Ron finds 9 books at a bookstore that he would like to buy, but he can afford only 5 of them. In how many ways can he make his selection? How many ways can he make his selection if he decides that one of the books is a must?
a. 3024; 1680
b. 7560; 840
c. 120; 1680
d. 126; 70

Answer:

a) [tex]P(t) = 2e^{0.275t}[/tex]

b) 54.225 square centimeters.

c) 2.52 hours

Step-by-step explanation:

The population growth law is:

[tex]P(t) = P_{0}e^{rt}[/tex]

In which P(t) is the population after t hours, [tex]P_{0}[/tex] is the initial population and r is the growth rate, as a decimal.

In this problem, we have that:

The colony occupied 2 square centimeters initially, so [tex]P_{0} = 2[/tex]

The colony triples every 4 hours. So

[tex]P(4) = 3P_{0} = 6[/tex]

(a) An expression for the size P(t) of the colony at any time t.

We have to find the value of r. We can do this by using the P(4) equation.

[tex]P(t) = P_{0}e^{rt}[/tex]

[tex]6 = 2e^{4r}[/tex]

[tex]e^{4r} = 3[/tex]

Applying ln to both sides, we get:

[tex]4r = 1.1[/tex]

[tex]r = 0.275[/tex]

So

[tex]P(t) = 2e^{0.275t}[/tex]

(b) The area occupied by the colony after 12 hours.

[tex]P(t) = 2e^{0.275t}[/tex]

[tex]P(12) = 2e^{0.275*12}[/tex]

[tex]P(12) = 54.225[/tex]

(c) The doubling time for the colony?

t when [tex]P(t) = 2P_{0} = 2*2 = 4[/tex].

[tex]P(t) = 2e^{0.275t}[/tex]

[tex]4 = 2e^{0.275t}[/tex]

[tex]e^{0.275t} = 2[/tex]

Applying ln to both sides

[tex]0.275t = 0.6931[/tex]

[tex]t = 2.52[/tex]

Answer:

II. This finding is significant for a two-tailed test at .01.

III. This finding is significant for a one-tailed test at .01.

d. II and III only

Step-by-step explanation:

1) Data given and notation    

[tex]\bar X=19.2[/tex] represent the battery life sample mean    

[tex]\sigma=2.5[/tex] represent the population standard deviation    

[tex]n=25[/tex] sample size    

[tex]\mu_o =18[/tex] represent the value that we want to test    

[tex]\alpha[/tex] represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)    

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

2) State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean battery life is equal to 18 or not for parta I and II:    

Null hypothesis:[tex]\mu = 18[/tex]    

Alternative hypothesis:[tex]\mu \neq 18[/tex]    

And for part III we have a one tailed test with the following hypothesis:

Null hypothesis:[tex]\mu \leq 18[/tex]    

Alternative hypothesis:[tex]\mu > 18[/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".    

3) Calculate the statistic    

We can replace in formula (1) the info given like this:    

[tex]z=\frac{19.2-18}{\frac{2.5}{\sqrt{25}}}=2.4[/tex]    

4) P-value    

First we need to calculate the degrees of freedom given by:  

[tex]df=n-1=25-1=24[/tex]  

Since is a two tailed test for parts I and II, the p value would be:    

[tex]p_v =2*P(t_{(24)}>2.4)=0.0245[/tex]

And for part III since we have a one right tailed test the p value is:

[tex]p_v =P(t_{(24)}>2.4)=0.0122[/tex]

5) Conclusion    

I. This finding is significant for a two-tailed test at .05.

Since the [tex]p_v <\alpha[/tex]. We reject the null hypothesis so we don't have a significant result. FALSE

II. This finding is significant for a two-tailed test at .01.

Since the [tex]p_v >\alpha[/tex]. We FAIL to reject the null hypothesis so we have a significant result. TRUE.

III. This finding is significant for a one-tailed test at .01.

Since the [tex]p_v >\alpha[/tex]. We FAIL to reject the null hypothesis so we have a significant result. TRUE.

So then the correct options is:

d. II and III only

Answer:

E. I and III only

Step-by-step explanation:

I. .05

III. one-tailed at .01

Green's theorem says the circulation of [tex]\vec F[/tex] along the rectangle's border [tex]C[/tex] is equal to the integral of the curl of [tex]\vec F[/tex] over the rectangle's interior [tex]D[/tex].

