A survey of 400 college seniors resulted in the following crosstabulation regarding their undergraduate major and whether or not they plan to go to graduate school. What percentage of the undergraduates surveyed are majoring in Engineering?

So, the required probabilities are,

Part(a):P(A)=0.6

Part(b):P(B)=0.1

Given that:

There are five Oklahoma state officials,

Governor (G), Lieutenant Governor (L), Secretary of State (S), Attorney General (A), and Treasurer (T).

All possible samples of size 3 are obtained from the population of five officials.

Here order does not matter so we use the combinations.

[tex]5_C_3=10[/tex] possible samples.

So, S={GLS,GLA,GLT,GSA,GST,GAT,LSA,LST,LAT,SAT}

Hence, n(s)=10

Part(a):

Let A denotes the event that the governor is included in the sample.

A={GLS,GLA,GLT,GSA,GST,GAT}

That is n(A)=6

So, the probability that the governor is included in the sample,

[tex]P(A)=\frac{n(A)}{n(S)} \\=\frac{6}{10}\\ =0.6[/tex]

Part(b):

Let B denotes the event that the government attorney general and the treasure are included in the sample.

B={GAT}

That is n(B)=1

Hence, the probability that the government attorney general and the treasure are included in the sample is,

[tex]P(B)=\frac{n(B)}{n(S)} \\=\frac{1}{10} \\=0.1[/tex]

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(a) Probability of governor included: 6/10 = 0.6.

(b) Probability of governor and attorney general included: 1/10 = 0.1.

let's break it down in detail:

(a) Probability that the governor is included in the sample:

To find this probability, we need to count how many of the possible samples include the governor. From the list of all possible samples:

1. {G, L, S}

2. {G, L, A}

3. {G, L, T}

4. {G, S, A}

5. {G, S, T}

6. {G, A, T}

We see that in 6 out of 10 samples, the governor is included. Thus, the probability of the governor being included in the sample is [tex]\( \frac{6}{10} = 0.6 \).[/tex]

(b) Probability that the governor and the attorney general are included in the sample:

To find this probability, we need to count how many of the possible samples include both the governor and the attorney general. Looking at the list of all possible samples again:

1. {G, L, S}

2. {G, L, A}

3. {G, L, T}

4. {G, S, A}

5. {G, S, T}

6. {G, A, T}

7. {L, S, A}

8. {L, S, T}

9. {L, A, T}

10. {S, A, T}

We see that only in one out of the 10 samples, both the governor and the attorney general are included (sample number 2). Thus, the probability of both the governor and the attorney general being included in the sample is [tex]\( \frac{1}{10} = 0.1 \).[/tex]

So, to summarize:

(a) Probability that the governor is included in the sample: 0.6

(b) Probability that the governor and the attorney general are included in the sample: 0.1

Final answer:

Researchers are testing the hypothesis that second-hand smoke increases the risk of low birthweight, with a null hypothesis of H0: p = 0.078, and an alternative hypothesis of Ha: p > 0.078.

Explanation:

The researchers are questioning whether second-hand smoke increases the risk of low birthweight in babies. They have a sample where 10.4% of babies born to mothers with extensive exposure to second-hand smoke are categorized as having low birth weight. This suggests the proportion might be higher than the national average of 7.8%. The null hypothesis (H0), which is the hypothesis to be tested, states that the proportion of low birth weight for the population is equal to the national average, thus H0: p = 0.078. The alternative hypothesis (Ha), representing the researchers' suspicion, is that the proportion is greater than the national average, hence Ha: p > 0.078. The last hypothesis option provided, H0: p = 0.104, is not relevant for the null hypothesis in this context since this value represents the sample proportion, not the population proportion they are testing against.

Answer:

The rope should be staked at 3ft  distance from bottom of  4 ft pole.

Step-by-step explanation:

Given that  the poles are 4 ft and 16 ft, separated by distance of 15 ft.

Refering the given figure, let the distance from 4 ft pole be "x".

consequently, distance from 16 ft pole is (15-x).

Refering the figure, in right angled triangle DCO, by pythagoras theorm, the the length OD is [tex]\sqrt{16-x^{2} }[/tex].

Similarly, the length OA is [tex]\sqrt{481+x^{2}-30x}[/tex].

thus the length of rope is AO + OD =  L = f(x) =

[tex]\sqrt{16-x^{2} }[/tex] + [tex]\sqrt{481+x^{2}-30x}[/tex]

now, for a function of one variable, to have maximum or minimum, differentiate once, and find value of x.

Now, f'(x)= [tex]x(\sqrt{16+x^{2} }^{-1} )+ (x-15)(\sqrt{481+x^{2}-30x }^{-1})[/tex]

f'(x)=0  gives,

x=3 or -5

but the length cannot be negetive.

thus x=3.

By applying the pythagroean theorem and differentiation, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.

The pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum the square of the otehr two legs of the right triangle.

The sketch of the problem is given in the diagram that is attached below.

Applying the pythagorean theorem in right triangle DCO, we would have:

DO = √(16 - x²)

AO would also be found using the pythagorean theorem:

AO = √(481 + x² - 30x)

Therefore, the length of the rope = AO + DO = f(x), which is:

f(x) = √(481 + x² - 30x) + √(16 - x²)

To find the value of x, take derivative and set equal to 0.

Differentiating, f(x) = √(481 + x² - 30x) + √(16 - x²), we would have:

f'(x) = 0 which will give us, x = 3 or -5.

