Interference for One Proportion Concepts in Statistics Research

Interference for One Proportion Concepts in Statistics

Research Questions Types of research questions we focus on in this module are bolded below. Type of Question Make an estimate about the population Test a claim about the population Compare two populations Examples Variable Type Unit What proportion of all U. S. adults support the death penalty? What is the average number of hours that community college students work each week? Do the majority of community college students qualify for federal student loans? Has the average birth weight in a town decreased from 3, 500 grams? Are teenage girls more likely to suffer from depression than teenage boys? In community colleges do female students have a higher average GPA than male students? Categorical variable Inference for One Proportion Quantitative variable Inference for Means Categorical variable Inference for One Proportion Quantitative variable Categorical variable Quantitative variable Inference for Means Inference for Two Proportions Inference for Means

Big Picture

Estimating a Population Proportion • When we select a random sample from the population of interest, we expect the sample proportion to be a good estimate of the population proportion. We expect some error. • For a given sample proportion, we will not know the amount of error, so we use the standard error as an estimate for the average amount of error we expect in sample proportions. • If a normal model is a good fit for the sampling distribution, then about 95% of sample proportions estimate the population proportion within 2 standard errors. We say that we are 95% confident that the following interval contains the population proportion.

Population Proportion •

Confidence Interval Formula •

Confidence Interval •

Hypothesis Testing In inference, we use a sample to draw a conclusion about a population. Two types of inference are the focus of our work in this course: • Estimate a population parameter with a confidence interval. • Test a claim about a population parameter with a hypothesis test. We can also use samples from two populations to compare those populations. In this situation, the two types of inference focus on differences in the parameters. • Estimate a difference in population parameters with a confidence interval. • Test a claim about a difference in population parameters with a hypothesis test.

Forming Hypotheses •

Hypothesis Testing General observations about null and alternative hypotheses: • • • The hypotheses are competing claims about the parameter or the comparison of parameters. Both hypotheses are statements about the same population parameter or same two population parameters. The null hypothesis contains an equal sign. The alternative hypothesis is always an inequality statement. It contains a “less than” or a “greater than” or a “not equal to” symbol. In a statistical investigation, we determine the research question, and thus the hypotheses, before we collect data.

Hypothesis Testing

Statistical Significance The P-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming the null hypothesis is true. A small P-value indicates that it’s unlikely the actual sample data came from the population described by the null hypothesis. We often compare the P-value to 0. 05. We reject the null hypothesis in favor of the alternative if the P-value is less than (or equal to) 0. 05. When the P-value is less than (or equal to) 0. 05, we also say that the difference between the actual sample statistic and the assumed parameter value is statistically significant.

Statistical Significance •

Statistical Significance •

Hypothesis Testing What Can Go Wrong: Two Types of Errors Statistical investigations involve making decisions in the face of uncertainty. In hypothesis testing, two types of wrong decisions can occur. • If the null hypothesis is true, but we reject it, the error is a type I error • If the null hypothesis is false, but we fail to reject it, the error is a type II error The following table summarizes type I and II errors:

What Is the Probability We Will Make a Type I Error •

What Is the Probability We Will Make a Type II Error The probability of a type I error, if the null hypothesis is true, is equal to the significance level. The probability of a type II error is much more complicated to calculate. We can reduce the risk of a type I error by using a lower significance level. The best way to reduce the risk of a type II error is by increasing the sample size.

Hypothesis Summary •

Hypothesis Test for a Population Proportion •

Important Note A hypothesis test can be onetailed or two-tailed. The previous example was a one-tailed hypothesis test. The P-value was the area of the right tail. If the inequality in the alternative hypothesis is < or >, the test is one-tailed. If the inequality is ≠, the test is two-tailed.

Quick Review • • What does a confidence interval estimate? What is a null hypothesis? What procedure do you use when our goal is to estimate a population proportion? What is a type I error in hypothesis testing? What is the formula used to calculate the confidence interval for a population proportion? What is the first step when hypothesis testing for a population proportion? What inference procedure do you use when our goal is to test a claim about a population proportion?