Questions and Answers
What is the standardized statistic for a sample statistic, assuming the sampling distribution is approximately normal?
z = (sample statistic - Population parameter) / s.d.(sample statistic)
What is the standardized statistic (z statistic) for a sample proportion p̂?
z = (p̂ - p) / s.d.(p̂) or (p̂ - p) / √(p(1-p)/n)
What do we use the standardized z statistic to find?
The difference between an observed sample proportion (p̂) and a possible value for the population proportion (p).
What is the formula for a standardized z statistic for a sample mean?
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When we can only approximate sigma with a small sample, what is the probability distribution called?
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What is degrees of freedom in a sample mean of x̄?
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As the number of degrees of freedom increases, what happens to the t-distribution?
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What is the formula for the t distribution?
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When do we use Student's t-distribution instead of z?
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When the parameter of interest is µ₁, µ₂, or µ₁ - µ₂, the standardized statistic is a t-statistic when the denominator is a standard deviation or standard error of the sample statistic?
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Why can't we summarize the probability for the Student's t-distribution in one table like we can for the standard normal distribution?
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What is the law of large numbers?
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Does a parameter have a changing value or a fixed value?
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Do we know the parameter?
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Three examples of what a parameter can be:
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If we cannot find out the numerical value of a parameter, how do we use it?
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Define statistic, or sample statistic.
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When is a sample estimate or estimate used?
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Can multiple samples of a population vary?
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What is the procedure we use for making conclusions about population parameters on the basis of sample statistics?
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What is a confidence interval?
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What is hypothesis testing used for?
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Which notion do you want to reject when you are trying to establish statistical significance?
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What value is necessary in hypothesis testing, and what does it mean if this value is true?
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What would the null value be for a weight loss clinic?
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Two basic types of variables?
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What are the Big Five scenarios for population parameters with categorical variables?
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What are the Big Five scenarios for sample statistics with categorical variables?
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What is the parameter of interest when you take paired differences?
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What is the population mean for paired differences?
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Are pairs of different populations taken as matched or unmatched pairs?
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What is the parameter of interest with quantitative data from independent samples?
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Population parameter symbols and equations for estimating the difference between two population proportions?
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Population parameter symbol and sample estimate symbol for estimating the mean of a quantitative variable?
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What are three conditions for which the approximate normality of the sampling distribution for a sample proportion applies?
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What does standard error describe?
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What does a fourfold increase in sample size do to the standard deviation of possible sample means?
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What does a ninefold increase in sample size do to the standard deviation of possible means?
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What is the symbol for the mean of the sampling distribution of paired differences?
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What is the standard deviation of the sampling distribution of d?
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What kind of experiments or data collection are used for inference of the difference in two population means?
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What does the z score measure?
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Study Notes
Parameters and Statistics
- Parameters are fixed, unchanging values describing a population.
- Parameters are often unknown because measuring every unit in a population is impractical.
- Examples of parameters include summary population characteristics, random situations, and comparisons of different populations.
- Sample statistics are computed from a subset of values taken from a larger population to estimate unknown parameters.
Statistical Methods
- Statistical inference involves drawing conclusions about population parameters based on sample statistics.
- Confidence intervals provide a range in which the true parameter value likely falls, e.g., "between 53% and 59%."
- Hypothesis testing aims to reject a null hypothesis, indicating that observed sample results aren't due to chance alone.
Types of Variables
- Two basic types of variables: categorical (e.g., gender, pet type) and quantitative (e.g., weight, time).
- In hypothesis testing, a null value represents the assumption that there's no effect or difference present in the population.
Big Five Scenarios
- Categories include population parameters for categorical variables (e.g., one population proportion, difference between two proportions) and quantitative variables (e.g., one population mean, paired differences).
Sampling Distributions
- A sampling distribution reflects the probability distribution of a sample statistic, derived from all possible samples from a population.
- The standard deviation of sample statistics, called standard error, describes the variability of sample means or proportions.
- Normal distribution of sample proportions requires specific conditions regarding sample size and independence.
Confidence Intervals and Estimation
- Confidence intervals estimate the range for population parameters based on sample statistics.
- The standard error formula differs, such as for sample proportions (√p(1-p)/n) and means (σ/√n).
- For paired differences, data measured in matched samples require specific standard deviation notations and conditions for normality.
Two Population Means
- Notation associated with differences between two means includes parameters (µ₁, µ₂) and sample statistics (x̄₁, x̄₂).
- The sampling distribution for the difference in two means describes the expected variability and standard errors.
- Randomized experiments help infer differences between populations effectively and involve ensuring independence among samples.
Standardized Statistics
- Z-scores measure how many standard deviations a sample statistic is from the population parameter.
- For small samples, when the population standard deviation is unknown, Student's t-distribution applies, becoming more normal as sample size increases.
- The t-statistic is specific for means and is computed using sample values to estimate parameters when the standard population deviation is unavailable.
Conclusion
- Understanding the distinctions between parameters and statistics, sampling methods, the importance of data independence, and the appropriate use of statistical tests ensures accurate conclusions in research.
- Mastery of formulas related to standard deviations, confidence intervals, and hypothesis testing is essential for applying statistical methods in practice.
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Test your understanding of key concepts from Chapter 9 of Statistics with these flashcards. Each card highlights important definitions related to parameters, their values, and examples. Perfect for reinforcing your knowledge and preparing for exams.