Since the length cannot be negative, therefore, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.

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Final answer:

By conducting a One-Sample t-test, with the data provided, it was found that there is a significant decrease in test scores for students who studied with an e-book compared to the overall class average. The calculated t-score exceeded the critical value in a one-tailed test at alpha level of .05, leading to the rejection of the null hypothesis.

Explanation:

To determine if there is a significant decrease in test scores for students who studied using an e-book compared to the overall class average, we need to conduct a hypothesis test. First, let's define our null hypothesis (H0) as there being no difference in means, implying that the e-book does not affect scores. The alternative hypothesis (H1) suggests that the e-book usage leads to lower scores. Given that the overall class average (μ) is 85 and the mean (M) for the 10 students who used the e-book is 77 with a standard deviation (s) of 6, we can use the One-Sample t-test to analyze the data.

To calculate the t-score, we use the formula t = (M - μ) / (s / √n), where n is the sample size, which is 10 in this case. Plugging in our values gives us t = (77 - 85) / (6 / √10), resulting in a t-score of approximately -4.216.

Consulting a t-table or using statistical software with a one-tailed test at α = .05 and df = 9 (n-1), we find that our critical t-value is around -1.833. Since our calculated t-score of -4.216 is more extreme than -1.833, we reject the null hypothesis. This indicates a significant decrease in test scores for students who studied with the e-book, as compared to the class average.

Answer: 28 * 1

Step-by-step explanation:

Answer:  2.  H0: μ = 40 vs. H1: μ < 40

Step-by-step explanation:

Let [tex]\mu[/tex] be the average number of tissues a person uses during the course of a cold.

Given : Researchers claim that 40 tissues is the average number of tissues a person uses during the course of a cold.

i.e. [tex]\mu=40[/tex]

The company who makes Puffs brand tissues thinks that fewer of their tissues are needed.

i.e. [tex]\mu<40[/tex]

Thus , the set of hypothesis would be :-

 [tex]H_0: \mu=40[/tex]

 [tex]H_a: \mu<40[/tex]

Thus , the correct answer is option 2. H0: μ = 40 vs. H1: μ < 40

The null hypothesis (H0) is that an average person uses 40 tissues during a cold (H0: μ = 40). The alternative hypothesis (H1), which Puffs brand wants to prove, is that less than 40 of their tissues are needed during a cold (H1: μ < 40).

The question is related to forming a null hypothesis and an alternative hypothesis in the context of statistics. Here, we are talking about the utilization of tissues during a cold. The researchers claim that on average a person uses 40 tissues (let's call this number 'μ') throughout a cold's duration. However, the Puffs brand makers, as per their assumption, think that less tissues of their brand are needed.

The null hypothesis, denoted by H0, is usually a claim of no effect or no difference. In this case, it is H0: μ = 40, denoting that the average number of tissues used is 40. This is also the claim that the Puffs brand intends to challenge or nullify.

On the other hand, the alternative hypothesis (H1) is the claim that the party interested in the outcome (here, the Puffs brand) wants to validate. Here, it would be H1: μ < 40, which signifies that fewer than 40 Puffs tissues are needed during the course of a cold.

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Answer:

We find the length of each subinterval dividing the distance between the endpoints of the interval by the quantity of subintervals that we want.

Then

Δx= [tex]\frac{0-(-2)}{4}=\frac{2}{4}=\frac{1}{2}[/tex]

Now, each [tex]x_i[/tex] is found by adding Δx iteratively from the left end of the interval.

So

[tex]x_0=-2\\x_1=-2+\frac{1}{2}=\frac{-3}{2}\\x_2=\frac{-3}{2}+\frac{1}{2}=-1\\x_3=-1+\frac{1}{2}=-\frac{1}{2}\\x_4=\frac{-1}{2}+\frac{1}{2}=0[/tex]

Each subinterval is

[tex]s_1=[-2,-3/2]\\s_2=[-3/2,-1]\\s_3=[-1,-1/2]\\s_4=[-1/2,0][/tex]

The midpoints of the subintervals are

[tex]m_1=\frac{-2-3/2}{2}=\frac{-7/2}{2}=\frac{-7}{4}\\m_2=\frac{-1-3/2}{2}=\frac{-5/2}{2}=\frac{-5}{4}\\m_3=\frac{-1/2-1}{2}=\frac{-3/2}{2}=\frac{-3}{4}\\m_4=\frac{0-1/2}{2}=\frac{-1}{4}[/tex]

The points used for the

1. left Riemann sums are the left endpoints of the subintervals, that is

[tex]x_0=-2, x_1=\frac{-3}{2}, x_2=-1, x_3= \frac{-1}{2}[/tex]

2. right Riemann sums are the right endpoints of the subinterval,

[tex]x_1=-\frac{3}{2}, x_2=-1, x_3=-\frac{1}{2}, x_4=0[/tex]

3. midpoint Riemann sums are the midpoints of each subinterval

[tex]m_1,m_2,m_3,m_4.[/tex]

The subinterval length in the given range is 0.5. The grid points are -2, -1.5, -1, -0.5, and 0. The points used for the left, right, and midpoint Riemann sums vary based on the position of the subinterval.

The interval specified is ​[negative 2​,0​] and is partitioned into n=4 subintervals. The length of each subinterval, Upper Delta x, is determined by subtracting the lower boundary from the upper boundary and dividing by the number of subintervals, in this case (0 - (-2)) / 4 = 0.5. Therefore, the subinterval length is 0.5.

