You want to know if there's a difference between the proportions of high-school students and college students who read newspapers regularly. Out of a random sample of 500 high-school students, 287 say they read newspapers regularly, and out of a random sample of 420 college students, 252 say they read newspapers regularly. For this question, think of high-school students as sample one and college students as sample two.

A. Construct a 95% confidence interval for the difference between the proportions of high-school students and college students who read newspapers regularly. Be sure to show that you've satisfied the conditions for using a z-interval. (5 points)

B. Draw a conclusion, based on your 95% confidence interval, about the difference between the two proportions. (2 points)

C. If you wanted to use a test statistic to determine whether the proportion of high-school students who read newspapers regularly is significantly lower than the proportion of college students who read newspapers regularly, what would you use as your null and alternative hypotheses? (2 points)

D. Calculate p ˆ, the pooled estimate of the population proportions you'd use for a significance test about the difference between the proportions of high-school students and college students who read newspapers regularly. (1 point)

E. Demonstrate that these samples meet the requirements for using a zprocedure for a significance test about the difference between two proportions. (2 points)

F. Calculate SEp ˆ , the pooled estimate of the standard errors of the proportions you'd use in a z-procedure for a significance test about the difference between two proportions. (1 point)

G. Calculate your test statistic and P-value for the hypothesis test H0 : p1 = p2 , Ha : p1 < p2 . (4 points)

H. Draw a conclusion about the difference between the two proportions using α = .05. Is the proportion of high-school students who read the newspaper on a regular basis less than the proportion of college students who read newspapers regularly?

Answer:

186

Step-by-step explanation:

The sample mean, x, is the central value in the confidence interval, that is, the average between the upper and lower bounds of the interval.

In this case, the Lower bound is 184.00 and the upper bound is 188.00. Therefore, the sample mean is given by:

[tex]x = \frac{184+188}{2}\\x=186.00[/tex]

The mean length, in seconds, of the sampled songs is 186.00.

The sample mean is 186.

The sample mean, denoted as [tex]\( x \)[/tex], can be found by taking the midpoint of the confidence interval. In this case, the confidence interval is (184.00, 188.00). To find [tex]\( x \)[/tex], we take the average of the lower and upper bounds of the interval. Thus,

[tex]\[ x = \frac{184.00 + 188.00}{2} = \frac{372.00}{2} = 186.00 \][/tex]

Therefore, the sample mean [tex]\( x \)[/tex] is 186.00 seconds. This means that, on average, the length of songs in the online database, based on the given sample, is 186.00 seconds.

Step-by-step explanation:

(a, b) → (a, -b)

This is a reflection across the x-axis.

(a, b) → (a, b+5)

This is a translation 5 units up.

(a, b) → (b, -a)

This is a rotation of 270° about the origin.

Answer:

Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2

Step-by-step explanation:

Set up hypotheses as

[tex]H_0: p1 = p2\\H_a: p1>p2[/tex]

(Right tailed t test)

Alpha = 0.05

Sample               I                II                Total

X                        117             141              248

N                        249           312            561

p                        0.4700     0.4519        0.4420

p difference = 0.0181    

Std dev = [tex]\sqrt{p(1-p)(\frac{1}{n_1} +\frac{1}{n_2} )}[/tex]

=0.0427

z statistic = 0.424

p value = 0.33724

Since p > alpha, we accept H0, there is no evidence to prove that p1 is greater than p2

Final answer:

To test whether p1 is greater than p2 using hypothesis testing, one needs to state the null and alternative hypotheses, calculate the test statistic and p-value, and then decide whether to reject or accept the null hypothesis based on the level of significance and the p-value. Calculating the test statistic and p-value requires specific formulas and is not provided here.

Explanation:

Conducting a Hypothesis Test for Two Population Proportions

A student would like assistance in conducting a hypothesis test for two population proportions. The steps to perform such a test include:

State the null and alternative hypotheses: The null hypothesis (H0) is that there is no difference between the two population proportions (p1 - p2 = 0), while the alternative hypothesis (Ha) posits that there is a difference (p1 > p2).

The random variable (P') represents the difference between the two sample proportions.

Calculate the test statistic: Using the provided sample sizes and successful outcomes, the test statistic is calculated using a formula based on the Z-distribution.

