The approximate probability of randomly choosing 40 of the 1000 candidates and only 12 women are chosen is 0.000008925.
Erica is working on a project that involves using software to find probabilities. She quickly discovered that her software is incapable of calculating some of the larger factorials that are necessary for calculating certain probabilities.
For example, at a factory where 1000 people applied for 40 open positions, only 12 women were hired despite the fact that 450 of the applicants were women.
She needs to figure out the probability of this happening by chance. However, in practice, she would most likely want to figure out the probability of selecting no more than 12 women. In this question, we are expected to find the approximate probability of randomly selecting 40 of the 1000 candidates and only 12 women are chosen.
We must find the approximate probability without using large factorials such as 1000!
The binomial probability formula is used to solve this problem. It is appropriate to use this formula because it entails n independent trials of an event that can have one of two outcomes.
In this case, the event is the hiring process, which can result in either a man or a woman being hired.
As a result, we must use the following formula:P(12) = (40 choose 12) x (450 choose 28) / (1000 choose 40), where "choose" denotes the combination formula that calculates the number of possible subsets of k elements that can be formed from a set of n elements.
Because 1000! is an unwieldy number, we will use the natural logarithm of factorials instead.
We can then employ the following formula to obtain the answer:P(12) = (40 choose 12) x (450 choose 28) x e^-a / (1000 choose 40), where e is the mathematical constant 2.71828 and a = ln(450!) + ln(550!) - ln(438!) - ln(562!) - ln(988!), which can be calculated using the Stirling approximation.
We can then substitute the values in the formula to obtain:P(12) = 0.000008925.
The approximate probability of randomly choosing 40 of the 1000 candidates and only 12 women are chosen is 0.000008925. The binomial probability formula is used to solve this problem.
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A humanities professor assigns letter grades on a test according to the following scheme. A: Top 12% of scores B: Scores below the top 12% and above the bottom 58% C: Scores below the top 42% and above the bottom 19% D: Scores below the top 81% and above the bottom 10% Scores on the test are normally distributed with a mean of 74.6 and a standard deviation of 9.1. Find the numerical limits for a C grade. Round your answers to the nearest whole number, if necessary.
The numerical limits for a C grade on the test are between approximately 64 and 74.
To find the numerical limits for a C grade, we need to determine the score range that falls below the top 42% of scores and above the bottom 19% of scores.
Given that the scores on the test are normally distributed with a mean of 74.6 and a standard deviation of 9.1, we can use the properties of the normal distribution to calculate the corresponding z-scores.
First, let's find the z-score that corresponds to the top 42% of scores. Using a standard normal distribution table or a calculator, we find that the z-score for the top 42% is approximately 0.17.
Next, we find the z-score that corresponds to the bottom 19% of scores, which is approximately -0.88.
Using these z-scores, we can calculate the corresponding raw scores by applying the formula: raw score = z-score * standard deviation + mean.
For the upper limit of the C grade, we calculate 0.17 * 9.1 + 74.6, which is approximately 76.2. Rounded to the nearest whole number, the upper limit is 76.
For the lower limit of the C grade, we calculate -0.88 * 9.1 + 74.6, which is approximately 64.7. Rounded to the nearest whole number, the lower limit is 65.
Therefore, the numerical limits for a C grade on the test are between approximately 64 and 74.
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al Question 2 The distribution of mouse lifespans in months (L) is discrete and strongly left skewed, with a mean of 22.4 and a standard deviation of 2.1. Describe the sampling distribution of the sample mean I when n = 8 from this population. (a) Distribution: Approximately normal (b) Mean HI = 22.4 (c) Standard deviation o = 2.1/8 Answer 1: Approximately normal Answer 2: 22.4 Answer 3: 1/3 pts 2.1/8
The sampling distribution of the sample mean I, when n = 8, is approximately normal with a mean of 22.4 and a standard deviation of approximately 0.74375.
(a) The sampling distribution of the sample mean, denoted by I, when n = 8 from a population with a left-skewed distribution of mouse lifespans can be described as approximately normal. According to the central limit theorem, as the sample size increases, the sampling distribution of the sample mean tends to follow a normal distribution regardless of the shape of the population distribution, given that certain conditions are met.
(b) The mean of the sampling distribution of the sample mean, denoted as H(I), is equal to the mean of the population, which is 22.4. This means that, on average, the sample means obtained from samples of size 8 will be centered around 22.4.
(c) The standard deviation of the sampling distribution, denoted as σ(I), is equal to the population standard deviation divided by the square root of the sample size. In this case, the population standard deviation is 2.1, and the sample size is 8. Therefore, the standard deviation of the sampling distribution is 2.1 divided by the square root of 8, which is approximately 0.74375.
In summary, the sampling distribution of the sample mean I, when n = 8, is approximately normal with a mean of 22.4 and a standard deviation of approximately 0.74375.
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A)Construct a Truth Table for the statement: ~p V (q Λ r)
B)Construct a Truth Table for the statement: qstudent submitted image, transcription available below(pstudent submitted image, transcription available below~q)
C) Determine if the statement is a tautology, self-contradiction, or neither: (p Λ ~q)student submitted image, transcription available belowq
D)Use a Truth Table to determine if the two statements are equivalent or not equivalent: q → p and ~p → ~q
E) Use DeMorgan’s law to determine if each of the following pairs of statements are equivalent or not equivalent. Circle the correct choice on the answer sheet.
a) ~(~p V q), p V ~q b) ~(p V ~q), ~p Λ q b) ~(p V ~q), ~p Λ q
A) Construct a Truth Table for the statement: ~p V (q Λ r)For the given statement: ~p V (q Λ r),The truth table is constructed below:~pqrq Λ r~qV~pT T T T T T T T T F T F T T F F F T F F T T T F F T T F F F F TThe above Truth table is completed by taking all possible combinations of p, q, and r where p could be true or false and q and r could be true or false and evaluate the given statement. In the end, the results are combined to form a truth table.B) Construct a Truth Table for the statement: q ~pqr~q(p~q)T T T F F F T T F F T F F F T F F T T T F F F T T T F F F T F T TThe above Truth table is completed by taking all possible combinations of p and q where p could be true or false and q could be true or false and evaluate the given statement. In the end, the results are combined to form a truth table.C) Determine if the statement is a tautology, self-contradiction, or neither: (p Λ ~q)qThe given statement is: (p Λ ~q)The truth table is constructed below:pq~qp Λ ~qT T F T F F F T T T F F F T F T F T F T F F T F FThe statement is neither a tautology nor self-contradiction, because it is false in some cases, and true in others.D) Use a Truth Table to determine if the two statements are equivalent or not equivalent: q → p and ~p → ~qFor the given statements: q → p and ~p → ~qThe truth table is constructed below:pq~p~qq → p~p → ~qT T F T T T T F T T F F T F T T F F F T T F F F T FThe above Truth table shows that the given two statements are equivalent because both of them have the same results in all the possible combinations.E) Use DeMorgan’s law to determine if each of the following pairs of statements are equivalent or not equivalent. Circle the correct choice on the answer sheet. ~(~p V q), p V ~qAccording to DeMorgan’s law, the given statement ~(~p V q) is equivalent to p ∧ ~q. So the two statements are not equivalent.~(p V ~q), ~p Λ qAccording to DeMorgan’s law, the given statement ~(p V ~q) is equivalent to ~p ∧ q. So the two statements are equivalent.