The grid points are calculated by adding the subinterval length to each previous point, starting from the lower boundary: x0​ is -2, for x1, add the subinterval length 0.5 to x0, resulting in -1.5. Repeat this process to find x2, x3 and x4 which are -1, -0.5 and 0 respectively.

Now, for the left Riemann sum, you use the left-hand side points of each subinterval, which are x0​, x1​, x2​, x3. The right Riemann sum uses the right-hand side points, x1​, x2​, x3​, x4. Lastly, the midpoint Riemann sum uses points in the middle of each subinterval. As the subintervals here each have a length of 0.5, their midpoints fall on x0.5, x1.5, x2.5 and x3.5 respectively.

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Answer:

f has relative maximum at t = [tex]+\frac{\sqrt2}{3}[/tex]

and

f has relative minimum at t = [tex]-\frac{\sqrt2}{3}[/tex]

Step-by-step explanation:

Data provided in the question:

f(t) = -3t³ + 2t

Now,

To find the  points of maxima or minima, differentiating with respect to t and putting it equals to zero

thus,

f'(t) = (3)(-3t²) + 2 = 0

or

-9t² + 2 = 0

or

t² = [tex]\frac{2}{9}[/tex]

or

t = [tex]\pm\frac{\sqrt2}{3}[/tex]

to check for maxima or minima, again differentiating with respect to t

f''(t) = 2(-9t) + 0 = -18t

substituting the value of t

at t = [tex]+\frac{\sqrt2}{3}[/tex]

f''(t) =  [tex](-18)\times\frac{\sqrt2}{3}[/tex]

=  - 6√2 < 0 i.e maxima

and at  t = [tex]-\frac{\sqrt2}{3}[/tex]

f''(t) =  [tex](-18)\times(-\frac{\sqrt2}{3})[/tex]  

= 6√2 > 0 i.e minima

Hence,

f has relative maximum at t = [tex]+\frac{\sqrt2}{3}[/tex]

and

f has relative minimum at t = [tex]-\frac{\sqrt2}{3}[/tex]

Answer: Option 'd' is correct.

Step-by-step explanation:

Since we have given that

n = 250

Average = 66.9 inches

standard deviation = 6.2 inches

We need to find the 95% confidence interval for the mean.

So, z = 1.96

Interval would be

[tex]\bar{x}\pm z\dfrac{\sigma}{\sqrt{n}}\\\\=66.9\pm 1.96\times \dfrac{6.2}{\sqrt{250}}\\\\=66.9\pm 0.77\\\\=(66.9-0.77,66.9+0.77)\\\\=(66.13,67.67)[/tex]

Hence, option 'd' is correct.

Answer:

B)1,1,1,4

Step-by-step explanation:

A.1,2,3,4

Mean=[tex]\bar x=\frac{1+2+3+4}{4}=2.5[/tex]

x    [tex]x-\bar x[/tex]   [tex](x-\bar x)^2[/tex]

1       -1.5                                2.25

2       -0.5                              0.25

3         0.5                              0.25  

4          1.5                               2.25

[tex]\sum(x-\bar x)^2=2.25+0.25+0.25+2.25=5[/tex]

B.

Mean=[tex]\bar x=\frac{1+1+1+4}{4}=1.75[/tex]

[tex](x-\bar x)^2[/tex]

0.5625

0.5625

0.5625

5.0625

[tex]\sum(x-\bar x)^2=0.5626+0.5625+0.5625+5.0625=6.75[/tex]

C.

Mean=[tex]\bar x=\frac{1+2+2+4}{4}=2.25[/tex]

[tex](x-\bar x)^2[/tex]

1.5625

0.0625

0.0625

3.0625

[tex]\sum (x-\bar x)=1.5625+0.0625+0.0625+3.0625=4.75[/tex]

D.[tex]Mean=\bar x=\frac{4+4+4+4}{4}=4[/tex]

[tex](x-\bar x)^2[/tex]

0

0

0

0

[tex]\sum (x-\bar x)^2=0+0+0+0=0[/tex]

We know that

S.D is directly proportional to  [tex]\sum (x-\bar x)^2[/tex].

When [tex]\sum (x-\bar x)^2[/tex] is highest  then the S.D is also highest.

We can see that  the value of  [tex]\sum (x-\bar x)^2[/tex] is highest  in option B.

Therefore, S.D of the date set of  option B is highest.

Final answer:

The data set with the numbers 1 through 4 (option A) likely has the highest standard deviation due to having a greater spread compared to others.

Explanation:

The student has asked which data set has the highest standard deviation without doing calculations. In looking at the provided options, we can determine that the higher the variability in the data set, the higher the standard deviation will be. Data set A has numbers 1 through 4, which are more spread out compared to the other sets. Therefore, option A) 1,2,3,4 is likely to have the highest standard deviation. Option D) 4,4,4,4 will have a standard deviation of zero since all the data points are the same. Standard deviation is a measure of how spread out numbers are and a zero standard deviation occurs when all values in a data set are identical.

Answer:

[tex]z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex]   (1)

[tex]z=\frac{0.34848-0.34782}{\sqrt{0.34831(1-0.34831)(\frac{1}{66}+\frac{1}{23})}}=0.0057[/tex]  

[tex]p_v =P(Z>0.0057)=0.4977[/tex]  

The p value is a very high value and using the significance level [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the percent of men who enjoy shopping for electronic equipment is NOT significantly higher than the percent of women who enjoy shopping for electronic equipment.  