Given [tex]\vec F(x,y)=2xy\,\vec\imath[/tex], its curl is the determinant

[tex]\det\begin{bmatrix}\frac\partial{\partial x}&\frac\partial{\partial y}\\2xy&0\end{bmatrix}=\dfrac{\partial(0)}{\partial x}-\dfrac{\partial(2xy)}{\partial y}=-2x[/tex]

So we have

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D-2x\,\mathrm dx\,\mathrm dy=-2\int_0^3\int_0^8x\,\mathrm dx\,\mathrm dy=\boxed{-192}[/tex]

The circulation of the vector field 2xyi around the given rectangle, as computed via Green's Theorem, is 0 due to the curl of F being 0.

To use Green's Theorem to calculate the circulation around a rectangle, first we should realize that Green's Theorem states that the line integral around a simple closed curve C of F.dr is equal to the double integral over the region D enclosed by C of the curl of F. Here, F is the vector field defined as 2xyi. The given rectangle is oriented counterclockwise and the values of x and y are given as 0≤x≤8 and 0≤y≤3 respectively. The line integral denotes the circulation of the field.

The circulation is thus the double integral over the rectangle of ∇ x F. But in this case, since F = 2xyi, we get ∇ x F = 0. Hence, the circulation of F around the given rectangle is 0.

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Final answer:

The degrees of freedom for the t-statistic are 18. The conclusion cannot be determined without the p-value. If the null hypothesis is true, there is not enough evidence to support the alternative hypothesis.

Explanation:

(a) The degrees of freedom for the t-statistic are found by subtracting 1 from the sample size. In this case, the sample size is 19 so the degrees of freedom would be 19 - 1 = 18.

(b) If the p-value is less than the alpha level (typically 0.05), we reject the null hypothesis. In this case, the p-value is not provided, so we cannot determine the conclusion.

(c) If the null hypothesis is true, we would not reject it and conclude that there is not enough evidence to support the alternative hypothesis.

In this exercise we have to use the knowledge of variance to calculate the value of n, so we have that:

the sample is n=306

Organizing the information given in the statement we have that:

So given by the equation we have:

[tex]Z' = t(0.075)= 1.44\\n = (Z'*S/E)^2\\n = ( 1.44 * 0.85/0.07)^2\\n = (17.4857)^2\\n = 305.75\\n = 306[/tex]

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The minimum number of males over age 25 they must include in their sample is 512.

Given, Desired confidence level: 85%

Maximum error (E): 0.07 liters

Variance ([tex]\sigma^{2}[/tex]): 1.21 liters

Standard deviation ([tex]\(\sigma\)[/tex]): [tex]\(\sigma\)[/tex] = [tex]\sqrt{1.21}[/tex] = 1.1

Z-value for 85% confidence level (lookup Z-value for 0.425 in the standard normal distribution): [tex]\[ Z \approx 1.44 \][/tex]

n= [tex]\left(\frac{Z \sigma}{E}\right)^2[/tex]

[tex]\[ n = \left(\frac{1.44 \times 1.1}{0.07}\right)^2 \][/tex]

n= (1.584/0.07)² = 511.986

n = 512

Answer: A. 14.02 to 15.98

Step-by-step explanation:

Let [tex]\mu[/tex] denotes the mean waiting time for population.

Given : Sample size : n= 64

Sample mean : [tex]\overline{x}=15[/tex]   (minutes)

Population standard deviation = [tex]\sigma= 4[/tex]

Confidence level : 95%

By z-table , the critical values for 95% confidence = z*=1.96

Confidence interval for population mean : [tex]\overline{x}\pm z^* \dfrac{\sigma}{\sqrt{n}}[/tex]

The 95% confidence interval for the population mean of waiting times will be :

[tex]15\pm (1.96)\dfrac{4}{\sqrt{64}}[/tex]

[tex]15\pm (1.96)\dfrac{4}{8}[/tex]

[tex]15\pm (1.96)(0.5)[/tex]

[tex]15\pm 0.98[/tex]

[tex](15-0.98,\ 15+0.98)=(14.02,\ 15.98)[/tex]

Hence, the 5% confidence interval for the population mean of waiting times is 14.02 to 15.98.

Thus , the correct answer is Option A.

Answer:

There is enough evidence to conclude that the sample is from a population of songs with a mean greater than 210 sec.

Step-by-step explanation:

We are given the following in the question:  

Population mean, μ = 210 seconds

Sample mean, [tex]\bar{x}[/tex] = 234.1 sec

Sample size, n = 40

Sample standard deviation, s = 54.52 sec

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu \leq 210\text{ seconds}\\H_A: \mu > 210\text{ seconds}[/tex]

We use one-tailed t test to perform this hypothesis.

Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{234.1 - 210}{\frac{54.52}{\sqrt{40}} } = 2.795[/tex]

Now, [tex]t_{critical} \text{ at 0.05 level of significance, 9 degree of freedom } = 1.684[/tex]

We calculate the p-value with the help of standard normal table.

P-value = 0.004005

Since,                    

[tex]t_{stat} > t_{critical}[/tex]

We fail to accept the null hypothesis and accept the alternate hypothesis.

Thus, there is enough evidence to conclude that the sample is from a population of songs with a mean greater than 210 sec.

The given data shows that  the sample mean length of 234.1 sec results in a low p-value, it reject the claim that songs must be no longer than 210 seconds.

Explanation of hypothesis testing for mean song length compared to a specific value with given data.

Null hypothesis (H0): The mean length of songs is 210 seconds or less (μ ≤ 210 sec).

Alternative hypothesis (Ha): The mean length of songs is greater than 210 seconds (μ > 210 sec).

Test statistic: Calculate the z-score using the sample mean, population mean, standard deviation, and sample size.

P-value: Using the z-score, find the p-value from the standard normal distribution table.

Conclusion: If the p-value is less than the significance level (0.05), reject the null hypothesis. In this case, if the sample mean length of 234.1 sec results in a low p-value, we reject the claim that songs must be no longer than 210 seconds.

Answer:

Option A)  k = 2

Step-by-step explanation:

Multimonial Experiment

A multimonial experiment is an experiment with n repeated trials and each trial has a discrete number of possible outcomes.

Binomial Experiment

Binomial experiment is an experiment with n repeated trials and each trial has only two possible outcomes.

Thus, if k  represents the number of possible outcomes, then for k = 2, a multimonial experiment will become a binomial experiment.

Option A)  k = 2

Answer:

a) [tex]\bar X= 19.2[/tex] represent the sample mean. And that represent the best estimator for the population mean since [tex]\hat \mu =\bar X=19.2[/tex]  b) The 90% confidence interval is given by (17.3;21.1)  

c) No, because 18 is above the lower limit of the confidence interval.

d) Because the parent population is assumed to be normally distributed.

The reason of this is because the t distribution is an special case of the normal distribution when the degrees of freedom increase.

Step-by-step explanation:

1) Notation and definitions  

n=17 represent the sample size

Part a  

[tex]\bar X= 19.2[/tex] represent the sample mean. And that represent the best estimator for the population mean since [tex]\hat \mu =\bar X=19.2[/tex]  Part b

[tex]s=4.4[/tex] represent the sample standard deviation  

m represent the margin of error  

Confidence =90% or 0.90

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".  

2) Calculate the critical value tc  

In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 90% of confidence, our significance level would be given by [tex]\alpha=1-0.90=0.1[/tex] and [tex]\alpha/2 =0.05[/tex]. The degrees of freedom are given by:  

[tex]df=n-1=17-1=16[/tex]  

We can find the critical values in excel using the following formulas:  

"=T.INV(0.05,16)" for [tex]t_{\alpha/2}=-1.75[/tex]  

"=T.INV(1-0.05,16)" for [tex]t_{1-\alpha/2}=1.75[/tex]  

The critical value [tex]tc=\pm 1.75[/tex]  

3) Calculate the margin of error (m)  

The margin of error for the sample mean is given by this formula:  

[tex]m=t_c \frac{s}{\sqrt{n}}[/tex]  

[tex]m=1.75 \frac{4.4}{\sqrt{17}}=1.868[/tex]  

4) Calculate the confidence interval  

The interval for the mean is given by this formula:  

[tex]\bar X \pm t_{c} \frac{s}{\sqrt{n}}[/tex]  

And calculating the limits we got:  

[tex]19.2 - 1.75 \frac{4.4}{\sqrt{17}}=17.332[/tex]  

[tex]19.2 + 1.75 \frac{4.4}{\sqrt{17}}=21.068[/tex]  

The 90% confidence interval is given by (17.332;21.068)  and rounded would be:  (17.3;21.1)

Part c

No, because 18 is above the lower limit of the confidence interval.

Part d

Because the parent population is assumed to be normally distributed.

The reason of this is because the t distribution is an special case of the normal distribution when the degrees of freedom increase.

