Główna zawartość

Kurs: Statystyka - program rozszerzony > Rozdział 16

Lekcja 1: Przekonaj się w jaki sposób nasz kurs spełnia wymagania AP®︎ Statistics Standards

Treść AP®︎ Statistics dostosowana do standardów

This page lists every piece of course content once and shows all the standards covered by that content. So, standards may appear more than once in this view. If you would like to quickly see all of the course content aligned to a particular standard, the Standards aligned to content page page may be better suited.

How to use this mapping

Each of our units is listed with a table of its contents, with the course content on the left side (Lessons ☰, Articles 📝, Videos 📺, and Exercises ✅) and the covered standards on the right side.

Place your cursor over a standard to reveal its description. Click a piece of course content to visit that Unit, Lesson, Article, Video, or Exercise.

If you’re looking for all of the course content that’s aligned with a particular standard, you can use the “Find” feature of your browser to search for the standard or use the Standards aligned to content page.

Analyzing categorical data

Content	Standards
☰ Welcome to AP Statistics
📺 Meet Jeff, a creator of AP Statistics on Khan Academy
☰ Analyzing one categorical variable	, , , , , , , , , , , , ,
📺 Identifying individuals, variables and categorical variables in a data set	, , , , ,
✅ Individuals, variables, and categorical & quantitative data	, , , , ,
📺 Creating a bar graph	, , ,
📺 Reading bar charts: comparing two sets of data	, ,
☰ Two-way tables	, , ,
📺 Two-way frequency tables and Venn diagrams	, ,
✅ Read two-way frequency tables	, ,
✅ Create two-way frequency tables	, ,
📺 Two-way relative frequency tables	, ,
✅ Read two-way relative frequency tables	, ,
📺 Interpreting two-way tables	, ,
✅ Interpret two-way tables	, ,
☰ Distributions in two-way tables	, , , , , , , ,
📺 Marginal and conditional distributions	, , ,
✅ Identifying marginal and conditional distributions	, , , , ,
✅ Marginal distributions	, ,
✅ Conditional distributions	, , , ,
📝 Conditional distributions and relationships	, ,

Displaying and describing quantitative data

Content	Standards
☰ Frequency tables and dot plots	, , ,
📺 Frequency tables & dot plots	, ,
✅ Creating dot plots	, ,
✅ Reading dot plots & frequency tables	, ,
☰ Histograms and stem-and-leaf plots	, , , ,
📺 Creating a histogram	, ,
📺 Interpreting a histogram	, ,
✅ Create histograms	, ,
✅ Read histograms	, ,
📺 Stem-and-leaf plots	, ,
📺 Reading stem and leaf plots	, ,
✅ Reading stem and leaf plots	, ,
☰ Describing and comparing distributions	, , , , , , , , , ,
📺 Classifying shapes of distributions	, , ,
✅ Shape of distributions	, ,
📺 Example: Describing a distribution	, , ,
✅ Describing distributions	, ,
📺 Example: Comparing distributions	, , , ,
✅ Comparing distributions	, , , ,

Summarizing quantitative data

Content	Standards
☰ Measuring center in quantitative data	, , , , ,
📺 Statistics intro: Mean, median, & mode	, , , ,
📺 Mean, median, & mode example	, , , ,
📺 Median in a histogram	, ,
✅ Calculating the mean	, ,
✅ Calculating the median	, ,
✅ Calculating mean and median from data displays	, , ,
☰ More on mean and median	, , , , , , ,
📝 Mean as the balancing point	, ,
📺 Missing value given the mean	, ,
✅ Missing value given the mean	, ,
📺 Impact on median & mean: increasing an outlier	, ,
📺 Impact on median & mean: removing an outlier	, ,
✅ Effects of shifting, adding, & removing a data point	, ,
📺 Estimating mean and median in data displays	, ,
✅ Estimating mean and median in data displays	, ,
☰ Measuring spread in quantitative data	, , , , , , ,
📺 Interquartile range (IQR)	, , ,
✅ Interquartile range (IQR)	, , ,
📺 Sample variance	, ,
📺 Sample standard deviation and bias	, ,
✅ Sample standard deviation	, ,
📺 Visually assessing standard deviation	, ,
✅ Visually assessing standard deviation	, ,
📺 Mean and standard deviation versus median and IQR	, ,
☰ More on standard deviation (optional)	, , , , , ,
📺 Review and intuition why we divide by n-1 for the unbiased sample variance	, , , , ,
📺 Why we divide by n - 1 in variance	, , , , ,
📺 Simulation showing bias in sample variance	, , , , ,
📺 Simulation providing evidence that (n-1) gives us unbiased estimate	, , , , ,
📝 Unbiased estimate of population variance	, , , ,
☰ Box and whisker plots	, , , , , ,
📺 Worked example: Creating a box plot (odd number of data points)	, , ,
📺 Worked example: Creating a box plot (even number of data points)	, , ,
✅ Creating box plots	, , ,
📺 Reading box plots	, , ,
✅ Reading box plots	, , ,
📺 Interpreting box plots	, , ,
✅ Interpreting quartiles	, , ,
📺 Judging outliers in a dataset	, ,
✅ Identifying outliers	, ,

