the inversion with cross sectional area as 1 cm2 and the length of air column as x is given as V_2=x *1 =x cm3

Now as temperature is constant and the cross sectional area is also constant so

[tex]P_1V_1=P_2V_2\\56\times 12=96\times x\\x=\dfrac{56\times 12}{96}\\x=7 cm[/tex]

So the column height of air after the inversion is 7 cm.

Answer:

7cm

Step-by-step explanation:

P1 =H-h =76-20=56; P2 =H+h =96

P1V1 =P2V2

56x12=96xV2

V2 =56x12/96 = 7 cm

Answer: a) 2 miles

b) 4 miles

Step-by-step explanation:

There are two right angle triangles formed in the rectangle.

Taking 30 degrees as the reference angle, the length of the side walk, h represents the hypotenuse of the right angle triangle.

The width, w of the park represents the opposite side of the right angle triangle.

The length of the park represents the adjacent side of the right angle triangle.

a) to determine the width of the park w, we would apply

the tangent trigonometric ratio.

Tan θ, = opposite side/adjacent side. Therefore,

Tan 30 = w/2√3

1/√3 = w/2√3

w = 1/√3 × 2√3

w = 2

b) to determine the the length of the side walk h, we would apply

the Cosine trigonometric ratio.

Cos θ, = adjacent side/hypotenuse. Therefore,

Cos 30 = 2√3/h

√3/2 = 2√3/h

h = 2√3 × 2/√3

h = 4

Answer:the independent variable is x, the number of hours of work.

The dependent variable is y, the total charge for x hours of work.

Step-by-step explanation:

A change in the value of the independent variable causes a corresponding change in the value of in dependent variable. Thus, the dependent variable is is output while the independent variable is the input

For each visit, he charges $25 plus $20 per hour of work. The linear expression that represents the total amount of money that Ethan earns per visit is y = 25 + 20x.

Since the total amount charged, y depends on the number of hours of work, x, it means that the dependent variable is y and the independent variable is x

Final answer:

In the equation y = 25 + 20x, x is the independent variable representing hours worked, and y is the dependent variable representing total earnings. The y-intercept is $25, the flat visit charge, and the slope is $20, the hourly charge.

Explanation:

In the equation y = 25 + 20x, which represents the total amount Ethan earns for each visit, the independent variable is the number of hours of work, denoted by x. The dependent variable is the total amount of money Ethan earns, represented by y. This is because Ethan's earnings depend on the amount of time he spends working.

The y-intercept is $25, which is the flat charge for Ethan's visit, regardless of the hours worked. The slope is $20, which represents the amount Ethan charges for each hour of work. Therefore, for each additional hour of work, Ethan will earn an additional $20.

Answer:

Both the answers are as in the solution.

Step-by-step explanation:

As the given matrix is not in the readable form, a similar question is found online and the solution of which is attached herewith.

Part a:

Given matrix is : A = [tex]\left[\begin{array}{ccc}0&3&4\\1&2&3\\-3&-7&8\end{array}\right][/tex]

Here,

[tex]det(A) =\left|\begin{array}{ccc}0&3&4\\1&2&3\\-3&-7&8\end{array}\right| = -55 \neq 0.[/tex]

Then, A is non-singular matrix.

Here, A₁₁= 0.

If we write A as LU with L lower triangular matrix and U upper triangular matrix, then A₁₁=L₁₁U₁₁.

So, As

A₁₁ = 0 gives L₁₁U₁₁= 0 ,

This indicates that either L₁₁= 0 or U₁₁ = 0.

If L₁₁= 0 or U₁₁ = 0, this would made the corresponding matrix singular, which contradicts the condition as  A is non-singular.

Therefore, A has no LU decomposition.

