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5-1 A HYPOTHESIS TEST FOR A POPULATION PROPORTION GETTING STARTED LEARNING GOALS Discern between binomial (categorical) and quantitative data. Test hypotheses about a population proportion. Assess hypothesis test conclusions in terms of practical significance and possible errors. Procedures in Module 4 were for quantitative data. If we collect categorical data, those formulas do not apply. Categorical data records the category for each case, like eye color or make of car. Binomial data are categorical data that meet four specific conditions. 1. Binary. Only two possible options, like yes or no, referred to as success or failure. 2. Independent. Observations are recorded independently, i.e., a random sample. 3. Number in the sample set in advance (sample size fixed in advance 4. Success probability constant. This means that if you select cases at random from the popula- tion, the probability of a "success" is the same for each selection. This is considered satisfied if in addition to random sampling, the population is much larger than the sample. The acronym BINS may help you remember these four conditions. If 1200 registered voters are randomly selected and asked if they would vote for a particular candidate, then x, the count of yes votes (successes), is a binomial random variable since there are only 2 options; yes, they plan to vote for this candidate or no, they don't; the voters are random sample; the sample size was set at 1200; and the population is much larger than the sample (We will use the rule of thumb population size at least 10 times the sample size.) If we have binomial data and count the number of successes, x, in a sample of size n, the sample proportion of successes is _ count of successes in the sample sample size If 720 of the 1200 voters selected say that they plan to vote for Candidate A, then the sample proportion of successes is p =- 720 10 = 0.6. 188 Learning Through Practice: Statistical Reasoning5-1 A Hypothesis Test for a Population Proportion: Getting Started the setting is binomial (BINS) and the sample size is large (# successes & failures both 2 10) then the sampling distribution of p (the distribution of all possible values of p) is approximately normal centered at p, the population proportion of successes, with standard deviation VP( - P) CAUTIONS Caution 1. A test conclusion to reject or fail to reject the null hypothesis is a statement about whether the evidence supports the alternative hypothesis or not. It is not proof of anything, and the conclusion could be wrong, which is a type I or type II error A type I error occurs if the null hypothesis is rejected but it is indeed true. If the significance level is 0.05 and we reject the null hypothesis, there is a 5% chance of a type I error. A type II error occurs if the null hypothesis is not rejected but the alternative is indeed true. If we consistently follow best practices, the chance of an error is limited. Caution 2. Each of the statistical procedures and formulas only applies in specific circum- stances. Before doing a procedure or interpreting results, determine: Is the data type categorical or quantitative? Does the question require a hypothesis test or a confidence interval (or neither)? Is the question about one group or two? So far, methods are for one group only. Are conditions met to use the chosen procedure Caution 3. A result can be statistically significant but not practically significant, which means that it may not matter in real life. This sometimes occurs with very large sample sizes. Hypoth- esis tests determine the extent that data supports an alternative hypothesis The test statistics we have calculated have the following form. Test statistic = (sample statistic - population parameter) (standard error of statistic) Hypotheses are about the population and are decided and written before data is collected They should contain no information provided by the sample Hypotheses must state the population parameter being tested. We tested u (a population mean) in Module 4, and now we'll test p (a population proportion). Learning Through Practice: Statistical Reasoning 1895-1 A HYPOTHESIS TEST FOR A POPULATION PROPORTION PRE-ACTIVITY 1. Which of the following are true? Select all that apply. a. If the collected data is categorical, it must be a binomial setting. b. If the collected data is categorical, it may be a binomial setting. c. If the collected data is quantitative, it can't be a binomial setting. 2. Decide whether each of the following settings is binomial. If so, verify all conditions, and if not, explain which condition is not satisfied. a. 100 Boise State students are randomly selected and their major is recorded. b. 100 Boise State students are randomly selected and asked whether they support a par- ticular candidate. c. A student asks 25 randomly selected students from their statistics class if they use tutor- ing services. d. From the medical records of a large multi-state health care organization, patients are randomly selected until 45 patients with a particular medical condition are obtained. 190 Learning Through Practice: Statistical Reasoning