Answer:

There is a 34% probability that a randomly selected UH student checks out a history book or a science book or both.

Step-by-step explanation:

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that a UH student checks out books on history.

B is the probability that a UH students checks out books on science.

We have that:

[tex]A = a + (A \cap B)[/tex]

In which a is the probability that a UH student checks a book on history but not on science and [tex]A \cap B[/tex] is the probability that a UH student checks books both on history and science.

By the same logic, we have that:

[tex]B = b + (A \cap B)[/tex]

What is the probability that a randomly selected UH student checks out a history book or a science book or both?

[tex]P = a + b + (A \cap B)[/tex]

We start finding these values from the intersection.

6% check out books on both history and science. So [tex]A \cap B = 0.06[/tex]

28% of all UH students check out books on science. So [tex]B = 0.28[/tex]

[tex]B = b + (A \cap B)[/tex]

[tex]0.28 = b + 0.06[/tex]

[tex]b = 0.22[/tex]

12% of all UH students check out books on history

[tex]A = a + (A \cap B)[/tex]

[tex]0.12 = a + 0.06[/tex]

[tex]a = 0.06[/tex]

So

[tex]P = a + b + (A \cap B) = 0.06 + 0.22 + 0.06 = 0.34[/tex]

There is a 34% probability that a randomly selected UH student checks out a history book or a science book or both.

Answer:

Step-by-step explanation:

The distance of each lap around pavia park is 1 7/8 miles. Converting

1 7/8 miles to improper fraction, it becomes 15/8 miles.

Ellen rode her bike for 3 1/2 laps before leaving the park. Converting

3 1/2 laps to improper fraction, it becomes 7/2 laps.

The total number of miles that Ellen rode her bike in pavia park would be the product of the distance of each lap and the number of laps that he covered. It becomes

15/8 × 7/2 = 105/16 = 6.5626 miles

Answer:x = 9

Step-by-step explanation:

The attached photo is that of the given diagram. b represents the angle adjacent 75 degrees.

If line m is parallel to line n, it means that angle b degrees and angle (10x + 15) are corresponding angles. Corresponding angles are equal.

Therefore,

b = 10x + 15

The sum of angles on a straight line is 180 degrees. It means that

b + 75 = 180

b = 180 - 75 = 105

Therefore

10x + 15 = 105

10x = 105 - 15 = 90

x = 90/10 = 9

Answer:

x = 9°

Step-by-step explanation:

105° must be equal to 10x + 15 ° for lines to be parallel.

> 105° = 10x + 15°

> 10x = 90°

> x = 9°

Answer:

Null hypothesis:[tex]\mu \leq 8000[/tex]  

Alternative hypothesis:[tex]\mu > 8000[/tex]  

[tex]z=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]  

[tex]p_v =P(Z>2)=0.0228[/tex]  

Step-by-step explanation:

1) Data given and notation  

[tex]\bar X=8300[/tex] represent the sample mean  

[tex]\sigma=1200[/tex] represent the population standard deviation  

[tex]n=64[/tex] sample size  

[tex]\mu_o =800[/tex] represent the value that we want to test  

[tex]\alpha[/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)  

2) State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 8000, the system of hypothesis are :  

Null hypothesis:[tex]\mu \leq 8000[/tex]  

Alternative hypothesis:[tex]\mu > 8000[/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".  

3) Calculate the statistic  

We can replace in formula (1) the info given like this:  

[tex]z=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]  

4) P-value  

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

[tex]p_v =P(Z>2)=0.0228[/tex]  

5) Conclusion  

If we compare the p value and the significance level assumed, for example [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the mean is significantly higher than $8000.  

Answer:

yes,def=(2399/3713)x100=64.61%

yes,probably= (774/3713)x100=20.85%

yes,probably no= (322/3713)x100=8.67%

yes,def.no= (218/3713)x100=5.87%

Step-by-step explanation:

Data given

We assume the following frequency table:

Type                       Frequency

Yes, definitely          2399

Yes, probably            774

No, probably not       322

No, probably not        218

Total                           3713

Solution to the problem

And for this case in order to find the percentages we can use the definition of percentage frequency with the following formula:

[tex] \% = \frac{Number of people with characteristic}{Total} x100[/tex]

And using this formula we can find the percentages like this:

yes,def=(2399/3713)x100=64.61%

yes,probably= (774/3713)x100=20.85%

yes,probably no= (322/3713)x100=8.67%

yes,def.no= (218/3713)x100=5.87%

Answer:

f(x) = 7x⁶ + 3x³ + 12

Step-by-step explanation:

The number of roots a function has corresponds to the highest exponent value.

