Homework Answers
3.5 (a) if we have a sufficiently large number of observations,
its sample mean is close to the population mean.
(According to the law of large numbers, when there are large number
of observations then sample mean is same as population mean)
3.6 (c) Control variable
(Control variable is any variable other than x that may affect
y)
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3.5 Which of the following is implied from the law of large numbers? (a) If we...
3. [Multiple Choice Questions (4 points per question, 32 points in total): Please answer the following...
3. [Multiple Choice Questions (4 points per question, 32 points in total): Please answer the following questions. 3.1 There is a random variable X with observations 3.5 Which of the following is implied from the law [X,,X2,...,X). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a c? of large numbers? (a) If we have a sufficiently large number of observations, its sample mean is close...
Law of Large Numbers We saw in the Theoretical and Experimental Probability Lab that as we do more and more repetitions or trials of an experiment, the closer the experimental probability gets to the...
Law of Large Numbers We saw in the Theoretical and Experimental Probability Lab that as we do more and more repetitions or trials of an experiment, the closer the experimental probability gets to the theoretical probability. This is called the Law of Large Numbers. Why is the Law of Large Numbers important? Why do we do experiments and find experimental probability when we could just use theoretical probability? Inferential statistics makes inferences about populations using data drawn from the population....
Assume that we have a finite population of 5 elements on which we measured a random...
Assume that we have a finite population of 5 elements on which we measured a random variable X. Below are the observations of Xover the population: {0,1,2,3,4). Assume that we want to select a random sample of size 3 form this population. Let S be the sample variance. a. Find the distribution of S2 b. Calculate E(S) c. Calculate Var(s)
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample...
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample mean (which is often used as an estimator of the population mean) (b) Find the sample variance (which is often used as an estimator of the population variance). (c) In general, the estimators above are both unbiased and consistent for the population mean and variance, respectively Bias and consistency are both measures of the central tendency of an estimators. One is more relevant in...
Law of Large Number↓ Led tin eperaje Theorem 9.11. (Central limit theorem) Suppose that we have...
Law of Large Number↓
Led tin eperaje Theorem 9.11. (Central limit theorem) Suppose that we have i.i.d. random variables Xi,X2. X3,... with finite mean EX and finite variance Var(X) = σ2. Let Sn-Xi + . . . + Xn. Then for any fixed - oo<a<b<oo we have lim Pax (9.6) Theorem 4.8. (Law of large numbers for binomial random variables) For any fixed ε > 0 we have (4.7) n-00
Law of Large Numbers, Central Limit Theorem, and Confidence Intervals 1. (15 points) In an exercise,...
Law of Large Numbers, Central Limit Theorem, and Confidence Intervals 1. (15 points) In an exercise, your Professor generated random numbers in Excel. The mean is supposed to be 0.5 because the numbers are supposed to be spread at randonm between 0 and 1. I asked the software to generate samples of 100 random numbers repeatedly. Here are the sample means x for 50 samples of size 100: 0.532 0.450 0.481 0.508 0.510 0.530 0.4990.4610.5430.490 0.497 0.5520.473 0.425 0.4490.507 0.472...
9) Suppose we have a sample of the heights of Davis students and want to use...
9) Suppose we have a sample of the heights of Davis students and want to use the sample meanto get a confidence interval for the mean height in the population. Which of the followingwould increase the width of this confidence interval? A) Switching from a 95% confidence interval to a 90% confidence interval B) Increasing the sample size used to calculate the sample mean. C) Switching from a 95% confidence interval to a 99% confidence interval D) All of the...
Benfords Law claims that numbers chosen from very large data files tend to have "1" as...
Benfords Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say...
Recall that Benford's Law claims that numbers chosen from very large data files tend to have...
Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file....
Benford's Law claims that numbers chosen from very large data files tend to have "l" as...
Benford's Law claims that numbers chosen from very large data files tend to have "l" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "l" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say...
3. [Multiple Choice Questions (4 points per question, 32 points in total): Please answer the following questions. 3.1 There is a random variable X with observations 3.5 Which of the following is implied from the law [X,,X2,...,X). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a c? of large numbers? (a) If we have a sufficiently large number of observations, its sample mean is close...
Assume that we have a finite population of 5 elements on which we measured a random variable X. Below are the observations of Xover the population: {0,1,2,3,4). Assume that we want to select a random sample of size 3 form this population. Let S be the sample variance. a. Find the distribution of S2 b. Calculate E(S) c. Calculate Var(s)
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample mean (which is often used as an estimator of the population mean) (b) Find the sample variance (which is often used as an estimator of the population variance). (c) In general, the estimators above are both unbiased and consistent for the population mean and variance, respectively Bias and consistency are both measures of the central tendency of an estimators. One is more relevant in...
Law of Large Number↓
Led tin eperaje Theorem 9.11. (Central limit theorem) Suppose that we have i.i.d. random variables Xi,X2. X3,... with finite mean EX and finite variance Var(X) = σ2. Let Sn-Xi + . . . + Xn. Then for any fixed - oo<a<b<oo we have lim Pax (9.6) Theorem 4.8. (Law of large numbers for binomial random variables) For any fixed ε > 0 we have (4.7) n-00
Law of Large Numbers, Central Limit Theorem, and Confidence Intervals 1. (15 points) In an exercise, your Professor generated random numbers in Excel. The mean is supposed to be 0.5 because the numbers are supposed to be spread at randonm between 0 and 1. I asked the software to generate samples of 100 random numbers repeatedly. Here are the sample means x for 50 samples of size 100: 0.532 0.450 0.481 0.508 0.510 0.530 0.4990.4610.5430.490 0.497 0.5520.473 0.425 0.4490.507 0.472...
9) Suppose we have a sample of the heights of Davis students and want to use the sample meanto get a confidence interval for the mean height in the population. Which of the followingwould increase the width of this confidence interval? A) Switching from a 95% confidence interval to a 90% confidence interval B) Increasing the sample size used to calculate the sample mean. C) Switching from a 95% confidence interval to a 99% confidence interval D) All of the...
Benfords Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say...
Benford's Law claims that numbers chosen from very large data files tend to have "l" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "l" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say...
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