A factorial experiment was designed to test for any significant differences in the time needed to perform English to foreign language translations with two computerized language translators. Because the type of language translated was also considered a significant factor, translations were made with both systems for three different languages: Spanish, French, and German. Use the following data for translation time in hours.

Language
Spanish French German
System 1 9 12 13
13 16 17
System 2 9 15 19
13 17 25
Test for any significant differences due to language translator system (Factor A), type of language (Factor B), and interaction. Use = .05.

Part(a):

Then the outcomes can be,

[tex]\{(1,1,1)(1,1,0)(1,0,1)(0,1,1)(1,0,0)(0,1,0)(0,0,1)(0,0,0) \}[/tex]

Part(b):

The correct option is (a).

Part(c):

The outcomes are,

[tex](1,1,1):x=3\\(1,1,0):x=2\\(1,0,1):x=2\\(0,1,1):x=2\\(1,0,0):x=1\\(0,1,0):x=1[/tex]

Experimental probability, also known as Empirical probability, is based on actual experiments and adequate recordings of the happening of events.

Let the ordered pair (a,b,c) denote the outcome with a,b,c taking either 1 if the position is offered

Or 0  if the position is not offered.

Then the outcomes can be,

[tex]\{(1,1,1)(1,1,0)(1,0,1)(0,1,1)(1,0,0)(0,1,0)(0,0,1)(0,0,0) \}[/tex]

Let, [tex]x[/tex] is number probability function offers made.

The variable is discrete taking 0 or 1 or 2 or 3 as values.

So, the correct option is (a)

The outcomes are,

[tex](1,1,1):x=3\\(1,1,0):x=2\\(1,0,1):x=2\\(0,1,1):x=2\\(1,0,0):x=1\\(0,1,0):x=1[/tex]

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There are 8 experimental outcomes for the interviews. The variable representing the number of offers is discrete, taking values between 0 and 3. For the given subsets of experimental outcomes, the value of this variable is the number of 'Y' present.

A) The total number of experimental outcomes is calculated by the rule of product. There are two possible results (Yes or No) for each of the three interviews. So the number of experimental outcomes is 2*2*2, which is 8.

B) The variable x, which is the number of students receiving an offer, is discrete. A variable is discrete if it can only take on a finite or countable number of values. In this case, x can take on only four possible values (0, 1, 2, or 3), depending on the number of students receiving an offer.

C) The value of the random variable X for the subset of experimental outcomes is as follows:
(Y,Y,Y) - X = 3
(Y,N,Y) - X = 2
(N,Y,Y) - X = 2
(N,N,Y) - X = 1
(N,N,N) - X = 0

The above experimental outcomes show the different possible results of the three interviews, with N signifying no job offer and Y signifying a job offer after the interview.

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Answer:

6435 different communities.

Step-by-step explanation:

This is a combination problem,

We need to know how many ways 8 species out of 15 can be selected for the new small island

15C8 = 15!/(15-8)!(8!) = 15!/(8!)(7!) = 6435 different communities.

Answer:

Pat's score therfore was the 20th percentiles

Step-by-step explanation:

When a value V is said to be in the xth percentile of a set, x% of the values in the set are lower than V and (100-x)% of the values in the set are higher than V.

In this problem, we have that:

34 scores lower than Pat's.

[tex]\frac{34}{170} = 0.20[/tex]

So Patrick's score was the 20th percentile.

Answer:

a) Total 16 possibilities

MMMM

FFFF

MMMF

MMFM

MFMM

FMMM

FFFM

FFMF

FMFF

MFFF

MMFF

MFMF

MFFM

FFMM

FMMF

FMFM

b) P(MMMM) = 1/16

Final answer:

The equally likely gender combinations for 4 children are listed by considering all possibilities, including MMMM, MMMF, and so on. The probability that all 4 children are male is 0.0625, or 1/16 as a fraction. The probability of having at least one female child is the complement, 0.9375 or 15/16 as a fraction.

Explanation:

Listing the Equally Likely Gender Combinations

To list the equally likely events for the gender of the 4 children in a family where 'M' represents a male child and 'F' represents a female child, consider all possible combinations. These combinations are: MMMM, MMMF, MMFM, MMFF, MFMM, MFMF, MFFM, MFFF, FMMM, FMMF, FMFM, FMFF, FFMM, FFMF, FFMM, FFFF.

Probability of All Male Children

To calculate the probability that all 4 children are male, note that the events are independent, and the probability of each child being male is 0.5. Since the events are independent, multiply the probabilities for each child: 0.5 * 0.5 * 0.5 * 0.5 = 0.0625 or 1/16 as a fraction.

Alternatively, the complement of having all male children is having at least one female child. The probability of at least one female child can be found by subtracting the probability of all male children from 1: 1 - 0.0625 = 0.9375 or 15/16 as a fraction.