Modeling data distributions

Content	Standards
☰ Percentiles (cumulative relative frequency)	, , , , ,
📺 Calculating percentile	, ,
✅ Calculating percentiles	, ,
📺 Analyzing a cumulative relative frequency graph	, , , ,
📝 Cumulative relative frequency graph problem	, , , ,
☰ Z-scores	, , , , , ,
📺 Z-score introduction	, , ,
✅ Calculating z-scores	, , ,
📺 Comparing with z-scores	, ,
✅ Comparing with z-scores	, ,
📝 Z-scores-problem	, , , , ,
☰ Effects of linear transformations	, , ,
📺 How parameters change as data is shifted and scaled	, ,
✅ Transforming data	, ,
📝 Transforming data problem	, ,
☰ Density curves	, , , , ,
📺 Density Curves	, , ,
📺 Median, mean and skew from density curves	, ,
📺 Density curve worked example	, ,
✅ Properties of density curves	, ,
📺 Worked example finding area under density curves
✅ Area under density curves
☰ Normal distributions and the empirical rule	, , , , ,
📺 Qualitative sense of normal distributions (from ck12.org)	, , ,
📺 Normal distribution problems: Empirical rule (from ck12.org)	, ,
✅ Empirical rule	, ,
📝 Basic normal calculations	, ,
☰ Normal distribution calculations
📺 Standard normal table for proportion below	, ,
📺 Standard normal table for proportion above	, ,
✅ Normal distribution: Area above or below a point	, ,
📺 Standard normal table for proportion between values	, ,
✅ Normal distribution: Area between two points	, ,
📺 Finding z-score for a percentile	, ,
📺 Threshold for low percentile	, ,
✅ Normal calculations in reverse	, ,

Exploring bivariate numerical data

Content	Standards
☰ Making and describing scatterplots	, , , , , , , , , , , , , ,
📺 Constructing a scatter plot	, , ,
✅ Making appropriate scatter plots	, , , ,
📺 Example of direction in scatterplots	, , , ,
✅ Positive and negative linear associations from scatter plots	, , , ,
✅ Describing trends in scatter plots	, ,
📝 Positive and negative associations in scatterplots	, , ,
📺 Bivariate relationship linearity, strength and direction	, , , , , , ,
📝 Describing scatterplots (form, direction, strength, outliers)	, ,
✅ Describing scatterplots	, ,
☰ Correlation coefficients	, , , , , ,
📺 Calculating correlation coefficient r	, , , , ,
📺 Example: Correlation coefficient intuition	, , , ,
✅ Correlation coefficient intuition	, , , ,
☰ Least-squares regression equations	, , , , , , , , , , , , , ,
📺 Introduction to residuals and least-squares regression	, , , , , , ,
📺 Calculating residual example	, ,
✅ Calculating and interpreting residuals	, ,
📺 Calculating the equation of a regression line	, , ,
✅ Calculating the equation of the least-squares line	, , ,
📺 Interpreting slope of regression line	, ,
📺 Interpreting y-intercept in regression model	, , , ,
✅ Interpreting slope and y-intercept for linear models	, , ,
📺 Using least squares regression output	,
✅ Using least-squares regression output	, ,
☰ Assessing the fit in least-squares regression	, , , , , , , , , , ,
📺 Residual plots	, , , , ,
✅ Residual plots	, , , , ,
📝 R-squared intuition	, ,
📺 R-squared or coefficient of determination	, ,
📺 Standard deviation of residuals or root mean square deviation (RMSD)	,
📺 Interpreting computer regression data	,
📝 Interpreting computer output for regression	,
📺 Impact of removing outliers on regression lines	, , ,
✅ Influential points	, ,

