Maryland, CMSC 12
Excerpt: ... standard deviation equal to 0.1 for all 6 components (cf. slide 17 on lecture about Tracking and Kalman filtering). c.) Write the state equation, assuming Gaussian noise with a standard deviation equal to 0.1 for each of the 6 state components. d.) Write the measurement equation, assuming Gaussian noise with standard deviation equal to 2 for each of the 3 measurement components. e.) Write a Matlab Kalman filter function to compute a position estimate of the missile (defined by 3 components) at each time step. f.) Run the Kalman filter (you can use an initial position estimate of (0, 0, 0), an initial velocity estimate of (0, 0, 0), and an initial covariance matrix for the prediction error equal to an identity matrix). Plot the estimates for the missile coordinates on the same figure as the measurements used in (a). 3. Read Chapter 19.4 of the book by Forsyth and Ponce on the condensation or particle filter (It is available on the web at http:/www.cs.berkeley.edu/~daf/book3draft/tracking.pdf) 4. After you fi ...
University of Florida, STA 4322
Excerpt: ... STA 4322 - HW # 10 - Due on Friday, March 30, 2007 1. Sports car owners in a town complain that the state vehicle inspection station judges their cars differently from family-style cars. Previous records indicate that 30% of all passenger cars fail the inspection on the first time through. In a random sample of 150 sports cars, 60 failed the inspection on the first time through. Is there sufficient evidence to indicate that the percentage of first failures for sports cars is higher than the percentage for all passenger cars? Use = 0.05. Please give details of your test. Give also the p-value for this test. 2. Amanufacturer of car batteries claims that the life of his batteries is approximately normally distributed with a standard deviation equal to 0.9 year. If a random sample of 10 of these batteries has a standard deviation of 1.2 years, (a) do we have sufficient evidence to conclude that > 0.9 year? Use = 0.05. Please give details of your test. (b) obtain the 95% confidence interval on . 3. A soft-drink ...
Washington, URBDP 520
Excerpt: ... list) selecting, for example, every tenth or hundredth individual for sampling might be sufficient. URBDP 520 Lecture 5 Page 2 of 9 The importance of good sampling technique cannot be over emphasized. If a sampling technique makes it more likely that some members of a population will be selected than others, this can seriously skew the resulting sample's characteristics and the implications of the sample. The sampling distribution of the sample mean Any time you select a sample, you are likely to get different members of a population. The resulting sample mean ( x ) will vary from sample to sample and, thus, is a random variable. As such, this random variable has a distribution we can discuss. The sample mean is a random variable with an expected value equal to the population mean () and a standard deviation equal to the population standard deviation divided by the square root of the sample size. Put another way: x = E(x) = x = n where is the population mean is the population standard deviation ...
Iowa State, STAT 503
Excerpt: ... Statistics 503 Homework 1 Due: in class on Friday, Jan 26 2007 Background: To complete this homework it would help for you to work through the notes on using R in the introduction part of the course material. This introductory code is available on the web page. Re-run the code, in small chunks, making sure that you understand what each line does, and how you would change the code to perform slightly different calculations. Purpose: The homeworks are assigned to help each person become proficient in the basic computational skills necessary to conduct the statistical analyses discussed in this course. Thus homework should be completed individually, with help from the instructor as needed. Hand in your solution on paper by the due date. This homework focuses on writing functions in R. Exercises: 1. Write an R function that takes a column of numbers and standardizes them to have mean equal to 0 and standard deviation equal to 1. Use this function to standardize the columns of this data: 100 110 90 115 80 2 ...
Iowa State, STAT 503
Excerpt: ... Statistics 503 Homework 1 Due: in class on Wednesday, Jan 19 2005 Background: To complete this homework it would help for you to work through the notes on using R in the introduction part of the course material. This introductory code is available on the web page. Re-run the code, in small chunks, making sure that you understand what each line does, and how you would change the code to perform slightly dierent calculations. Purpose: The homeworks are assigned to help each person become procient in the basic computational skills necessary to conduct the statistical analyses discussed in this course. Thus homework should be completed individually, with help from the instructor as needed. Hand in your solution on paper by the due date. This homework focuses on writing functions in R. Exercises: 1. Write an R function that takes a column of numbers and standardizes them to have mean equal to 0 and standard deviation equal to 1. Use this function to standardize the columns of this data: ...
SUNY Albany, GOG 496
Excerpt: ... 8 4 Which level of data specification differentiates by type but not amount? *nominal ordinal interval ratio Which level of data specification differentiates by rank but not amount? nominal *ordinal interval ratio The Fahrenheit scale measures temper ...