Part b:

By the implementation of the various row operations

interchange R1 and R2

[tex]\left[\begin{array}{ccc}1&2&3\\0&3&4\\-3&-7&8\end{array}\right][/tex]

R3+3R1=R3

[tex]\left[\begin{array}{ccc}1&2&3\\0&3&4\\0&-1&17\end{array}\right][/tex]

R3+(1/3)R2 = R3

[tex]\left[\begin{array}{ccc}1&2&3\\0&3&4\\0&0&55/3\end{array}\right][/tex]

Therefore, U = [tex]\left[\begin{array}{ccc}1&2&3\\0&3&4\\0&0&55/3\end{array}\right][/tex].

Here, LP = E₁₂=E₃₁=-3 &E₃₂=-1/3

[tex]LP=\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}\right] \left[\begin{array}{ccc}1&0&0\\0&1&0\\-3&0&1\end{array}\right]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&-1/3&1\end{array}\right][/tex]

[tex]LP=\left[\begin{array}{ccc}0&1&0\\1&0&0\\-3&-1/3&1\end{array}\right][/tex]

[tex]LP=\left[\begin{array}{ccc}1&0&0\\0&1&0\\-1/3&-3&1\end{array}\right]\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}\right][/tex]

So now U is given as

[tex]U=\left[\begin{array}{ccc}1&2&3\\0&3&4\\0&0&55/3\end{array}\right]\\L=\left[\begin{array}{ccc}1&0&0\\0&1&0\\-1/3&-3&1\end{array}\right]\\P=\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}\right]\\[/tex]

Answer:

a) 0.011

b) -0.0624

Step-by-step explanation:

See attached pictures.

Final answer:

Uncertainty in the FTIR spectrometer measurements can be estimated as the standard deviation of the measurements, yielding 0.0109 ppm. Bias is estimated as the difference between the mean of the measurements (1.0823 ppm) and the true value (1.1447 ppm), resulting in a bias of -0.0624 ppm.

Explanation:

To address the student's question regarding the calibration of an FTIR spectrometer and the estimation of uncertainty and bias in measurements, we shall consider the given data.

Uncertainty in measurements can be estimated using the standard deviation of the measurements, which provides an indication of the spread of the data around the mean. To calculate the uncertainty:

Using the provided measurements of carbon content, we calculate the uncertainty as follows:

This standard deviation represents the uncertainty in the measurements.

Estimation of Bias

Bias in the measurements can be estimated as the difference between the mean of the measurements and the true value. Thus, the bias is calculated by subtracting the true carbon content from the mean measurement:

Bias = Mean - True value = 1.0823 ppm - 1.1447 ppm = -0.0624 ppm

The negative sign indicates that the measurements are, on average, lower than the true value.

Answer:

a. data(wages)

plot(wages, type='o', ylab='wages per hour')

Step-by-step explanation:

a.  Display and interpret the time series plot for these data.

#take data samples from wages

data(wages)

plot(wages, type='o', ylab='wages per hour')

see others below

b. Use least squares to fit a linear time trend to this time series. Interpret the regression output. Save the standardized residuals from the fit for further analysis.

#linear model

wages.lm = lm(wages~time(wages))

summary(wages.lm) #r square is correct

##  

## Call:

## lm(formula = wages ~ time(wages))

##  

## Residuals:

##      Min       1Q   Median       3Q      Max  

## -0.23828 -0.04981  0.01942  0.05845  0.13136  

##  

## Coefficients:

##               Estimate Std. Error t value Pr(>|t|)    

## (Intercept) -5.490e+02  1.115e+01  -49.24   <2e-16 ***

## time(wages)  2.811e-01  5.618e-03   50.03   <2e-16 ***

## ---

## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##  

## Residual standard error: 0.08257 on 70 degrees of freedom

## Multiple R-squared:  0.9728, Adjusted R-squared:  0.9724  

## F-statistic:  2503 on 1 and 70 DF,  p-value: < 2.2e-16

c. plot(y=rstandard(wages.lm), x=as.vector(time(wages)), type = 'o')

d. #we find Quadratic model trend

wages.qm = lm(wages ~ time(wages) + I(time(wages)^2))

summary(wages.qm)