Study design

Content	Standards
☰ Sampling and observational studies	, , , , , , , , , , , , , , , , , ,
📺 Identifying a sample and population	, , ,
✅ Identifying the population and sample	, , ,
📺 Generalizabilty of survey results example	, , ,
✅ Generalizability of results	, , ,
📺 Examples of bias in surveys	, , , , , , ,
📺 Example of undercoverage introducing bias	, , , , , , , ,
📝 Identifying bias in samples and surveys	, , , , , , , , , ,
✅ Bias in samples and surveys	, , , , , , , , , ,
☰ Sampling methods	, , , , , ,
📺 Techniques for generating a simple random sample	, , ,
✅ Simple random samples	, , ,
📺 Techniques for random sampling and avoiding bias	, , ,
✅ Sampling methods	, , , ,
☰ Types of studies (experimental vs. observational)	, , , , , , , , ,
📺 Types of statistical studies	, , , , , , , ,
📺 Worked example identifying experiment	, ,
📺 Worked example identifying observational study	, , , ,
📝 Observational studies and experiments	, , ,
✅ Observational studies versus experiments	, , ,
☰ Experiments
📺 Introduction to experiment design	, , , , , , , , , , , , , ,
📝 The language of experiments	, , , ,
📝 Principles of experiment design	, , , , , ,
📝 Random sampling vs. random assignment (scope of inference)	, , , , ,
📺 Matched pairs experiment design	, ,
✅ Experiment designs	, , , , ,
📺 Invalid conclusions from studies example	, ,
📺 Can causality be established from this study?	, ,
✅ Finding errors in study conclusions	, , , , , , , ,
✅ Experiment design considerations	, , , , , , , ,

Probability

Content	Standards
☰ Randomness, probability, and simulation	, , , , , , , , , , , , , , , ,
📺 Intro to theoretical probability	, , , , , , , , ,
📺 Experimental versus theoretical probability simulation	, , , , , , ,
📝 Theoretical and experimental probability: Coin flips and die rolls	, , , ,
📺 Random number list to run experiment	, , , ,
📺 Random numbers for experimental probability	, , , ,
✅ Interpreting results of simulations	, , ,
📺 Statistical significance of experiment
☰ Addition rule	, , , , ,
📺 Probability with Venn diagrams	, ,
📺 Addition rule for probability	, , , ,
📝 Addition rule for probability (basic)	, , , ,
✅ Two-way tables, Venn diagrams, and probability	, , , ,
☰ Multiplication rule	, , , , , , , ,
📺 Compound probability of independent events	,
📺 Independent events example: test taking	, , ,
📺 Free-throw probability	, , ,
📺 Three-pointer vs free-throw probability	, , ,
✅ Independent probability	, , ,
✅ Probabilities of compound events	, , ,
📺 Dependent probability introduction	, ,
✅ Dependent probability	, ,
📝 The general multiplication rule	, , , , ,
📺 Coin flipping probability	, , , ,
📝 Probabilities involving "at least one" success	, , , ,
✅ Probability of "at least one" success	, , , ,
☰ Conditional probability	, , , , , ,
📺 Conditional probability and independence	, , ,
📺 Conditional probability with Bayes' Theorem	, ,
✅ Calculating conditional probability	, ,
📝 Conditional probability using two-way tables	, ,
📝 Conditional probability and independence	, , , , ,
📺 Conditional probability tree diagram example	, , ,
📝 Tree diagrams and conditional probability	, , ,