Step-by-step explanation:

1) Data given and notation  

[tex]X_{M}=23[/tex] represent the number of men that said they enjoyed the activity of Saturday afternoon shopping

[tex]X_{W}=8[/tex] represent the number of women that said they enjoyed the activity of Saturday afternoon shopping

[tex]n_{M}=66[/tex] sample of male selected

[tex]n_{W}=23[/tex] sample of demale selected

[tex]p_{M}=\frac{23}{66}=0.34848[/tex] represent the proportion of men that said they enjoyed the activity of Saturday afternoon shopping

[tex]p_{W}=\frac{8}{23}=0.34782[/tex] represent the proportion of women with red/green color blindness  

z would represent the statistic (variable of interest)  

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

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment , the system of hypothesis would be:  

Null hypothesis:[tex]p_{M} \leq p_{W}[/tex]  

Alternative hypothesis:[tex]p_{M} > p_{W}[/tex]  

We need to apply a z test to compare proportions, and the statistic is given by:  

[tex]z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex]   (1)

Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{23+8}{66+23}=0.34831[/tex]

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

[tex]z=\frac{0.34848-0.34782}{\sqrt{0.34831(1-0.34831)(\frac{1}{66}+\frac{1}{23})}}=0.0057[/tex]  

4) Statistical decision

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

Since is a one side right tail test the p value would be:  

[tex]p_v =P(Z>0.0057)=0.4977[/tex]  

So the p value is a very high value and using the significance level [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the percent of men who enjoy shopping for electronic equipment is NOT significantly higher than the percent of women who enjoy shopping for electronic equipment.  

A two-proportion z-test shows there is insufficient evidence at the 5% significance level to conclude that a higher proportion of men enjoy shopping for electronic equipment compared to women. The p-value is greater than the alpha value, leading to not rejecting the null hypothesis.

A hypothesis test can be used to determine if the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy the same activity. We will perform a two-proportion z-test for this purpose. The hypotheses are as follows:

Null Hypothesis (H0): pm = pw

Alternative Hypothesis (Ha): pm > pw

Here, pm is the proportion of men who enjoy shopping for electronic equipment, and pw is the proportion of women who enjoy shopping for electronic equipment.

Given:

The test statistic for the difference in proportions is calculated as:

z = (pm - pw) / [tex]\sqrt{(p*(1 - p*)(1/nm + 1/nw))}[/tex]

Where p* = (xm + xw) / (nm + nw)

Substituting the values:

The calculated z-value is approximately 0.0130.

The corresponding p-value for a z-value of 0.0130 is approximately 0.4948. Since p-value > alpha (0.4948 > 0.05), we do not reject the null hypothesis.

Conclusion:

At the 5% significance level, there is insufficient evidence to conclude that the proportion of men who enjoy shopping for electronic equipment is significantly higher than that of women.

Answer:

Hence the value of damping coefficient = 1.49071.

Step-by-step explanation:

Answer:

Probability that we have an ordered pair (x,y) representing two points that satisfy the conditions is 328/400.

Step-by-step explanation:

This is a geometric probability.

set of all possible x in [0,20) and y in (20,40] we obtain a square in the x-y plane (in the first quadrant).

Area of this square is 20x20 = 400.

Set of all (x,y), that satisfy the distance being greater than 12

y > x, this means that (y - x) > 12 which is the same as the inequality y > x + 12.

For inequality, we obtain the region in the plane above the line y = x + 12.  The area of this region is

(1/2)bh = (1/2)(12)(12) = 72.

Thus the area of the set of all points in our square that satisfy the condition (y - x) > 12 is 400 - 72 = 328.

 Probability that we have an ordered pair (x,y) representing two points that satisfy the conditions is 328/400.

Answer:

41 is correct

Step-by-step explanation:

90 + 49 = 139

180 - 139 = 41

Answer:

744

Step-by-step explanation:

The alphabet has 26 letters. Since each character of the password can be a capital letter or one of the four character of the set {*, $, &, #, %}, then each space in the password has 26+5=31 options to select a character.

a) If the password has length 7, then there are 31*7=217 different passwords.

b) If the password has length 8, then there are 31*8=248 different passwords and,

c) if the password has length 9, then there are 31*9=279 different passwords.

Then the total of different passwords is 217+248+279=744.

There are different numbers of possible passwords based on the length of the password.

To determine the number of possible different passwords, we need to consider the different options for each character in the password and multiply them together.

For a password of length 7, each character has 5 possible options (capital letters or special characters). Therefore, there are 5 options for each of the 7 characters, giving us a total of 5^7 = 78,125 possible passwords.

Similarly, for a password of length 8, there are 5 options for each of the 8 characters, giving us a total of 5^8 = 390,625 possible passwords.

Finally, for a password of length 9, there are 5 options for each of the 9 characters, giving us a total of 5^9 = 1,953,125 possible passwords.

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Answer:

Step-by-step explanation:

To compute the tSTAT, plug the given values into the formula tSTAT = (X1 - X2) / sqrt(Sp2 * (1/n1 + 1/n2)). For the degrees of freedom, calculate df = n1 + n2 - 2. Whether to reject or not the null hypothesis H0 depends on whether the absolute value of tSTAT is greater than the critical value tCritical.