Answer:

1. 90% 2. 10% 3. 50%

Step-by-step explanation:

Standard Deviation (σ) = 50 days

Average/Mean (μ) = 300days

Probability that it would last more than 300 days = P(Bulb>300 days)

We will assume there are 365 days in a year.

P(Bulb>300 days)  implies that the bulb would

Using the normal equation;

z = standard/normal score = (x-μ)/σ where x is the value to be standardized

P(Bulb>300 days) implies x = 365 days

Therefore z = (365-300)/50 = 1.3

Using the normal graph for z=1.3, probability = 90%

2. P(Bulb<300 days) = 1 - P(Bulb>300 days)\

P(Bulb<300 days) = 1 - 0.9

P(Bulb<300 days)  = 10%

3. P(Bulb=300 days) implies z=0 since x=300

Using the normal graph for z=0, probability =50%

1. The probability that an Acme light bulb will last more than 300 days is 50%. 2. The probability that an Acme light bulb will last less than 300 days is 50%. 3. The probability that an Acme light bulb will last exactly 300 days is zero.

1. To find the probability that an Acme light bulb will last more than 300 days, we need to determine the area under the normal distribution curve to the right of 300 days. We use the z-score formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation. Substituting the values, we get: z = (300 - 300) / 50 = 0. The area to the right of 300 days is equal to the area to the left of 0. Using a standard normal distribution table, we find that the area to the left of 0 is 0.5. Therefore, the probability that an Acme light bulb will last more than 300 days is 0.5 or 50%.

2. To find the probability that an Acme light bulb will last less than 300 days, we need to determine the area under the normal distribution curve to the left of 300 days. Using the same z-score formula, we get: z = (300 - 300) / 50 = 0. The area to the left of 0 is 0.5. Therefore, the probability that an Acme light bulb will last less than 300 days is 0.5 or 50%.

3. To find the probability that an Acme light bulb will last exactly 300 days, we need to determine the area under the normal distribution curve at 300 days. Since the normal distribution is continuous, the probability of any single value is zero. Therefore, the probability that an Acme light bulb will last exactly 300 days is zero.

Answer:

6/8

Step-by-step explanation:

The Fraction of mowers that fail the functional performance test is the ratio of the number of mowers that fail to the total number of mowers. Hence the fraction of mowers that fail [tex] \frac{9}{500}[/tex]

The probability of failure can be defined thus :

Therefore, the probability of having x failures in the next 100 mowers is (0.018x)

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Answer:

a) 75%

b) 84%

c) 95%                                                    

Step-by-step explanation:

We are given the following in the question:

Mean, μ = 6.7 hours

Standard Deviation, σ = 1.8 hours

Chebyshev's Theorem:

Empirical Formula:

a) minimum percentage of individuals who sleep between 3.1 and 10.3 hours

[tex]10.3 = 6.7 + 2(3.1) = \mu + 2(\sigma)\\3.1 = 6.7 - 2(3.1) = \mu - 2(\sigma)[/tex]

Minimum percentage:

[tex]1-\dfrac{1}{4} = 75%[/tex]

Thus, minimum 75% of individuals who sleep between 3.1 and 10.3 hours.

b) minimum percentage of individuals who sleep between 2.2 and 11.2 hours.

[tex]11.2 = 6.7 + 2.5(3.1) = \mu + 2.5(\sigma)\\2.2 = 6.7 - 2.5(3.1) = \mu - 2.5(\sigma)[/tex]

Minimum percentage:

[tex]1-\dfrac{1}{(2.5)^2} = 84\%[/tex]

Thus, minimum 84% of individuals who sleep between 2.2 and 11.2 hours.

c) percentage of individuals who sleep between 3.1 and 10.3 hours per day

According to Empirical rule about 95% of the data lies within 2 standard deviations from the mean.

Thus, 95% of individuals who sleep between 3.1 and 10.3 hours.

d) Comparison with Chebyshev's theorem

This is greater than the results obtained from the Chebyshev's theorem in part (a)

According to Chebyshev's theorem, the minimum percentage of individuals who sleep between 3.1 and 10.3 hours is approximately 93.75%. This minimum percentage also applies to individuals who sleep between 2.2 and 11.2 hours. However, using the empirical rule, approximately 95% of individuals are expected to sleep between 3.1 and 10.3 hours per day. The result from the empirical rule suggests that the distribution of sleep hours is closer to a normal distribution than what Chebyshev's theorem predicts.

(a) According to Chebyshev's theorem, at least 75% of the individuals will sleep between 3.1 and 10.3 hours. The formula for Chebyshev's theorem is P(X - μ < kσ), where μ is the mean, σ is the standard deviation, and k is the number of standard deviations from the mean.