Random variables

Content	Standards
☰ Discrete random variables	, , , , , , , , , , ,
📺 Constructing a probability distribution for random variable	, , , ,
📺 Valid discrete probability distribution examples	, , , ,
📺 Probability with discrete random variable example	, , , ,
✅ Probability with discrete random variables	, , , ,
📺 Mean (expected value) of a discrete random variable	, , , , ,
✅ Mean (expected value) of a discrete random variable	, , , , ,
📺 Variance and standard deviation of a discrete random variable	, , , , , ,
✅ Standard deviation of a discrete random variable	, ,
📝 Mean and standard deviation of a discrete random variable	, , , , ,
☰ Continuous random variables	, , , , , , ,
📺 Probabilities from density curves	, , , , ,
✅ Probability in density curves
✅ Probability in normal density curves	, , , , , ,
☰ Transforming random variables	, , ,
📺 Impact of transforming (scaling and shifting) random variables	, ,
📺 Example: Transforming a discrete random variable	, ,
✅ Transforming random variables	, ,
☰ Combining random variables	, , , , ,
📺 Mean of sum and difference of random variables	, ,
📺 Variance of sum and difference of random variables	, , ,
📺 Intuition for why independence matters for variance of sum	, , ,
📺 Deriving the variance of the difference of random variables	, , ,
📝 Combining random variables	, , , ,
✅ Combining random variables	, , , ,
📺 Example: Analyzing distribution of sum of two normally distributed random variables	, , ,
📺 Example: Analyzing the difference in distributions	, , ,
📝 Combining normal random variables	, , ,
✅ Combining normal random variables	, , ,
☰ Binomial random variables	, , , , ,
📺 Binomial variables	, ,
📺 Recognizing binomial variables	, ,
📺 10% Rule of assuming "independence" between trials
✅ Identifying binomial variables	, ,
📺 Binomial probability example	, ,
📺 Generalizing k scores in n attempts	, ,
📺 Free throw binomial probability distribution	, ,
📺 Graphing basketball binomial distribution
📺 Binompdf and binomcdf functions	,
📝 Binomial probability (basic)	, ,
✅ Binomial probability formula	, ,
✅ Calculating binomial probability	, ,
☰ Binomial mean and standard deviation formulas	, , , , , , ,
📺 Expected value of a binomial variable	, ,
📺 Variance of a binomial variable	, ,
📺 Finding the mean and standard deviation of a binomial random variable
✅ Mean and standard deviation of a binomial random variable	, ,
☰ Geometric random variables	, , , , , ,
📺 Geometric random variables introduction	, ,
✅ Binomial vs. geometric random variables	, ,
📺 Probability for a geometric random variable	, ,
✅ Geometric probability	, ,
📺 Cumulative geometric probability (greater than a value)	, ,
📺 Cumulative geometric probability (less than a value)	, ,
📺 TI-84 geometpdf and geometcdf functions	, ,
✅ Cumulative geometric probability	, ,
📺 Proof of expected value of geometric random variable	, ,

Sampling distributions

Content	Standards
☰ What is a sampling distribution?	, , , , ,
📺 Introduction to sampling distributions	, ,
📺 Sample statistic bias worked example	, , , ,
✅ Biased and unbiased estimators	, ,
☰ Sampling distribution of a sample proportion	, , , , , , , ,
📺 Sampling distribution of sample proportion part 1	, , ,
📺 Sampling distribution of sample proportion part 2	, , ,
📺 Normal conditions for sampling distributions of sample proportions	, ,
✅ The normal condition for sample proportions	, ,
✅ Mean and standard deviation of sample proportions	, ,
📺 Probability of sample proportions example	, ,
✅ Finding probabilities with sample proportions	, ,
📝 Sampling distribution of a sample proportion example	, ,
☰ Sampling distribution of a sample mean	, , , , , , , , , , , , ,
📺 Central limit theorem	, ,
📺 Sampling distribution of the sample mean	, , , ,
📺 Sampling distribution of the sample mean 2	, , , ,
📺 Standard error of the mean	, ,
✅ Mean and standard deviation of sample means	, , ,
✅ Sample means and the central limit theorem	, , ,
📺 Example: Probability of sample mean exceeding a value	, ,
✅ Finding probabilities with sample means	, ,
📝 Sampling distribution of a sample mean example	, , , , , , , ,