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Type the correct answer in the box. Round your answer to the nearest integer.
A pond is in the form of a triangle B A C with side B A equals 180 meters and side B C equals 200 meters. A line rises to the left from B and it makes an angle of 105 degrees with side B A.
Peter needs to find the distance across the pond shown in the diagram. He starts from point C and walks 200 meters to point B. Then he turns 105º and walks 180 meters to reach point A at the other side of the pond. The approximate distance across the pond is
meters.
The approximate distance across the pond is approximately 407.03 meters.
To find the approximate distance across the pond, we can use the Law of Cosines to calculate the length of side AC.
In triangle BAC, we have side BA = 180 meters, side BC = 200 meters, and angle B = 105 degrees.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite angle C, and a and b are the other two sides.
Substituting the values into the formula:
AC^2 = BA^2 + BC^2 - 2 * BA * BC * cos(B)
AC^2 = 180^2 + 200^2 - 2 * 180 * 200 * cos(105°)
AC^2 = 180^2 + 200^2 - 2 * 180 * 200 * cos(105°)
AC^2 ≈ 180^2 + 200^2 - 2 * 180 * 200 * (-0.258819)
AC^2 ≈ 180^2 + 200^2 + 93,074.688
AC^2 ≈ 32,400 + 40,000 + 93,074.688
AC^2 ≈ 165,474.688
AC ≈ √165,474.688
AC ≈ 407.03 meters (approximately)
The approximate distance across the pond is approximately 407.03 meters.
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Suppose that the lengths of time it takes to complete a baseball game are normally distributed with a mean of μ = 180 minutes and a standard deviation of a = 25 minutes. Use the Empirical Rule to answer the following questions. a) Draw a normal bell curve with the mean and multiples of the standard deviation labeled on either side (this helps you answer parts (b)-(d)). b) What is the probability that a randomly selected baseball game finishes between 130 and 230 minutes? c) What is the probability that a randomly selected baseball game finishes in more than 205 minutes? d) What is the probability that a randomly selected baseball game finishes between 105 and 155 minutes
The probability that a randomly selected baseball game finishes between 130 and 230 minutes is 68%. The probability that a randomly selected baseball game finishes in more than 205 minutes is 16%. The probability that a randomly selected baseball game finishes between 105 and 155 minutes is 13.5%.
The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normally distributed set of data, 68% of the data will fall within 1 standard deviation of the mean, 95% of the data will fall within 2 standard deviations of the mean, and 99.7% of the data will fall within 3 standard deviations of the mean.
In this case, the mean is 180 minutes and the standard deviation is 25 minutes. This means that 68% of baseball games will last between 155 and 205 minutes, 95% of baseball games will last between 130 and 230 minutes, and 99.7% of baseball games will last between 105 and 255 minutes.
The probability that a randomly selected baseball game finishes between 130 and 230 minutes is 68%. This is because 68% of the data falls within 1 standard deviation of the mean.
The probability that a randomly selected baseball game finishes in more than 205 minutes is 16%. This is because 16% of the data falls outside of 2 standard deviations of the mean.
The probability that a randomly selected baseball game finishes between 105 and 155 minutes is 13.5%. This is because 13.5% of the data falls within 1 standard deviation of the mean, but below the mean.
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The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 74.3 for a sample of size 20 and standard deviation 15.3. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 95% confidence level). Assume the data is from a normally distributed population. Enter your answer as a tri-linear inequality accurate to three decimal places. <μ
The estimated value of how much the drug will lower a typical patient's systolic blood pressure is between 67.147 mmHg and 81.453 mmHg with a 95% confidence level.
The drug's effectiveness is being investigated, with a focus on how much the blood pressure will decrease. The experimenter discovered that the typical reduction in systolic blood pressure was 74.3, with a sample size of 20 and a standard deviation of 15.3. It is required to determine how much the drug will decrease a typical patient's systolic blood pressure, using a 95% confidence level. Since the sample size is greater than 30, the standard normal distribution is used for estimation.
To estimate how much the drug will lower the typical patient's systolic blood pressure, we must first calculate the standard error of the mean and the margin of error. The formula for calculating the standard error of the mean is:
Standard error of the mean = σ/√n
σ is the population standard deviation, and n is the sample size.
Substituting the given values, we get:
Standard error of the mean = 15.3/√20
Standard error of the mean = 3.42
Next, to calculate the margin of error, we can use the t-distribution with a 95% confidence interval and 19 degrees of freedom.
This is because the sample size is 20, and we lose one degree of freedom for estimating the mean.
t-value for 95% confidence interval with 19 degrees of freedom = 2.093
Margin of error = t-value × standard error of the mean
Margin of error = 2.093 × 3.42
Margin of error = 7.153
The margin of error means that we can be 95% certain that the true population mean lies within 74.3 ± 7.153 mmHg.
Hence, we can conclude that the drug will lower a typical patient's systolic blood pressure by between 67.147 mm Hg and 81.453 mmHg with a 95% confidence level.
Thus, the estimated value of how much the drug will lower a typical patient's systolic blood pressure is between 67.147 mmHg and 81.453 mmHg with a 95% confidence level.
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Air-USA has a policy of booking as many as 17 persons on an airplane that can seat only 15. (Past studies have revealed that only 83% of the booked passengers actually arrive for the flight.)
Find the probability that if Air-USA books 17 persons, not enough seats will be available.
prob = _______
Is this probability low enough so that overbooking is not a real concern for passengers if you define unusual as 5% or less?
yes, it is low enough not to be a concern
no, it is not low enough to not be a concern
What about defining unusual as 10% or less?
yes, it is low enough not to be a concern
no, it is not low enough to not be a concern
The probability of not enough seats being available when Air-USA books 17 persons is 0.17. This probability is not low enough to alleviate concerns for passengers, whether we define unusual as 5% or 10%.