The question pertains to statistical hypothesis testing, specifically the t-test for comparing two independent samples. To solve this:

a. First, calculate the tSTAT as follows: tSTAT = (X1 - X2) / sqrt(Sp2 * (1/n1 + 1/n2)). Using your given values, this becomes tSTAT = (42 - 34) / sqrt(776 * (1/8 + 1/15)). Evaluating the right side will give you the tSTAT.

b. For this t-test, the degree of freedom (df) is calculated as df = n1 + n2 - 2, which gives df = 8+15-2 = 21.

c. The decision whether to reject or fail to reject the null hypothesis H0 depends on the comparison of the calculated tSTAT with the critical value (from the t-distribution table for df = 21). If |tSTAT| > tCritical, reject H0; otherwise, fail to reject H0.

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Answer:

[tex]t=\frac{\pm\sqrt{dk}}{k}[/tex]

Step-by-step explanation:

Given relation:

[tex]d=k\ t^2[/tex]

We need to solve for [tex]t[/tex]

The given relation can be rearranged by isolating [tex]t[/tex] on one side.

Swapping the sides of the equation we have,

[tex]k\ t^2=d[/tex]

Dividing both sides by [tex]k[/tex] on order to cancel out [tex]k[/tex] on left side.

[tex]\frac{k\ t^2}{k}=\frac{d}{k}[/tex]

[tex]t^2=\frac{d}{k}[/tex]

We have got an expression for [tex]t^2[/tex] but we need to solve for [tex]t[/tex]

So, we take square root both sides to change [tex]t^2[/tex] to [tex]t[/tex]

[tex]\sqrt{t^2}=\sqrt{\frac{d}{k}}[/tex]

[tex]t=\pm\sqrt{\frac{d}{k}}[/tex]

So, we have successfully isolated [tex]t[/tex] on left side.

But the expression we got is a fraction with a square root  in the denominator. Thus we need to rationalize it to make it in simplest form.

The expression can be written by taking square root separately for numerator and denominator :

[tex]t=\pm{\frac{\sqrt{d}}{\sqrt{k}}}[/tex]

Multiplying the numerator and denominator by [tex]\sqrt{k}[/tex]

[tex]t=\pm\frac{\sqrt{d}}{\sqrt{k}}\times\frac{\sqrt{k}}{\sqrt{k}} [/tex]

[tex]t=\pm\frac{\sqrt{dk}}{(\sqrt{k})^2}[/tex]

Square of a square root will remove the square root. Thus we have,

∴ [tex]t=\frac{\pm\sqrt{dk}}{k}[/tex]

Thus we have successfully got the expression in the simplest form.

Answer:

a) 25%

b) 50%

c) 87.5%

Step-by-step explanation:

There are [tex]2^4 = 16[/tex] number of outcome in total.

a)First and last flips come up heads. There are 4 outcomes here. Probability of the event is 4/16 = 1/4, or 25%.

- HHHH

- HHTH

- HTHH

- HTTH

b) at least 2 Consecutive flips that is heads. There are 8 outcomes. Probability of event is 8/16 = 1/2, or 50%

- HHHH

- HHHT

- HHTH

- HHTT

- THHH

- THHT

- HTHH

- TTHH

c) At least 2 consecutive flips that are the same. There are 14 outcomes. Probability of 14/16 = 7/8, or 87.5%

- HHHH

- HHHT

- HHTH

- HHTT

- THHH

- THHT

- HTHH

- TTHH

- TTTT

- TTTH

- TTHT

- HTTT

- HTTH

- THTT

Final answer:

In a) The probability is 1/4 and the event can be expressed as {HH**}, b) The probability is 7/16 and the event can be expressed as {HH**, *HH*, **HH}, c) The probability is 9/16 and the event can be expressed as {HH**, TT**, *HH*, *TT*}

Explanation:

To express the events as sets in roster notation, we need to list all the outcomes that satisfy each event.

(a) The first and last flips come up heads:

This event can be expressed as {HH**} (where ** represents any outcome for the two middle flips). The probability of this event is 1/2 * 1 * 1/2 = 1/4, since the first flip must be heads (1/2 probability), the second and third flips can be either heads or tails (1/2 probability each), and the last flip must be heads (1/2 probability).

(b) There are at least two consecutive flips that come up heads:

This event can be expressed as {HH**, *HH*, **HH}, where * represents any outcome for a single flip. The probability of this event is 1/2 * 1 * 1/2 * 1/2 + 1/2 * 1/2 * 1 * 1/2 + 1/2 * 1/2 * 1/2 * 1 = 7/16, since there are three possible patterns that satisfy this event.

(c) There are at least two consecutive flips that are the same:

This event can be expressed as {HH**, TT**, *HH*, *TT*}, where * represents any outcome for a single flip. The probability of this event is 1/2 * 1 * 1 + 1/2 * 1 * 1 + 1/2 * 1/2 * 1 + 1/2 * 1/2 * 1 = 9/16, since there are four possible patterns that satisfy this event.

Answer: Our required probability is 0.351.

Step-by-step explanation:

Since we have given that

Probability of getting classroom 1 = [tex]\dfrac{1}{2}[/tex]

Probability of getting classroom 2 = [tex]\dfrac{1}{2}[/tex]

Probability of getting girl from classroom 1 = [tex]\dfrac{13}{18}[/tex]

Probability of getting girl from classroom 2 = [tex]\dfrac{9}{23}[/tex]

Using "Bayes theorem".

so, the probability that classroom 2 was chosen given that a girl was chosen would be

[tex]\dfrac{\dfrac{1}{2}\times \dfrac{9}{23}}{\dfrac{1}{2}\times \dfrac{9}{23}+\dfrac{1}{2}\times \dfrac{13}{18}}\\\\=\dfrac{0.195}{0.195+0.361}\\\\=0.351[/tex]

Hence, our required probability is 0.351.