Calculate the p-value: The p-value is determined from the test statistic, indicating the probability of observing such a result if the null hypothesis were true.

At the 5 percent level of significance, compare the p-value to the alpha value of 0.05 to make a decision. If the p-value is less than alpha, reject the null hypothesis.

The Type I error would occur if the null hypothesis is incorrectly rejected when it is actually true.

The Type II error would occur if the null hypothesis is not rejected when it is actually false.

The test statistic and p-value cannot be calculated from the information provided without the appropriate formulas or statistical tools.

Decision Making and Errors

If the alpha level is greater than the p-value, the null hypothesis should be rejected, indicating there is evidence to suggest p1 is greater than p2.

Failure to reject the null hypothesis when it is false constitutes a Type II error.

Answer:

a discrete random variable

Step-by-step explanation:

You can only have a natural number of clients entering the store.

For example, 0 clients, 1 client, 2 clients, 100 clients, ...

You cannot have a decimal value, for example, 0.5 clients.

So the correct answer is:

a discrete random variable

The probability that the fisherman stays for more than two hours is approximately 0.4512. The probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043. The expected number of fish that the fisherman catches is 0.6.

To solve this problem, we can use the concept of a Poisson process and the properties of the Poisson distribution.

(a) We want to find the probability that the fisherman stays for more than two hours. Since the fisherman quits at the end of the second hour if he has caught at least one fish, he will stay for more than two hours only if he hasn't caught any fish in the first two hours. Using the Poisson distribution, the probability of catching zero fish in two hours is given by P(X=0)=e^(-lambda*t)*(lambda^0)/(0!)=e^(-0.6*2)≈0.5488. Therefore, the probability that the fisherman stays for more than two hours is 1 - P(X=0) = 1 - 0.5488 ≈ 0.4512.

(b) The total time the fisherman spends fishing can be between two and five hours. This can happen in three ways: fishing for 2 hours without catching fish (P(X=0)) and then fishing for 3 more hours until catching the first fish (P(X=1)); fishing for 3 hours without catching fish (P(X=0)) and then fishing for 2 more hours until catching the first fish (P(X=1)); fishing for 4 hours without catching fish (P(X=0)) and then fishing for 1 more hour until catching the first fish (P(X=1)). Using the Poisson distribution, we can calculate the probabilities for each case: P(X=0) = e^(-0.6*2) ≈ 0.5488, P(X=1) = e^(-0.6*3)*(0.6^1)/(1!) ≈ 0.3293. Adding up these probabilities, we get P(X=0)*P(X=1) + P(X=0)*P(X=1) + P(X=0)*P(X=1) = 0.5488*0.3293 + 0.5488*0.3293 + 0.5488*0.3293 ≈ 0.5043. Therefore, the probability that the total time the fisherman spends fishing is between two and five hours is approximately 0.5043.

(c) The expected number of fish that the fisherman catches can be found using the formula for the mean of a Poisson distribution, which is equal to the rate parameter lambda. In this case, lambda = 0.6, so the expected number of fish is 0.6.

(d) To find the expected total fishing time given that the fisherman has been fishing for four hours, we need to condition on the event that the fisherman hasn't caught any fish in the first four hours. Using the Poisson distribution, the probability of catching zero fish in four hours is given by P(X=0) = e^(-lambda*t)*(lambda^0)/(0!) = e^(-0.6*4) ≈ 0.3012. Therefore, the expected total fishing time given that the fisherman has been fishing for four hours is 4 + (1/P(X=0)) = 4 + (1/0.3012) ≈ 7.32 hours.

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Answer:

12 feet

Step-by-step explanation:

Draw a diagram (see picture below). The tree and its shadow is one triangle, the man and its shadow is another triangle. We assume both are right triangles because people and trees stand vertical.

Create a proportion to solve. Put the missing value in a numerator.

Tree height / Tree shadow = Man height / Man shadow

[tex]\frac{x}{8} =\frac{6}{4}[/tex]

Solve using cross multiplication. Multiply x by 4. Multiply 6 by 8.

4x = 48    Divide both sides by 4 to isolate x.

x = 12       Height of tree

The tree is 12 feet tall.