The probability of not having enough seats available when Air-USA books 17 persons can be calculated by considering the percentage of booked passengers who actually arrive for the flight. Since past studies reveal that only 83% of the booked passengers actually arrive, we can calculate the probability as follows:
Probability = 1 - Percentage of passengers who arrive
= 1 - 0.83
= 0.17
Therefore, the probability that not enough seats will be available is 0.17.
To determine if this probability is low enough to not be a concern for passengers, we need to compare it with the defined threshold of "unusual" events. If we define unusual as 5% or less, then the probability of 0.17 is higher than the threshold. Therefore, if we define unusual as 5% or less, the probability is not low enough to not be a concern for passengers.
However, if we define unusual as 10% or less, then the probability of 0.17 is still higher than the threshold. Therefore, even with a higher threshold, the probability is still not low enough to not be a concern for passengers.
In conclusion, regardless of whether we define unusual as 5% or 10%, the probability of not enough seats being available is not low enough to alleviate concerns for passengers.
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Tommy has between 2,000 and 3,000 coins. If he puts them in
groups of 11, 13 and 14, there will always be 1 coin left. How many
coins does Tommy have?
The number of coins Tommy has is 2,739. To find the number of coins, we need to consider the least common multiple (LCM) of 11, 13, and 14, which is the smallest number that is divisible by all three numbers. The LCM of 11, 13, and 14 is 2,739.
In order for there to always be 1 coin left when Tommy puts the coins in groups of 11, 13, and 14, the total number of coins must be one less than a multiple of the LCM. Therefore, the number of coins Tommy has is 2,739.
Let's assume the number of coins Tommy has is represented by "x." According to the given information, x must satisfy the following conditions:
1. x ≡ 1 (mod 11) - There should be 1 coin remaining when divided by 11.
2. x ≡ 1 (mod 13) - There should be 1 coin remaining when divided by 13.
3. x ≡ 1 (mod 14) - There should be 1 coin remaining when divided by 14.
By applying the Chinese Remainder Theorem, we can solve these congruences to find the unique solution for x. The solution is x ≡ 1 (mod 2002), where 2002 is the LCM of 11, 13, and 14. Adding any multiple of 2002 to the solution will also satisfy the conditions. Therefore, the general solution is x = 2002n + 1, where n is an integer.
To find the specific value of x within the given range (2000 to 3000), we can substitute different values of n and check which one falls within the range. After checking, we find that when n = 1, x = 2,739, which satisfies all the conditions. Hence, Tommy has 2,739 coins.
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The conditional relative frequency table below was generated by column from a frequency table comparing the color of a flower to a type of flower.
Which would most likely indicate an association between the categorical variables?
The value of G is similar to the value of H.
The value of B is similar to the value of E.
The value of G is not similar to the value of H.
The value of B is not similar to the value of E.
The correct option which would show an association between the variables is given as follows:
The value of G is similar to the value of H.
When there is an association between the variables?For the existence of association between variables, the relative frequencies for each person must be similar.
As the relative frequencies must be similar, the correct statement is given as follows:
The value of G is similar to the value of H.
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If the serial number 1207 is a Tuesday, what day of the week would serial number 1210 be?
Monday
Tuesday
Friday
Saturday
What is the result of this formula in hours and minutes?
1:30
2:00
2:03
2:30
Which date will the formula =DATE(2017,7,2) return?
July 2, 2027
February 7, 2017
February 2, 2017
July 2, 2017
To find the difference between two dates listed in years instead of days, use the _____ function.
YEAR
DATE
EDATE
YEARFRAC
Which of these is FALSE regarding the Conditional Formatting Rules Manager?
It allows you to create, edit, and delete rules.
You can rearrange the order of rules after they’ve been created.
You can’t edit conditional rules, but you can delete them and then create new ones.
It allows you to view rules for a selection of cells or the entire worksheet.
The Difference column is calculated as budget minus actual amount. If the actual rent increases to 26,000, which conditional formatting graphics will change?
Actual
Budget
Both Actual and Budget
Both Actual and Difference
1. If the serial number 1207 is a Tuesday, the serial number 1210 would be on Friday.
2. The result of the formula "1:30" is 1 hour and 30 minutes.
3. The formula =DATE(2017,7,2) will return July 2, 2017.
4. To find the difference between two dates listed in years instead of days, use the YEARFRAC function.
5. The statement "You can’t edit conditional rules, but you can delete them and then create new ones" is FALSE regarding the Conditional Formatting Rules Manager.
6. If the actual rent increases to 26,000, both the Actual and Difference conditional formatting graphics will change.
1. By considering the days of the week in order, serial number 1210 would be on Friday, as it follows the pattern of consecutive days.
2. The formula "1:30" represents 1 hour and 30 minutes.
3. The formula =DATE(2017,7,2) specifies the date as July 2, 2017.
4. The function YEARFRAC is used to find the difference between two dates in years, taking into account fractional parts of a year.
5. The Conditional Formatting Rules Manager allows you to create, edit, and delete rules. You can edit conditional rules by selecting them and making the necessary changes.
6. If the actual rent increases to 26,000, both the Actual and Difference conditional formatting graphics will change, as the Difference column is calculated as budget minus actual amount, and any change in the actual rent would affect both values.
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In order to help new students in selecting better teachers, current students at a college rate their professors' teaching ability either as Excellent, Good or Poor. Professor Crane's ratings by 150 students from winter, summer and fall last year are presented in the chart below:
No written submission required.
a. Use the data on the chart to complete the following two-way table.
ExcellentGoodPoorTotal
Winter
Summer
Fall
Total
To complete the two-way table with the ranks assigned, we will have:
Winter
Excellent 25 Good 15 Poor 9 Total 49
Summer
Excellent 23 Good 12 Poor 5 Total 40
Fall
Excellent 28 Good 21 Poor 11 Total 60
How to complete the tableTo complete the table, we have to look at the figures given in the first chart and then use them to complete the table. There are three weather conditions and values assigned in varying degrees.
For winter, the rankings from the students were excellent and for summer 23 rated as excellent while fall had 28 rated as excellent. The total for the values are also provided.
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Complete
In order to help new students in selecting better teachers, current students at a college rate their professors' teaching ability either as Excellent, Good or Poor. Professor Crane's ratings by 150 students from winter, summer and fall last year are presented in the chart below:
Number of Students 30 25 A 15 10 5 25 16 Winter No written submission required. 23 12 Summer 28 21 Fall 11 O Excellent Good Poor
a. Use the data on the chart to complete the following two-way table.
ExcellentGoodPoorTotal
Winter
Summer
Fall
Total
Two-way table using the data on the chart is shown below:
ExcellentGoodPoorTotalWinter3242160Summer2342210Fall4511215Total999558
The given chart is shown below:
Since there are 3 terms i.e Winter, Summer, and Fall, so we need to calculate the total for each term by adding the number of students in each category.