Confidence intervals

Content	Standards
☰ Introduction to confidence intervals	, , , , , , , ,
📺 Confidence intervals and margin of error	, ,
📺 Confidence interval simulation	, , ,
📺 Interpreting confidence level example	, , , ,
📝 Interpreting confidence levels and confidence intervals	, , , ,
☰ Confidence intervals for proportions	, , , , , , , , , , , , , , , , , , , , ,
📺 Conditions for valid confidence intervals	, , ,
📺 Conditions for confidence intervals worked examples	, , ,
📝 Reference: Conditions for inference on a proportion	, , , , , ,
✅ Conditions for a z interval for a proportion	, , ,
📺 Critical value (z*) for a given confidence level	, , ,
✅ Finding the critical value z* for a desired confidence level	, ,
📺 Example constructing and interpreting a confidence interval for p	, , ,
✅ Calculating a z interval for a proportion	, , ,
📝 Interpreting a z interval for a proportion	, , , ,
📺 Determining sample size based on confidence and margin of error	, , , , ,
✅ Sample size and margin of error in a z interval for p	, , , , , , , ,
☰ Confidence intervals for means	, , , , , , , , , , , , , , , ,
📺 Introduction to t statistics	, ,
📺 Simulation showing value of t statistic	, ,
📺 Conditions for valid t intervals	, ,
📝 Reference: Conditions for inference on a mean	, ,
✅ Conditions for a t interval for a mean	, ,
📺 Example finding critical t value	, ,
✅ Finding the critical value t* for a desired confidence level	, ,
📺 Example constructing a t interval for a mean	, , , , , , ,
✅ Calculating a t interval for a mean	, , , , , , ,
📺 Confidence interval for a mean with paired data	, , , , ,
📝 Making a t interval for paired data	, , , , ,
📝 Interpreting a confidence interval for a mean	, ,
📺 Sample size for a given margin of error for a mean	,
✅ Sample size and margin of error in a confidence interval for a mean

Significance tests (hypothesis testing)

Content	Standards
☰ The idea of significance tests	, , , , , , , , ,
📺 Idea behind hypothesis testing
📺 Examples of null and alternative hypotheses	, , , , , ,
✅ Writing null and alternative hypotheses	, , , , , ,
📺 P-values and significance tests	, ,
📺 Comparing P-values to different significance levels	,
📺 Estimating a P-value from a simulation
✅ Estimating P-values from simulations
📝 Using P-values to make conclusions
☰ Error probabilities and power	, , , , , , , , , , , , ,
📺 Introduction to Type I and Type II errors	, , ,
📺 Examples identifying Type I and Type II errors	, , ,
✅ Type I vs Type II error	, , ,
📺 Introduction to power in significance tests	, , , ,
📺 Examples thinking about power in significance tests	, ,
✅ Error probabilities and power	, , , ,
📝 Consequences of errors and significance	, , , , , , , ,
☰ Testing hypotheses about a proportion	, , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
📺 Constructing hypotheses for a significance test about a proportion	, , , , , ,
✅ Writing hypotheses for a test about a proportion	, , , , , ,
📺 Conditions for a z test about a proportion	, ,
📝 Reference: Conditions for inference on a proportion	, , , , , ,
✅ Conditions for a z test about a proportion	, ,
📺 Calculating a z statistic in a test about a proportion	, , , ,
✅ Calculating the test statistic in a z test for a proportion	, , , ,
📺 Calculating a P-value given a z statistic	, , , , , ,
✅ Calculating the P-value in a z test for a proportion	, , , , ,
📺 Making conclusions in a test about a proportion	, , , , , , ,
✅ Making conclusions in a z test for a proportion	, , , , , , ,
📺 Significance test for a proportion free response example	, , , , , , , , , , , , , , , , , , , , , ,
📺 Significance test for a proportion free response (part 2 with correction)	, , , , ,
☰ Testing hypotheses about a mean	, , , , , , , , , , , , , ,
📺 Writing hypotheses for a significance test about a mean	, ,
✅ Writing hypotheses for a test about a mean	, ,
📺 Conditions for a t test about a mean	, ,
📝 Reference: Conditions for inference on a mean	, , , , ,
✅ Conditions for a t test about a mean	, ,
📺 When to use z or t statistics in significance tests	, ,
📺 Example calculating t statistic for a test about a mean	, ,
✅ Calculating the test statistic in a t test for a mean	, ,
📺 Using TI calculator for P-value from t statistic
📺 Using a table to estimate P-value from t statistic
✅ Calculating the P-value in a t test for a mean
📺 Comparing P-value from t statistic to significance level	, , ,
✅ Making conclusions in a t test for a mean	, , ,
📺 Free response example: Significance test for a mean	, , , , , , , , , , , ,