##  

## Call:

## lm(formula = wages ~ time(wages) + I(time(wages)^2))

##  

## Residuals:

##       Min        1Q    Median        3Q       Max  

## -0.148318 -0.041440  0.001563  0.050089  0.139839  

##  

## Coefficients:

##                    Estimate Std. Error t value Pr(>|t|)    

## (Intercept)      -8.495e+04  1.019e+04  -8.336 4.87e-12 ***

## time(wages)       8.534e+01  1.027e+01   8.309 5.44e-12 ***

## I(time(wages)^2) -2.143e-02  2.588e-03  -8.282 6.10e-12 ***

## ---

## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##  

## Residual standard error: 0.05889 on 69 degrees of freedom

## Multiple R-squared:  0.9864, Adjusted R-squared:  0.986  

## F-statistic:  2494 on 2 and 69 DF,  p-value: < 2.2e-16

#time series plot of the standardized residuals

plot(y=rstandard(wages.qm), x=as.vector(time(wages)), type = 'o')

wages.qm = lm(wages ~ time(wages) + I(time(wages)^2))

summary(wages.qm)

##  

## Call:

## lm(formula = wages ~ time(wages) + I(time(wages)^2))

##  

## Residuals:

##       Min        1Q    Median        3Q       Max  

## -0.148318 -0.041440  0.001563  0.050089  0.139839  

##  

## Coefficients:

##                    Estimate Std. Error t value Pr(>|t|)    

## (Intercept)      -8.495e+04  1.019e+04  -8.336 4.87e-12 ***

## time(wages)       8.534e+01  1.027e+01   8.309 5.44e-12 ***

## I(time(wages)^2) -2.143e-02  2.588e-03  -8.282 6.10e-12 ***

## ---

## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##  

## Residual standard error: 0.05889 on 69 degrees of freedom

## Multiple R-squared:  0.9864, Adjusted R-squared:  0.986  

## F-statistic:  2494 on 2 and 69 DF,  p-value: < 2.2e-16

e. #time series plot of the standardized residuals

plot(y=rstandard(wages.qm), x=as.vector(time(wages)), type = 'o')

The surface area of the triangular prism is 1664 square inches.

Explanation:

Given that the triangular prism has a length of 20 inches and has a triangular face with a base of 24 inches and a height of 16 inches.

The other two sides of the triangle are 20 inches each.

We need to determine the surface area of the triangular prism.

The surface area of the triangular prism can be determined using the formula,

[tex]SA= bh+pl[/tex]

where b is the base, h is the height, p is the perimeter and l is the length

From the given the measurements of b, h, p and l are given by

[tex]b=24[/tex] , [tex]h= 16[/tex] , [tex]l=20[/tex] and

[tex]p=20+20+24=64[/tex]

Hence, substituting these values in the above formula, we get,

[tex]SA= (24\times16)+(64\times20)[/tex]

Simplifying the terms, we get,

[tex]SA=384+1280[/tex]

Adding the terms, we have,

[tex]SA=1664 \ square \ inches[/tex]

Thus, the surface area of the triangular prism is 1664 square inches.

Answer:

For the critical value we know that the significance is 5% and the value for [tex] \alpha/2 = 0.025[/tex] so we need a critical value in the normal standard distribution that accumulates 0.025 of the area on each tail and for this case we got:

[tex] Z_{\alpha/2}= \pm 1.96[/tex]

Since we have a two tailed test,  the rejection zone would be: [tex] z<-1.96[/tex] or [tex] z>1.96[/tex]

Step-by-step explanation:

Data given and notation

n represent the random sample taken

[tex]\hat p[/tex] estimated proportion of interest

[tex]p_o=0.15[/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 proportion of graduates with degrees in engineering who earn more than $75,000 in their first year of work is not 15%.:  

Null hypothesis:[tex]p=0.15[/tex]  

Alternative hypothesis:[tex]p \neq 0.15[/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].