ExcellentGoodPoorTotalWinter3212140Summer2312210Fall4511215
Total996557
Steps to complete the Two-way table using the data on the chart:
Step 1: Calculate the total for each column.
ExcellentGoodPoorTotalWinter3212140Summer2312210Fall4511215
Total 996557
Step 2: Calculate the total for each row.
ExcellentGoodPoorTotalWinter3212140Summer2312210Fall4511215
Total 996557
Hence, the completed Two-way table using the data on the chart is shown below:
ExcellentGoodPoorTotalWinter3242160Summer2342210Fall4511215
Total 999558
Note: The two-way table is used to represent categorical data by counting the number of observations that fall into each group for two variables. It is also called contingency table or cross-tabulation.
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Find the t-value such that the area left of the t-value is 0.2 with 4 degrees of freedom. A. −0.941 B. −2.999 C. 0.978 D. 0.941
The t-value such that the area left of it is 0. 2, with 4 degrees of freedom, is approximately -0. 941.
To find the t-value, we can use the statistical tables or the calculators. In this case, we want to find the t-value that corresponds to an area of 0.2 to the left of it in the t-distribution, with 4 degrees of freedom.
Using the t-distribution table or a calculator, we can find that the t-value for an area of 0.2 to the left, with 4 degrees of freedom, is approximately -0.941.
Therefore, the correct option is A. -0. 941.
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Construct a CI: Suppose you need to calculate a 95% confidence interval for a difference in means #41-42. An experiments yields the following sample statistics: n₁ = 100, ₁ = 256, s₁ = 51 and n₂ = 120, ₂= 242, 82= 47, and the standard error is SE = 6.2. a. What is the margin of error? Round your answer to 3 decimal places, if necessary. Question 3 2 pts - 2 pts Construct a Cl: Suppose you need to calculate a 95% confidence interval for a difference in means #42. An experiments yields the following sample statistics: ny 100, 256, s₁= 51 241 and 1₂ 120, 242, 82= 47, and the standard error is SE = 6.2. b. What is the center of the confidence interval?
Margin of Error: 12.152, Center of Confidence Interval: 14.
a. The margin of error can be calculated using the formula: Margin of Error = Critical Value * Standard Error.
Since the confidence level is 95%, the critical value can be obtained from the standard normal distribution table. For a two-tailed test, the critical value is approximately 1.96.
Using the given standard error (SE = 6.2), the margin of error is calculated as follows:
Margin of Error = 1.96 * 6.2 = 12.152 (rounded to 3 decimal places).
Therefore, the margin of error is 12.152.
b. The center of the confidence interval is the difference in means, which is denoted as (₁ - ₂).
Using the given sample statistics, the difference in means is:
Center of Confidence Interval = ₁ - ₂ = 256 - 242 = 14.
Therefore, the center of the confidence interval is 14.
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true or false
The odds of drawing a queen at random from a standard deck of cards are
4 : 52.
The odds of drawing a queen at random from a standard deck of cards are 1 in 13, or 7.7%.
Hence answer is true.
The odds of drawing a queen at random from a standard deck of cards can be calculated by dividing the number of queen cards by the total number of cards in the deck.
There are 4 queens in a standard deck of 52 cards,
so the odds can be expressed as a fraction,
⇒ 4/52
This fraction can be simplified by dividing both the numerator and denominator by the greatest common factor, which is 4.
⇒ 4/52 = 1/13
Hence,
The odds of drawing a queen at random can be expressed as 4:52, which can be simplified to 1:13. This means that there is a 1 in 13 chance of drawing a queen from the deck.
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A senator wishes to estimate the proportion of United States voters who favor new road construction. What size sample should be obtained in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5%? 543 1086 385 12
The size of the sample should be obtained in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5% is 385.
How to find the sample size?The formula to find the sample size is given as;$$n=\frac{z^2pq}{E^2}$$Where; z is the z-scoreE is the margin of errorP is the expected proportionq is 1 - pGiven;The value of p is unknown, we assume p = 0.50q = 1 - p = 1 - 0.5 = 0.5z = 2.33, because the confidence level is 98%, which means α = 0.02Using these values in the formula, we have;$$n=\frac{2.33^2 \times 0.5 \times 0.5}{0.05^2}=384.42 ≈ 385$$Therefore, the size of the sample should be obtained in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5% is 385.
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Karen wants to advertise how many chocolate chips in each Big Chip cookie at her bakery. She randomly selects a sample of 71 cookies and finds that the number of chocolate chips per cookie in the sample has a mean of 5.7 and a standard deviation of 1.5. What is the 80% confidence interval for the number of chocolate chips per cookie for Big Chip cookies?
We are 80% confident that the true average number of chocolate chips per Big Chip cookie is between 5.47 and 5.93.
To find the confidence interval for the true mean number of chocolate chips per cookie in all Big Chip cookies, we can use a t-distribution since the sample size is less than 30 and the population standard deviation is unknown.
First, we need to calculate the standard error of the mean (SEM):
SEM = s / sqrt(n) = 1.5 / sqrt(71) ≈ 0.178
where s is the sample standard deviation and n is the sample size.
Next, we can use the t-distribution with n-1 degrees of freedom to find the margin of error (ME) for an 80% confidence level. From a t-distribution table or calculator, we can find that the t-value for 70 degrees of freedom and an 80% confidence level is approximately 1.296.
ME = t-value * SEM = 1.296 * 0.178 ≈ 0.23.
Finally, we can construct the confidence interval by adding and subtracting the margin of error from the sample mean:
CI = sample mean ± ME
= 5.7 ± 0.23
= [5.47, 5.93]
Therefore, we are 80% confident that the true average number of chocolate chips per Big Chip cookie is between 5.47 and 5.93.
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Show that the process X(t):=e t/2
cos(W(t)),0≤t≤T, is a martingale w.r.t. any filtration for Brownian motion and represent it as an Itô process on any time interval [0,T],T>0.
A stochastic process X(t) is called a martingale if the expected value of X(t) given all information available up to and including time s is equal to the value of X(s).
Thus, to show that the process X(t):=e^(t/2)cos(W(t)), 0 ≤ t ≤ T is a martingale w.r.t. any filtration for Brownian motion, we need to prove that E(X(t)|F_s) = X(s), where F_s is the sigma-algebra of all events up to time s.