Answer:

Step-by-step explanation:

the sign here means division

7 divided by the number in the box equals to 8

simply meaning 7 multiplied by 8 will give the number

7 x 8 = 56  

Formula for monthly payment is:

A = P x (r(1+r)^t)/((1+r)^t-1) where P is the amount financed, r is the interest rate divided by 12 and t is the amount of time for the loan in months.

P = 33714 x 0.85 = 28656.90

A = 28656.90 x (0.07/12 (1+0.07/12)^48) / (1 +0.07/12)^48 - 1)

A = $686.23

Answer:

[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]  

[tex]p_v =P(Z<-17.527) \approx 0[/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.  

We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.

Step-by-step explanation:

Data given and notation  

[tex]\bar X=120[/tex] represent the sample mean  

[tex]\sigma=25[/tex] represent the population standard deviation  

[tex]n=30[/tex] sample size  

[tex]\mu_o =200[/tex] represent the value that we want to test  

[tex]\alpha=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 population mean is less than 200, the system of hypothesis are :  

Null hypothesis:[tex]\mu \geq 200[/tex]  

Alternative hypothesis:[tex]\mu < 200[/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) the info given like this:  

[tex]z=\frac{120-200}{\frac{25}{\sqrt{30}}}=-17.527[/tex]  

P-value  

Since is a one-side left tailed test the p value would given by:  

[tex]p_v =P(Z<-17.527) \approx 0[/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.  

We can say that at 5% of significance the mean average waiting time is significantly less than 200 seconds per call.

Answer:

1) n=114

2) n=59

3) On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.

Step-by-step explanation:

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".  

The population proportion have the following distribution

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 80% of confidence, our significance level would be given by [tex]\alpha=1-0.80=0.2[/tex] and [tex]\alpha/2 =0.1[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.28, z_{1-\alpha/2}=1.28[/tex]

Part 1

The margin of error for the proportion interval is given by this formula:  

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

And on this case we have that [tex]ME =\pm 0.06[/tex] or 6% points, and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

Since we don't have a prior estimate of [tex]\het p[/tex] we can use 0.5 as a good estimate, replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.5(1-0.5)}{(\frac{0.06}{1.28})^2}=113.77[/tex]  

And rounded up we have that n=114

Part 2

On this case we have a prior estimate for the population proportion and is [tex]\hat p =0.85[/tex] so replacing the values into equation (b) we got:

[tex]n=\frac{0.85(1-0.85)}{(\frac{0.06}{1.28})^2}=58.027[/tex]

And rounded up we have that n=59

Part 3

On this case no, because if we survey just the adults of the nearest college that would be a convenience sample. And when we use "convenience sample" we have some problems associated to bias. This methodology it's not appropiate in order to have a good estimation of the parameter of interest. It's better use a random, cluster or stratified sampling.

Answer: d. random variable.

Step-by-step explanation:

A random variable is a numerical description of outcomes of an experiment, it can be used to represent the possible values of a past experiment or yet-to-be-performed experiments. It is a variable whose values depends on outcome of a random occurrence. Random variables also allows the calculation of probability of an occurrence or result in a particular experiment.

The numerical description of the outcome of an experiment is best described as a random variable. It is not referred to as a descriptive statistic, a probability function, or variance.

In statistics, a numerical description of the outcome of an experiment is referred to as a random variable. The term random variable refers to a function that assigns a real number to each outcome of an experiment conducted according to a certain probability distribution. This term is central to probability theory and statistics, in which numerical results of random variables are analyzed to understand underlying processes or to make predictions.

On the other hand, descriptive statistics summarize and organize characteristics of a data set. A probability function is a mathematical function that provides the probabilities of occurrence of different possible outcomes. Variance is a measurement of spread between numbers in a data set.

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This ensures that if both project 2 and project 4 are selected, project 5 must also be selected. Since x5 is a binary variable, 2x5 is equivalent to x5, ensuring that project 5 is selected when the condition is met. (option d)

Let's formulate constraints for each of the given statements:

a. Out of projects 1, 2, 4, and 6, CRT’s management wants to select exactly two projects.

Let [tex]\(x_i\)[/tex] be a binary decision variable representing whether project i is selected or not, where [tex]\(i = 1, 2, 4, 6\).[/tex]

The constraint can be formulated as:

[tex]\[x_1 + x_2 + x_4 + x_6 = 2\][/tex]

This ensures that exactly two out of the listed projects are selected.

b. Project 2 can be selected only if project 3 is selected and vice-versa.