(b) Using the same steps as in part (a), the minimum percentage of individuals who sleep between 2.2 and 11.2 hours is also approximately 93.75%.

(c) According to the empirical rule, for a normal distribution, approximately 68% of the individuals will sleep between μ - σ and μ + σ, and approximately 95% will sleep between μ - 2σ and μ + 2σ. Since the interval 3.1 and 10.3 hours falls within 2 standard deviations from the mean, we can estimate that around 95% of the individuals will sleep between 3.1 and 10.3 hours per day.

The result obtained using the empirical rule in part (c) is slightly higher than the value obtained using Chebyshev's theorem in part (a), indicating that the distribution of sleep hours is closer to a normal distribution than a general distribution predicted by Chebyshev's theorem.

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Answer: the due date would be 92 weeks

Step-by-step explanation:

Since the time required to complete a project is normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = number of weeks.

µ = mean

σ = standard deviation

From the information given,

µ = 80 weeks

σ = 10 weeks

If a construction company bidding on this contract wishes to be 90 percent sure of finishing by the due date, the z score corresponding to 90%(90/100 = 0.9) is 1.29

Therefore,

1.29 = (x - 80)/10

x - 80 = 1.2 × 10

x - 80 = 12

x = 80 + 12 = 92

To determine the due date the construction company should negotiate for a 90 percent confidence level of finishing on time, we use the z-score formula with the mean and standard deviation.

The negotiated due date is approximately 93 project week .

To determine the due date the construction company should negotiate, we need to find the project week number that corresponds to being 90 percent sure of finishing by that time. To do this, we use the z-score formula to find the z-score associated with a 90 percent confidence level. We then use this z-score to find the corresponding project week number using the mean and standard deviation of the project time.

The z-score formula is: z = (X - μ) / σ, where X is the project week number, μ is the mean (80 weeks), and σ is the standard deviation (10 weeks).

By substituting the values into the formula, we get: z = (X - 80) / 10.

Next, using a z-table or a calculator, we find the z-score associated with a 90 percent confidence level, which is approximately 1.2816.

Substituting this value for z in the formula, we get: 1.2816 = (X - 80) / 10.

Now, we can solve for X by multiplying both sides of the equation by 10 and then adding 80 to both sides. This gives us: 12.816 = X - 80.

Finally, adding 80 to both sides of the equation, we find that the negotiated due date (project week #) should be approximately 92.816. However, since project week numbers are typically whole numbers, we can round the negotiated due date up to the nearest whole number, which is 93.

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Answer: The disadvantage of using a related sample is that "A study that uses related samples to compare two drugs (specifically, one sample of participants with repeated measures) can have a carryover and/or order effect such that the efects of the drug taken before the first measurement may not wear off before the second measurement".

Step-by-step explanation: Related sample is when a particular sample or two sample that are the same is used to study an effect.

The carryover effect is one of the major disadvantages of using a related sample. Because when a first treatment is applied, their is a tendency of it not being fully consumed or it's effect not being fully neutralized before the second treatment is applied. This will increase error in the result, because in a drugs intake for instance, the drug taken at a particular time may not be the one that cure a sickness, but the drug that was previously taken. But the study will assume the drug taken a that particular moment to be the cure of that sickness.

Answer:

Related samples (especially, one sample of Participants with repeated measures) can have a carryover effect such that Participants can leam from their first measurement and therefore do better on their second Measurement.

Explanation:

Related samples/groups (i.e., dependent measurements) The subjects in each specimen, or organization, are identical. This indicates that the subjects in the first group are also in the second group.

Various Disadvantages of using a related sample versus using two independent samples:

Thus we can say using a related sample versus using two independent samples has various disadvanages.

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Answer: increase the sample size

Step-by-step explanation: the margin of error for any confidence interval is given by the formulae below.

Margin of error = critical value × standard deviation/√n

From what we can see the critical value and standard deviation are constants, the only variables here are the margin of error and sample size which are inversely proportional to each other, that is the margin of error is inversely proportional to the square of the sample size.

Hence, reducing the sample size will increase the margin of error while increasing the sample size will reduce the margin of error.

Answer:

The First-In, First-Out (FIFO) inventory costing method assumes that the inventory items ordered first are the first ones sold.

Step-by-step explanation:

The First-In, First-Out inventory costing method assumes that the inventory items ordered first are the first sold. This is ideal for goods that are highly perishable, for example fresh milk. Since no figures or dates are given, we will assume that the month is March 2019 and use any figures to make the example.