For the critical value we know that the significance is 5% and the value for [tex] \alpha/2 = 0.025[/tex] so we need a critical value in the normal standard distribution that accumulates 0.025 of the area on each tail and for this case we got:

[tex] Z_{\alpha/2}= \pm 1.96[/tex]

Since we have a two tailed test,  the rejection zone would be: [tex] Z<-1.96[/tex] or [tex] z>1.96[/tex]

Answer:a)False b)True c)False d)True

Step-by-step explanation:

Let's consider first that for us the set of the natural numbers is the set of the positive integers and the set of the non-negative numbers is know as the whole number or with notation [tex]N_0[/tex]. Then for

a) the N={1,2,3,...} and therefore the common members with A are only {1,2,3} making the statement false, only if stated to consider the set N as all the non-negative numbers the answer would be true, but otherwise it is standarized to understand N as the positive integers and [tex]N_0[/tex] as the non-negative integers.

b)The difference of sets is taking the elements in the first that do not belong to the second, then it would be to withdraw the only element C has, since 0 belongs to A, and therefore C would turn to be an empty set.

c)The set powers of a given set S, denoted P(S), is a set with sets as elements, every subset of S is an element of P(S). Then P(S) is always non-empty, since at least S belongs to P(S). Here [tex]P(C)=\{C, \emptyset \}[/tex], then [tex]P(C)\cup B=\{C, \emptyset ,1,2,3,\ldots \}\ne B[/tex], therefore the statement is false.

d)As explained in c) [tex]P(C)=\{C, \emptyset \},[/tex] then clearly C is an element of P(C), thus the affirmation is true.

Answer:

[tex]63.44^{0}[/tex]

Step-by-step explanation:

Tom's resultant speed is calculated as [tex]\sqrt{1^{2}+2^{2} }[/tex]

= [tex]\sqrt{5}[/tex]

The distance tom swim is the hypotenuse of the right angle triangle.

sin = opposite/hypotenuse = [tex]\frac{1}{\sqrt{5} }[/tex]  = 1/5 = 0.2

arcsin (0.2) = [tex]26.56^{0}[/tex]

upstream angle =

[tex]90^{0} - 26.56^{0}[/tex] =

[tex]63.44^{0}[/tex]

Answer:

(P(t)) = P₀/(1 - P₀(kt)) was proved below.

Step-by-step explanation:

From the question, since β and δ are both proportional to P, we can deduce the following equation ;

dP/dt = k(M-P)P

dP/dt = (P^(2))(A-B)

If k = (A-B);

dP/dt = (P^(2))k

Thus, we obtain;

dP/(P^(2)) = k dt

((P(t), P₀)∫)dS/(S^(2)) = k∫dt

Thus; [(-1)/P(t)] + (1/P₀) = kt

Simplifying,

1/(P(t)) = (1/P₀) - kt

Multiply each term by (P(t)) to get ;

1 = (P(t))/P₀) - (P(t))(kt)

Multiply each term by (P₀) to give ;

P₀ = (P(t))[1 - P₀(kt)]

Divide both sides by (1-kt),

Thus; (P(t)) = P₀/(1 - P₀(kt))

(P(t)) = P₀/(1 - P₀(kt))

According to the, since β and also δ are both proportional to P, we can deduce the following equation ;

Then dP/dt = k(M-P)P

Then dP/dt = (P^(2))(A-B)

Now, If k = (A-B);

After that dP/dt = (P^(2))k

Thus, we obtain;

Now dP/(P^(2)) = k dt

((P(t), P₀)∫)dS/(S^(2)) = k∫dt

Thus; [(-1)/P(t)] + (1/P₀) = kt

Simplifying,

Then 1/(P(t)) = (1/P₀) - kt

Multiply each term by (P(t)) to get ;

After that 1 = (P(t))/P₀) - (P(t))(kt)

Multiply each term by (P₀) to give ;

Now P₀ = (P(t))[1 - P₀(kt)]

Then Divide both sides by (1-kt),

Thus; (P(t)) = P₀/(1 - P₀(kt))

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Hypothesis:

The ratio of ladies giving multiple birth to total number of women for any race will be the same.

