...total area within any continuous probability distribution is equal to 1.00 Answer: True Difficulty: Easy Goal: 1 AACSB: REF 5. For any continuous probability distribution, the probability, P(x), of any value of the random variable, X, can be computed. Answer: False Difficulty: Medium Goal: 1 6. For any discrete probability distribution, the probability, P(x), of any value of the random variable, X, can be computed. Answer: True Difficulty: Medium Goal: 1 AACSB: AS 7. The uniform probability distribution's standard deviation is proportional to the distribution's range. Answer: True Difficulty: Medium Goal: 2 8. For any uniform probability distribution, the mean and standard deviation can be computed by knowing the maximum and minimum values of the random variable. Answer: True Difficulty: Medium Goal: 2 9. In a uniform probability distribution, P(x) is constant between the distribution's minimum and maximum values. Answer: True Difficulty: Easy Goal: 3 10. For a uniform probability distribution, the probability of any event is equal to 1/(b-a). Answer: False Difficulty: Hard Goal: 3 11. The uniform probability distribution is symmetric about the mode. Answer: False Difficulty: Easy Goal: 3 12. The uniform probability distribution's...
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...CHAPTER 6: THE NORMAL DISTRIBUTION AND OTHER CONTINUOUS DISTRIBUTIONS 1. In its standardized form, the normal distribution a) has a mean of 0 and a standard deviation of 1. b) has a mean of 1 and a variance of 0. c) has an area equal to 0.5. d) cannot be used to approximate discrete probability distributions. ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: standardized normal distribution, properties 2. Which of the following about the normal distribution is NOT true? a) Theoretically, the mean, median, and mode are the same. b) About 2/3 of the observations fall within 1 standard deviation from the mean. c) It is a discrete probability distribution. d) Its parameters are the mean, , and standard deviation, . ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 3. If a particular batch of data is approximately normally distributed, we would find that approximately a) 2 of every 3 observations would fall between 1 standard deviation around the mean. b) 4 of every 5 observations would fall between 1.28 standard deviations around the mean. c) 19 of every 20 observations would fall between 2 standard deviations around the mean. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 4. For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770. The value of Z is a) 0.18. b) 0.81. c) 1.16. d) 1...
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...C H A P T E R 6 The Normal Distribution Objectives After completing this chapter, you should be able to Outline Introduction 6–1 Normal Distributions 1 2 3 4 5 6 7 Identify distributions as symmetric or skewed. Identify the properties of a normal distribution. Find the area under the standard normal distribution, given various z values. Find probabilities for a normally distributed variable by transforming it into a standard normal variable. Find specific data values for given percentages, using the standard normal distribution. Use the central limit theorem to solve problems involving sample means for large samples. Use the normal approximation to compute probabilities for a binomial variable. 6–2 Applications of the Normal Distribution 6–3 The Central Limit Theorem 6–4 The Normal Approximation to the Binomial Distribution Summary 6–1 300 Chapter 6 The Normal Distribution Statistics Today What Is Normal? Medical researchers have determined so-called normal intervals for a person’s blood pressure, cholesterol, triglycerides, and the like. For example, the normal range of systolic blood pressure is 110 to 140. The normal interval for a person’s triglycerides is from 30 to 200 milligrams per deciliter (mg/dl). By measuring these variables, a physician can determine if a patient’s vital statistics are within the normal interval or if some type of treatment is needed to correct a condition and avoid future illnesses. The question then is,...
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...10 and 100. Construct a mean for each sample and look at the distribution of the sample means (third row). Record the mean and SD in the population for the original random variable. Make a table and record the mean of the means, and the standard deviation of the mean for each sample size (2, 10 and 100). For a bell-shaped distribution we showed that the mean of the sample means is very similar to the mean in the population for all population sizes. This is the property of unbiasedness. From your results, does the mean appear to be unbiased when the original distribution of the data is skewed? For a bell-shaped distribution we showed that the standard deviation of the mean, or the standard error of the mean, decreases with sample size. Is this true for skewed distributions? Calculate the standard error of the mean that is expected for each sample size and record in the table. How did the observed result compared to your calculation? * * Mean of the Population: 15.5297 * Median of the Population: 12.2861 * Standard Deviation of the Population: 12.5078 * * Sample Size | * Mean of the means | * Standard Deviation | * Standard Error of the Mean | * 2 | * 15.7741 | * 8.8115 | * 6.23067 | * 10 | * 15.4568 | * 3.847 | * 1.21653 | * 100 | * 15.5223 | * 1.2636 | * 0.12636 | * * Taking the mean of the population as the true value, we can find the percent errors for the means of the different sample sizes...