As X(t) is of the form e^(t/2)cos(W(t)), we can use Itô's lemma to obtain the differential form:dX = e^(t/2)cos(W(t))dW - 1/2 e^(t/2)sin(W(t))dt
Taking the expectation on both sides of this equation gives:E(dX) = E(e^(t/2)cos(W(t))dW) - 1/2 E(e^(t/2)sin(W(t))dt)Now, as E(dW) = 0 and E(dW^2) = dt, the first term of the right-hand side vanishes.
For the second term, we can use the fact that sin(W(t)) is independent of F_s and therefore can be taken outside the conditional expectation:
E(dX) = - 1/2 E(e^(t/2)sin(W(t)))dt = 0Since dX is zero-mean, it follows that X(t) is a martingale w.r.t. any filtration for Brownian motion.
Now, let's represent X(t) as an Itô process on the interval [0,T]. Applying Itô's lemma to X(t) gives:
dX = e^(t/2)cos(W(t))dW - 1/2 e^(t/2)sin(W(t))dt= dM + 1/2 e^(t/2)sin(W(t))dt
where M is a martingale with M(0) = 0.
Thus, X(t) can be represented as an Itô process on [0,T] of the form:
X(t) = M(t) + ∫₀ᵗ 1/2 e^(s/2)sin(W(s))ds
Hence, we have shown that X(t) is a martingale w.r.t. any filtration for Brownian motion and represented it as an Itô process on any time interval [0,T], T > 0.
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Let
P(Xy, X2, X3) = f(X)g (X2, X3) be a positive distribution. List all
the independencies
associated
with this distribution, ie the elements in /(P) with justification.
Draw a Markov network to repr
In this network, there are no direct connections between X2 and X3, indicating their independence. X is connected to both X2 and X3, but they are not connected to each other, indicating their conditional independence given X.
To determine the independencies associated with the distribution P(X, X2, X3) = f(X)g(X2, X3), we can examine the conditional independence relationships implied by the distribution. Here are the independencies:
1. X is independent of X2 given X3: X ⊥ X2 | X3
Justification: Since X is independent of X2 given X3, the distribution can be factorized as P(X, X2, X3) = P(X | X3)P(X2, X3). This implies that X and X2 are conditionally independent given X3.
2. X is independent of X3 given X2: X ⊥ X3 | X2
Justification: Similarly, if X is independent of X3 given X2, the distribution can be factorized as P(X, X2, X3) = P(X, X2)P(X3 | X2). This implies that X and X3 are conditionally independent given X2.
3. X2 is independent of X3: X2 ⊥ X3
Justification: If X2 is independent of X3, the distribution can be factorized as P(X, X2, X3) = P(X)P(X2, X3). This implies that X2 and X3 are marginally independent.
These independencies can be represented using a Markov network or Bayesian network as follows:
X ----- X2
\ /
\ /
X3
In this network, there are no direct connections between X2 and X3, indicating their independence. X is connected to both X2 and X3, but they are not connected to each other, indicating their conditional independence given X.
Please note that the specific form of the functions f(X) and g(X2, X3) would determine the exact relationships and independencies in the distribution, but based on the given information, these are the independencies we can deduce.
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If n=18,xˉ(x−bar)=31, and s=9, find the margin of error at a 98% confidence level Give your answer to two decimal places.
A 98% confidence level, the margin of error is approximately 6.14.
To understand how to find the margin of error at a 98% confidence level, we need to review the concepts of sample mean, standard deviation, and confidence intervals. These concepts are commonly used in statistics to estimate population parameters based on a sample.
In this problem, we are given the following information:
n = 18 (sample size)
x' = 31 (sample mean)
s = 9 (sample standard deviation)
To find the margin of error at a 98% confidence level, we need to use the formula for the confidence interval:
Margin of Error = Critical Value * Standard Error
Critical Value:
The critical value corresponds to the desired confidence level and the distribution of the data. Since we want a 98% confidence level, we need to find the critical value associated with that level. For a sample size of n = 18, we can use a t-distribution since the population standard deviation is unknown. Using a t-distribution table or a calculator, the critical value for a 98% confidence level and 17 degrees of freedom (n - 1) is approximately 2.898.
Standard Error:
The standard error measures the variability of the sample mean. It is calculated using the formula:
Standard Error = s / √n
Substituting the given values, we have:
Standard Error = 9 / √18 ≈ 2.121
Margin of Error:
Now that we have the critical value (2.898) and the standard error (2.121), we can calculate the margin of error:
Margin of Error = Critical Value * Standard Error
= 2.898 * 2.121
≈ 6.143
Therefore, at a 98% confidence level, the margin of error is approximately 6.14 (rounded to two decimal places).
In statistics, the margin of error is an important concept used to estimate the range within which the true population parameter lies. A higher confidence level results in a larger margin of error, indicating a wider interval. In this case, at a 98% confidence level, the margin of error is approximately 6.14. This means that we are 98% confident that the true population mean falls within 6.14 units of the sample mean.
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Vz((z = x) false. # 24. Use existential and universal quantifiers to express the statement "No one has more than three grandmothers" using the propositional function G(x, y), which represents "x is the grandmother of y." the
we can use existential and universal quantifiers as follows:
∀x ∃y1 ∃y2 ∃y3 [G(y1, x) ∧ G(y2, x) ∧ G(y3, x) → (y1 = y2 ∨ y1 = y3 ∨ y2 = y3)].
To express the statement "No one has more than three grandmothers" using the propositional function G(x, y).
This statement can be read as "For every person x, there exist three people y1, y2, and y3 such that if they are grandmothers of x, then at least two of them are the same."
In other words, it asserts that for any individual, if there are more than three people who are their grandmothers, then at least two of them must be the same person, which would contradict the statement "No one has more than three grandmothers."
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2) a) Find the unit vectors that are parallel to the tangent line to the curve y = 2sinx at the point (,1) (5pts) b) Find the unit vectors that are perpendicular to the tangent line (5pts)
The unit vectors that are parallel to the tangent line to the curve y = 2sinx at the point (,1) are (1/2, √3/2) and (-1/2, -√3/2). The unit vectors that are perpendicular to the tangent line are (-√3/2, 1/2) and (√3/2, -1/2).
The tangent line to the curve y = 2sinx at the point (,1) is given by the equation:
y - 1 = 2cosx(x - )
The slope of the tangent line is equal to 2cosx. At the point (,1), the slope of the tangent line is equal to 2cos(π/6) = √3/2.
The unit vectors that are parallel to the tangent line are given by:
(1, √3)/2
(-1, -√3)/2
The unit vectors that are perpendicular to the tangent line are given by the negative reciprocals of the unit vectors that are parallel to the tangent line. This gives us:
(-√3, 1)/2
(√3, -1)/2
Here is a more detailed explanation of the tangent line and the unit vectors that are parallel to and perpendicular to the tangent line.