CSU Northridge, MR 31841
Excerpt: ... distribution of sample means is normally distributed for any sample size n. In either case, the sampling distribution of sample means has a mean equal to the population mean. = 3. And the sampling distribution of sample means has a variance equal to 1/n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n. The standard deviation of the sample means is often called the standard error of the mean. Probability and the Central Limit Theorem We can find the probability that a sample mean, , will fall in a given interval of the sampling distribution. To transform to a z score, you can use the equation Example 14: The mean rent of an apartment in a professionally managed apartment building is $780. You randomly select nine professionally managed apartments. What is the probability that the mean rent is less than $825? Assume that the rents are normally distributed, with a standard deviation of $150. Example 15: Credit card balanc ...
Georgia Tech, CEE 3770
Excerpt: ... e you want to test a randomly selected sample of n water specimens and estimate the mean daily rate of pollution produced by the mining operation. If you want 95% condence interval estimate of width 2 milligrams, how many specimens you need to sample? Assume prior knowledge indicates that pollution readings in water samples taken during a day are approximately normally distributed with a standard deviation equal to 5 milligrams. Solution: 95% half width = 1 = z/2 2 n 52 1 = z0.025 n 2 2 n = 1.96 5 n = 1.962 52 n = 96 1 3. The average speed of vehicles on a highway is being studied. (a) Suppose that observations on 50 vehicles yielded a sample mean of 65 mph. Assume that the standard deviation of vehicle speed is known to be 6 mph. Determine two-sided 99% condence intervals of the mean speed. Solution: 99%CI = X z/2 2 n 62 = 65 z0.005 50 6 = 65 2.58 50 = 65 2.189 (b) In part (a), how many additional vehicles speed should be observed such that the mean speed can be es ...
SUNY Stony Brook, AMS 310.01
Excerpt: ... e told that the probability of a defective equals 0.2. (a) Write an expression for the exact probability that in a sample of 100 we observe less than 15% defective. (b) Find the approximate probability that in a sample of n=100 we observe less than 15% defective. (c) For the sample of n=100, find the value of c such that Pr(X c) 0.01. 3. System breakdowns are distributed, according to a Poisson law at an average of 0.5 per hour. (a) What is the probability that the time between system breakdowns is less than 1.5 hours? (b) We do a study to see if this assumption of 0.5 per hour is correct by recording the time between system breakdowns for 100 days. Use the Central limit theorem to determine the approximate probability that the sample mean will be less than 1.5 hour. 4. Suppose that the weight of people getting on the elevator has mean 150 lb and standard deviation equal to 30 lb. and is normally distributed. The elevator contains a sign saying that "No more than 8 people Allowed on this Elevator". (a) Su ...
North Texas, RSS 5700
Excerpt: ... the mean: Is just the standard deviation of the sampling distribution. i.e. it is a particular standard deviation X Sampling error The sample cannot be fully representative of the population As such, there is variability due to chance We could have a thousand sample means and none of them equal exactly the population mean. However. CLT (continued) Properties of the sampling distribution of the mean random has a mean of has a standard error n Distributed approximately normal for large samples Normal for all samples if the variable X is normal The Central Limit Theorem For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution as the sample size (N) gets larger. This of course begs the question of what is `large enough' Furthermore, the sampling distribution of the mean will have a mean equal to (the population mean), and a standard deviation equal to N Central Limit Theore ...
Utah, ECON 3640
Excerpt: ... and the standard error of the mean. a. = 10, = 3 b. = 100, = 25 c. = 20, = 40 Ex. 4.136 A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and standard deviation equal to 16. a. Give the mean and standard deviation of the (repeated) sampling distribution of x. b. Describe the shape of the sampling distribution of x. Does your answer depend on the sample size? c. Calculate the standard normal z-score corresponding to a value of x = 16. d. Calculate the standard normal z-score corresponding to a value of x = 23. e. Find P(x < 16). f. Find P(x > 23). g. Find P(16 < x < 23). Eg. 4.28 A manufacturer of automobile batteries claims that the distribution of the lengths of life of its best battery has a mean of 54 months and a standard deviation of 6 months. Suppose a consumer group decides to check the claim by purchasing a sample of 50 of these batteries and subjecting them to tests that determine battery life. a. Assuming that the manufacturer's claim is true, describe t ...