In the function f(x) = 7x⁶ + 3x³ + 12 ,

the number 7x⁶ has the highest exponent value of 6, so it'll have 6 roots.

Answer:

Step-by-step explanation:

Given that Student scores on exams given by a certain instruc-tor have mean 74 and standard deviation 14.

                                      Group I  X             Group II Y

Sample mean                     74                          74

n                                         25                          64

Std error (14/sqrtn)             2.8                        1.75

a) P(X>80) =[tex]1-0.9839\\= 0.0161[/tex]

b) P(Y>80) = [tex]1-0.9997\\=0.0003[/tex]

c) X-Y is Normal with mean = 0 and std deviation = [tex]\sqrt{2.8^2+1.75^2} \\=3.302[/tex]

P(X-Y>2.2) = [tex]1-0.8411\\=0.1589[/tex]

d) [tex]P(\bar x -\bar Y>2.2) = 0.1589[/tex]

Answer:

1.86 < t < 2.31

Step-by-step explanation:

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean height actually is higher\lower than an specified value, the system of hypothesis would be:  

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

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

Or like this:

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

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

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

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

t-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 don't know on this case the calculated value and with the p value we can find it.

P-value

The first step is calculate the degrees of freedom, on this case:  

[tex]df=n-1=9-1=8[/tex]  

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

[tex]p_v =P(t_{(8)}>t_o)=0.035[/tex]

And we can find the critical value with the following excel code:

" =T.INV(1-0.035,8)" and we got [tex]t_p =2.09[/tex]

And cannot be a lower tail test since all the options have positive values for the statistic. So on this case the best option is:

1.86 < t < 2.31

Answer:

The dimensions of the newly formed rectangle is length is 41 cm and width is 9 cm.

Step-by-step explanation:

Given,

A 1 meter steel rod is bent into a rectangle.

That means the perimeter of the rectangle is 1 meter.

So, Perimeter = [tex]1\ m=100\ cm[/tex]

Let the width of the rectangle be 'x'.

Now According to question, length exceeds quadruple its width by 5 cm.

Hence framing the above sentence in equation form, we get;

[tex]Length=4x+5[/tex]

Now we use the formula of  perimeter of rectangle,

[tex]Perimeter=2(Length+width)[/tex]

On substituting the values, we get;

[tex]2(4x+5+x)=100\\\\(5x+5)=\frac{100}{2}\\\\5x+5=50\\\\5x=50-5\\\\5x=45\\\\x=\frac{45}{5}=9[/tex]

Now, we substitute the value of x to get the value of length;

[tex]Length=4x+5=4\times9+5=36+5=41\ cm[/tex]

Thus, Width=9 cm    Length=41 cm

Hence the dimensions of the newly formed rectangle is length is 41 cm and width is 9 cm.

Answer:

Step-by-step explanation:

Since the human body temperatures are normally distributed, the formula for normal distribution is expressed as

z = (x - u)/s

Where

x = human body temperatures

u = mean body temperature

s = standard deviation

From the information given,

u = 98.20oF

s = 0.62oF

We want to find the probability that their mean body temperature will be less than 98.50oF. It is expressed as

P(x lesser than 98.50)

For x = 98.50,

z = (98.50 - 98.20)/0.62 = 0.48

Looking at the normal distribution table, the corresponding probability to the z score is 0.6844

P(x lesser than 98.50) = 0.6844

Answer:

Option D) The prisoner is actually guilty and the jury sets him free.

Step-by-step explanation:

We are given the following in the question:

Null Hypothesis:

The null hypothesis states that the prisoner is innocent

Alternate Hypothesis:

The alternate hypothesis states that the prisoner is guilty and not innocent.

Type II error:

It is the type error made when we fail to reject the ll hypothesis when it is actually false.

That is we accept a false null hypothesis.

Thus a type II error in the above scenario will be to accept that the prisoner is innocent (accepting the null hypothesis) when actually he is guilty( the alternate hypothesis)

Thus, type II error would be setting free a guilty prisoner.

Option D) The prisoner is actually guilty and the jury sets him free.

"The prisoner is actually guilty and the jury sets him free" represents the risk involved in making a Type II error. The correct option is d) The prisoner is actually guilty and the jury sets him free.

In statistical hypothesis testing, particularly in the context of a trial, we have two competing hypotheses:

Null Hypothesis (H₀): The prisoner is innocent (presumed innocent until proven guilty).