Date Item      Quantity of stock Cost Price

01  Opening stock bought on Feb 28  10   100

05  Sale of 5 goods (cost is $10 each)  (5)   50

15 Purchase of stock (20 goods at $20 each) 20   400

25 Sale of 15 goods                     (15)   250  

(5 at $10 each & 10 at $20 each)

31 Closing Stock               10   200

       (20 goods bought on 15th - 10 goods sold on 25th)

The quantity on hand at the end of the month is 10 units.  

Total cost of goods on hand at end of the month = 10 units * $20 = 200.

Total cost of goods purchased during the month = $20 * 20 units = $400

Total cost of goods sold during the month = [($10 *5) + ($10 * 5)+ ($20 * 10)] = $200

Answer:

The null hypothesis was rejected.

Conclusion: The bartender uses incorrect proportions in more than 50% of margaritas.

Step-by-step explanation:

The hypothesis for this test can be defined as:

H₀: The bartender uses incorrect proportions in less than 50% of margaritas, i.e. p < 0.50.

Hₐ: The bartender uses incorrect proportions in more than 50% of margaritas, i.e. p > 0.50.

Given:

[tex]\hat p=\frac{10}{30} =0.33\\n=30[/tex]

The test statistic is:

[tex]z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}} }=\frac{0.33-0.50}{\sqrt{\frac{0.50(1-0.50)}{30}}} =-1.862[/tex]

Decision Rule:

If the p-value of the test statistic is less than the significance level, α = 0.05, then the null hypothesis is rejected.

The p-value of the test is:

[tex]p-value=P(Z<-1.86) = 0.0314[/tex]

*Use the z-table.

The p-value = 0.0314 < α = 0.05.

The null hypothesis will be rejected.

Conclusion:

As the null hypothesis was rejected it can be concluded that the bartender uses incorrect proportions in more than 50% of margaritas.

Using the z-distribution, it is found that since the test statistic is less than the critical value for the left-tailed test, there is enough evidence to conclude that the manager's suspicion is correct.

At the null hypothesis, we test if the manager's suspicion is incorrect, that is, the proportion is correct in at least 50% of the margaritas.

[tex]H_0: p \geq 0.5[/tex]

At the alternative hypothesis, we test if the suspicion is correct, that is, the proportion is less than 50%.

[tex]H_1: p < 0.5[/tex]

The test statistic is given by:

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

In which:

For this problem, the parameters are: [tex]p = 0.5, n = 30, \overline{p} = \frac{10}{30} = 0.3333[/tex]

Then, the value of the test statistic is:

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

[tex]z = \frac{0.3333 - 0.5}{\sqrt{\frac{0.5(0.5)}{30}}}[/tex]

[tex]z = -1.83[/tex]

The critical value for a left-tailed test, as we are testing if the mean is less than a value, with a significance level of 0.05, is [tex]z^{\ast} = -1.645[/tex].

Since the test statistic is less than the critical value for the left-tailed test, there is enough evidence to conclude that the manager's suspicion is correct.

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Final answer:

The probability of a student making more than 20 mistakes on the exam is 0.19, making 6 or more mistakes is 0.72, and making at most 20 mistakes is 0.81. Events making at most 20 mistakes and making more than 20 mistakes are complementary.

Explanation:

Given a driving exam consisting of 30 multiple-choice questions, we have the following probabilities:

Now, we can find the probabilities for each scenario using these probabilities:

The two complementary events are:

These events are complementary because their probabilities sum up to 1.

Answer:

α= 133.6 degrees

(a)Sin(α/2)=0.9191

(b)cos(α/2)=0.3939

(c)Tan(α/2)=2.3332

Step-by-step explanation:

If Tan α= [tex]-\frac{21}{20}[/tex]

90<α<180

We determine first the value of α in the first quadrant

α=[tex]Tan^{-1}\frac{21}{20}[/tex]

=46.4

Since 90<α<180

α=180-46.4=133.6 degrees

(a)Sin(α/2)=Sin(133.6/2)=Sin 66.8 =0.9191

(b)cos(α/2)=cos(133.6/2)=cos 66.8 =0.3939

(c)Tan(α/2)=Tan(133.6/2)=Tan 66.8 =2.3332

Answer:

D) No, because either np or n(1−p) are less than 15.

Step-by-step explanation:

Percentage of students who had tattoos = 36%

Percentage of students who were smokers = 4%

Sample size = n = 100

The condition to use the Normal distribution as an approximation to construct the confidence interval for population proportion is:

Both np and n(1-p) must be equal to or greater than 15.

Since, we are interested in smokers only, so p = 4% = 0.04

np = 100  x 0.04 = 4

n (1 - p) = 100 x 0.96 = 96

Since, np < 15, we cannot use the Normal distribution as an approximation here.

Therefore, the correct answer is:

No, because either np or n(1−p) are less than 15.

Answer: option 3 is the correct answer.