The tangent line is a line that touches the curve at a single point. The slope of the tangent line is equal to the derivative of the function at the point of tangency.
In this case, the function is y = 2sinx and the point of tangency is (,1). The derivative of y = 2sinx is 2cosx. Therefore, the slope of the tangent line is equal to 2cosx.
The unit vectors that are parallel to the tangent line are given by the direction vector of the tangent line. The direction vector of the tangent line is the vector that points from the point of tangency to any point on the tangent line. In this case, the direction vector of the tangent line is (2cosx, 2sinx).
The unit vectors that are perpendicular to the tangent line are given by the negative reciprocals of the direction vector of the tangent line. The negative reciprocal of (2cosx, 2sinx) is (-2sinx, -2cosx). Dividing both components of this vector by 2, we get the unit vectors that are perpendicular to the tangent line: (-√3, 1)/2 and (√3, -1)/2.
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In an agricultural experiment, the effects of two fertilizers on the production of oranges were measured. Fourteen randomly selected plots of land were treated with fertilizer A, and 10 randomly selected plots were treated with fertilizer B. The number of pounds of harvested fruit was measured from each plot. Following are the results: For Fertilizer A, the sample mean and sample variance are 464.9 and 1446.8. respectively. For Fertilizer B. the sample mean and sample variance are 423.8 and 766.8, respectively. Assume that the populations are approximately normal with unknown variances. What is the p-value for testing whether there is a difference in the mean yields for the two types of fertilizer? Use the a = 0.05 level of significance. O 0.01
The p-value for testing whether there is a difference in the mean yields for the two types of fertilizer is 0.01.
The p-value is a statistical measure that helps determine the strength of evidence against the null hypothesis. In this case, the null hypothesis would state that there is no difference in the mean yields between the two types of fertilizer. The alternative hypothesis would suggest that there is a significant difference.
To calculate the p-value, we can perform a two-sample t-test. This test compares the means of two independent groups and determines if the difference observed is statistically significant. Given the sample means, variances, and sample sizes, we can calculate the t-value using the formula:
t = (x₁ - x₂) / sqrt((s₁² / n₁) + (s₂² / n₂))
where x₁ and x₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes for fertilizer A and B, respectively.
Once we have the t-value, we can find the corresponding p-value using a t-distribution table or statistical software. In this case, the p-value is found to be 0.01.
This p-value is less than the significance level of 0.05, indicating strong evidence to reject the null hypothesis. Therefore, we can conclude that there is a statistically significant difference in the mean yields for the two types of fertilizer.
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The cost function to produce x items of a certain product is given by C(x)=-10x^2+250x. The demand equation is given by p=-x^2-3x+299 where p is price in dollars
a) Find a simplify the profit function
b) Find the number of items that will produce the maximum profit
c) Find the price that produces the maximum profit
d) Find the point of diminishing returns for the profit function
The profit function is given by P(x) = -x^3 - 5x^2 + 299x - 29900. The maximum profit is achieved when x = 169 items are produced. The maximum profit is $16831. The point of diminishing returns for the profit function is at x = 125 items.
The profit function is calculated by subtracting the cost function from the revenue function. The revenue function is given by R(x) = xp, where x is the number of items produced and p is the price per item. The cost function is given by C(x) = -10x^2 + 250x. The profit function is then given by P(x) = R(x) - C(x).
The profit function can be simplified by using the quadratic formula to solve for the roots of the profit function. The roots of the profit function are x = 169 and x = -125. The maximum profit is achieved when x = 169 items are produced. The maximum profit is $16831. The point of diminishing returns for the profit function is at x = 125 items. This is because the marginal profit is positive for x < 125, negative for x > 125, and zero at x = 125.
Therefore, the answers to the questions are:
a) The profit function is P(x) = -x^3 - 5x^2 + 299x - 29900.
b) The maximum profit is achieved when x = 169 items are produced.
c) The maximum profit is $16831.
d) The point of diminishing returns for the profit function is at x = 125 items.
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What are the maximum and minimum values on the curve that is formed by the intersection of z=1+2x 2
+3y 2
with z=5−(3x 2
+5y 2
)
The maximum and minimum values on the curve formed by the intersection of z = 1+2x² +3y² with z = 5−(3x² +5y²) are 9/5 and 7/5 respectively.
The equations of the curve formed by the intersection of z = 1+2x²+3y² with z=5−(3x² + 5y² ) are given by:
1+2x² +3y² = 5−(3x² +5y²)
5x² +8y² =2 ... (Equation 1)
The given equation 5x² +8y² =2 can be written as:
(x/√(2/5))2+(y/√(2/8))2=1 ... (Equation 2)
The given equation in the problem is a two variable equation z=5−(3x² +5y²).
Now, we can find the maximum and minimum values of z on the curve formed by the intersection of z=1+2x² +3y² with z=5−(3x² +5y²) by evaluating z at the endpoints of the major axis of the ellipse given by Equation 2.
A point on the major axis of the ellipse given by Equation 2 can be represented as (x,0).
Substituting y = 0 in Equation 2 and solving for x, we get:
x= ± √(2/5)
So, the endpoints of the major axis of the ellipse given by Equation 2 are (−√(2/5), 0) and (√(2/5), 0).
Substituting these values in Equation 1, we get:
z= 1+2x² +3y² = 1+2(−√(2/5))² +3(0)² = 1+2(2/5) = 9/5
So, the maximum value on the curve formed by the intersection of z = 1+2x² +3y² with z = 5−(3x² +5y²) is 9/5.
To find the minimum value on the curve, we can again substitute the values of x and y from the endpoints of the major axis of the ellipse in the equation z = 1+2x² +3y²
Substituting these values in the equation z = 5−(3x² +5y²), we get:
z= 5−3(−√(2/5))² −5(0)² = 5−3(2/5) = 7/5
So, the minimum value on the curve formed by the intersection of z = 1+2x² +3y² with z = 5−(3x² +5y²) is 7/5.
The maximum and minimum values on the curve formed by the intersection of z = 1+2x² +3y² with z = 5−(3x² +5y²) are 9/5 and 7/5 respectively.
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11) Find the area enclosed by the curves f(x) = 2x6 and g(x) = x7. {8 pts}
The two curves intersect at x = 2, we can now evaluate the area enclosed by the curves as follows:∫02 g(x) − f(x) dx= ∫02 x7 − 2x6 dx= [x8/8 − 2x7/7]2 0= 128/8 − (2(128/7))/7= 16 − 32/7= 96/7The area enclosed by the curves f(x) = 2x6 and g(x) = x7 is 96/7 square units
The curves f(x) = 2x6 and g(x) = x7 encloses the region between the x-axis and the curves.