Let [tex]\(x_2\) and \(x_3\)[/tex] be binary decision variables representing whether project 2 and project 3 are selected, respectively.

The constraints can be formulated as:

[tex]\[x_2 \leq x_3\]\[x_3 \leq x_2\][/tex]

These constraints ensure that if project 2 is selected, project 3 must also be selected, and vice versa.

c. Project 5 cannot be undertaken unless both projects 3 and 4 are also undertaken.

Let [tex]\(x_3\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 3, project 4, and project 5 are selected, respectively.

The constraint can be formulated as:

[tex]\[x_5 \leq x_3 + x_4\][/tex]

This ensures that if project 5 is selected, both project 3 and project 4 must also be selected.

d. If projects 2 and 4 are undertaken, then project 5 must also be undertaken.

Let [tex]\(x_2\), \(x_4\), and \(x_5\)[/tex] be binary decision variables representing whether project 2, project 4, and project 5 are selected, respectively.

The constraint can be formulated as:

[tex]\[x_2 + x_4 \leq 2x_5\][/tex]

Answer:

We need to check the conditions in order to use the normal approximation.

[tex]np=100*0.2=20 \geq 10[/tex]

[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]

If we check the conditions with the estimated proportion we got:

[tex]n\hat p=100*0.41=41 \geq 10[/tex]

[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]

So we see that we satisfy the conditions and then we can apply the approximation.

Step-by-step explanation:

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

[tex]X \sim Binom(n=100, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

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

Where (nCx) means combinatory and it's given by this formula:

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

We need to check the conditions in order to use the normal approximation.

[tex]np=100*0.2=20 \geq 10[/tex]

[tex]n(1-p)=100*(1-0.2)=80 \geq 10[/tex]

If we check the conditions with the estimated proportion we got:

[tex]n\hat p=100*0.41=41 \geq 10[/tex]

[tex]n(1-\hat p)=100*(1-0.41)=59 \geq 10[/tex]

So we see that we satisfy the conditions and then we can apply the approximation.

To form a larger 4-inch cube, Clare will need 512 wooden cubes with an edge length of 1/2 inch. We find this by dividing the volume of the large cube (64 cubic inches) by the volume of the small one (1/8 cubic inch).

Clare is building a larger cube with an edge length of 4 inches, consisting of smaller wooden cubes each with an edge length of 1/2 inch. To solve this problem, we need to find out how many smaller cubes make up the volume of the larger cube. The volume of a cube is found by multiplying the length of an edge by itself three times, or cube the edge length.

So first, we calculate the volume of the large cube which is 4in * 4in * 4in = 64 cubic inches.

Next, we calculate the volume of a small cube which is (1/2in) * (1/2in) * (1/2in) = 1/8 cubic inch.

Finally, we divide the volume of the large cube by the volume of the small cube to find out how many small cubes are needed. Thus, 64 cubic inches / 1/8 cubic inch = 512. So, Clare will need 512 small wooden cubes to construct her larger cube.

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To find the number of little cubes needed, divide the volume of the larger cube by the volume of each little cube.

To find the number of little cubes Clare needs to build a larger cube with an edge length of 4 inches, we need to

determine the volume of the larger cube and divide it by the volume of each little cube.

The volume of the larger cube is calculated by multiplying the length of one side by itself three times (4 x 4 x 4 = 64 cubic inches).

The volume of each little cube is calculated by multiplying the length of one side by itself three times (1/2 x 1/2 x 1/2 = 1/8 cubic inches).

To find the number of little cubes needed, we divide the volume of the larger cube by the volume of each little cube (64 / (1/8) = 512 little cubes).

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Answer: Houston residences

Step-by-step explanation:

Here , The objective of the researcher is to determine the average number of vehicles are registered to a typical Houston residence.

Clearly , the population is this situation is "Houston residences" having vehicles.

Note : 250 randomly selected residences are defining the sample of the entire population of Houston residences which is a subset of population.

Hence, the correct answer is Houston residences.