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...+ q5 5. a. 0.015 b. 0.13` 6. a. 0.004 b. 0.33 7. 0.17 8. 0.65 8 – 7 Fitting to a Normal Distribution A normal curve is used in a wide variety of situations to estimate probabilities. Before we examine exactly what a normal curve is, we will recap how it is related to what we’ve already learned. Say you were rolling a die for a binomial experiment. There is a random variable associated with the outcomes of the experiment that we can calculate the probabilities for using the equations from the last section. A probability distribution shows the probabilities that correspond to the possible values of a random variable. Probability Distribution In a binomial experiment, there are only a finite number of outcomes to the experiment. We can graph the probability of 0 successes, 1 success, 2 successes…etc to get a binomial distribution. However, in continuous probability distribution, from more complex experimental data, would be a smooth curve where the outcome can be any number. Normal curves You may be familiar with the bell-shaped curve called the normal curve. A normal distribution is a function of the mean and standard deviation of a data set that assigns probabilities to intervals of real numbers associated with continuous random variables. How the Normal Curve Works In a normal distribution, the probability of a value occurring is based its position in the curve relative to the mean. The curve is evenly distributed so that a certain amount of the data will lie under a...
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...MAT 540 Quiz Answers 1) Deterministic techniques assume that no uncertainty exists in model parameters. Answer: TRUE Diff: 1 Page Ref: 489 Main Heading: Types of Probability Key words: deterministic techniques 2) Probabilistic techniques assume that no uncertainty exists in model parameters. Answer: FALSE Diff: 1 Page Ref: 489 Main Heading: Types of Probability Key words: probabilistic techniques 3) Objective probabilities that can be stated prior to the occurrence of an event are classical or a priori. Answer: TRUE Diff: 2 Page Ref: 489 Main Heading: Types of Probability Key words: objective probabilities, classical probabilities 4) Objective probabilities that are stated after the outcomes of an event have been observed are relative frequencies. Answer: TRUE Diff: 2 Page Ref: 489 Main Heading: Types of Probability Key words: relative frequencies 5) Relative frequency is the more widely used definition of objective probability. Answer: TRUE Diff: 1 Page Ref: 490 Main Heading: Types of Probability Key words: relative frequencies 6) Subjective probability is an estimate based on personal belief, experience, or knowledge of a situation. Answer: TRUE Diff: 2 Page Ref: 490 Main Heading: Types of Probability Key words: subjective probability 7) An experiment is an activity that results in one of several possible outcomes. Answer: TRUE Diff: 1 Page Ref: 491 Main Heading: Fundamentals of Probability Key words: experiment ...
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...word from a reference. You need to explain the concept so I know you understand what it means. For questions requiring material from Statdisk, make sure to turn labels on, take a screen capture (CTRL-Print Screen on most Windows-based computers), and paste the image into the worksheet. Crop the image as appropriate. 1. Describe a normal distribution in no more than 100 words (5 point). Answer: A normal distribution is a continuous random variable distribution with a bell shape, and has only two parameters: the mean, and the variance. A normal distribution can be represented by the formula: y=e^(-1/2)(x-μ/σ)/(σ√2pi). The mean can be any positive number and variance can be any positive number, so there are an infinite number of normal distributions. The shape of the distribution when graphed is symmetrical and bell-shaped. Use this information to answer questions 2-4. Following a brushfire, a forester takes core samples from the ten surviving Bigcone Douglas-fir trees in a test plot within the burn area, and a dendrochronologist determines the age of the source trees to be as follows (in years): 15 38 48 67 81 83 94 102 135 167 2. Construct a normal quantile plot in Statdisk, show the regression line, and paste the image into your response. Based on the normal quantile plot, does the data above appear to come from a population of Bigcone Douglas-fir tree ages that has a normal distribution? Explain (5 point). [pic] [pic] Answer: The data does...