Step-by-step explanation:

The formula for determining the area of a sector is expressed as

Area = θ/360 × πr²

Where

θ represents the central angle.

r represents the radius of the circle.

π is a constant whose value is 3.14

From the information given,

r = 4 miles

θ = 120°

Area of sector = 120/360 × 3.14 × 4²

= 16.75 square miles

To determine the area of ∆ABC, we would apply the formula,

Area of triangle = 1/2abSinC

From the information given,

a = 4

b = 4

C = 120°

Therefore,

Area = 1/2 × 4 × 4 × Sin120

Area = 8 × Sin120 = 6.92 square miles. Therefore area of the segment is

16.75 - 6.92 = 9.83 square miles

Answer:

0.0273

Step-by-step explanation:

np  n

10  100

9    100

11   100

7    100

3   100

12  100

8   100

4    100

6    100

11   100

pbar=sumnp/sumn

pbar=10+9+11+7+3+12+8+4+6+11/10+10+10+10+10+10+10+10+10+10

pbar=81/1000

pbar=0.081

nbar=sumn/k=1000/10=100

[tex]Standard deviation for pbar chart=\sqrt{\frac{pbar(1-pbar)}{nbar} }[/tex]

[tex]Standard deviation for pbar chart=\sqrt{\frac{0.081(0.919)}{100} }[/tex]

[tex]Standard deviation for pbar chart=\sqrt{\frac{0.0744}{100} }[/tex]

[tex]Standard deviation for pbar chart=\sqrt{0.0007444 }[/tex]

Standard deviation for p-chart=0.0273

Answer:

(a) The total number of combinations that can be applied for making potato chips is 36.

(b) The number of combinations that will be used for each type of oil is 12.

(c) Permutations are not an issue because order does not matter.

Step-by-step explanation:

The effects of cooking temperature, cooking time and cooking oil is studied for making potato chips.

The number of different temperatures applied is, n (T) = 3.

The number of different times taken is, n (t) = 4.

The number of different oils used is, n (O) = 3.

If an assignment can be done in n₁ ways and if for this assignment another assignment can be done in n₂ ways then these two assignments can be performed in (n₁ × n₂) ways.

(a)

Compute the total number of combinations that can be applied for making potato chips as follows:

Total number of combinations for making chips = n (T) × n (t) × n (O)

                                                                               [tex]=3\times4\times 3\\=36[/tex]

Thus, the total number of combinations that can be applied for making potato chips is 36.

(b)

To make potato chips 4 different temperatures are used and 3 different oils are used.

Each of the oil type is cooked in 4 different temperatures.

So the number of ways to select each oil type is,

n (T) × n (O) = [tex]4\times3=12[/tex]

Thus, the number of combinations that will be used for each type of oil is 12.

(c)

Permutation is the arrangement of objects in a specified order.

Since in this case ordering of the the three effects, i.e. temperature. time and oil type is not important, permutations are not an issue.

Answer:

A) The brand of cars driven by Harvard instructors

C) The types of plants growing in yellowstone  

Step-by-step explanation:

Categorical and numerical variables:

A) The brand of cars driven by Harvard instructors

Since the brands cannot be expressed in numerical, it is a categorical variable.

B) The age of randomly selected biology students.

The age of student are expressed in whole numbers or decimals. Thus, it is a numerical data.

C) The types of plants growing in yellowstone

Again, the type of plants cannot be expressed with the help of numerical. Thus, it is a categorical variable.

Variables A ('The brand of cars driven by Harvard instructors') and C ('The types of plants growing in Yellowstone') are categorical. Variable B ('The age of randomly selected biology students') is numerical.

In statistical terms, variables are characteristics or properties that can take different values or categories. They can be either categorical (qualitative) or numerical (quantitative).

For the variables provided:

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John Cacioppo's study uses independent samples while the car cleanliness study uses matched subjects because the samples are dependent as couples are paired.

In two given scenarios, the research designs differ in whether they use related samples or not. Related samples refer to a study design where the same subjects are measured more than once, or their measures are compared to those of individuals with whom they share a distinct relationship (e.g., married couples).

In John Cacioppo's study, the research design uses independent samples. Participants were randomly assigned to either the group of lonely people or nonlonely people and the similarities or differences in sleep quality were observed. As there is no mention of matching based on particular characteristics, and individuals from one group are not directly linked to those in the other, these samples are not related.

In contrast, the car cleanliness study deals with matched subjects, as the samples are paired together based on marriage. The husbands' ratings are compared directly with their respective wives' ratings, giving rise to related, paired measurements within each couple.

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The correct answer for the first scenario is that the design uses a related sample with matched subjects. For the second scenario, the design uses a related sample with repeated measures.