Answer:

F=y+z=4/6.25

Step-by-step explanation:

First, we have to consider that in the problem model we have only two possible states: sunny and cloudy. Now, according to the information given in the statement, we also have the behavior of the last two days. In any case, we can have four possible transitional states:

Today-Yesterday(S=Sunny, C=cloudy)

1) ST and SY (Sunny today and sunny yesterday)

2) ST and CY.

3) CT and SY.

4) CT and CY.

Now, according to the statement, the probabilities given for the four states can be expressed by the following matrix:

[tex]\left[\begin{array}{cccc}0.6&0&(1-0.6)&0\\0.5&0&(1-0.5)&0\\0&0.4&0&(1-0.4)\\0&0.2&0&(1-0.2)\end{array}\right][/tex]

Now, making w, x, y, z as the transition probabilities for the four states mentioned, we then have that:

x=0.6w+0.5x

w=1.25x (1)

x=0.4y+0.2z (2)

y=0.4w+0.5x

y= 0.4(1.25x)+0.5x=x

y=x (3)

replacing 3 in 2:

y=0.4y+0.2x

x=3y (4)

And as w+x+y+z= 1 (no more possible combinations):

w+x+y+z=1 (5)

So, replacing the expressions obtained previously in equation 5, we have finally that:

1.25x+x+x+3x=1

x=1/6.25=y

z=3x=3/6.25

So, the fraction of sunny days is given by:

F=y+z=4/6.25

Answer:

If the 62 year old man makes a lump-sum withdrawal from the plan or tax structure, his investments would start incurring ordinary income taxes without attracting any other form of penalties. However, it has to be noted that prior to withdrawal of the lump-sum, his investments would grow without incurring income taxes.

Answer:

Radius of convergence of power series is [tex] \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{1}{108}[/tex]

Step-by-step explanation:

Given that:

n!! = 1⋅3⋅5⋅⋅⋅⋅(n−2)⋅n        n is odd

n!! = 2⋅4⋅6⋅⋅⋅⋅(n−2)⋅n       n is even

(-1)!! = 0!! = 1

We have to find the radius of convergence of power series:

[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]

Power series centered at x = a is:

[tex]\sum_{n=1}^{\infty}c_{n}(x-a)^{n}[/tex]

[tex]\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}](8x+6)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}n!(3n+3)!(2n)!!}{2^{n}[(n+9)!]^{3}(4n+3)!!}]2^{n}(4x+3)^{n}\\\\\sum_{n=1}^{\infty}[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}](x+\frac{3}{4})^{n}\\[/tex]

[tex]a_{n}=[\frac{8^{n}4^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}n!(3(n+1)+3)!(2(n+1))!!}{[(n+1+9)!]^{3}(4(n+1)+3)!!}]\\\\a_{n+1}=[\frac{8^{n+1}4^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}][/tex]

Applying the ratio test:

[tex]\frac{a_{n}}{a_{n+1}}=\frac{[\frac{32^{n}n!(3n+3)!(2n)!!}{[(n+9)!]^{3}(4n+3)!!}]}{[\frac{32^{n+1}(n+1)!(3n+6)!(2n+2)!!}{[(n+10)!]^{3}(4n+7)!!}]}[/tex]

[tex]\frac{a_{n}}{a_{n+1}}=\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]

Applying n → ∞

[tex]\lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}= \lim_{n \to \infty}\frac{(n+10)^{3}(4n+7)(4n+5)}{32(n+1)(3n+4)(3n+5)(3n+6)+(2n+2)}[/tex]

The numerator as well denominator of [tex]\frac{a_{n}}{a_{n+1}}[/tex] are polynomials of fifth degree with leading coefficients:

[tex](1^{3})(4)(4)=16\\(32)(1)(3)(3)(3)(2)=1728\\ \lim_{n \to \infty}\frac{a_{n}}{a_{n+1}}=\frac{16}{1728}=\frac{1}{108}[/tex]

Answer:

Step-by-step explanation:

Given

There are 52 cards in total

there are total of 13 pairs of same cards with each pair containing 4 cards

Probability of getting a pair or three of kind card=1-Probability of all three cards being different

Probability of selecting all three different cards can be find out by selecting a card from first 13 pairs and remaining 2 cards from remaining 12 pairs i.e.

[tex]=\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]

for first card there are 52 options after choosing first card one pair is destroyed as we have to select different card .

