9.6: Chapter 9 - Key Terms and Symbols
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Glossary of Key Terms and Symbols
Key Terms
Confidence interval: a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter, expressed with a specified level of confidence (such as 90%, 95%, or 99%).
Confidence interval for the difference of two means: estimates the true difference between two population means by using the difference of sample means, the standard error of the difference, and a critical value from the z or t distribution.
Confidence interval for the difference of two proportions: estimates the difference between two population proportions using the sample proportions, the standard error of their difference, and a critical z value.
Hypothesis testing: a statistical procedure used to evaluate claims or questions about population parameters using sample data. It requires setting up a null hypothesis (usually a statement of no effect or no difference) and an alternative hypothesis (a statement indicating the presence of an effect or difference), then using sample evidence to decide whether to reject the null hypothesis.
The null hypothesis (H₀): the default assumption that there is no effect or no difference between groups or variables.
The alternative hypothesis (H₁): the statement that there is a true effect or difference in the population.
Hypothesis testing for the difference between two means: evaluates whether there is a statistically significant difference between the means of two independent groups, based on sample data. This test is used when comparing, for example, average scores, measurements, or outcomes.
Z-test for the difference of two means: a hypothesis test that is used when the population standard deviations are known or the sample sizes are large. It uses the standard normal distribution to assess statistical significance.
T-test for the difference of two means: a hypothesis test that is used when population standard deviations are unknown and sample sizes are small. It relies on the t-distribution and takes into account the variability in both samples.
Hypothesis testing for the difference between two proportions: evaluates whether the proportion of a certain outcome in one population is significantly different from that in another. It uses the sample proportions and a pooled estimate to calculate the test statistic.
Test statistic: a standardized value computed from sample data, used to decide whether to reject the null hypothesis. For differences in means or proportions, this often takes the form of a z or t value.
P-value: the probability, assuming the null hypothesis is true, of obtaining a result as extreme as or more extreme than the observed sample result. A small p-value indicates strong evidence against the null hypothesis.
Level of significance (\(\alpha\)): is the threshold for rejecting the null hypothesis, often set at 0.05. If the p-value is less than alpha, the null hypothesis is rejected.
Standard error: the estimated standard deviation of a sampling distribution. It measures how much a sample statistic is expected to vary from the true population parameter.
Key Symbols
\( \alpha \) - Significance level. The probability of rejecting the null hypothesis when it is true (Type I error).
CL - Confidence level. The probability that a confidence interval contains the true population parameter.
\( H_0 \) - Null hypothesis. The default assumption that there is no effect or no difference between groups or variables.
\( H_1 \) - Alternative hypothesis. The claim that there is a true effect or difference in the population.
\( \hat{p}_1 \) - Sample proportion from the first group. The observed proportion of successes in the sample from the first population.
\( \hat{p}_2 \) - Sample proportion from the second group. The observed proportion of successes in the sample from the second population.
\( \mu_1 \) - Population mean of the first group. The true average value in the first population.
\( \mu_2 \) - Population mean of the second group. The true average value in the second population.
\( n_1 \) - Sample size of the first group. The number of observations in the first sample.
\( n_2 \) - Sample size of the second group. The number of observations in the second sample.
\( p_1 \) - Population proportion of the first group. The true proportion of successes in the first population.
\( p_2 \) - Population proportion of the second group. The true proportion of successes in the second population.
\(\bar{p}\) - The pooled sample proportion is the combined proportion of successes from both samples, used in hypothesis testing for the difference between two proportions.
p-value - The probability of observing the sample result, or more extreme, assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against \( H_0 \).
\(\bar{q}\) -The pooled complement of the sample proportions is the complement of the pooled proportion, representing the proportion of failures.
\( t \) - T-test statistic. Used in hypothesis tests when the population standard deviation is unknown and the sample size is small.
\( t_{\alpha/2} \) - Critical t-value. The value from the t-distribution corresponding to the desired confidence level and degrees of freedom.
Test statistic - A standardized value (such as \( z \) or \( t \)) calculated from sample data and used to determine whether to reject the null hypothesis.
\( z \) - Z-test statistic. Used when the population standard deviation is known or the sample size is large.
\( z_{\alpha/2}\) - Critical z-value. The z-score corresponding to the desired confidence level (e.g., 1.96 for 95\%).
\( \bar{x}_1 \) - Sample mean from the first group. The average of the sample data from the first group.
\( \bar{x}_2 \) - Sample mean from the second group. The average of the sample data from the second group.
Authors
"9.6: Chapter 9 - Key Terms and Symbols" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0