ECON 371 Fall 2026 Class Recording 9/3

Overview

Professor Lantis reviews Stata-based standardization and sampling distributions, then develops hypothesis testing using test statistics, critical values, and p-values. Simulations of NBA player heights show how sample-mean variance falls as sample size increases and why samples of roughly 30 or more support a normal approximation; an IQ example then demonstrates rejection decisions at different confidence levels.

Key takeaways

Standardizing a variable with (value − mean) / standard deviation gives it a mean near zero and standard deviation near one, but does not by itself make the distribution normal.
The variance of the sampling distribution of a sample mean is the population variance divided by sample size; with NBA height variance 12.6, a sample size of 10 implies a mean variance of about 1.26.
The Central Limit Theorem explains why sample means become approximately normal for sufficiently large samples, commonly around n = 30 or more, even when the original data are not normal.
A two-tailed test at 95% confidence uses alpha = 0.05 and critical values of approximately ±1.96; reject the null when the absolute test statistic exceeds the critical value.
A p-value is calculated under the assumption that the null hypothesis is true and measures the probability of observing a result at least as extreme as the sample result; it is not the probability that the null itself is true.
For the IQ example, a test statistic of approximately -1.77 permits rejection of a hypothesized mean of 105 at the 10% significance level, but not at the 5% or 1% level.

Chapters

0:00 Course Logistics and a Correction to the Grouped Covariance Example

Professor Lantis cautions that missing class will make it difficult to keep up as ECON 371 moves beyond statistics review.
He corrects the violent-crime grouped-data example: the two values should be 400 and 300.
The first quiz is postponed until after the following Tuesday class and will be due Friday at the end of the day.

3:00 Using Stata to Plot Data and Standardize Variables

Stata do-files use a file path to locate data; keeping course files together avoids repeatedly changing that path.
A scatter plot of state cigarette tax and cigarette cost per pack shows a positive relationship.
A z-score is calculated as a value minus its mean, divided by its standard deviation; Stata can create it manually or with the built-in `egen` standardization function.

10:00 Checking Z-Scores and Reading Standard Normal Probabilities

Summarizing a standardized variable should produce a mean near zero and a standard deviation near one.
Standardizing changes a variable's scale, but does not necessarily make its distribution normal.
A z-score of 1 means an observation is one standard deviation above the mean; standard normal tables give probabilities for z-scores.

15:00 Why Small Samples Use Student’s t Distributions

With small samples, sample statistics vary more, so Student’s t distributions have heavier tails than the standard normal distribution.
The t distribution is symmetric, and its shape depends on sample size; as sample size grows, it becomes closer to the standard normal.
The sample mean is calculated by summing sample observations and dividing by sample size; Professor Lantis distinguishes the sample mean notation, y-bar, from the population mean, mu.

19:00 How Sample Size Changes the Distribution of Sample Means

Across repeated random samples, the expected value of the sample mean equals the population mean.
The variance of the sampling distribution of the mean is the population variance divided by sample size.
Larger samples reduce the spread of possible sample means, making an individual sample mean more informative about the population mean.

25:00 Simulating 1,000 Samples of NBA Player Heights in Stata

Professor Lantis uses NBA player heights as population data, with a mean near 79.1 inches and variance near 12.6.
A Stata program repeatedly samples players, calculates each sample mean, and stores 1,000 means for sample sizes of 10, 30, and 100.
The simulated means stay near 79.1, while their variance decreases as sample size rises; for n = 10, the theoretical variance is about 12.6/10 = 1.26.

30:00 The Central Limit Theorem and Small-Sample Uncertainty

A histogram of the simulated sample means is approximately bell-shaped, illustrating the Central Limit Theorem.
For sample sizes around 30 or greater, the distribution of sample means is often approximated by a normal distribution, even if the original data are not normal.
When population variance is unknown and samples are small, inference relies on Student’s t distribution; smaller samples have heavier tails and make extreme standardized values more likely.

34:00 Null and Alternative Hypotheses for Testing a Population Mean

A hypothesis test starts with a null hypothesis, H₀, representing the assumed population value, and an alternative hypothesis, H₁ or Hₐ, representing the claim being tested.
For a two-sided test of whether IU students’ average IQ differs from 100, the null is that the mean equals 100 and the alternative is that it does not.
One-sided tests instead ask whether the population value differs in a specified direction; the choice depends on the research question.

38:30 Test Statistics Measure Distance from the Assumed Mean

A test statistic expresses how many standard errors the sample mean lies from the mean assumed by the null hypothesis.
Its numerator is the sample mean minus the hypothesized mean; its denominator is the standard error of the sample mean.
When population variance is unknown, the sample variance estimates it, and the standard error is based on the square root of estimated variance divided by sample size.

43:00 Two-Tailed Critical Values and Rejection Regions

For a two-tailed test, the total significance level alpha is split between the two tails of the distribution.
At 95% confidence, alpha is 0.05, with 0.025 in each tail and standard normal critical values of approximately -1.96 and 1.96.
The rejection region consists of test statistics beyond the critical values; for a two-tailed test, compare the absolute test statistic with the positive critical value.

48:00 Using P-Values to Make the Same Decision as Critical Values

A p-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed.
Reject the null when the p-value is below alpha; this decision matches checking whether the test statistic falls in the critical-value rejection region.
The common confidence levels of 90%, 95%, and 99% correspond to significance levels alpha = 0.10, 0.05, and 0.01.

55:00 Interpreting Rejection, Failure to Reject, and Significance

Rejecting a null hypothesis does not establish the sample mean as the true population mean; it rules out the hypothesized value at the selected significance level.
Failing to reject means the sample did not provide strong enough evidence against the null, not that the null has been proven true.
Professor Lantis defines significance level as 100% minus confidence level and outlines the test sequence: state hypotheses, choose alpha, calculate the statistic and p-value, then reject or fail to reject.

1:05:00 In-Class IQ Test Example and Quiz Submission

The class example tests a hypothesized mean IQ of 105 using a sample mean of 102, sample size 50, and stated population variance 144.
The test statistic is approximately -1.77; it falls beyond the 90% two-tailed critical value of about 1.65, but not beyond the 95% value of 1.96.
The result supports rejection at the 10% significance level, but not at 5% or 1%; students access the Canvas quiz with the in-class code 2.13.

1:10:00 Upcoming Confidence Intervals, Quiz Deadline, and Course Support

The next Tuesday class will cover confidence intervals and additional Stata work; confidence intervals are closely related to hypothesis tests.
Professor Lantis says the quiz will be posted after Tuesday’s class and due Friday at the end of the day; practice problems are already available.
Students with connectivity or software problems are advised to use eduroam or campus IT support, and Professor Lantis will post updated office-hour information on Canvas.

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.