For second card we have to select from remaining 12 pairs i.e. 48 cards and so on for third card.

Required Probability is [tex]=1-\frac{52\times 48\times 44}{52\times 51\times 50}[/tex]

[tex]=\frac{22776}{132600}[/tex]

Answer:

after 15%, discount = $5,015

after 10%, discount = $5,310

after 4%, discount=  $5,664

Step-by-step explanation:

1.discount of  15 % of the price listed

so 15 % of $5,900 will be=    15/100      x 5900  =    885 $

 so after discounting the price =$5,900 - $885    = $5,015

2. discount of  10 % of the price listed

so 10 % of $5,900 will be=    10/100      x 5900  =    590 $

 so after discounting the price =$5,900 - $590    = $5,310

3. discount of  4 % of the price listed

so 4 % of $5,900 will be=    4/100      x 5900  =    236 $

 so after discounting the price =$5,900 - $236  = $5,664

For this case we must simplify the following expression:

[tex](\frac {3} {4} + \frac {4} {5} i) - (\frac {1} {2} - \frac {3} {10} i) =[/tex]

By law of multiplication signs we have to:

[tex]- * + = -\\- * - = +\\\frac {3} {4} + \frac {4} {5} i- \frac {1} {2} + \frac {3} {10} i =[/tex]

We add similar terms:

[tex]\frac {3} {4} - \frac {1} {2} + \frac {4} {5} i + \frac {3} {10} i =\\\frac {3 * 2-4 * 1} {4 * 2} + \frac {4 * 10 + 5 * 3} {10 * 5} i =\\\frac {2} {8} + \frac {55} {50} i =[/tex]

We simplify:

[tex]\frac{1}{4}+\frac{11}{10}i[/tex]

Answer:

[tex]\frac{1}{4}+\frac{11}{10}i[/tex]

Answer:

We have a strong negative relationship between the variables.

Step-by-step explanation:

Given two random variables X and Y, it is possible to calculate the covariance as Cov(X, Y) = E(XY)-E(X)E(Y). We have E(X)=5, E(Y)=6 and E(XY)=21. Therefore Cov(X,Y)=21-(5)(6)=21-30=-9. On the other hand, we know that the correlation of X and Y is the number defined by [tex]Cov(X,Y)/\sqrt{Var(X)}\sqrt{Var(Y)}[/tex] and because in this particular case we have V(X)=9 and V(Y)=10, we have [tex]-9/\sqrt{9}\sqrt{10}[/tex] = -0.9487. Therefore, we have a strong negative relationship between the variables.

Final answer:

The X and Y variables have a strong negative relationship.

Explanation:

The X and Y variables have a strong negative relationship. This can be determined by analyzing the correlation coefficient, which indicates the strength and direction of the relationship between two variables.

In this case, since the correlation coefficient is significantly different from zero (positive or negative), we can conclude that there is a significant linear relationship between X and Y. The fact that the correlation coefficient is negative indicates that as X increases, Y tends to decrease, and vice versa.

Therefore, the correct answer is strong negative relationship.

In hypothesis testing, it is critical to state the null and alternative hypotheses, choose an appropriate significance level, and conclude in the context of the problem. Decisions must reflect the probabilistic nature of the tests, with careful consideration of Type I and Type II errors.

In hypothesis testing, the following statements are indeed true:

You must state null and alternative hypotheses in the context of the problem.

You must state a significance level so you can decide if a given P-value provides evidence to reject the null hypothesis.

You must state a conclusion in the context of the problem.

When conducting a hypothesis test, one must also be mindful not to claim that a hypothesis is definitively proven true or false due to the probabilistic nature of hypothesis testing. Instead, you can infer whether there is sufficient evidence to support the alternative hypothesis if the null hypothesis is rejected. However, remember that making a decision at a certain significance level involves a trade-off between Type I and Type II errors.