Final answer:

The distance the pitcher must travel to get to the ball is approximately 36 ft.

Explanation:

The distance the pitcher must travel to get to the ball can be found using trigonometry. We can break down the motion of the ball into horizontal and vertical components. The horizontal component of the motion is the distance the ball rolls down the first base line, which is given as 33 ft. The vertical component of the motion can be found using the angle of 25 degrees. We can use the sine function to find the vertical distance:

Vertical distance = sin(25 degrees) * 33 ft = 14.04 ft

The distance the pitcher must travel is the hypotenuse of a right triangle formed by the horizontal and vertical distances. Using the Pythagorean theorem, we can find the distance:

Distance = sqrt((33 ft)^2 + (14.04 ft)^2) = 35.88 ft

Therefore, the pitcher must run approximately 36 ft to get to the ball.

8x^4y – 16x^2y^2=

=8x^4y-16x^4y

=-8x^4y

=-8*(x^4y)

Answer:

8\,x^2\,y\,(x-\sqrt{2} )(x+\sqrt{2} )

Step-by-step explanation:

Let's start by extracting all common factors from the two terms of this binomial. These common factors are: 8, [tex]x^2[/tex], and [tex]y[/tex].

The extraction renders:

[tex]8\,x^2\,y\,(x^2-2)[/tex]

In the real number system, the binomial in parenthesis can still be factored out considering that 2 is the perfect square of [tex]\sqrt{2}[/tex], that is:

[tex]2=(\sqrt{2} )^2[/tex]

We can then forwards with the factoring of this binomial using the factorization of a difference of squares as:

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

Thus giving the complete factorization as:

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

The volume of the original cone is multiplied by 27

We have been given that the height and radius of a cone are each multiplied by 3.

We need to find what effect it has on the volume of the cone.

The volume of the original cone is given as follows:

[tex]\text {Volume of cone}=\frac{1}{3} \pi r^{2} h[/tex]

Now, the radius and height are multiplied by 3.

Therefore, the new radius is ‘3r’ and the new height is ‘3h’.

Hence the new volume is  

[tex]\text {Volume of new cone}=\frac{1}{3} \pi(3 r)^{2}(3 h)=9 \pi r^{2} h[/tex]

original volume of cone x 27 = volume of new cone

Therefore, the original volume of cone is multiplied by 27

Final answer:

When the height and radius of a cone are each multiplied by 3, the volume of the cone is multiplied by 27.

Explanation:

When the height and radius of a cone are each multiplied by 3, the effect on the volume of the cone is that the volume is multiplied by 27.

The formula for the volume of a cone is V = (1/3)πr²h. When the height and radius are multiplied by 3, the new volume would be V' = (1/3)π(3r)²(3h) = 27(1/3)πr²h = 27V.

Therefore, the volume of the cone is multiplied by 27.

Answer:

40.72 gallons

Step-by-step explanation:

Since [tex]x = 18cos(\theta)[/tex] and [tex]y = 18sin(\theta)[/tex]. We can say this base has a shape of a circle of radius r = 18.

We can also calculate h in term of angle θ:

[tex]h(\theta) = 12 + \frac{36cos(\theta) - 18sin(\theta)}{6}[/tex]

[tex]h(\theta) = 12 + 6cos(\theta) - 3sin(\theta)[/tex]

We can calculate the area of the fence if we integrate this h function over θr, which is the chord length. θ ranges from 0 to 2π

A = ∫h(θ)rdθ

A = ∫(12 + 6cos(θ) - 3sin(θ))18d

A = (12θ + 6sin(θ) + 3cos(θ))18

As θ ranges from 0 to 2π as a full circle

A = (12*0 + 6sin(0) + 3cos(0))18 - (12*2π + 6sin(2π) + 3cos(2π))18

A = 3*18 - 18(24π + 3) = 1357.17 square yard

As 1 yard = 3 feet then 1 square yard = 9 square feet

1 gallon of paint can cover 300 square feet or 300 / 9 = 33.33 square yard

So to cover square yard Bob would need

1357.17 / 33.33 = 40.72 gallons of paint.

(a) The displacement of the particle is [tex]\(-\frac{21}{2}\)[/tex] meters.

(b) The distance travelled by the particle during the given time interval is [tex]\(\frac{169}{6}\)[/tex] meters.

(a) To find the displacement of the particle over the time interval [0, 3], we need to calculate the definite integral of the velocity function v(t) over this interval. The displacement is given by the integral of velocity with respect to time.

The velocity function is v(t) = 3t - 8.

The displacement S over the interval [a, b] is given by:

[tex]\[ S = \int_{a}^{b} v(t) \, dt \][/tex]

For our interval [0, 3], we have:

[tex]\[ S = \int_{0}^{3} (3t - 8) \, dt \][/tex]

To compute this integral, we find the antiderivative of the integrand:

[tex]\[ \int (3t - 8) \, dt = \frac{3}{2}t^2 - 8t + C \][/tex]

Now, we evaluate this antiderivative at the endpoints of the interval and subtract:

[tex]\[ S = \left(\frac{3}{2}(3)^2 - 8(3)\right) - \left(\frac{3}{2}(0)^2 - 8(0)\right) \][/tex]

[tex]\[ S = \left(\frac{3}{2} \cdot 9 - 24\right) - (0) \][/tex]

[tex]\[ S = \frac{27}{2} - 24 \][/tex]

[tex]\[ S = \frac{27}{2} - \frac{48}{2} \][/tex]

[tex]\[ S = -\frac{21}{2} \][/tex]

So the displacement of the particle is [tex]\(-\frac{21}{2}\)[/tex] meters.