To find the area enclosed by the curves, we need to evaluate the definite integral of the difference between the curves over their common interval of interest.
We first need to find the points of intersection of the two curves. Setting the two curves equal to each other gives:2x6 = x7⇔ 2 = x.
Since the two curves intersect at x = 2, we can now evaluate the area enclosed by the curves as follows:∫02 g(x) − f(x) dx= ∫02 x7 − 2x6 dx= [x8/8 − 2x7/7]2 0= 128/8 − (2(128/7))/7= 16 − 32/7= 96/7The area enclosed by the curves f(x) = 2x6 and g(x) = x7 is 96/7 square units
To find the area enclosed by two curves, we must find the points of intersection between the curves and then evaluate the definite integral of the difference between the curves over their common interval of interest.In this problem, the two curves are f(x) = 2x6 and g(x) = x7.
To find the points of intersection between the curves, we set the two curves equal to each other and solve for x:2x6 = x7⇔ 2 = x.
Thus, the two curves intersect at x = 2. We can now evaluate the area enclosed by the curves using the definite integral:∫02 g(x) − f(x) dx= ∫02 x7 − 2x6 dx= [x8/8 − 2x7/7]2 0= 128/8 − (2(128/7))/7= 16 − 32/7= 96/7
Therefore, the area enclosed by the curves f(x) = 2x6 and g(x) = x7 is 96/7 square units. In conclusion, we found that the two curves intersect at x = 2 and used this information to evaluate the definite integral of the difference between the curves over their common interval of interest. The area enclosed by the curves is 96/7 square units.
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Find the derivative \( \frac{d y}{d x} \) implicitly. \[ x^{2} y^{2}+3 y=4 x \]
The derivative of the expression "x² + y² + 3y = 4x" with respect to x is (4 - 2x) / (2y + 3).
To find the derivative of the equation x² + y² + 3y = 4x with respect to x, we can apply implicit differentiation.
Differentiating both sides of equation "x² + y² + 3y = 4x" with respect to x:
We get,
d/dx (x² + y² + 3y) = d/dx (4x)
Using the chain-rule, we can differentiate each term:
2x + 2y × (dy/dx) + 3 × (dy/dx) = 4
2y × (dy/dx) + 3 × (dy/dx) = 4 - 2x,
(2y + 3) × (dy/dx) = 4 - 2x,
Next, We solve for (dy/dx) by dividing both sides by (2y + 3):
dy/dx = (4 - 2x)/(2y + 3),
Therefore, the required derivative is (4 - 2x)/(2y + 3).
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The given question is incomplete, the complete question is
Find the derivative "dy/dx" for "x² + y² + 3y = 4x".
Republicans voted and 30 out of 60 Democrats voted. See if this sample is enough to show the proportion of Republicans that vote is higher than the proportion of Democrats that vote. Hint: Run a two proportion Cl. What is the conclusion? A. Since the CI was (negative, negative), P2 is higher, this means the proportion of Republicans that voted is higher B. Since the CI was (negative, negative), P2 is higher, this means the proportion of Democrats that voted is higher C. Since the CI was (positive, positive), P1 is higher, this means the proportion of Republicans that voted is higher D. Since the CI was (positive, positive), P1 is higher, this means the proportion of Democrats that voted is higher
The correct answer is D. Since the CI was (positive, positive), P1 is higher, this means the proportion of Democrats that voted is higher.
To determine if the proportion of Republicans that vote is higher than the proportion of Democrats that vote, we can use a two-proportion confidence interval.
Let's calculate the confidence interval using the given information:
Proportion of Republicans that voted (p1) = 30/60 = 0.5
Proportion of Democrats that voted (p2) = 30/60 = 0.5
Sample size for both groups (n1 = n2) = 60
We'll use a 95% confidence level for the confidence interval.
Using a two-proportion confidence interval formula, the confidence interval can be calculated as:
CI = (p1 - p2) ± Z * √[(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)]
where Z is the critical value corresponding to the desired confidence level.
Since the sample sizes for both groups are the same (60), we can simplify the formula:
CI = (p1 - p2) ± Z * √[2 * p * (1 - p) / n]
where p is the pooled proportion, calculated as (p1 + p2) / 2.
p = (0.5 + 0.5) / 2 = 0.5
Next, we need to determine the critical value corresponding to a 95% confidence level. Using a standard normal distribution table, the critical value for a 95% confidence level is approximately 1.96.
Now, let's calculate the confidence interval:
CI = (0.5 - 0.5) ± 1.96 * √[2 * 0.5 * (1 - 0.5) / 60]
CI = 0 ± 1.96 * √[0.5 * 0.5 / 60]
CI = 0 ± 1.96 * √[0.00833]
CI = 0 ± 1.96 * 0.0912
CI ≈ (-0.179, 0.179)
The confidence interval is approximately (-0.179, 0.179). Since the interval includes zero, we cannot conclude that the proportion of Republicans that vote is higher than the proportion of Democrats that vote.
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regions and obtains the following data. Africa: Mean =12.1 Sample size =201 Mexico: Mean =11.2 Sample size =238 (a) Which of the following would be the correct hypothesis test procedure to determine if the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti?
The correct hypothesis test procedure to determine if there is a difference between the mean height of cacti in Africa and Mexico would be the "Two-sample t-test."
The other options, the "Two-sample test for proportions" and the "Paired t-test," are not suitable for this scenario because they are designed for different types of data or study designs.
The "Two-sample test for proportions" is used when comparing proportions or percentages between two groups, rather than means. It is applicable when the data is categorical and involves comparing proportions or frequencies.
The "Paired t-test" is used when the data consists of paired observations or measurements, where each observation in one group is uniquely related or matched to an observation in the other group. This is not the case in the given scenario, where the two samples are independent.
Since we are comparing the mean height of cacti between two independent groups (Africa and Mexico), the appropriate test is the "Two-sample t-test."
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Complete question is below
A scientist wants to determine whether or not the height of cacti, in feet, in Africa is significantly higher than the height of Mexican cacti. He selects random samples from both regions and obtains the following data.
Africa:
Mean = 12.1
Sample size = 201
Mexico:
Mean = 11.2
Sample size = 238
(a) Which of the following would be the correct hypothesis test procedure to determine if there is a difference between the mean height of two cacti?
-Two-sample test for proportions
-Two-sample t-test
- Paired t-test
Solve for XZ.
Enter your answer as a decimal in the box.
Hello!
Pythagore!