Answer:

a) (y-16)/y = -7*e∧(-0.016946*t)

b) y = 6.34

c) t = 129.66 years in 2055

Step-by-step explanation:

a) dy/dt = ky*(16-y)

Solving the differential equation we have

dy / (y*(y-16)) = -k dt

∫ dy / (y*(y-16)) = ∫ -k dt

(-1/16)*Ln (y) + (1/16)*Ln (y-16) = -k*t + C

(1/16) Ln ((y-16)/y) = -k*t + C

Ln ((y-16)/y) = -16*k*t + C  

(y-16)/y = C*e∧(-16*k*t)

If  t = 0  and y = 2

(2-16)/2 = C*e∧(0)  

C = -7   then we have

(y-16)/y = -7*e∧(-16*k*t)

In 1975 we have t = 1975 - 1925= 50 years   and  y = 4

(4-16)/4 = -7*e∧(-16*k*50)

k= - Ln (3/7) / 800 = 0.001059

Finally, the differential equation will be

(y-16)/y = -7*e∧(-16*0.001059*t)

(y-16)/y = -7*e∧(-0.016946*t)

b)  In 2015 we have t = 2015 – 1925 = 90 years

(y-16)/y = -7*e∧(-0.016946*90)

Solving the equation we get

y = 6.34

c) If  y = 9

(9-16)/9 = -7*e∧(-0.016946*t)

t = 129.66 years  in 2055

The solution for (a)[tex]a) (y-16)/y = -7*e^{(-0.016946*t)}[/tex]b) y = 6.34 and (c) the value of t is  129.66 years  in 2055

We have given that,

[tex]a) dy/dt = ky*(16-y)[/tex]

By using variable separable form we have,

A variable separable differential equation is any differential equation in which variables can be separated

Therefore by solving the differential equation we have

[tex]dy / (y*(y-16)) = -k dt[/tex]

integrating both side with respect to t

[tex]\int dy / (y*(y-16)) = \int -k dt[/tex]

Solve the integration of the above

[tex](-1/16)*ln (y) + (1/16)*ln (y-16) = -k*t + C[/tex]

[tex](1/16) ln ((y-16)/y) = -k*t + C[/tex]

[tex]ln ((y-16)/y) = -16*k*t + C[/tex]

[tex](y-16)/y = C*e^{(-16*k*t)}[/tex]

If  t = 0  and y = 2

[tex](2-16)/2 = C*e^{0}[/tex]

C = -7   then we have

[tex](y-16)/y = -7*e^{(-16*k*t)}[/tex]

In 1975 we have t = 1975 - 1925= 50 years   and  y = 4

[tex](4-16)/4 = -7*e^{(-16*k*50)[/tex]

[tex]k= - Ln (3/7) / 800 = 0.001059[/tex]

Finally, the differential equation will be

[tex](y-16)/y = -7*e^{(-16*0.001059*t)}[/tex]

[tex](y-16)/y = -7*e^{(-0.016946*t)}[/tex]

b)  In 2015 we have t = 2015 – 1925 = 90 years

[tex](y-16)/y = -7*e^{(-0.016946*90)}[/tex]

Solving the equation we get

y = 6.34

c) If  y = 9

[tex](9-16)/9 = -7*e^{(-0.016946*t)}[/tex]

Therefore we get the value of t is  129.66 years  in 2055

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The Triangle Angle Sum Theorem states that the sum of interior angles in any triangle is always 180 degrees. Represented by the equation x + y + z = 180°, it allows for solving missing angles in a triangle using x = 180° - y - z or similar expressions.

Understanding the Triangle Angle Sum Theorem:

In any triangle, regardless of its shape or size, the sum of the interior angles always equals 180 degrees. This is known as the Triangle Angle Sum Theorem.

This theorem is a fundamental property of triangles and has numerous applications in geometry and other mathematical fields.

Representing the Angle Sum with an Equation:

Let's use variables to represent the angle measures of a triangle:

Angle 1 = x

Angle 2 = y

Angle 3 = z

According to the Triangle Angle Sum Theorem, the equation becomes:

x + y + z = 180°

Solving for Missing Angles:

This equation can be used to solve for any missing angle if we know the values of the other two angles.

For example, if we know the measures of angles y and z, we can find x using:

x = 180° - y - z

Similarly, we can find y or z if we know x and the other angle.

Example:

Consider a triangle with angles x = 50°, y = 70°, and z unknown.

Using the equation:

z = 180° - x - y = 180° - 50° - 70° = 60°

The equation that represents the sum of the angle measures of the triangle is 2y + x = 198.

The value of x is 86 and the value of y is 56.

A)

The sum of the interior angles of a triangle adds up to 180 degrees.