Wisc La Crosse, MATH 145
Excerpt: ... Math 145 - Elementary Statistics Chapter 7 Exercises July 12, 2007 1. A soft-drink machine is regulated so that the amount of drink dispensed is approximately normally distributed with a standard deviation equal to 1.5 deciliters. A random sample of 36 drinks had an average content of 22.5 deciliters. a. Construct a 95% confidence interval estimate for the mean of all drinks dispensed by this machine. b. Give a practical interpretation for the interval estimate you obtained in part (a). c. Determine how large a sample is needed if we wish to be 95% confident that our estimate will be within 0.1 deciliters of the true mean? d. Construct a 99% confidence interval estimate for the mean of all drinks dispensed by this machine. Interpret your answer. e. Determine how large a sample is needed if we wish to be 99% confident that our estimate will be within 0.1 deciliters of the true mean? 2. The contents of 10 similar containers of a commercial soap are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9 ...
Wisc La Crosse, MATH 145
Excerpt: ... Math 145 - Elementary Statistics Chapter 7 Exercises March 26, 2008 1. A soft-drink machine is regulated so that the amount of drink dispensed is approximately normally distributed with a standard deviation equal to 1.5 deciliters. A random sample of 36 drinks had an average content of 22.5 deciliters. a. Construct a 95% confidence interval estimate for the mean of all drinks dispensed by this machine. b. Give a practical interpretation for the interval estimate you obtained in part (a). c. Determine how large a sample is needed if we wish to be 95% confident that our estimate will be within 0.1 deciliters of the true mean? d. Construct a 99% confidence interval estimate for the mean of all drinks dispensed by this machine. Interpret your answer. e. Determine how large a sample is needed if we wish to be 99% confident that our estimate will be within 0.1 deciliters of the true mean? 2. The contents of 10 similar containers of a commercial soap are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and ...
Wisc La Crosse, MATH 145
Excerpt: ... Math 145 - Elementary Statistics Chapter 7 Exercises March 25, 2009 1. A soft-drink machine is regulated so that the amount of drink dispensed is approximately normally distributed with a standard deviation equal to 1.5 deciliters. A random sample of 36 drinks had an average content of 22.5 deciliters. a. Construct a 95% condence interval estimate for the mean of all drinks dispensed by this machine. b. Give a practical interpretation for the interval estimate you obtained in part (a). c. Determine how large a sample is needed if we wish to be 95% condent that our estimate will be within 0.1 deciliters of the true mean? d. Construct a 99% condence interval estimate for the mean of all drinks dispensed by this machine. Interpret your answer. e. Determine how large a sample is needed if we wish to be 99% condent that our estimate will be within 0.1 deciliters of the true mean? 2. The contents of 10 similar containers of a commercial soap are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, ...
Wisc La Crosse, MATH 145
Excerpt: ... Math 145 - Elementary Statistics Chapter 7 Exercises October 22, 2008 1. A soft-drink machine is regulated so that the amount of drink dispensed is approximately normally distributed with a standard deviation equal to 1.5 deciliters. A random sample of 36 drinks had an average content of 22.5 deciliters. a. Construct a 95% confidence interval estimate for the mean of all drinks dispensed by this machine. b. Give a practical interpretation for the interval estimate you obtained in part (a). c. Determine how large a sample is needed if we wish to be 95% confident that our estimate will be within 0.1 deciliters of the true mean? d. Construct a 99% confidence interval estimate for the mean of all drinks dispensed by this machine. Interpret your answer. e. Determine how large a sample is needed if we wish to be 99% confident that our estimate will be within 0.1 deciliters of the true mean? 2. The contents of 10 similar containers of a commercial soap are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, an ...
Georgia Tech, CEE 3770
Excerpt: ... the mining operation. If you want 95% condence interval estimate of width 2 milligrams, how many specimens you need to sample? Assume prior knowledge indicates that pollution readings in water samples taken during a day are approximately normally distributed with a standard deviation equal to 5 milligrams. 3. The average speed of vehicles on a highway is being studied. (a) Suppose that observations on 50 vehicles yielded a sample mean of 65 mph. Assume that the standard deviation of vehicle speed is known to be 6 mph. Determine two-sided 99% condence intervals of the mean speed. (b) In part (a), how many additional vehicles speed should be observed such that the mean speed can be estimated to within 1 mph with 99% condence? (c) Suppose that Jake and Alan are assigned to collect data on the speed of vehicles on this highway. After each person has separately observed 10 vehicles, what is the probability that Jakes sample mean will exceed Alans sample mean by 2 mph? The standard deviation of ...