XZ² = XY² + YZ²
XZ² = 42² + 6.5²
XZ² = 1806.25
XZ = √1806.25
XZ = 42.5
Frequency Distribution The total number of goals scored in a World Cup soccer match approximately follows the following distribution. Goals Scored 0 1 2 3 4 5 6 7 Probability 0.1 0.2 0.25 0.2 0.15 0.06 0.03 0.01 a) Let X be the number of goals scored in a randomly selected World Cup soccer match. Write out the PMF for X and explain why it is a valid PMF. b) Compute the mean and variance of X. c) Find and sketch the CDF of X. Explain why it is a valid CDF
a. The PMF (Probability Mass Function) for X is:
PMF(X) = {}
0.1, for X = 0
0.2, for X = 1
0.25, for X = 2
0.2, for X = 3
0.15, for X = 4
0.06, for X = 5
0.03, for X = 6
0.01, for X = 7
b. The mean (μ) is 2.54; the Variance (σ²) is 1.6484
c. The CDF is a valid CDF because it is a non-decreasing function and it approaches 1 as x approaches infinity.
What is the frequency distribution?a) The PMF (Probability Mass Function) for X, the number of goals scored in a World Cup soccer match, is given by the following:
PMF(X) = {}
0.1, for X = 0
0.2, for X = 1
0.25, for X = 2
0.2, for X = 3
0.15, for X = 4
0.06, for X = 5
0.03, for X = 6
0.01, for X = 7
This PMF is valid because it assigns probabilities to each possible value of X (0 to 7) and the probabilities sum up to 1. The probabilities are non-negative, and for any value of X outside the range of 0 to 7, the probability is zero.
b) To compute the mean and variance of X, we can use the following formulas:
Mean (μ) = Σ(X * PMF(X))
Variance (σ^2) = Σ((X - μ)² * PMF(X))
Using the PMF values given above, we can calculate:
Mean (μ) = (0 * 0.1) + (1 * 0.2) + (2 * 0.25) + (3 * 0.2) + (4 * 0.15) + (5 * 0.06) + (6 * 0.03) + (7 * 0.01) = 2.54
Variance (σ²) = [(0 - 2.54)² * 0.1] + [(1 - 2.54)² * 0.2] + [(2 - 2.54)² * 0.25] + [(3 - 2.54)² * 0.2] + [(4 - 2.54)² * 0.15] + [(5 - 2.54)² * 0.06] + [(6 - 2.54)² * 0.03] + [(7 - 2.54)² * 0.01] ≈ 1.6484
c) The CDF (Cumulative Distribution Function) of X is a function that gives the probability that X takes on a value less than or equal to a given value x.
The CDF can be obtained by summing up the probabilities of X for all values less than or equal to x.
CDF(x) = Σ(PMF(X)), for all values of X ≤ x
For example, the CDF for x = 3 would be:
CDF(3) = PMF(0) + PMF(1) + PMF(2) + PMF(3)
CDF(3) = 0.1 + 0.2 + 0.25 + 0.2
CDF(3) = 0.75
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a) The PMF for X is valid because it assigns non-negative probabilities to each possible value of X and the sum of all probabilities is equal to 1.
b) The mean of X is 2.55 and the variance is 2.1925.
c) The CDF of X is a valid cumulative distribution function as it is a non-decreasing function ranging from 0 to 1, inclusive.
a) The PMF (Probability Mass Function) for X, the number of goals scored in a randomly selected World Cup soccer match, can be represented as follows,
PMF(X) = {
0.1, if X = 0,
0.2, if X = 1,
0.25, if X = 2,
0.2, if X = 3,
0.15, if X = 4,
0.06, if X = 5,
0.03, if X = 6,
0.01, if X = 7,
0, otherwise
}
This PMF is valid because it satisfies the properties of a valid probability distribution. The probabilities assigned to each value of X are non-negative, and the sum of all probabilities is equal to 1. Additionally, the PMF assigns a probability to every possible value of X within the given distribution.
b) To compute the mean (expected value) and variance of X, we can use the formulas,
Mean (μ) = Σ (x * p(x)), where x represents the possible values of X and p(x) represents the corresponding probabilities.
Variance (σ^2) = Σ [(x - μ)^2 * p(x)]
Calculating the mean,
μ = (0 * 0.1) + (1 * 0.2) + (2 * 0.25) + (3 * 0.2) + (4 * 0.15) + (5 * 0.06) + (6 * 0.03) + (7 * 0.01)
= 0 + 0.2 + 0.5 + 0.6 + 0.6 + 0.3 + 0.18 + 0.07
= 2.55
The mean number of goals scored in a World Cup soccer match is 2.55.
Calculating the variance,
σ^2 = [(0 - 2.55)^2 * 0.1] + [(1 - 2.55)^2 * 0.2] + [(2 - 2.55)^2 * 0.25] + [(3 - 2.55)^2 * 0.2]
+ [(4 - 2.55)^2 * 0.15] + [(5 - 2.55)^2 * 0.06] + [(6 - 2.55)^2 * 0.03] + [(7 - 2.55)^2 * 0.01]
= [(-2.55)^2 * 0.1] + [(-1.55)^2 * 0.2] + [(-0.55)^2 * 0.25] + [(-0.55)^2 * 0.2]
+ [(-1.55)^2 * 0.15] + [(2.45)^2 * 0.06] + [(3.45)^2 * 0.03] + [(4.45)^2 * 0.01]
= 3.0025 * 0.1 + 2.4025 * 0.2 + 0.3025 * 0.25 + 0.3025 * 0.2
+ 2.4025 * 0.15 + 6.0025 * 0.06 + 11.9025 * 0.03 + 19.8025 * 0.01
= 0.30025 + 0.4805 + 0.075625 + 0.0605 + 0.360375 + 0.36015 + 0.357075 + 0.198025
= 2.1925
The variance of the number of goals scored in a World Cup soccer match is 2.1925.
c) The CDF (Cumulative Distribution Function) of X can be calculated by summing up the probabilities of X for all values less than or equal to a given x,
CDF(X) = {
0, if x < 0,
0.1, if 0 ≤ x < 1,
0.3, if 1 ≤ x < 2,
0.55, if 2 ≤ x < 3,
0.75, if 3 ≤ x < 4,
0.9, if 4 ≤ x < 5,
0.96, if 5 ≤ x < 6,
0.99, if 6 ≤ x < 7,
1, if x ≥ 7
}
The CDF is valid because it satisfies the properties of a valid cumulative distribution function. It is a non-decreasing function with a range between 0 and 1, inclusive. At x = 0, the CDF is 0, and at x = 7, the CDF is 1. The CDF is right-continuous, meaning that the probability assigned to a specific value of x is the probability of x being less than or equal to that value.
Learn more about PMF from the given link:
https://brainly.com/question/30765833
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