(b) To find the distance travelled by the particle, we need to consider the intervals where the velocity is positive and where it is negative. The distance travelled is the sum of the distances travelled during each interval, without regard to direction.

First, we find the times when the velocity is zero by setting v(t) equal to zero:

[tex]\[ 3t - 8 = 0 \][/tex]

[tex]\[ 3t = 8 \][/tex]

[tex]\[ t = \frac{8}{3} \][/tex]

Since [tex]\(0 \leq t \leq 3\),[/tex] the velocity changes sign at [tex]\(t = \frac{8}{3}\)[/tex]. This divides the interval [0, 3] into two subintervals: [0, 8/3] and [8/3, 3].

We calculate the distance travelled in each subinterval and sum them:

For [0, 8/3]:

[tex]\[ d_1 = \int_{0}^{\frac{8}{3}} |3t - 8| \, dt \][/tex]

Since the velocity is negative on this interval, we have:

[tex]\[ d_1 = \int_{0}^{\frac{8}{3}} (8 - 3t) \, dt \][/tex]

[tex]\[ d_1 = \left(8t - \frac{3}{2}t^2\right) \Bigg|_{0}^{\frac{8}{3}} \][/tex]

[tex]\[ d_1 = \left(8\left(\frac{8}{3}\right) - \frac{3}{2}\left(\frac{8}{3}\right)^2\right) - (0) \][/tex]

[tex]\[ d_1 = \frac{64}{3} - \frac{3}{2}\left(\frac{64}{9}\right) \][/tex]

[tex]\[ d_1 = \frac{64}{3} - \frac{32}{3} \][/tex]

[tex]\[ d_1 = \frac{32}{3} \][/tex]

For [8/3, 3]:

[tex]\[ d_2 = \int_{\frac{8}{3}}^{3} |3t - 8| \, dt \][/tex]

Since the velocity is positive on this interval, we have:

[tex]\[ d_2 = \int_{\frac{8}{3}}^{3} (3t - 8) \, dt \][/tex]

[tex]\[ d_2 = \left(\frac{3}{2}t^2 - 8t\right) \Bigg|_{\frac{8}{3}}^{3} \][/tex]

[tex]\[ d_2 = \left(\frac{3}{2}(3)^2 - 8(3)\right) - \left(\frac{3}{2}\left(\frac{8}{3}\right)^2 - 8\left(\frac{8}{3}\right)\right) \][/tex]

[tex]\[ d_2 = \left(\frac{27}{2} - 24\right) - \left(\frac{32}{3} - \frac{64}{3}\right) \][/tex]

[tex]\[ d_2 = \frac{27}{2} - 24 + \frac{32}{3} \][/tex]

[tex]\[ d_2 = \frac{81}{6} - \frac{144}{6} + \frac{64}{6} \][/tex]

[tex]\[ d_2 = -\frac{105}{6} \][/tex]

[tex]\[ d_2 = -\frac{35}{2} \][/tex]

Since we cannot have a negative distance, we take the absolute value:

[tex]\[ d_2 = \frac{35}{2} \][/tex]

The total distance travelled is the sum of the distances in each interval:

[tex]\[ D = d_1 + d_2 \][/tex]

[tex]\[ D = \frac{32}{3} + \frac{35}{2} \][/tex]

[tex]\[ D = \frac{64}{6} + \frac{105}{6} \][/tex]

[tex]\[ D = \frac{169}{6} \][/tex]

So the distance travelled by the particle during the given time interval is [tex]\(\frac{169}{6}\)[/tex] meters.

The result when the number 30 is decreased by 10% is 27, calculated by finding 10% of 30, which is 3, and subtracting it from the original number.

When the number 30 is decreased by 10%, this implies we need to find 10% of 30 and then subtract it from 30. To calculate 10% of a number, you can simply divide the number by 10. Hence, 10% of 30 is 30 ÷ 10 = 3. Now, to decrease 30 by this value, we do the subtraction: 30 - 3 = 27. Therefore, the result when the number 30 is decreased by 10% is 27.

The number 27 is obtained by subtracting 10% from 30.

Step 1: Identify the initial number.

Given that the initial number is 30.

Step 2: Determine the percentage to be decreased.

The percentage to be decreased is 10%.

Step 3: Calculate the amount to be decreased.

To find 10% of 30, multiply 30 by 0.10.

10% of 30 = 30 × 0.10 = 3.

Step 4: Subtract the amount to be decreased from the initial number.

30 - 3 = 27.

Step 5: Interpret the result.

So, when 30 is decreased by 10%, the result is 27.

Therefore, the result of decreasing the number 30 by 10% is 27.