Alternative Hypothesis (H₁): The prisoner is guilty.

A Type II error, in this scenario, occurs when the jury fails to reject the null hypothesis (fails to convict the prisoner) when the alternative hypothesis is true (the prisoner is actually guilty).

Let's analyze the options in the context of Type II error:

(a) The prisoner is actually guilty and the jury sends him to jail. This is not a Type II error. This describes a correct decision where the jury convicts the guilty prisoner.

(b) The prisoner is actually innocent and the jury sends him to jail. This is a Type I error, where the jury convicts an innocent person (rejects the null hypothesis when it is true).

(c) The prisoner is actually innocent and the jury sets him free. This is the correct outcome when the jury correctly fails to reject the null hypothesis (the prisoner is innocent).

(d) The prisoner is actually guilty and the jury sets him free. This is a Type II error. Here, the jury fails to reject the null hypothesis (prisoner is innocent) when it is false (prisoner is guilty).

(e) The prisoner is sent to jail. This could happen in both correct decisions (prisoner is guilty) and Type I errors (prisoner is innocent but convicted).

Therefore, the answer that represents the risk involved in making a Type II error is d. The prisoner is actually guilty and the jury sets him free.

Answer:

Step-by-step explanation:

Given that a sample of 161children was selected from fourth and fifth graders at elementary schools in Philadelphia. In addition to recording the grade level, the researchers determined whether each child had a previously undetected reading disability

a) The two qualitative variables are disability and not having disability and secondly the grades of children

b) Contingency table:

Grade                  4                5                      Total

Normal read.       32              34                        66

Not normal read.  23                6                        29  

Total                      55             40                         95

H0: Reading disability is independent of grade.

Ha: There is association between the two

c)  4 5 Total

Nor read 38.21052632 27.78947368 66

Not norm 16.78947368 12.21052632 29

Expected cells are obtained using the formula

row total*col total/grand total

Answer:[tex]\pm2.131[/tex]

Step-by-step explanation:

The critical value in the t-distribution for a two-tailed hypothesis test is the t-value in the students's t-distribution table corresponding to the [tex]\alpha/2[/tex] and df , where [tex]\alpha=[/tex] significance level and df = Degree of freedom.

We are given , [tex]\alpha=0.05[/tex]

Then,  [tex]\alpha/2=0.025[/tex]

df = 15

Now , from the students's t-distribution table

The  critical value in the t-distribution for a two-tailed hypothesis test is [tex]t=\pm2.131449\approx\pm2.131[/tex]

Hence, the required t-value = [tex]\pm2.131[/tex]

Final answer:

The expected number of girls in a family with five children, given a 51% probability of having a girl, is 2.6 when rounded to the nearest tenth.

Explanation:

The question involves calculating the expected number of girls in a family with five children, given that the probability of having a girl is 51%. The expected value in probability theory is calculated by multiplying each possible outcome by its probability and then summing these values. In this case, since there are five children and each child has a 51% chance of being a girl, the expected number of girls can be found using the formula for the expectation:

Expected number of girls = Total number of children × Probability of having a girl

Therefore, the expected number of girls = 5 × 0.51 = 2.55. Rounding this to the nearest tenth, we get 2.6 girls.

Answer:

Step-by-step explanation:

Let X(t) denote the number of machines breakdown at time t.

The givenn problem follows birth-death process with finite space

S={0, 1, 2, 3} with

[tex] \lambda_0=\frac{3}{10}, \mu_1=\frac{1}{8}\\\\ \lambda_1=\frac{2}{10}, \mu_2=\frac{2}{8}\\\\ \lambda_2=\frac{1}{10}, \mu_3=\frac{2}{8}[/tex]

The birth-death process having balance equations [tex]\lambda_sP_i=\mu_{s+1}P_{i+1},i=0,1,2[/tex]

since, state  rate at which leave = rate at which enter

            0      [tex]\lambda_0P_0=\mu_1P_1[/tex]

             1     [tex](\lambda_1+\mu_1)P_1= \mu_2P_2 + \lambda_0P_0[/tex]

             2   [tex](\lambda_2+\mu_2)P_2= \mu_3P_3 + \lambda_1P_1[/tex]

[tex]P_1=\frac{12}{5}P_0=P_0=\frac{5}{12}P_1\\\\P_2=\frac{48}{25}P_0=P_0=\frac{25}{48}P_2\\\\P_3=\frac{192}{250}P_0=P_0=\frac{250}{192}P_3[/tex]