Wisc La Crosse, MATH 145
Excerpt: ... Geometric Distribution Poisson Distribution Hypergeometric Distribution 5. Continuous Random Variable: A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. 6. Practice: In a survey study of 200 college students, list 3 continuous random variables. a. b. c. 7. Examples of Continuous Probability Distribution: a. b. c. d. e. f. Normal Distribution Uniform Distribution 2 -squared Distribution t-Distribution F -Distribution Gamma Distribtion 8. A coffee machine is regulated so that it discharges an average of 200 ml per cup. If the amount of coffee discharged is normally distributed with a standard deviation equal to 10 ml, a. What is the probability that a cup will contain less than 188 ml? b. How many cups will overflow if 225 ml cups are used for the next 1000 discharges? c. What is the probability that a c ...
Penn State, STAT 100
Excerpt: ... re a rule about this? 2. Variability of a proportion There is a mathematical rule that tells us how the sample proportion will behave over repeated samples from a population. This rule is called the Central Limit Theorem. It was demonstrated first in 1733 and again in 1812. But it was not really known or used much until the Russian mathematician Lyapunov proved it in general terms in 1901. The Central Limit Theorem is the cornerstone of modern statistical inference. Now we will learn what the Central Limit Theorem says about the behavior of a sample proportion. Central Limit Theorem for a proportion Suppose that you take many random samples of size n from a large population in which the true proportion of Yes is p. And, for each sample, suppose you find the estimated value of p in the usual way (number of yes divided by n). These sample estimates of p will be approximately normally distributed, with mean equal to p, and standard deviation equal to the square root of p(1p) / n. ...
Penn State, STAT 100
Excerpt: ... and standard deviation equal to the square root of p(1p) / n. Example Earlier, we took 100 random samples of size 500 from a population in which the true proportion of "yes" was 57% or 0.57. What does the Central Limit Theorem say about what should happen? The CLT says that, over many samples, the sample proportions will be approximately normally distributed with mean equal to p = 0.57 or 57% standard deviation equal to the square root of ( 0.57 0.43 ) / 500 = square root of .0004702 = 0.0221 = 2.21% Histogram of 100 sample proportions Is that what actually happened? The histogram of the 100 sample proportions does look like a normal curve. Frequency 0 50 5 10 15 20 25 30 52 54 56 p 58 60 62 64 The average of the 100 sample percentages was 56.8, which is quite close to the population value of 57%. The standard deviation of the 100 sample percentages was 2.4, which is close to what the CLT said it should be (which was 2.2). ...
Marietta, HEM 001
Excerpt: ... on that has density f(x) = 63,574,540(x/100)29(1 (x/100)9 on the range 0 to 100. Using this density, it is possible to find the expectation and variance for any particular score is E(X) = 75 and V(X) = 45.73171. Suppose 60 students take the exam. Let X be the average of the 60 scores. What is P( X < 77)? 7. The probability is approximately .1 that a person aged 80 will die within 1 year. An insurance company insures 900 eighty-year-olds. What is the probability that less than 9% of these people will die in the following year? 8. The distribution of annual salaries of full-time carpenters has a mean equal to $24,000 and a standard deviation equal to $2,500. The distribution of annual salaries of full-time welders has a mean equal to $25,000 and a standard deviation equal to $3,000. Suppose we take random samples of 50 carpenters and 50 welders. a) What is the probability that, in the samples, the mean salary for the welders exceeds the mean salary for the carpenters? b) What is the probability that, in th ...
Cal Poly, STAT 321
Excerpt: ... are to your guess in (a)? Now calculate these probabilities exactly from the binomial distribution. (h) Use Minitab (Calc> Probability distributions> Binomial, with the "cumulative probability" option) to determine these exact probabilities. [Hint: Remember that with the binomial distribution, P(X>k) = 1-P(X<k-1).] Hospital A: Hospital B: (i) Are these probabilities close to the predictions of your simulations? (j) Judging from the histograms or dotplots of your simulation results, would you say that a normal distribution would approximate the binomial distribution well in these cases? With which case (A or B) do you think the approximation will be closer? Explain. (k) Determine the expected (mean) value and standard deviation of X and of Y. [Hint: Use what you know about the binomial distribution.] E(X): SD(X): E(Y): SD(Y): Fall, 2004 Thursday, October 28 (l) Let V be a normal random variable with mean and standard deviation equal to those of X. Determine P(V>12). Is it reasonably close to P(X>12)? ( ...