Answer:

Length = 20 ft

Height = 10 ft

Step-by-step explanation:

Let 'X' be the total width of the billboard and 'Y' the total height of the billboard. The total area and printed area (excluding margins) are, respectively:

[tex]200 = x*y\\A_p = (x-4)*(y-2)[/tex]

Replacing the total area equation into the printed area equation, gives as an expression for the printed area as a function of 'X':

[tex]y=\frac{200}{x} \\A_p = (x-4)*(\frac{200}{x} -2)\\A_p=208 -2x -\frac{800}{x}[/tex]

Finding the point at which the derivate for this expression is zero gives us the value of 'x' that maximizes the printed area:

[tex]\frac{dA_p(x)}{dx} =\frac{d(208 -2x -\frac{800}{x})}{dx}=0\\0=-2 +\frac{800}{x^2} \\x=\sqrt{400}\\x=20\ ft[/tex]

If x = 20 ft, then y=200/20. Y= 10 ft.

The dimensions that maximize the printed area are:

Length = 20 ft

Height = 10 ft

The dimensions that maximize the printed area of the billboard are 12 ft by 12 ft (excluding the margins) which gives a printed area of 144 square feet.

To solve this problem, we use the relationship between area, length, and width. The total area, including marginal spaces is given as 200 square feet. The top and bottom margins sum to 2 feet, and the side margins sum to 4 feet.

Let's denote the width of the printed area (excluding the margins) as x and the length as y. Therefore, the width of the whole billboard is x + 4 ft and the length is y + 2 ft.

We know that Area = length * width, so we can establish the equation (x + 4)(y + 2) = 200.

To maximize the printed area, it's best when x and y are equal (a principle in mathematics), making the printed area a square. Also, x + y being constant, the square gives the maximum product.

Solving equation (x + 4) = (y + 2), we find that x = y = 12 feet, which maximizes the printed area giving us 144 square feet.

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Final answer:

The demand for tie-dyed T-shirts will drop by approximately 23 T-shirts when the price is increased from $15 to $17. This calculation is based on the function q = 510 − 90[tex]p^{0.5}[/tex], which represents the demand in relation to the price.

Explanation:

The student is asking how fast the demand for tie-dyed T-shirts will drop if the price increases by $2 when the demand is represented by the function q = 510 − 90[tex]p^{0.5}[/tex]. Since the current price is $15, we need to calculate the demand at both $15 and $17 to find the rate of change in demand when the price is raised by $2.

To find the demand at the current price of $15, substitute p = 15 into the demand function:
q = 510 − 90[tex](15)^{0.5}[/tex]
q = 510 − 90(3.87)
q = 510 − 348.3
q = 161.7

To find the demand when the price is $17, substitute p = 17 into the demand function:
q = 510 − 90[tex](17)^{0.5}[/tex]
q = 510 − 90(4.12)
q = 510 − 371
q = 139

The change in demand is the difference between the two quantities:
Δq = 161.7 - 139 = 22.7
The demand drops by approximately 23 T-shirts when the price increases by $2.

The demand will drop by approximately [tex]\( \boed{23} \)[/tex] T-shirts per month.

Step 1

To determine how fast the demand for tie-dyed T-shirts will drop when the price increases by $2, we need to calculate the rate of change of the demand q with respect to the price p. The given demand function is:

[tex]\[ q = 510 - 90p^{0.5} \][/tex]

We need to find the derivative of q with respect to p, which gives us the rate of change of demand with respect to price.

Step 2

First, let's find the derivative [tex]\( \frac{dq}{dp} \):[/tex]

[tex]\[ q = 510 - 90p^{0.5} \]\[ \frac{dq}{dp} = -90 \cdot \frac{1}{2} p^{-0.5} \]\[ \frac{dq}{dp} = -45 p^{-0.5} \]\[ \frac{dq}{dp} = -45 \cdot \frac{1}{\sqrt{p}} \][/tex]

Step 3

Now, we need to evaluate this derivative at the current price [tex]\( p = 15 \)[/tex]:

[tex]\[ \frac{dq}{dp} \bigg|_{p=15} = -45 \cdot \frac{1}{\sqrt{15}} \][/tex]

Calculate \[tex]( \sqrt{15} \)[/tex]:

[tex]\[ \sqrt{15} \approx 3.872 \][/tex]

So,

[tex]\[ \frac{dq}{dp} \bigg|_{p=15} = -45 \cdot \frac{1}{3.872} \approx -11.62 \][/tex]

This means the rate of change of demand with respect to price at [tex]\( p = 15 \)[/tex] is approximately [tex]\(-11.62\)[/tex] T-shirts per dollar.

Given that the price is increased by $2, we need to find how much the demand drops for this price increase:

[tex]\[ \frac{dq}{dp} \times 2 \approx -11.62 \times 2 \approx |-23.24| \][/tex]

Rounded to the nearest whole number, the demand drops by approximately 23 T-shirts per month when the price is raised by $2.

Answer:

Yes it is below 10% of the population

Step-by-step explanation:

10% of 12 million is 1,200,00

One of the conditions is as 2,500

The population is at least 10 times as large as the sample.

here 10*n = 10* 2500 = 25000 < 12000000

So the correct option is "Yes, 10(2,500) = 25,000, which is less than the total population."

5. The Lakeland Post polled 1,231 adults in the city to determine whether they wash their hair before their body while in the shower. Of the respondents, 63% said they washed their hair first. Suppose 51% of all adults actually wash their hair first. What are the mean and standard deviation of the sampling distribution?

Sampling distribution of sample proportion ( \hat p ) is approximately normal for large n with mean and standard deviations are as

mean = p = 0.51

for finding the standard deviation we need to use p = 0.51

б = √p·(1 - p )/ n = √0.51 . 0.49/ 1234 = 0.014248

So the correct choice is "The mean is 0.51 and the standard deviation is 0.0142"

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