Statistics: A Step-by-step Introduction

Explore why we study probability and how observed samples relate to population parameters, highlighting population versus sample, and the parameter versus statistic distinction with p-hat as an estimate.

Define outcomes and the sample space through rolling a die, explain events as sets of outcomes, and use the complement rule to calculate probabilities like even versus odd numbers.

Learn how to compute the probability of unions of events, the inclusion-exclusion formula, and when events are disjoint or mutually exclusive using a die example.

Explore how to compute the probability of A and B using intersections and conditional probability, illustrated with dice events, and distinguish between independent and dependent cases.

Explore random variables and probability models by comparing probability density functions, probability mass functions, and the cumulative distribution function, with examples like die rolls and tail probabilities.

Practice probability mass functions and cumulative distribution functions for a discrete random variable with values 1, 2, 3, comparing observed and theoretical distributions via pmf, bar charts, and formulas.

Learn how cumulative distribution functions describe probabilities for random variables, using the uniform [0,1] example to show zero probability for exact values and the cdf as area under the pdf.

Learn how the binomial distribution models the number of heads in n coin flips, with X ~ binomial(n, p), and derive its probability mass function using n choose k.

Compute expected values for a discrete random variable using a 0, 1, 2 eggs distribution and a binomial example to show E[X] = np and E(p-hat) = p.

Learn the standard normal distribution with mean zero and unit standard deviation, and apply the 68-95-99.7 empirical rule to locate data within one, two, or three standard deviations.

Learn how the empirical rule characterizes the standard normal distribution, estimating probabilities within one, two, or three standard deviations and using symmetry to solve interval questions.

Learn to use the z-table for standard normal distribution to compute probabilities, tails, percentiles, and critical values in statistics, and verify the empirical rule with cdf and inverse cdf methods.

Compute z-scores by subtracting the mean and dividing by the standard deviation; use the z-table for probabilities. Mean 120, std dev 10; the 5th percentile is 103.55 (z = -1.645).

Use the z-table as a standard normal cdf to find probabilities for z less than or greater than values, and use inverse cdf for 95% confidence intervals and hypothesis tests.

Compute z-scores for a normal with mean 50 and standard deviation 5 using z = (x - mu)/sigma. Map to the standard normal and use the z-table to find probabilities.

Practice calculating probabilities for means and applying the central limit theorem, showing X is normal with mu 70 and sigma 3, while Xbar with n=100 has sd 0.3.

Explore the central limit theorem and how averages like p-hat and x-bar become approximately normal as sample size grows. Use the np and n(1-p) rules to justify this normal approximation.

Apply the central limit theorem to approximate a binomial with n=100 and p=0.7 by a normal, using P hat ≥ 0.79 and z ≈ 1.96 to estimate about 0.025.

Examine hypothesis testing by formulating null and alternative hypotheses about population parameters, using p=0.5 and p-hat, and exploring one-sided and two-sided tests and decision rules.

Explore how we set a null hypothesis and use the null distribution. Learn about critical values and the rejection region in a one-sided test of heads bias with six flips.

Explore p-values as the probability of observing results as extreme or more extreme under the null distribution, guiding decisions to reject the null hypothesis in one- or two-sided tests.

Use the normal approximation to the binomial when np and n(1-p) are at least ten, convert observed counts to a z score, and estimate the p-value (about 0.025).

Perform a full one-proportion z-test on a 164-person poll to see if the blue preference changed from 30%, using null p=0.3, two-sided testing, normal approximation, and p-values leading to rejection.

Compare two proportions with a two-sample z test: null p1 equals p2, alternative p1 greater than p2, using a pooled proportion and normal approximation.

Perform a two-proportion hypothesis test comparing men and women on liking action movies, using p-hat, pooled proportion, and z statistic to test equality with alpha 0.05 for a one-sided alternative.

Learn a z-test for means with quantitative data, using x-bar, mu0, and sigma, to compare athletes' blood pressure to the average, and why t-test is used when sigma is unknown.

Explore the one-sample t-test when sigma is unknown, estimate sigma with the sample standard deviation (n-1), and use the t distribution with appropriate degrees of freedom to assess the mean.

Conduct a two-sample t-test to compare mean heights of boys and girls, testing H0: mu_B = mu_G against H1: mu_B > mu_G.

Learn to analyze paired data by creating a difference column and applying a one-sample t test with zero mean difference. This reframes paired data from a two-sample perspective.

Explore how to construct and interpret confidence intervals for parameters using point estimates and margins of error, with 95% confidence and connections to hypothesis testing.

Pivoting the one-sample mean test to construct a two-sided confidence interval by inverting the t-test, using x̄, mu, se, n, df = n−1, and a margin of error.

Construct confidence intervals using four formulas for means and proportions, applying z or t critical values, point estimates and standard errors, and margins of error to interpret population parameters.

Construct a 95% confidence interval for a single proportion using p-hat and z-crit 1.96, and interpret it as a 95% confident range for the true p, linking to hypothesis testing.

Construct a 95% confidence interval for the mean height of four basketball players using a t distribution with three degrees of freedom, giving 177.2 to 197.8.

Learn the chi-square goodness of fit test using a six-face die, compare observed counts to the ten-per-face expectation, compute the statistic, and assess significance with five degrees of freedom.

Compare observed 40 ice cream sales to expected counts across chocolate, vanilla, and strawberry in a chi-square goodness-of-fit test; compute the statistic and assess significance at alpha 0.05.

Learn how two way tables reveal relationships between gender and ice cream choice, compute observed versus expected counts, and apply the chi square test for independence.

Perform a chi-square test for a two-way table to test independence of gender and ice cream preference, comparing observed to expected counts; with df=1, 6.66 exceeds 3.841, so not independent.

Distinguish independence from homogeneity in chi-square tests, following identical steps to compute expected counts and the chi-square statistic, with examples like ice cream flavor and gender.

Apply a two-way chi-square test for independence between college major and music preference, comparing observed and expected counts in a 3x3 table to determine significance.

Explore how two quantitative variables relate using scatterplots, linear form, and strength. Learn about correlation and regression, predicting y from x, and describing direction, form, and strength.

Learn how the correlation coefficient, or Pearson's r (R), measures the strength and direction of a linear relationship, ranges from -1 to 1, with zero indicating no linear relation.

Learn how regression predicts Y from X with a line Y = beta0 + beta1 X, and estimate beta0 and beta1 from data, interpreting the intercept, slope, and error.

Explains linear regression to predict y from x, introduces the line of best fit using least squares, defines residuals, slope, intercept, and goes through the x bar, y bar point.

Compute the correlation coefficient, and estimate the slope and intercept for a Celsius–Fahrenheit data set with measurement error, yielding the regression equation and discussing residuals.

explore how residuals and the error term assess model fit in linear regression, estimate residual variance as sigma squared, and compute the standard deviation of residuals (s) from data.

Explore how r-squared, the coefficient of determination, measures the percent of y's variability explained by x in simple linear regression, illustrated by r=0.99 and r^2 approximately 0.98.

Learn regression inference with hypothesis tests and confidence intervals for the slope, using p values, standard errors, and t distributions to assess x-y relationships.