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...will concentrate on estimating the mean from a normal distribution, and the proportion of success in a binomial distribution. To talk about estimation, we first need to know something about distributions and sampling. Most data used for decision making exhibit variation. We are interested in drawing conclusions from such data. Probability and statistics give us the necessary tools. We will again make use of the concept of a random variable. We have reviewed discrete random variables already. Before moving on, we need to review continuous random variables. Continuous random variables take on continuous or interval values (there are an infinite number of possibilities). If you are measuring, the distribution of the result will almost always be continuous. For example, the width of an extruded bar is a continuous random variable. The distribution of a continuous random variable is represented by a continuous curve (called the probability density function (pdf) and often denoted f(y)). The height of the curve does not represent probabilities; instead, the area under the curve between two points tells us about the probability. As a result, the probability that a continuous random variable is exactly equal to a single value is 0 (there is no area under a single point). The Binomial and Normal Distributions As noted earlier, we will be talking about estimating the mean of a normal distribution and the proportion...
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...parameters nor statistics. 2. The Central Limit Theorem is important in statistics because a) for a large n, it says the population is approximately normal. b) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. c) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. d) for any sized sample, it says the sampling distribution of the sample mean is approximately normal. 3. Which of the following statements about the sampling distribution of the sample mean is incorrect? a) The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large ( n ≥ 30 ). b) The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means. c) The mean of the sampling distribution of the sample mean is equal to µ . d) The standard deviation of the sampling distribution of the sample mean is equal to σ . 4. Which of the following is true about the sampling distribution of the sample mean? a) The mean of the sampling distribution is always µ . b) The standard deviation of the sampling distribution is always σ . c) The shape of the sampling distribution is always approximately normal. d) All of the above are true. 1 5. Sales prices of baseball...
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...Statistics for Business [pic] Discrete and Continuous Probability Distributions Business Statistics With Canadian Applications Hummelbrunner Rak Gray Third Edition Week6 Pages 261-263 chapter 8 Pages 288-314, 320-325 chapter 9 Arranged by: Neiloufar Aminneia Probability distribution A probability distribution is a list of all events of an experiment together with the probability associated with each event in a tabular form. It is used for business and economic problems. We learned frequency distribution to classify data, relating to actual observations and experiments but probability distribution describes how outcomes are expected to vary. Probability distribution for rolling a true die x P(x) Events Frequencies Probability 1 1 1/6 2 1 1/6 3 1 1/6 4 1 1/6 5 1 1/6 6 1 1/6 Total: 1 [pic] Probability distribution for tossing 3 coins x P(x) Events(# of Heads) Frequencies Probability 0 1 (TTT))...
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...parameters nor statistics. 2. The Central Limit Theorem is important in statistics because a) for a large n, it says the population is approximately normal. b) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. c) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. d) for any sized sample, it says the sampling distribution of the sample mean is approximately normal. 3. Which of the following statements about the sampling distribution of the sample mean is incorrect? a) The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large ( n ≥ 30 ). b) The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means. c) The mean of the sampling distribution of the sample mean is equal to µ . d) The standard deviation of the sampling distribution of the sample mean is equal to σ . 4. Which of the following is true about the sampling distribution of the sample mean? a) The mean of the sampling distribution is always µ . b) The standard deviation of the sampling distribution is always σ . c) The shape of the sampling distribution is always approximately normal. d) All of the above are true. 1 5. Sales prices of baseball...
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...QNT-351 Discussion Question Responses * DQ#1: What is the importance of statistics in business decision making? Describe a business situation where statistics was used in making a decision. 1. Using statistics to evaluate the performance of your business. Taking all factors into account, determine whether you are making or losing money. In addition, determine the trend of your business. For example, determine whether, over time, you are making more or less profit (or loss). Track the share of the market that your business holds, and how this changes over time. Evaluate all these factors for each product type, even each model, in your company's line. The statistics to use here are simple line and bar graphs of data. These data can alert you to where you need to change aspects of your business, and how quickly you have to make that change. Furthermore, statistics are an excellent way to determine how you should allocate your resources to increase sales. Sales in a given market are a result of a number of factors, such as pricing, the size of the sales force, and the type and number of advertisements placed. However, these factors will not be equally important for every product. You would use multiple regressions to determine which of these factors are most important, and how much changing your asset allocation (for example, the size of the sales force) will change sales. 2. Statistics: The science of collecting, organizing, and analyzing data. Descriptive Statistics...