Hence, the equation that represents the sum of the angle measures of the given triangle is:

( y - 18 ) + y + x = 180

Simplifying; we get:

y + y + x = 180 + 18

2y + x = 198

B)

To solve for the values of x and y, we solve the system of equations:

2y + x = 198

3x - 5y = -22

Solve for x in equation 1:

2y + x = 198

x = -2y + 198

Plug x = -2y + 198 into equation 2 and solve for y:

3( -2y + 198 ) - 5y = -22

-6y + 594 - 5y = -22

-11y + 594 = -22

11y = 594 + 22

11y = 616

y = 616/11

y = 56

Now, plug y = 56 into equation 3 and solve for x:

x = -2y + 198

x = -2( 56 ) + 198

x = -112 + 198

x = 86

Therefore, the x = 86 and y = 56.

The missing image is uploaded below:

Answer:

C. No. In warm weather, more children will go outside and play

Step-by-step explanation:

The correct answer is option C: No. In warm weather, more children will go outside and play.

Correlation means that two variables are related, but it does not necessarily mean that one causes the other. In this case, there is a strong correlation between temperature and the number of skinned knees on playgrounds, but it does not tell us that warm weather causes children to trip.

Option A is incorrect because warm weather does not directly cause children to go outside and play. It may be a factor, but it is not the sole reason. Option B is incorrect because warm weather does not cause fewer children to trip and suffer skinned knees. In fact, the correlation suggests that more children are likely to be outside playing in warm weather, which could potentially increase the number of skinned knees. Option D is incorrect because warm weather does not directly cause more children to trip and suffer skinned knees.

The correlation suggests that the increase in skinned knees is likely due to more children being outside and playing, rather than the warm weather itself. To summarize, the strong correlation between temperature and the number of skinned knees on playgrounds indicates that in warm weather, more children will go outside and play. However, it does not tell us that warm weather causes children to trip.

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Answer: [tex]x=\dfrac{25}{3}[/tex]

Step-by-step explanation:

The given equation : [tex]\log_2(3x+7)=5[/tex]

Using logarithmic property : [tex]\log_a N=M\to N=a^M[/tex]

The given equation will be equivalent to [tex](3x+7)=2^5[/tex]

[tex]\Rightarrow\ 3x+7=32[/tex]

Subtract 7 from both sides , we get

[tex]3x=25[/tex]

Divide both sides by 3 , we get

[tex]x=\dfrac{25}{3}[/tex]

Hence, the solution is [tex]x=\dfrac{25}{3}[/tex]

The probability density function (pdf) is;

f(x)=[tex]\frac{1}{b - a}[/tex] for a ≤ x ≤ b.

while the mean is given as [tex]\frac{a + b}{2}[/tex]

And the standard deviation given as [tex]\sqrt{\frac{(b - a)^{2} }{12} }[/tex]

Answer: -None of these choices.

PDF = 1/(b-a)

Step-by-step explanation:

The probability density function f(x) of a random variable X that has a uniform distribution between a and b is given by:

PDF = 1/(b-a) for X€[a,b]

otherwise zero.

Step-by-step explanation:

∑ (4ⁿ⁺¹ / 5ⁿ)

Rewrite 4ⁿ⁺¹ as 4 (4ⁿ).

∑ 4 (4ⁿ / 5ⁿ)

∑ 4 (4/5)ⁿ

This is a geometric series.  The sum of an infinite geometric series is:

S = a / (1 −r)

where a is the first term and r is the common ratio.

Here, the first term is 16/5 (because n starts at 1), and the common ratio is 4/5.

S = 16/5 / (1−4/5)

S = 16/5 / (1/5)

S = 16

Answer:

Δy = 1

dy = 1

Step-by-step explanation:

Data provided in the question:

dx = Δx

y = x

x = 1,

Δx = 1

Now,

we know,

Δy = f( x + Δx ) - f(x)

also, we have

y = f(x) = x

thus,

f( x + Δx ) =  x + Δx

Therefore,

Δy = ( x + Δx ) - x

on substituting the respective values, we get

Δy = ( 1 + 1 ) - 1

or

Δy = 1

and,

dy = f'(x) = [tex]\frac{d(x)}{dx}[/tex]

or

dy = 1