Since [tex]\sum\limits^3_{i=0} {P_i=1}\\\\p_0=[1+\frac{5}{12}+\frac{48}{25}+\frac{192}{250}]^{-1}=\frac{250}{1522}[/tex]

a)

Average number not in use equals the mean of the stationary distribution [tex]P_1+2P_2+3P_3=\frac{2136}{751}[/tex]

b)

Proportion of time both repairmen are busy [tex]P_2+P_3=\frac{672}{1522}=\frac{336}{761}[/tex]

The average number of machines not in use is 0.5, and the repairmen are both busy 64% of the time. This has been found under the assumption of exponential distribution for both longevity of machines and repair time. The scenario represents a M/M/2 queue in operations research.

In this scenario, the life of the machines and the repair time are governed by exponential distributions. Exponential distribution is often used to model the amount of time until an event occurs, such as machine failure in this case.

(a) To find the average number of machines not in use, we need to consider the rate of machine breakdown and repair. A machine works an average of 10 hours before failure, which translates to a failure rate of 1/10. A single repairman can fix a machine in an average of 8 hours, meaning a rate of 1/8 per repairman, or 1/4 for two repairmen combined. As there are three machines, the average number of machines in use is the ratio of the arrival rate to the service rate: (1/10) / (1/4) = 2.5 machines. This implies that on average, 0.5 machines are not in use.

(b) Both repairmen will be busy when there are at least two machines that require fixing. The proportion of time in which this is the case is obtained by calculating the probability that the number of machines failed is two or more. This is a problem of queueing theory, in particular an M/M/2 queue. The formula for this probability is P(X >= k) = (1 - rho) * rho^(k) / (1 - rho^(c+1)), where rho = arrival rate / service rate, k = 2, and c = 2 (service channels). Substituting rho = 2.5, we obtain P(X >= 2) = 0.64, meaning that the repairmen are both busy 64% of the time.

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To find the closest point on a line to another point, convert the problem into finding a minimum of a distance function, then use calculus (derivatives) to find the minimum.

To solve this problem, you need to identify a goal and an approach using calculus. The goal is to find a point (x, y) on the given line 5x + y = 7 that is closest to the point (−3, 2). This translates into minimizing the distance between (x, y) and (-3, 2) with respect to (x, y).

The distance formula between two points (x1, y1) and (x2, y2) is: √[(x2-x1)²+(y2-y1)²]. For our purposes x1 = -3, y1 = 2 and x2 = x, y2 = y. However, since y is from our given line (5x + y = 7), we can substitute for y = 7 - 5x. Thus, our new distance will become a function in terms of x: d(x) = √[(x - -3)²+(7 - 5x - 2)²].

To minimize it, we should take the derivative of this function, and set it equal to zero, which will give an extreme point (either minimum or maximum). Using the chain-rule results in the minimizing x value, and substituting it back to the line gives us the y value. This resultant point (x, y) would be our answer to the problem.

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Answer:

b. 1.71

[tex]z=\frac{0.5296 -0.5}{\sqrt{\frac{0.5(1-0.5)}{827}}}=1.71[/tex]  

Step-by-step explanation:

1) Data given and notation

n=827 represent the random sample taken

X=438 represent the  people that indicate that they would like to see the new show in the lineup

[tex]\hat p=\frac{438}{827}=0.5296[/tex] estimated proportion of people that indicate that they would like to see the new show in the lineup

[tex]p_o=0.5[/tex] is the value that we want to test

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

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true proportion is higher than 0.5:  

Null hypothesis:[tex]p \leq 0.5[/tex]  

Alternative hypothesis:[tex]p > 0.5[/tex]  

When we conduct a proportion test we need to use the z statisitc, 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].

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.5296 -0.5}{\sqrt{\frac{0.5(1-0.5)}{827}}}=1.71[/tex]  

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level assumed is [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.  

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

[tex]p_v =P(Z>1.71)=0.044[/tex]  

If we compare the p value obtained and the significance level assumed [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the true proportion is higher than 0.5.  

Answer:

21.75 square units

Step-by-step explanation:

Draw a radial line from O to the point where AB intersects the circle.  We'll call this point P.

Draw another radial line from O to the point where arc BC intersects the circle.  We'll call this point Q.

OQ is equal to the radius of the circle, 2.  And AQ is equal to the radius of the sector, 8.  Therefore, the length of AO is 6.

