Local Instructional Theory Topik Dua Tipe Kesalahan dalam Pengujian Hipotesis

Two types of errors are one of the important topics in hypothesis testing. Studying the two types of errors is not only studying how the procedure determines the probability of making an error, but it is very important to study the theoretical concepts of the two types of errors. To achieve this goal, a learning design is needed that can facilitate students to construct their own concepts of the two types of errors. The learning design developed is local instructional theory resulting from the cyclic process of hypothetical learning trajectory. The type of research used is design research using the model developed by Gravemeijer and Cobb. The test subjects used in this study were students of the Ma thematics Education Study Program, FMIPA UNP who took the Elementary Statistics Course in the July – December 2019 semester. This research resulted in a very practical local instructional theory used to facilitate students in carrying out horizontal and vertical mathematization processes, so that students are able to construct their own concepts of two types of errors in hypothesis testing.

The Composite Covariance of a Kalman Filter Estimation

A Kalman filter estimation of the state of a system is merely a random vector that has a normal, also called Gaussian, distribution. Elementary statistics teaches any Gaussian distribution is completely and uniquely characterized by its mean and covariance (variance if univariate). Such characterization is required for statistical inference problems on a Gaussian random vector. This mean and composite covariance of a Kalman filter estimate of a system state will be derived here. The derived covariance is in recursive form. One must not confuse it with the “error covariance” output of a Kalman filter. Potential applications, including geological ones, of the derivation are described and illustrated with a simple example.

The Optimal Correlation Detector?

Correlation detectors are now used routinely in seismology to detect occurrences of signals bearing close resemblance to a reference waveform. They facilitate the detection of low-amplitude signals in significant background noise that may elude detection using energy detectors, and they associate a detected signal with a source location. Many seismologists use the fully normalized correlation coefficient $C$ between the template and incoming data to determine a detection. This is in contrast to other fields with a longer tradition for matched filter detection where the theoretically optimal statistic $C^2$ is typical. We perform a systematic comparison between the detection statistics $C$ and $C|C|$, the latter having the same dynamic range as $C^2$ but differentiating between correlation and anti-correlation. Using a database of short waveform segments, each containing the signal on a 3-component seismometer from one of 51 closely spaced explosions, we attempt to detect P- and S- phase arrivals for all events using short waveform templates from each explosion as a reference event. We present empirical statistics of both $C$ and $C|C|$ traces and demonstrate that $C|C|$ detects confidently a higher proportion of the signals than $C$ without evidently increasing the likelihood of triggering erroneously. We recall from elementary statistics that $C^2$, also called the coefficient of determination, represents the fraction of the variance of one variable which can be explained by another variable. This means that the fraction of a segment of our incoming data that could be explained by our signal template decreases almost linearly with $C|C|$ but diminishes more rapidly as $C$ decreases. In most situations, replacing $C$ with $C|C|$ in operational correlation detectors may improve the detection sensitivity without hurting the performance-gain obtained through network stacking. It may also allow a better comparison between single-template correlation detectors and higher order multiple-template subspace detectors which, by definition, already apply an optimal detection statistic.

Confidence Intervals of COVID-19 Vaccine Efficacy Rates

This tutorial uses publicly available data from drug makers and the Food and Drug Administration to guide learners to estimate the confidence intervals of COVID-19 vaccine efficacy rates with a Bayesian framework. Under the classical approach, there is no probability associated with a parameter, and the meaning of confidence intervals can be misconstrued by inexperienced students. With Bayesian statistics, one can find the posterior probability distribution of an unknown parameter, and state the probability of vaccine efficacy rate, which makes the communication of uncertainty more flexible. We use a hypothetical example and a real baseball example to guide readers to learn the beta-binomial model before analyzing the clinical trial data. This note is designed to be accessible for lower-level college students with elementary statistics and elementary algebra skills.

Using COVID-19 Vaccine Efficacy Data to Teach One-Sample Hypothesis Testing

In late November 2020, there was a flurry of media coverage of two companies’ claims of 95% efficacy rates of newly developed COVID-19 vaccines, but information about the confidence interval was not reported. This paper presents a way of teaching the concept of hypothesis testing and the construction of confidence intervals using numbers announced by the drug makers Pfizer and Moderna publicized by the media. Instead of a two-sample test or more complicated statistical models, we use the elementary one-proportion z-test to analyze the data. The method is designed to be accessible for students who have only taken a one-semester elementary statistics course. We will justify the use of a z-distribution as an approximation for the confidence interval of the efficacy rate. Bayes’s rule will be applied to relate the probability of being in the vaccine group among the volunteers who were infected by COVID-19 to the more consequential probability of being infected by COVID-19 given that the person is vaccinated.

Attitudes toward Using and Teaching Confidence Intervals: A Latent Profile Analysis on Elementary Statistics Instructors

The use of confidence intervals (CIs) for making a statistical inference is gaining popularity in research communities. To evaluate college statistics instructors’ readiness to teach CIs, this study explores their attitudes toward teaching CIs in elementary statistics courses, and toward using CIs in inferential statistics. Data were collected with a survey that classifies instructors’ attitudes on the basis of three previously established pedagogical components: affective, cognitive, and behavioral. Based on the survey responses from 270 participants, we created three profiles (subgroups) via latent profile analysis, and identified each profile’s pattern of attitudes toward CIs and common characteristics of the instructors that fit each profile. In addition, we compared the profiles across groupings created by six variables: gender, academic background, statistics teaching experience, subject preference, degree level, and desire to improve teaching. The results of the latent profile analysis support three profiles within the population of statistics instructors, and the results of the comparative analysis of teacher characteristics indicate that the six variables are moderate to strong predictors of the grouping of the sample into three profiles.