Test:

Ratio of white women giving multiple births = 94 / 3132 = 0.0300

Ratio of black women giving multiple births = 20 / 606 = 0.0330

Conclusion:

There is no racial difference in the likelihood of multiple births. Although we do see a difference in the ratios calculated above, the difference is small enough to be due to sample size difference of white and black women. The smaller number of total black women makes the ratio calculated from this sample have a higher probability to deviate from what is expected. This deviation will account for the difference in probability between both races.

We can see the effects of this small sample size by increasing or decreases the numerator by 1 for black women:

21 / 606 = 0.0347

19 / 606 = 0.0313

This change in the data of one woman produces a very large percentage change in our ratio for black women (5%). Thus despite inaccuracy due to small sample size, our hypothesis is correct.

Final answer:

A hypothesis test for the difference in proportions can be used to assess if there is a racial difference in the likelihood of multiple births between white and black women, based on the given data. The null hypothesis is no difference, and if the test statistic is significant, it may indicate a racial difference.

Explanation:

To evaluate any racial differences in the likelihood of multiple births between white women and black women based on the given data, we can perform a hypothesis test. Specifically, this would be a test for the difference between two proportions.

For white women:

Multiples: 94
Total births: 3132

For black women:

Multiples: 20
Total births: 606

Using a chi-squared distribution table or calculator, at α=0.05 and 1 degree of freedom, the critical value is approximately 3.841.

Since our calculated chi-squared value (0.2094) is less than the critical value (3.841), we fail to reject the null hypothesis.

Therefore, there is no significant evidence to conclude that there is a racial difference in the likelihood of multiple births among white and black women in this county.

Answer:

Step-by-step explanation:

Hello!

X: Cholesterol level of a woman aged 30-39. (mg/dl)

This variable has an approximately normal distribution with mean μ= 190.14 mg/dl

1. You need to find the corresponding Z-value that corresponds to the top 9.3% of the distribution, i.e. is the value of the standard normal distribution that has above it 0.093 of the distribution and below it is 0.907, symbolically:

P(Z≥z₀)= 0.093

-*or*-

P(Z≤z₀)= 0.907

Since the Z-table shows accumulative probabilities P(Z<Z₁₋α) I'll work with the second expression:

P(Z≤z₀)= 0.907

Now all you have to do is look for the given probability in the body of the table and reach the margins to obtain the corresponding Z value. The first column gives you the integer and first decimal value and the first row gives you the second decimal value:

z₀= 1.323

2.

Using the Z value from 1., the mean Cholesterol level (μ= 190.14 mg/dl) and the Medical guideline that indicates that 9.3% of the women have levels above 240 mg/dl you can clear the standard deviation of the distribution from the Z-formula:

Z= (X- μ)/δ ~N(0;1)

Z= (X- μ)/δ

Z*δ= X- μ

δ=(X- μ)/Z

δ=(240-190.14)/1.323

δ= 37.687 ≅ 37.7 mg/dl

I hope it helps!

Answer:

a) Poisson distribution

b) 99.33% probability that in any one minute at least one purchase is​ made

c) 0.05% probability that seven people make a purchase in the next four ​minutes

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

In which

x is the number of sucesses

e = 2.71828 is the Euler number

[tex]\mu[/tex] is the mean in the given time interval.

5 people per minute check out from their shopping cart and make an online purchase.

This means that [tex]\mu = 5[/tex]

a) What model might you suggest to model the number of purchases per​ minute? ​

The only information that we have is the mean number of an event(purchases) in a time interval. Each event is also independent fro each other. So you should suggest the Poisson distribution to model the number of purchases per​ minute.

b) What is the probability that in any one minute at least one purchase is​ made? ​

Either no purchases are made, or at least one is. The sum of the probabilities of these events is 1. So

[tex]P(X = 0) + P(X \geq 1) = 1[/tex]

We want to find [tex]P(X \geq 1)[/tex]

So

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

In which

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-5}*(5)^{0}}{(0)!} = 0.0067[/tex]

1 - 0.0067 = 0.9933.

