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Questions and Answers
What is the standardized statistic for a sample statistic, assuming the sampling distribution is approximately normal?
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?
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?
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?
What is the formula for a standardized z statistic for a sample mean?
When we can only approximate sigma with a small sample, what is the probability distribution called?
When we can only approximate sigma with a small sample, what is the probability distribution called?
What is degrees of freedom in a sample mean of x̄?
What is degrees of freedom in a sample mean of x̄?
As the number of degrees of freedom increases, what happens to the t-distribution?
As the number of degrees of freedom increases, what happens to the t-distribution?
What is the formula for the t distribution?
What is the formula for the t distribution?
When do we use Student's t-distribution instead of z?
When do we use Student's t-distribution instead of z?
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?
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?
Why can't we summarize the probability for the Student's t-distribution in one table like we can for the standard normal distribution?
Why can't we summarize the probability for the Student's t-distribution in one table like we can for the standard normal distribution?
What is the law of large numbers?
What is the law of large numbers?
Does a parameter have a changing value or a fixed value?
Does a parameter have a changing value or a fixed value?
Do we know the parameter?
Do we know the parameter?
Three examples of what a parameter can be:
Three examples of what a parameter can be:
If we cannot find out the numerical value of a parameter, how do we use it?
If we cannot find out the numerical value of a parameter, how do we use it?
Define statistic, or sample statistic.
Define statistic, or sample statistic.
When is a sample estimate or estimate used?
When is a sample estimate or estimate used?
Can multiple samples of a population vary?
Can multiple samples of a population vary?
What is the procedure we use for making conclusions about population parameters on the basis of sample statistics?
What is the procedure we use for making conclusions about population parameters on the basis of sample statistics?
What is a confidence interval?
What is a confidence interval?
What is hypothesis testing used for?
What is hypothesis testing used for?
Which notion do you want to reject when you are trying to establish statistical significance?
Which notion do you want to reject when you are trying to establish statistical significance?
What value is necessary in hypothesis testing, and what does it mean if this value is true?
What value is necessary in hypothesis testing, and what does it mean if this value is true?
What would the null value be for a weight loss clinic?
What would the null value be for a weight loss clinic?
Two basic types of variables?
Two basic types of variables?
What are the Big Five scenarios for population parameters with categorical variables?
What are the Big Five scenarios for population parameters with categorical variables?
What are the Big Five scenarios for sample statistics with categorical variables?
What are the Big Five scenarios for sample statistics with categorical variables?
What is the parameter of interest when you take paired differences?
What is the parameter of interest when you take paired differences?
What is the population mean for paired differences?
What is the population mean for paired differences?
Are pairs of different populations taken as matched or unmatched pairs?
Are pairs of different populations taken as matched or unmatched pairs?
What is the parameter of interest with quantitative data from independent samples?
What is the parameter of interest with quantitative data from independent samples?
Population parameter symbols and equations for estimating the difference between two population proportions?
Population parameter symbols and equations for estimating the difference between two population proportions?
Population parameter symbol and sample estimate symbol for estimating the mean of a quantitative variable?
Population parameter symbol and sample estimate symbol for estimating the mean of a quantitative variable?
What are three conditions for which the approximate normality of the sampling distribution for a sample proportion applies?
What are three conditions for which the approximate normality of the sampling distribution for a sample proportion applies?
What does standard error describe?
What does standard error describe?
What does a fourfold increase in sample size do to the standard deviation of possible sample means?
What does a fourfold increase in sample size do to the standard deviation of possible sample means?
What does a ninefold increase in sample size do to the standard deviation of possible means?
What does a ninefold increase in sample size do to the standard deviation of possible means?
What is the symbol for the mean of the sampling distribution of paired differences?
What is the symbol for the mean of the sampling distribution of paired differences?
What is the standard deviation of the sampling distribution of d?
What is the standard deviation of the sampling distribution of d?
What kind of experiments or data collection are used for inference of the difference in two population means?
What kind of experiments or data collection are used for inference of the difference in two population means?
What does the z score measure?
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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