Inference comparing two groups or populations

Content	Standards
☰ Confidence intervals for the difference between two proportions	, , , , , , ,
📺 Confidence intervals for the difference between two proportions	, ,
📺 Examples identifying conditions for inference on two proportions	, ,
✅ Conditions for inference on two proportions	, ,
📺 Calculating a confidence interval for the difference of proportions	, ,
✅ Two-sample z interval for the difference of proportions	, ,
☰ Testing the difference between two proportions	, , , , , , , , , , , , , , , , , , , , , ,
📺 Hypothesis test for difference in proportions	, , , , , , , , , , ,
📺 Constructing hypotheses for two proportions	, , , ,
✅ Writing hypotheses for testing the difference of proportions	, , , ,
📺 Hypothesis test for difference in proportions example	, , , , , , , , , , , , ,
✅ Test statistic in a two-sample z test for the difference of proportions	, ,
✅ P-value in a two-sample z test for the difference of proportions	, ,
📺 Comparing P value to significance level for test involving difference of proportions	, , , , ,
📺 Confidence interval for hypothesis test for difference in proportions	, ,
✅ Making conclusions about the difference of proportions	, , , , , ,
☰ Confidence intervals for the difference between two means	, , , , , , , , , , , ,
📺 Conditions for inference for difference of means	, ,
✅ Conditions for inference on two means	, ,
📺 Constructing t interval for difference of means	, , , , , , , , ,
📺 Calculating confidence interval for difference of means	, , , , , , , , ,
✅ Two-sample t interval for the difference of means	, , , , , , , ,
☰ Testing the difference between two means	, , , , , , , , , , , , , , , , , , , ,
📺 Hypotheses for a two-sample t test	, , , ,
📺 Example of hypotheses for paired and two-sample t tests	, , , , , ,
✅ Writing hypotheses to test the difference of means	, , , , , ,
📺 Two-sample t test for difference of means	, , , , , ,
✅ Test statistic in a two-sample t test	, ,
✅ P-value in a two-sample t test
📺 Conclusion for a two-sample t test using a P-value	, , ,
📺 Conclusion for a two-sample t test using a confidence interval	, ,
✅ Making conclusions about the difference of means	, , , , , ,

Chi-square tests for categorical data

Content	Standards
☰ Chi-square goodness-of-fit tests	, , , , ,
📺 Chi-square statistic for hypothesis testing	, , , ,
📺 Chi-square goodness-of-fit example
✅ Expected counts in a goodness-of-fit test
✅ Conditions for a goodness-of-fit test
✅ Test statistic and P-value in a goodness-of-fit test
✅ Conclusions in a goodness-of-fit test	, ,
☰ Chi-square tests for relationships	, , , , , , , , ,
📺 Introduction to the chi-square test for homogeneity	, ,
📺 Chi-square test for association (independence)	, , , ,
✅ Expected counts in chi-squared tests with two-way tables
✅ Test statistic and P-value in chi-square tests with two-tables
✅ Making conclusions in chi-square tests for two-way tables	, ,

More on regression

Content	Standards
☰ Inference about slope	, , , , , , , , , , , , , , , , , , , ,
📺 Introduction to inference about slope in linear regression	, , , , , , , ,
📺 Conditions for inference on slope	, ,
📺 Confidence interval for the slope of a regression line	, , ,
✅ Confidence interval for slope	, , ,
📺 Calculating t statistic for slope of regression line
✅ Test statistic for slope
📺 Using a P-value to make conclusions in a test about slope	, , , ,
📺 Using a confidence interval to test slope	, ,
✅ Making conclusions about slope	, , , , ,
☰ Transformations to achieve linearity	, , ,
📺 Transforming nonlinear data	, ,
📺 Worked example of linear regression using transformed data	, ,

Prepare for the 2019 AP Statistics Exam

Content	Standards
☰ Prepare for the exam
📝 Submit your questions about the 2019 AP Statistics exam
📺 Significance test for a proportion free response example	, , , , , , , , , , , , , , , , , , , , , ,
📺 Significance test for a proportion free response (part 2 with correction)	, , , , ,
📺 Free response example: Significance test for a mean	, , ,
✅ Choosing an appropriate inference procedure

Chcesz dołączyć do dyskusji?

Zaloguj się

Sortuj według

Na razie brak głosów w dyskusji

Rozumiesz angielski? Kliknij tutaj, aby zobaczyć więcej dyskusji na angielskiej wersji strony Khan Academy.