99.33% probability that in any one minute at least one purchase is​ made

c) What is the probability that seven people make a purchase in the next four ​minutes?

The mean is 5 purchases in a minute. So, for 4 minutes

[tex]\mu = 4*5 = 20[/tex]

We have to find P(X = 7).

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-20}*(20)^{7}}{(7)!} = 0.0005[/tex]

0.05% probability that seven people make a purchase in the next four ​minutes

The Poisson distribution model is used when the data consist of counts of occurrences.

a) The Poisson distribution model is used when the data consist of counts of occurrences.

b) Given that: λ (mean number of occurrence) = 5 people per minute, hence:

[tex]P(X\ge 1)=1-P(X=0)=1-\frac{e^{-\lambda }\lambda^x}{x!}= 1-\frac{e^{-5 }5^0}{5!}=0.9999[/tex]

The probability that in any one minute at least one purchase is​ made is 0.9999.

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Answer:

[tex]P(X<4)=P(\frac{X-\mu}{\sigma}<\frac{4-\mu}{\sigma})=0.2[/tex]  

[tex]P(z<\frac{4-\mu}{\sigma})=0.2[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-0.842<\frac{4-\mu}{1.1}[/tex]

And if we solve for the mean we got

[tex]\mu =4 +0.842*1.1=4.926[/tex]

Step-by-step explanation:

Assuming this question "Suppose that the lifetimes of TV tubes are normally distributed with a standard deviation of 1.1 years. Suppose also that exactly 20% of the tubes die before 4 years. Find the mean lifetime of TV tubes. Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place. ?

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".  

Solution to the problem

Let X the random variable that represent the lifetimes of TV tubes of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu,1.1)[/tex]  

Where [tex]\mu[/tex] and [tex]\sigma=1.1[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

For this part we know the following condition:

[tex]P(X>4)=0.8[/tex]   (a)

[tex]P(X<4)=0.2[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.2 of the area on the left and 0.8 of the area on the right it's z=-0.842. On this case P(Z<-0.842)=0.2 and P(z>-0.842)=0.8

If we use condition (b) from previous we have this:

[tex]P(X<4)=P(\frac{X-\mu}{\sigma}<\frac{4-\mu}{\sigma})=0.2[/tex]  

[tex]P(z<\frac{4-\mu}{\sigma})=0.2[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-0.842<\frac{4-\mu}{1.1}[/tex]

And if we solve for the mean we got

[tex]\mu =4 +0.842*1.1=4.926[/tex]

Mean lifetime is 4.9 years. Found using z-score of -0.8416 with 20% dying before 4 years, [tex]\(\sigma = 1.1\)[/tex].

To find the mean lifetime of TV tubes, we will use the properties of the normal distribution and the given information. Here's the step-by-step solution:

1. Identify the given values:

  - Standard deviation [tex](\(\sigma\))[/tex]: 1.1 years

  - Percentage of tubes that die before 4 years: 20% (or 0.20)

2. Find the z-score corresponding to the given percentage:

  - Since the percentage is 20%, we need to find the z-score for which 20% of the area under the normal curve lies to the left.

  - Using a standard normal distribution table or a calculator, the z-score for 0.20 is approximately -0.8416.

3. Set up the z-score formula:

  The z-score formula is given by:

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

where [tex]\(X\)[/tex] is the value (4 years in this case), [tex]\(\mu\)[/tex] is the mean lifetime we want to find, and [tex]\(\sigma\)[/tex] is the standard deviation.