Explanation for the first scenario:

In the first scenario, John Cacioppo is comparing the sleep quality of lonely people to nonlonely people. Since the researcher is comparing two different groups of people based on their loneliness status, this is an example of a matched subjects design. The subjects are matched on the characteristic of loneliness, meaning that each lonely individual is paired with a nonlonely individual who is similar in other relevant characteristics that could affect sleep quality, such as age, gender, or health status. This allows the researcher to control for these extraneous variables and focus on the effect of loneliness on sleep quality.

Explanation for the second scenario:

In the second scenario, the researcher is interested in the difference in car cleanliness importance ratings between husbands and wives. By collecting data from both partners within each of the 100 married couples, the researcher is using a repeated measures design. Each couple serves as their own control, with the husband and wife providing two sets of measurements (ratings) on the same variable (importance of car cleanliness). This design allows for the examination of within-subject differences, controlling for the influence of other variables that might differ between different families, such as socioeconomic status or cultural background.

In summary, the first scenario is an example of a related sample with matched subjects, while the second scenario is an example of a related sample with repeated measures.

The probability that [tex]\( R_2 \)[/tex] exceeds[tex]\( R_1 \)[/tex] by more than 30Ω in series connection is approximately 0.04%, calculated using their normal distributions' means, standard deviations, and the difference's Z-score.

To find the probability that[tex]\( R_2 \)[/tex] exceeds [tex]\( R_1 \)[/tex] by more than 30Ω when they are connected in series, we can use the properties of normal distributions and their differences.

Given:

[tex]\( R_1 \)[/tex] has a mean [tex]\( \mu_1 = 1000 \)[/tex] and standard deviation [tex]\( \sigma_1 = 50 \).[/tex]

[tex]\( R_2 \)[/tex] has a mean[tex]\( \mu_2 = 1202 \)[/tex] and standard deviation [tex]\( \sigma_2 = 10.2 \)[/tex].

The difference [tex]\( R_2 - R_1 \)[/tex] will follow a normal distribution with the mean[tex]\( \mu_2 - \mu_1 = 1202 - 1000 = 202 \)[/tex] and the standard deviation [tex]\( \sqrt{(\sigma_1)^2 + (\sigma_2)^2} = \sqrt{50^2 + 10.2^2} \).[/tex]

Now, we want to find the probability that [tex]\( R_2 - R_1 > 30 \).[/tex]

Using Z-score:

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

[tex]\[ Z = \frac{30 - 202}{\sqrt{50^2 + 10.2^2}} \][/tex]

[tex]\[ Z = \frac{-172}{\sqrt{2500 + 104.04}} \][/tex]

[tex]\[ Z = \frac{-172}{\sqrt{2604.04}} \][/tex]

Z ≈ -172 / 51.02 ≈ -3.37

Consulting a standard normal distribution table or calculator for the probability associated with[tex]\( Z = -3.37 \)[/tex] we find that the probability is approximately 0.0004 or 0.04%.

Therefore, the probability that \( R_2 \) exceeds \( R_1 \) by more than 30Ω when they are connected in series is approximately 0.04%.

The probability [tex]\( P(R_2 > R_1) \approx 0.9631 \)[/tex], found by standardizing and using the normal distribution table.

To compute [tex]\( P(R_2 > R_1) \)[/tex], we need to find the probability that the resistance [tex]\( R_2 \)[/tex] is greater than [tex]\( R_1 \)[/tex]. Given that [tex]\( R_1 \) and \( R_2 \)[/tex] are normally distributed and independent, we can use properties of the normal distribution.

First, we define the random variables:

- [tex]\( R_1 \sim N(100, 5^2) \)[/tex]

- [tex]\( R_2 \sim N(120, 10^2) \)[/tex]

We are interested in the distribution of [tex]\( R_2 - R_1 \)[/tex].

1. Find the mean and standard deviation of [tex]\( R_2 - R_1 \)[/tex]:

  - The mean of [tex]\( R_2 - R_1 \)[/tex]:

    [tex]\[ \mu_{R_2 - R_1} = \mu_{R_2} - \mu_{R_1} = 120 - 100 = 20 \][/tex]

  - The variance of [tex]\( R_2 - R_1 \)[/tex]:

    [tex]\[ \sigma_{R_2 - R_1}^2 = \sigma_{R_2}^2 + \sigma_{R_1}^2 = 10^2 + 5^2 = 100 + 25 = 125 \][/tex]

  - The standard deviation of [tex]\( R_2 - R_1 \)[/tex]:

    [tex]\[ \sigma_{R_2 - R_1} = \sqrt{125} = 5\sqrt{5} \][/tex]

2. Standardize the variable [tex]\( R_2 - R_1 \)[/tex]:

  We want to find [tex]\( P(R_2 > R_1) \)[/tex], which is equivalent to [tex]\( P(R_2 - R_1 > 0) \)[/tex].

