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.

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.

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.

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.

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.

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).

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.

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.

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.

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 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.