Statistics Chapter 9 Flashcards
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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?

<p>z = (x̄ - µ) / s.d.(x̄) = (x - µ) / (sigma/√n) = √n(x̄ - µ) / sigma</p> Signup and view all the answers

When we can only approximate sigma with a small sample, what is the probability distribution called?

<p>A student's t-distribution.</p> Signup and view all the answers

What is degrees of freedom in a sample mean of x̄?

<p>df = n - 1 where n is the sample size.</p> Signup and view all the answers

As the number of degrees of freedom increases, what happens to the t-distribution?

<p>The t-distribution gets closer to the standard normal curve.</p> Signup and view all the answers

What is the formula for the t distribution?

<p>t = (x̄ - µ) / (s/√n)</p> Signup and view all the answers

When do we use Student's t-distribution instead of z?

<p>When we replace population standard deviation sigma with its estimate, the sample standard deviation s.</p> Signup and view all the answers

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?

<p>Standard error.</p> Signup and view all the answers

Why can't we summarize the probability for the Student's t-distribution in one table like we can for the standard normal distribution?

<p>Because we would need a separate table for each possible df value.</p> Signup and view all the answers

What is the law of large numbers?

<p>For any specific population, the larger the sample size, the closer x̄ becomes an accurate representation of µ.</p> Signup and view all the answers

Does a parameter have a changing value or a fixed value?

<p>True</p> Signup and view all the answers

Do we know the parameter?

<p>Usually we do not know the parameter because we cannot measure every unit in the population.</p> Signup and view all the answers

Three examples of what a parameter can be:

<ol> <li>A summary characteristic of a population, 2. A random situation, 3. A comparison of different populations.</li> </ol> Signup and view all the answers

If we cannot find out the numerical value of a parameter, how do we use it?

<p>We use statistical methods to make a good guess at the parameter.</p> Signup and view all the answers

Define statistic, or sample statistic.

<p>A number computed from a sample of values taken from a larger population.</p> Signup and view all the answers

When is a sample estimate or estimate used?

<p>When the statistic is used to estimate the unknown value of a population parameter.</p> Signup and view all the answers

Can multiple samples of a population vary?

<p>True</p> Signup and view all the answers

What is the procedure we use for making conclusions about population parameters on the basis of sample statistics?

<p>Statistical inference.</p> Signup and view all the answers

What is a confidence interval?

<p>An interval in which the researcher is fairly sure will cover the true, unknown value of the parameter.</p> Signup and view all the answers

What is hypothesis testing used for?

<p>To reject a hypothesis about a population.</p> Signup and view all the answers

Which notion do you want to reject when you are trying to establish statistical significance?

<p>You want to reject the hypothesis that chance alone can explain the sample results.</p> Signup and view all the answers

What value is necessary in hypothesis testing, and what does it mean if this value is true?

<p>A null value is necessary; if true, it means nothing of interest is happening, and chance can explain the sample results.</p> Signup and view all the answers

What would the null value be for a weight loss clinic?

<p>That the average weight loss for the population of clinic patrons is 0.</p> Signup and view all the answers

Two basic types of variables?

<p>Categorical and quantitative.</p> Signup and view all the answers

What are the Big Five scenarios for population parameters with categorical variables?

<p>One population proportion = p; difference in two population proportions = p₁ - p₂.</p> Signup and view all the answers

What are the Big Five scenarios for sample statistics with categorical variables?

<p>One population proportion (or probability) = p̂; difference in two population proportions = p̂₁ - p̂₂.</p> Signup and view all the answers

What is the parameter of interest when you take paired differences?

<p>The population mean for paired differences.</p> Signup and view all the answers

What is the population mean for paired differences?

<p>The mean we would get if we took differences for the entire population of possible pairs.</p> Signup and view all the answers

Are pairs of different populations taken as matched or unmatched pairs?

<p>True</p> Signup and view all the answers

What is the parameter of interest with quantitative data from independent samples?

<p>The difference in two population means.</p> Signup and view all the answers

Population parameter symbols and equations for estimating the difference between two population proportions?

<p>Population parameter = p₁ - p₂.</p> Signup and view all the answers

Population parameter symbol and sample estimate symbol for estimating the mean of a quantitative variable?

<p>Population parameter = µ; Sample estimate = x̄.</p> Signup and view all the answers

What are three conditions for which the approximate normality of the sampling distribution for a sample proportion applies?

<ol> <li>The physical situation must be a fixed proportion; 2. Random sample or repeatable situation; 3. Sample size must be large enough.</li> </ol> Signup and view all the answers

What does standard error describe?

<p>The estimated standard deviation for a sampling distribution.</p> Signup and view all the answers

What does a fourfold increase in sample size do to the standard deviation of possible sample means?

<p>Cuts it in half.</p> Signup and view all the answers

What does a ninefold increase in sample size do to the standard deviation of possible means?

<p>Cuts the standard deviation of possible means to a third of what it was.</p> Signup and view all the answers

What is the symbol for the mean of the sampling distribution of paired differences?

<p>µd.</p> Signup and view all the answers

What is the standard deviation of the sampling distribution of d?

<p>s.d(đ) = sigma sub d / √n.</p> Signup and view all the answers

What kind of experiments or data collection are used for inference of the difference in two population means?

<p>Randomized experiments.</p> Signup and view all the answers

What does the z score measure?

<p>The number of standard deviations that the raw score falls above or below the mean.</p> Signup and view all the answers

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.

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