  Standardize [tex]\( R_2 - R_1 \)[/tex] by subtracting the mean and dividing by the standard deviation:

  [tex]\[ Z = \frac{R_2 - R_1 - \mu_{R_2 - R_1}}{\sigma_{R_2 - R_1}} = \frac{R_2 - R_1 - 20}{5\sqrt{5}} \][/tex]

  We want to find:

  [tex]\[ P(R_2 - R_1 > 0) = P\left(\frac{R_2 - R_1 - 20}{5\sqrt{5}} > \frac{0 - 20}{5\sqrt{5}}\right) \][/tex]

  Simplify the right-hand side:

  [tex]\[ P\left(Z > \frac{-20}{5\sqrt{5}}\right) = P\left(Z > \frac{-20}{5 \cdot 2.2361}\right) = P\left(Z > \frac{-20}{11.1803}\right) = P\left(Z > -1.7889\right) \][/tex]

3. Find the probability:

  - Using standard normal distribution tables or a calculator, find the probability [tex]\( P(Z > -1.7889) \)[/tex].

  - The standard normal distribution table provides the cumulative probability for [tex]\( Z \leq z \)[/tex]. For [tex]\( Z > -1.7889 \)[/tex]:

    [tex]\[ P(Z > -1.7889) = 1 - P(Z \leq -1.7889) \][/tex]

  Using standard normal distribution tables or a calculator:

  [tex]\[ P(Z \leq -1.7889) \approx 0.0369 \][/tex]

  Therefore:

  [tex]\[ P(Z > -1.7889) = 1 - 0.0369 = 0.9631 \][/tex]

So, the probability that [tex]\( R_2 \)[/tex] is greater than [tex]\( R_1 \)[/tex] is approximately [tex]\( 0.9631 \)[/tex].

Answer:

49 cars

Step-by-step explanation:

Probability of cars to be sold in a week,a = 0.2

Probabiity of cars not sold in a week, b = 0.8

Number of cars estimated to be sold in a week = 20 and 60 cars in 3 weeks

Using, P(x) = nCx *(a)∧x * (b)∧n - x, where n = 3 weeks, x = 1 week

P(x=1) = 3C1 * (0.2) * (0.8)² = 3 X 0.2 X 0.64 X 60 cars = 23 cars

P(x=2) = 3C2 * (0.2)² * (0.8) = 3 X 0.04 X 0.8 X 60 cars =  6 cars

Number of cars three weeks from now: 20 + 23 + 6 = 49 cars

Answer:

The forecast of the number of cars 3 weeks from now is 52 cars.

Step-by-step explanation:

As per the trent the number of the cars per week is 5 cars

The current level of cars is 40 cars per week

Number of cars sold in current week=20 cars

Forecast of the cars sold 3 weeks from now is given as

[tex]L_t=\alpha Y_t+(1-\alpha)(L_{t-1}+T_{t-1})\\[/tex]

From the data

Y_t=20 cars

L_t-1=40 cars

T_t-1=5 cars

α=β=0.2

So the equation becomes

[tex]L_t=\alpha Y_t+(1-\alpha)(L_{t-1}+T_{t-1})\\L_t=0.2*20+(1-0.2)(40+5)\\L_t=40[/tex]

Now the trend is calculated as

[tex]T_t=\beta(L_{t}-L_{t-1})+(1-\beta)T_{t-1}[/tex]

By putting the values the equation becomes

[tex]T_t=\beta(L_{t}-L_{t-1})+(1-\beta)T_{t-1}\\T_t=0.2(40-40)+(1-0.2)5\\T_t=0+0.8*5\\T_t=4[/tex]

Now the forecast of the cars sale 3 weeks from now is given as

[tex]L_{t+k}=L_t+kT_t[/tex]

where k is 3 so

[tex]L_{t+k}=L_t+kT_t\\L_{t+3}=40+3*4\\L_{t+3}=40+12\\L_{t+3}=52\\[/tex]

So the forecast of the number of cars 3 weeks from now is 52 cars.

Final answer:

Relation R defines a specific type of relationship within set theory, where subsets of set A are related if one is a proper subset of the other within the power set of A.

Explanation:

The student's question pertains to the concept of a relation in set theory, particularly concerning the power set of a given finite non-empty set A. In this context, the domain for the relation R is the power set of A, which includes all subsets of A. The relation R is defined such that if X and Y are subsets of A, then X is related to Y if and only if X is a proper subset of Y, denoted as X ⊂ Y. This means that every element in subset X is also contained in subset Y, and Y contains at least one additional element that is not in X.

