ECON E371 Class Recording 9/8

Overview

Professor Lantis teaches hypothesis testing and confidence intervals, connecting test statistics, p-values, critical values, and sample size to decisions about population means. The class then applies those ideas in Stata to county unemployment and 2020 Trump vote-share data, while reviewing the Friday problem set and quiz requirements and previewing linear regression.

Key takeaways

For samples larger than 30, the course approximates the sampling distribution with the standard normal; a 95% two-sided test uses cutoffs of approximately ±1.96.
The p-value and critical-value approaches are equivalent: reject the null when the p-value is below alpha or the test statistic lies beyond the critical cutoff.
A 95% confidence interval and a two-tailed test at alpha 0.05 produce the same decision about a hypothesized population mean: reject when the value falls outside the interval.
Higher confidence widens a confidence interval and makes rejection harder, while a larger sample reduces the standard error, narrows the interval, and can make rejection easier.
In the Stata exercise, counties classified as having low unemployment showed higher average 2020 Trump vote share than the comparison group, motivating a formal difference-in-means test rather than establishing causation.
The first problem set and first quiz are due Friday; the quiz uses LockDown Browser and covers material through confidence intervals.

Chapters

0:00 Course Files, Stata Preparation, and Friday Deadlines

Professor Lantis says updated Zoom recordings are being moved to Kaltura, accessible through Canvas, while he troubleshoots compatibility issues.
The statistics-review do-file repeats the prior code; students should practice opening data, saving do-files, and working in Stata before the first problem set.
The first problem set and quiz are due by end of day Friday; the quiz covers material through confidence intervals and requires LockDown Browser.
Office hours are shifted to 10 a.m.–noon the next day, with virtual office hours Friday morning for last-minute questions.

3:30 Null and Alternative Hypotheses for Population Means

A hypothesis test asks whether a sample mean is plausible if the population mean equals an assumed value, such as 100.
For a one-sided test of whether the mean is greater than a value, the null represents the opposite direction; two-sided tests ask whether the mean differs in either direction.
The sample mean varies across samples according to sample size and the underlying data’s variation.

7:30 How Sample Size Changes the Test Statistic

The test statistic standardizes the difference between the observed sample mean and the hypothesized population mean by the standard deviation of sample means.
The standard deviation of sample means shrinks as sample size increases, so the same difference from the hypothesized mean produces a larger-magnitude test statistic.
A larger test statistic means the observed sample mean is farther from the null value in standard-deviation units, which can count as stronger evidence against the null.

12:00 Critical Values and the Standard Normal Approximation

A critical value marks a cutoff: test statistics farther into the relevant tail fall in the rejection region.
For the course’s samples larger than 30, Professor Lantis uses standard-normal critical values as an approximation to the Student t distribution.
A right-tail critical value of about 1.65 leaves 0.05 probability in the upper tail; left-tail cutoffs have the corresponding negative sign.

17:24 P-Values, Alpha, and the Rejection Decision

A p-value is the probability, assuming the null is true, of observing the test statistic found or one more extreme in the relevant direction.
Alpha is the chosen rejection threshold; common levels are 0.10, 0.05, and 0.01, corresponding to 90%, 95%, and 99% confidence.
The critical-value and p-value methods give the same decision: reject when the test statistic enters the rejection region or when p-value is below alpha.

21:42 Four Steps for a Hypothesis Test

Specify the null and alternative hypotheses, choosing a left-tail, right-tail, or two-tail test to match the question.
Choose alpha, calculate the test statistic from the sample, and compare it with the critical value.
Alternatively, compare the p-value with alpha; a p-value below alpha means the result is sufficiently unlikely under the null to reject it.

23:23 Two-Tailed Tests Split Alpha Across Both Ends

A two-tailed test treats equally distant results above and below the null mean as evidence against the null—for example, sample means of 110 and 90 when testing a mean of 100.
The total alpha is divided between the two tails, so each tail contains alpha divided by two and a one-tail p-value must be doubled.
For the same overall alpha, two-tailed critical values are farther from zero than one-tailed values because each tail must contain less probability.

28:14 Type I and Type II Errors in Hypothesis Testing

A Type I error is rejecting a true null hypothesis; its probability is alpha, the significance level chosen before the test.
A Type II error is failing to reject a false null—for example, treating a sample mean near 101 as support for a null of 100 when the true mean is 105.
Professor Lantis says the course will emphasize Type I errors and alpha rather than calculating Type II error probabilities.

31:04 Building a 95% Confidence Interval Around a Sample Mean

A 95% confidence interval leaves 5% outside its bounds, split into 2.5% in each tail for a two-sided interval.
The standard-normal cutoff is approximately 1.96, so the interval extends 1.96 standard errors below and above the sample mean.
The standard error of the sample mean is based on the sample variance divided by sample size, with the square root taken to obtain the standard deviation.

36:55 Why Confidence Intervals Match Two-Tailed Tests

The interval is centered on the observed sample mean because the unknown population mean cannot serve as its center.
For a 95% interval, a hypothesized population mean outside the bounds is rejected by the corresponding two-tailed test at alpha 0.05.
If the hypothesized mean lies inside the interval, the test fails to reject it; the sample mean itself is always at the center of its interval.

44:27 Margin of Error, Confidence Level, and Sample Size

The margin of error is the amount added to and subtracted from the sample mean to form the interval’s upper and lower bounds.
Increasing confidence from 95% to 99% requires a more extreme critical value, widening the interval and making rejection harder.
Increasing sample size lowers the standard error and narrows the interval at a fixed confidence level, making it easier to rule out values outside the narrower range.

47:46 Extending the Test to Differences Between Two Means

The same hypothesis-testing and confidence-interval procedures apply when the statistic is the difference between two sample means.
The variance of the difference is formed from the variances of the two sample statistics, so its test statistic uses the standard error of that difference.
After calculating the appropriate statistic, the critical-value comparison or p-value-versus-alpha decision works as before.

49:19 Practice Quiz: Confidence Levels, P-Values, and Intervals

Students open the in-class quiz using access code 52.16 and practice interpreting a two-tailed test of whether a population mean equals 50.
Rejecting at 95% implies rejection at 90%, but does not establish rejection at 99%; the 99% critical value is more demanding.
A two-tailed rejection does not reveal whether a one-tailed test would reject unless the direction of the sample result is known.
For a p-value of 0.075, reject at the 90% level but fail to reject at 95% and 99%; the corresponding higher-confidence intervals include the null value.

1:00:45 Importing Election Data and Comparing Unemployment Groups in Stata

For an Excel file, use Stata’s File > Import rather than File > Open, select the option treating the first row as variable names, and set the global path to the folder containing the data.
The election dataset includes county unemployment and 2020 Trump vote share; Professor Lantis demonstrates renaming variables, plotting unemployment with `hist`, and generating a high-unemployment indicator equal to 1 above the mean.
A second low-unemployment indicator illustrates why comparison operators matter: values exactly equal to the mean can be coded as 0 in both groups if the inequalities are not complementary.
Conditional histograms and `bysort` summaries show that counties with low unemployment had a higher average Trump vote share in this initial comparison; the analysis is descriptive and leads into testing a difference in means.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Lantis.