Elementary Statistics Lesson #23A

The instructor welcomes students to the last lesson of the semester, lesson 23, and introduces a change in the lesson plan. Instead of the usual hypothesis tests for population means or proportions, the focus shifts to a final hypothesis test known as the goodness of fit test. This test utilizes the chi-square distribution model, which is new to the students. The lesson includes a quick review of t-tests and z-tests, explaining their reliance on the t-distribution and standard normal z-distribution respectively. The chi-square test is introduced for qualitative or categorical data, and the instructor promises to delve into the chi-square distribution's characteristics and how it differs from the t-distribution.

The instructor provides an overview of the chi-square distribution, highlighting its similarities to and differences from the t-distribution. Key points include the distribution's right skewness, especially with smaller degrees of freedom, and the fact that chi-square values are always non-negative. The lesson explains that the chi-square distribution is used to determine p-values in hypothesis testing. The instructor introduces the concept of degrees of freedom in the context of the chi-square distribution and shows different chi-square distributions for various degrees of freedom. The importance of using a chi-square table or calculator to find critical values or p-values is emphasized.

The instructor presents a practical example of a goodness of fit test using a dice-rolling scenario. A gambler wants to test if a die is fair by rolling it 60 times and recording the outcomes. The null hypothesis posits that each face of the die has an equal probability of 1/6. The alternate hypothesis suggests that the die is not fair. The instructor explains the process of calculating expected counts based on the null hypothesis and compares them with the observed counts. The chi-square test statistic is introduced as a method to quantify the difference between observed and expected frequencies.

The lesson continues with the dice example, explaining how to calculate the chi-square test statistic. The formula for the test statistic is presented, and the calculation involves taking the squared difference between observed and expected counts, dividing by the expected count, and summing these values across all categories. The degrees of freedom for the test are explained as one less than the number of categories. The instructor emphasizes the importance of ensuring that the sample size is large enough and that the expected counts meet certain conditions for the chi-square test to be valid.

The instructor demonstrates the computation of the chi-square test statistic for the dice example. The observed counts for each face of the die are compared with the expected counts, and the differences are squared and weighted by the expected counts. The sum of these values gives the test statistic. The process is illustrated step by step, and the resulting test statistic is used as part of the hypothesis test to determine if there is enough evidence to reject the null hypothesis that the die is fair.

The instructor outlines the steps involved in a goodness of fit hypothesis test, using a template that includes setting up null and alternate hypotheses, determining the significance level, calculating expected counts, computing the test statistic, and making a decision based on the p-value or critical value. The template is applied to the dice example, with the test statistic being compared to a critical value from the chi-square distribution to decide whether to reject the null hypothesis. The p-value approach and critical value approach are both discussed.

The lesson moves on to another example involving FBI crime data from 2012, where the proportions of different types of violent crimes in the U.S. are compared to those in California. The null hypothesis is that the proportions in California match the U.S. proportions, while the alternate hypothesis suggests at least one category differs. The instructor guides through the calculation of expected counts based on the U.S. proportions and the sample size of 500 crimes in California. The goodness of fit test is set up to determine if the national distribution is a good fit for California's crime data.

The instructor explains how to use a calculator program to compute the chi-square test statistic for the FBI crime data example. The program, named 'Chi-square Goftest', is used to input observed and expected counts and calculate the test statistic and its associated probability. The program's steps are detailed, and the resulting test statistic and p-value are used to make a decision regarding the null hypothesis about the crime proportions.

The final example presented is a Gallup poll from May 2013, which surveyed adult Americans on their views regarding the availability of abortion. The poll results are compared with the proportions of people holding these views in 2010. The four categories of opinion are outlined, and the data is set up for a goodness of fit test to determine if the 2010 proportions are a good fit for the 2013 poll results. The null hypothesis posits that the proportions have remained the same, while the alternate hypothesis suggests a change in public opinion.