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...values from the uniform distribution on the interval [3,7] as follows: random 10000 c1; uniform 3 7. [3] a. Use mean command to find the sample mean x of these data———————– ¯ [2] b. What is the mean µ of the uniform distribution on the interval [3,7]?————[1] c. Compare µ to the value x you found in part a). ———————– ¯ Generate and store in column c2 1,000 values from exponential distribution with parameter λ = .125 as follows: random 1000 c2; exponential 8. Note: The mean µ and the standard deviation σ of such distribution are both equal to 1/λ = 8 and this is the value you are asked to enter in the command above. [3] d. Use desc command to find the sample mean x and sample standard deviation s for ¯ these 1,000 data —————– and —————— Are x and s close to the value 1/λ = 8?———————– Why?——————————¯ [3] e. Print (and include in your assignment) the histogram of the 1,000 values you generated from this exponential distribution. What is the shape of this distribution?———————– 2. Normal distribution: Generate and store in column c3 10,000 values from the standard normal distribution as follows: random 10000 c3; normal. [3] a. Print (and include in your assignment) the histogram for these data. What is the shape of this histogram?———————————– [3] b. What is the value on the horizontal axis around which the histogram seems to be symmetric? x=—— [3] c. Use Minitab to find the sample mean x=———-, and the sample standard...
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...By the way, parametric distribution means normal, binomial distribution etc.. Probability Distribution A listing of all the outcomes of an experiment and the probability associated with each outcome. Mean of probability Distribution μ= Σ[xP(x) Variance of probability Distribution σ2= Σ[x-μ2Px] Number of Cars (x) | Probability (x) | (x-μ) | (x-μ)2 | x-μ2P(x) | 0 | 0.1 | 0 – 2.1 | 4.41 | 0.441 (4.41*0.1) | 1 | 0.2 | 1 – 2.1 | 1.21 | 0.242 | 2 | 0.3 | 2 – 2.1 | 0.01 | 0.003 | 3 | 0.3 | 3 – 2.1 | 0.81 | 0.243 | 4 | 0.3 | 4 – 2.1 | 3.61 | 0.361 | | 1.0 (total) | | | σ2=1.290 | To derive μ=00.10+10.20+30.30+40.10 = 2.1 Discrete Random Variable A random variable that can assume only certain clearly separated values. Usually the result of counting something. Clearly separated means e.g. judge scoring points can be 8.8 and 8.9. they are clearly separated. It is NOT clearly separated if the score is 8.34144134134134134 or 8.351512312312. This will be infinite!! Continuous Random Variable A random variable that assume one of an infinitely large number of values, within certain limitations. E.g. The tiems of commercial flights between Singapore and Msia are 4.67 hours, 5.13 hours….so on. The random variable is the number of hours. Limitation rage is between Singapore and Msia. Binomial Probability Distribution * There are only two possible outcome on a trial of an experiment. – Success or failure * The random variable counts the...
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...Normality What does it mean to be normal? Well if you ask Webster’s dictionary about the concept of normality you get: “the state or fact of being the way things usually are”. But is it really that simple? Can you just define what it means to be normal with only a few lines and dots on a piece of paper? Is it actually possible to define normality as things being as they usually are? If everything is as it usually is there would never be any change. Even if you look at normality seen from a mathematical perspective which is perhaps the most simple way of looking at normality things are almost never as they usually are. In mathematics you add all the numbers up and find an average. Most people are usually either above or below the average. But if we look back at our definition of normality it suddenly becomes normal to be either above or below average. However this may seem a bit confusing because in mathematics we accept that the average is what is normal but it turns out that it is not necessarily the usual. This means that it might be a lot harder to define normality than just using a few words. For example if we look at someone who is physically handicapped they might not be “normal” in the way they look but there are lots of other people in their situation and it is socially acceptable. We have made it a part of our daily lives and there are even parking spots specifically for handicapped people and they have all the same rights as the rest of us which leads me...
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