Next, OP is equal to the radius of the circle, 2.  Since AB is tangent to the circle, it is perpendicular to OP.

So we have a right triangle with a hypotenuse of 6 and a short leg of 2.  Finding the angle ∠PAO:

sin ∠PAO = 2/6

∠PAO = asin(1/3)

That means the angle ∠BAC is double that:

∠BAC = 2 asin(1/3)

∠BAC ≈ 38.94°

Therefore, the area of the sector is:

A = (θ/360) πr²

A = (38.94/360) π(8)²

A ≈ 21.75

The area of the sector is approximately 21.75 square units.

Answer:

No. He would need $1,080 to perform a test with this error.

Step-by-step explanation:

To answer this question, we have to calculate the sample size. This is the sample size that allow to estimate the average amount of sugar per can with 95% confidence (95% CI).

The difference between the upper and lower limit of the CI have to be equal or less to e=2 mg. The z value for a 95% CI is z=1.96.

[tex]e\leq z\sigma/\sqrt{n}[/tex]

[tex]e=z\sigma/\sqrt{n}\\\\\sqrt{n}=z\sigma/e=\\\\n=(z\sigma/e)^2=(1.96*15/2)^2=14.7^2=216\\\\n=216[/tex]

The minimum sample size needed for this error is 216. At a cost of $5/test, this sample size would cost [tex]n*p=216*5=\$ 1,080[/tex].

This is over the budget for this experiment ($1000).

Answer:

A solution curve pass through the point (0,4) when [tex]c_{1} = -\frac{4}{3}[/tex].

There is not a solution curve passing through the point(0,1).

Step-by-step explanation:

We have the following solution:

[tex]P(t) = \frac{c_{1}e^{t}}{1 + c_{1}e^{t}}[/tex]

Does any solution curve pass through the point (0, 4)?

We have to see if P = 4 when t = 0.

[tex]P(t) = \frac{c_{1}e^{t}}{1 + c_{1}e^{t}}[/tex]

[tex]4 = \frac{c_{1}}{1 + c_{1}}[/tex]

[tex]4 + 4c_{1} = c_{1}[/tex]

[tex]c_{1} = -\frac{4}{3}[/tex]

A solution curve pass through the point (0,4) when [tex]c_{1} = -\frac{4}{3}[/tex].

Through the point (0, 1)?

Same thing as above

[tex]P(t) = \frac{c_{1}e^{t}}{1 + c_{1}e^{t}}[/tex]

[tex]1 = \frac{c_{1}}{1 + c_{1}}[/tex]

[tex]1 + c_{1} = c_{1}[/tex]

[tex]0c_{1} = 1[/tex]

No solution.

So there is not a solution curve passing through the point(0,1).

Answer:

(D) 21

Explanation:

3(2x−1)−10=8+5x

Step 1: Simplify both sides of the equation.

3(2x−1)−10=8+5x

(3)(2x)+(3)(−1)+−10=8+5x(Distribute)

6x+−3+−10=8+5x

(6x)+(−3+−10)=5x+8(Combine Like Terms)

6x+−13=5x+8

6x−13=5x+8

Step 2: Subtract 5x from both sides.

6x−13−5x=5x+8−5x

x−13=8

Step 3: Add 13 to both sides.

x−13+13=8+13

x=21

Answer:

D. 21

Step-by-step explanation:

Given equation: [tex]\[3(2x – 1) – 10 = 8 + 5x\] [/tex]

Simplifying:

[tex]\[6x – 3 – 10 = 8 + 5x\] [/tex]

=> [tex]\[6x – 13 = 8 + 5x\] [/tex]

=> [tex]\[6x – 5x = 8 + 13\] [/tex]

=> [tex]\[(6 – 5)x = 21\] [/tex]

=> [tex]\[x = 21\] [/tex]

Verifying by substituting the value of x in the equation:

Left Hand Side: [tex]\[3(2x – 1) – 10\][/tex]

[tex]\[3(2*21 – 1) – 10\][/tex]

= [tex]\[3(42 – 1) – 10\][/tex]

= [tex]\[3(41) – 10\][/tex]

= [tex]\[123 – 10\][/tex]

= [tex]\[113\][/tex]

Right Hand Side: [tex]\[8 + 5x\][/tex]

= [tex]\[8 + 5*21\][/tex]

= [tex]\[8 + 105\][/tex]

= [tex]\[113\][/tex]

So, Left Hand Side = Right Hand Side when x=21.