Questions and Answers
What is the name of the plot that represents the probabilities of each possible outcome?
Probability Distribution
What is the range of probability of each value of a discrete random variable?
0 ≤ P(x) ≤ 1
What is the sum of the probabilities of all possible outcomes of a discrete random variable?
∑ P(x) = 1
What type of plot is used to visualize a discrete probability distribution?
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What is an example of a discrete random variable?
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What is the classification of the distribution of a continuous random variable?
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What is the x-axis of a histogram or probability distribution plot used to visualize a discrete probability distribution?
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What are two types of discrete random variables mentioned in the text?
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Why did William Sealy Gossett publish his work under the pseudonym 'Student'?
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What is the primary method of communication for scholars to share information about statistical work?
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What is the name of the distribution that is defined by the degrees of freedom?
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What is the sum of a marginal probability distribution?
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What is the range of temperatures when it is said that the temperature is 39°C?
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What is the significance of the degrees of freedom in statistical analysis?
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What is used to describe a continuous probability distribution?
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How are conditional probability distributions obtained from a crosstab?
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Who first introduced the concept of degrees of freedom in statistics?
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What type of calculus is used to find the area under the curve in a continuous probability distribution?
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What is the purpose of a mean comparison test?
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What is the characteristic shape of the t-distribution?
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What is the importance of hypothesis testing, which includes the use of t-tests?
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What is the most common type of continuous probability distribution?
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What is the name of the nonparametric test used to compare medians?
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What is the main limitation of using sample means to make inferences about the population?
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What is the purpose of the probability density function in a continuous probability distribution?
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What is the correct definition of degrees of freedom in applied statistics?
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What is the purpose of studying the rules of crosstabulation tables?
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What is a common scenario where a nonparametric test is used?
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What is the purpose of the x-axis in a continuous probability distribution graph?
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What is the primary purpose of the z-table?
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What is the main concern when comparing two sample means or medians?
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What is the advantage of using a probability density function to describe a continuous probability distribution?
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What is the percentage probability of randomly selecting a score between -2 and +2 standard deviations from the mean?
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What is the primary difference between classical and quantum physics in terms of probability?
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What is the assumption of the classical probability method?
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What is the significance of the discovery of ancient societies playing probability games with primitive dice-like objects?
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What is the primary purpose of calculating probabilities in day-to-day life?
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What is the name of the curve that represents the probabilities under the standard normal distribution?
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What is the primary field of study that has changed our understanding of probability?
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What is the defining characteristic of a continuous probability distribution?
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Study Notes
Standard Normal Distribution
- The z-table represents the probabilities under the standard normal distribution, allowing researchers to calculate the probability of randomly obtaining a score from the sample distribution.
- There is a 95.45% probability of randomly selecting a score between -2 and +2 standard deviations from the mean.
Probability Calculation
- The study of probability began with ancient societies playing games with primitive dice-like objects made from animal bones.
- Classical physics states that true probability exists due to unknown inputs, while quantum physics suggests that truly probabilistic phenomena exist at a subatomic perspective.
- There are two methods of calculating probabilities: classical and non-classical.
Classical Probability Method
- In classical probability, every event has an equal likelihood of occurring.
- Marginal probability distributions are provided by the row and column totals of a crosstab, with the sum of a marginal distribution being one.
- Conditional probability distributions describe the distribution of one variable given the levels of the other variable, with the sum of a conditional distribution being one.
Mean Comparison Test
- A nonparametric test, such as the Wilcoxon signed-rank test, can be used to compare medians.
- The probability distribution (PD) represents the probabilities of each possible outcome.
- The height of each bar in the graph for X gives the probability of that particular outcome.
Discrete Probability Distributions
- A discrete probability distribution describes the distribution of a discrete random variable.
- A discrete random variable is a variable resulting from an experiment with discrete or countable values.
- Each value of a discrete random variable has a probability between zero and one, inclusive, and the sum of all probabilities is equal to one.
- Discrete probability distributions can be visualized via a histogram or probability distribution plot.
Continuous Probability Distributions
- A continuous probability distribution is classified as such when the random variable is continuous.
- Continuous probability distributions are expressed using a formula known as the probability density function.
- The probability density function is used to find the area under the curve, which represents the probability.
Normal Distribution
- The normal distribution is the most common type of continuous probability distribution.
- The t-distribution is defined by the degrees of freedom, which is a correction related to sample size.
- The t-distribution is bell-shaped but has heavier tails than the standard normal distribution.
Degrees of Freedom
- The concept of degrees of freedom was first mentioned by Carl Frederick Gauss in the 1880s and was brought into modern statistics by William Sealy Gossett in the early 1900s.
- Degrees of freedom are the number of independent values that can vary in an analysis without violating any constraints.
- This concept is important in techniques such as the t-distribution and t-tests and can impact the precision and power of analyses.
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