Practice Set 2 QNT275 Version 61 Univers

Titleabc123 Version X1practice Set 2qnt275 Version 61university Of P

List the simple events for each of the following statistical experiments in a sample space. a) One roll of a die. Note: Separate your response with a comma (,). For example 22, 23, 24 b) Three tosses of a coin. Note: Use this notation for your answer. heads = H. tails = T. For example HT, TH c) One toss of a coin and one roll of a die. Note: Use this notation. Heads = H or numbers 1, 2, 3, 4, 5, 6 for the dice. For example H1 indicates heads and dice roll equal to 1. 2.

Two students are randomly selected from a statistics class, and it is observed whether or not they suffer from math anxiety. Indicate which are simple and which are compound events. a) Both students suffer from math anxiety. b) Exactly one student suffers from math anxiety. c) The first student does not suffer and the second suffers from math anxiety. d) None of the students suffers from math anxiety.

A hat contains 40 marbles. Of them, 18 are red and 22 are green. If one marble is randomly selected out of this hat: a) What is the probability that this marble is red (round to two decimal places)? b) What is the probability that this marble is green (round to two decimal places)?

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. Have Shopped, Have Never Shopped, Male, Female. a) If one adult is selected at random from these 2000 adults, find the probability that this adult has never shopped on the Internet. b) If one adult is selected at random from these 2000 adults, find the probability that this adult is a male. c) If one adult is selected at random from these 2000 adults, find the probability that this adult has shopped on the Internet given that this adult is a female. d) If one adult is selected at random from these 2000 adults, find the probability that this adult is a male given that this adult has never shopped on the Internet.

Find the joint probability of AA and BB for the following. a) P(A)=.36 and P(B|A)=.87 b) P(B)=.53 and P(A|B)=.

Classify each of the following random variables as discrete or continuous. a) The time left on a parking meter b) The number of bats broken by a major league baseball team in a season c) The number of cars in a parking lot at a given time d) The price of a car e) The number of cars crossing a bridge on a given day f) The time spent by a physician examining a patient g) The number of books in a student’s bag

The following table gives the probability distribution of a discrete random variable x. x, P(x): .11, .19, .28, .15, .12, .09, .06. Find the following probabilities. a) P(1≤x≤4) b) Probability that x assumes a value less than 4. c) Probability that x assumes a value greater than 2.

A limousine has eight tires on it. A fleet of such limos was fit with a batch of tires that mistakenly passed quality testing. The following table lists the probability distribution of the number of defective tires on this fleet of limos where x represents the number of defective tires on a limo and P(x) is the corresponding probability. x, P(x): .0454, .1723, .2838, .2669, .1569, .0585, .0139, .0015, .0008. Calculate the mean and standard deviation of this probability distribution. Give a brief interpretation of the values of the mean and standard deviation.

Let x be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a) p(5) for n=8 and p=.70. b) p(3) for n=4 and p=.40. Verify your answers by using Table I of Appendix B.

Let x be a discrete random variable that possesses a binomial distribution. If n = 5 and p = 0.8, then… a) What is the mean (round to three decimal places)? b) What is the standard deviation of the probability distribution (round to three decimal places)?

Paper For Above instruction

The fundamental understanding of probability and statistics involves analyzing random experiments, classifying events, calculating probabilities, and understanding the behavior of discrete and continuous variables. This paper discusses the computation of sample spaces, classification of events, probability calculations, statistical distributions, and basic properties of random variables, with practical examples drawn from everyday experiments and data analysis scenarios.

Sample Space and Simple Events

The sample space for a random experiment includes all possible outcomes. For example, in a single die roll, the simple events are the six outcomes represented as 1, 2, 3, 4, 5, 6. When rolling a die, the sample space is {1, 2, 3, 4, 5, 6}. For three coin tosses, each toss can result in heads or tails, leading to 2^3 = 8 possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. If one coin is tossed and one die is rolled, each outcome combines the result of the coin and the die, such as H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6, making 12 simple events.

Events and Their Classifications

When selecting students and observing their math anxiety, the events can be classified into simple or compound. A simple event involves a single outcome, while a compound event involves multiple outcomes. For example, both students suffering from math anxiety (event) is a compound because it involves two variables. Similarly, exactly one student suffering from anxiety is also a compound event, as it encompasses two possible outcomes: either student one or student two, but not both.

Probability Calculations

In scenarios involving marbles, the probability that a randomly selected marble is red is calculated as the ratio of red marbles to total marbles: 18/40 = 0.45, rounded to two decimal places as 0.45. Likewise, the probability of selecting a green marble is 22/40 = 0.55. In a survey of 2000 adults regarding Internet shopping, probabilities are computed based on the counts or percentages for each category. For instance, the probability that one adult has never shopped online can be derived from the table proportions, and similarly for other categories.

Joint and Conditional Probabilities

Joint probability involves calculating the probability of two events occurring together, such as the occurrence of events A and B simultaneously. For example, if P(A) = 0.36 and P(B|A) = 0.87, then the joint probability P(AA and BB) is P(A) P(B|A) = 0.36 0.87 = 0.3132. Conversely, if P(B) = 0.53 and P(A|B) needs to be determined, then P(A and B) = P(B) * P(A|B). Conditional probabilities quantify the likelihood of one event given that another has occurred.

Random Variables: Discrete vs. Continuous

Decision about whether a random variable is discrete or continuous depends on the nature of the data. The time left on a parking meter is continuous because it can take any value within a range. The number of bats broken or the number of books in a bag are discrete because they involve countable quantities. The price of a car or number of cars crossing a bridge can be continuous or discrete depending on the measurement precision. Time spent examining a patient is continuous, while the number of cars crossing a bridge in a day is discrete.

Probability Distribution and Calculations

Given a discrete probability distribution, such as probabilities associated with the variable x, cumulative probabilities can be computed by summing individual probabilities over specified ranges. For example, if P(x) is given for various values, P(1≤x≤4) is the sum of P(x) for x=1, 2, 3, 4. The complement or probabilities exceeding a certain value are also calculated similarly. Understanding how to interpret and utilize probability distributions is key to analyzing discrete random variables.

Expected Value and Variance of Discrete Distributions

The mean (expected value) of a distribution provides a measure of its central tendency and is calculated as the sum of each value of x multiplied by its probability. The standard deviation measures spread or variability and is the square root of variance. For probabilities of defective tires or other items, these calculations help assess average performance and variability, essential in quality control. For example, if the probability distribution indicates an average of 1.5 defective tires per limo, with a certain spread, managers can evaluate quality assurance measures accordingly.

Binomial Distribution Applications

The binomial distribution models scenarios with fixed number of trials, each trial having two outcomes (success or failure), and constant probability of success. Using the binomial formula, P(x) = C(n, x) p^x (1-p)^{n-x}, probabilities are calculated for specific successes. For example, computing the probability of exactly 5 successes in 8 trials with a success probability of 0.70 yields P(5) = C(8,5) 0.7^5 0.3^3. Similarly, probabilities for different parameters are calculated and verified with tables. The mean and standard deviation of binomial distributions are n p and √(n p * (1 - p)), respectively, providing insights into expected successes and variability.

Conclusion

Understanding the concepts of sample spaces, event types, basic probability computations, and distribution characteristics forms the backbone of statistical reasoning. Proper classification of variables as discrete or continuous and calculating their distributions allows analysts to make informed decisions based on data. In real-world applications, such as quality control, consumer behavior studies, and data analysis, these fundamental principles underpin the interpretation of results and the formulation of strategies for improvement or decision-making.

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