To illustrate this, consider a simple set A = {1, 2}. Its power set, which is the domain of R, will include { }, {1}, {2}, and {1, 2}. In this case, the set {1} is a proper subset of {1, 2}, since it contains all elements of {1} and A contains an additional element, which is 2. Hence, the ordered pair ({1}, {1, 2}) is part of the relation R.

Answer: 804.25cm

Step-by-step explanation:

V=πr2h/3

pi= 3.14

r=8cm

h= 12cm

Slot the values

3.14× (8x8) × 12/3

3.14 × 64 x 4

V= 804.25cm

Answer:

Since it's a removable discontinuity, we will remove the discontinuity by creating a new function defined by x=3;

So we have;

F(x) = {[x² - 2x - 3]/(x-3); x ≠ 3

         {4, x = 3

Step-by-step explanation:

I have attached my explanation as the system here is not allowing me to save my answer.

To remove the discontinuity at x=3 in the function f(x) = (x^2 - 2x - 3) / (x - 3), simplify the function to f(x) = x + 1, then calculate f(3) = 3 + 1 = 4. Defining f(3) = 4 makes the function continuous at x=3.

To remove the discontinuity of a function such as f(x) = (x^2 - 2x - 3) / (x - 3), you need to find a value for f(3) that would make the function continuous at x = 3. First, let's manipulate the function:

The function f(x) = (x^2 - 2x - 3) / (x - 3) simplifies to f(x) = (x - 3)(x + 1) / (x - 3).

As you can see, the (x - 3) terms cancel out, leaving f(x) = x + 1.

However, this is only valid for x ≠ 3. To find f(3), you can now simply substitute 3 into the simplified function:

f(3) = 3 + 1 = 4.

So, if we define f(3) = 4, then the function becomes continuous at x = 3.

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Answer:

a) 16.7% probability that both dice show the same number

b) 83.3% probability that both dice show different numbers

c) 41.67% probability that the second die lands on a lower value than does the first.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have these possible outcomes:

Format(Dice A, Dice B)

(1,1), (1,2), (1,3), (1,4), (1,5),(1,6)

(2,1), (2,2), (2,3), (2,4), (2,5),(2,6)

(3,1), (3,2), (3,3), (3,4), (3,5),(3,6)

(4,1), (4,2), (4,3), (4,4), (4,5),(4,6)

(5,1), (5,2), (5,3), (5,4), (5,5),(5,6)

(6,1), (6,2), (6,3), (6,4), (6,5),(6,6)

There are 36 possible outcomes.

(a) What is the probability that both dice show the same number?

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

6 outcomes in which both dice show the same number.

6/36 = 0.167

16.7% probability that both dice show the same number

(b) What is the probability that both dice show different numbers?

The other 30 outcomes

30/36 = 0.833

83.3% probability that both dice show different numbers

(c) What is the probability that the second die lands on a lower value than does the first?

(2,1)

(3,1), (3,2)

(4,1), (4,2), (4,3)

(5,1), (5,2), (5,3), (5,4)

(6,1), (6,2), (6,3), (6,4), (6,5)

15 outcomes in which the second die lands on a lower value than does the first.

15/36 = 0.4167

41.67% probability that the second die lands on a lower value than does the first.

Answer:

5.96%

7.6%

38 days

Step-by-step explanation:

15 months = 1.25 years

5138.5 = 4780 × R^1.25

R^1.25 = 43/40

1.25 lgR = lg(43/40)

lg R = 0.0251267714

R = 1.059562969

R - 1 = 0.059562969

Interest: 5.96%

(150/365) × (r/100) × 12000 = 375

r/100 = 0.0760416667

r = 7.6%

40 = (n/365) × (6/100) × 6400

(n/365) = 5/48

n = 38.02083333 = 38

Answer:

The correct option is option C max 10A +15B +7C+8D

Step-by-step explanation:

As the complete question is not given here thus the complete question is found online and is attached herewith.

From the data the profit for 1 unit of A is 10,  B is 15, C is 7 and D is 8, so the profit function is given as 10A+15B+7C+8D. and as profit is to be maximized so the correct option is option C.

Answer:

343 pieces of wood.

Step-by-step explanation:

In order to determine the starting number of pieces of wood that yield 330 railings, the easiest course of action is to work your way back through step 3, step 2, and step 1. The number of pieces that must arrive at each step are given by:

[tex]n_f=330\\n_3 = \frac{330}{1-0.016}\\n_2 = \frac{n_3}{1-0.011}\\n_1 = \frac{n_2}{1-0.011} \\\\n_3 = 335.3658\\n_2=339.09591\\n_1=342.86745=343[/tex]

Rounding up to the next whole piece, 343 pieces are needed.

Therefore, you must start with 343 pieces of wood to have 330 railings that can be sold.