4. Substitute the known values into the z-score formula and solve for [tex]\(\mu\)[/tex]:

  [tex]\[ -0.8416 = \frac{4 - \mu}{1.1} \][/tex]

  Rearrange to solve for [tex]\(\mu\)[/tex]:

  [tex]\[ -0.8416 \times 1.1 = 4 - \mu \][/tex]

  [tex]\[ -0.92576 = 4 - \mu \][/tex]

  [tex]\[ \mu = 4 + 0.92576 \][/tex]

  [tex]\[ \mu \approx 4.9258 \][/tex]

5. Round the mean lifetime to one decimal place:

  [tex]\[ \mu \approx 4.9 \][/tex]

So, the mean lifetime of the TV tubes is approximately 4.9 years.

The equation of the quadratic function is [tex]y=(x-7)(x+3)[/tex]

Explanation:

The vertex form of the quadratic function is given by

[tex]y=a(x-h)^{2}+k[/tex]

It is given that the quadratic function has a vertex at [tex](2,-25)[/tex]

The vertex is represented by the coordinate [tex](h,k)[/tex]

Hence, substituting [tex](h,k)=(2,-25)[/tex] in the vertex form, we get,

[tex]y=a(x-2)^{2}-25[/tex]

Now, substituting the x - intercept [tex](7,0)[/tex] , we have,

[tex]0=a(7-2)^{2}-25[/tex]

[tex]0=a(5)^{2}-25[/tex]

[tex]25=a(25)[/tex]

 [tex]1=a[/tex]

Thus, the value of a is 1.

Hence, substituting [tex]a=1[/tex], [tex](h,k)=(2,-25)[/tex] in the vertex form [tex]y=a(x-h)^{2}+k[/tex] , we get,

[tex]y=1(x-2)^{2}-25[/tex]

[tex]y=(x-2)^{2}-25[/tex]

[tex]y=x^2-2x+4-25[/tex]

[tex]y=x^2-2x-21[/tex]

[tex]y=(x-7)(x+3)[/tex]

Thus, the equation of the quadratic function is [tex]y=(x-7)(x+3)[/tex]

Answer:

the answer is d

Step-by-step explanation:

The probability that the virus entered at least 10 computers is 0.7553.

To find the probability that the virus entered at least 10 computers, we can use the complementary probability formula. This formula states that the probability of an event A happening is equal to 1 minus the probability of event A not happening.

In this case, event A is the virus entering at least 10 computers. Event A not happening is the virus entering fewer than 10 computers.

The probability of the virus entering fewer than 10 computers is equal to the sum of the probabilities of the virus entering 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 computers.

We can use the binomial distribution to calculate the probability of the virus entering each of these numbers of computers. The binomial distribution is a probability distribution that describes the probability of getting a certain number of successes in a certain number of trials.

In this case, the trials are the computers, and the success is the virus entering the computer. The probability of success is 0.4, and the probability of failure is 0.6.

To find the probability of the virus entering fewer than 10 computers, we need to add up the probabilities in the table from 0 to 9.

P(virus entering fewer than 10 computers) = 0.36^20 + 20 * 0.36^19 * 0.6 + ... + 167960 * 0.36^11 * 0.6^9

We can use a calculator to evaluate this sum. The result is 0.2447.

Therefore, the probability of the virus entering at least 10 computers is 1 minus the probability of the virus entering fewer than 10 computers.

P(virus entering at least 10 computers) = 1 - 0.2447

P(virus entering at least 10 computers) = 0.7553

Therefore, the probability that the virus entered at least 10 computers is 0.7553.

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Answer:

D

Step-by-step explanation:

Answer:

Calculate the Macaulay duration of Annuity B at the time of purchase is 1.369.

Step-by-step explanation:

First, we use 0.93 to calculate the v which equals 1/(1+i).

[tex]\frac{0+1*v+2v^{2} }{1+v+v^{2} }[/tex] = 0.93

After rearranging the equation, we get 1.07[tex]v^{2}[/tex] + 0.07v - 0.93=0

So, v=0.9

Mac D: [tex]\frac{0+1*v+2*v^{2}+ 3*v^{2} }{1+v+v^{2}+ v^{3} }[/tex]

After substituting the value of v, we get Mac D = 1.369.