Exam 14: Learning From Categorical Data

The row and column marginal totals provide information on the distribution of the observed values for each of the two variables defining the contingency table.

As every airline passenger knows, there are never enough armrests! Over the course of 20 flights occurring over a variety of weekdays, nights, and weekends, researchers selected a random sample of passengers that had been seated next to individuals of opposite gender. The passengers were surveyed as they left the boarding area. The researchers were interested in the level of agitation felt when their "seat-mate" used the common armrest. Only one person was randomly selected to be interviewed from a seat-mate pair, and couples were excluded from the survey. The question of interest was whether males and females are equally bothered by their opposite gender's use of the common armrest. The table below summarizes data gathered from interviewing the passengers. ​ a)What is the appropriate null hypothesis for this investigation? (You may state this hypothesis in ordinary English if you wish.)b)Using the hypothesis from part (a), test the null hypothesis using the appropriate chi-square procedure.

Scandinavian researchers investigated the effects of hunting male bears on succeeding generations. (The effects of hunting females on successive generations are well documented.) They hypothesized that when a male is killed a new male will move in, and maximize his reproductive success by killing existing cubs (so that the females will conceive sooner.) A random sample of radio tagged females was monitored and data on the number of bear cubs surviving 1.5 years were gathered. In the southern area, hunters killed bears legally; in the northern area, no hunting took place. The data on cub survival in this sample are reproduced below. The researcher suspects an association between the location of the females and cub survival. ​ a)What is the appropriate null hypothesis for this investigation? (You may state this hypothesis in ordinary English if you wish.)b)Using the hypothesis from part (a), test the null hypothesis using the appropriate chi-square procedure.

Each person in a random sample of 105 ice cream eaters was asked to "vote for" a preferred flavor from the flavors listed in the table below. ​ ​ ​ What is the value of the test statistic for the hypothesis that, given these flavor choices, ice cream eaters have no flavor preference? ​ ​

Each person in a random sample of 54 students living in the dorms at a large university was asked whether he or she preferred to study in the dorm room, in the dorm's study lounge, or at the university library. The resulting data is summarized in the table below. In a hypothesis test to determine if there is evidence that the three locations are not equally preferred by students at the university who live in the dorms, what is the expected count for the library category?

Some people believe that criminals who plead guilty will, on average, get lighter sentences. The following table summarizes data from a random sample of San Francisco defendants in burglary cases. If the null hypothesis is that the plea and the sentencing fate are independent, perform the calculations to find the expected number of individuals who plead not guilty and are sent to prison. Show your work.

The chi-squared test statistic for testing independence in a two-way tables has rc − 1 degrees of freedom.

Black bears (Ursus americanus), and their use of National Parks can be a problem for humans. Bear travels are governed by the need for food, as well as a high degree of natural curiosity. Some humans inadvertently or purposefully feed the bears, leading them to harass the human population. When bears become troublesome, one strategy is to transplant them--that is, take them far from home and hope they don't find their way back. In a study of the success of this transplantation strategy, researchers kept track of the troublesome bears' fate over the course of a 10-year period of transplantations. One question of interest was whether the strategy worked equally well for male and female bears. The relevant data are presented in the table below. Test the hypothesis that the transplant outcome and the gender of the population of troublesome bears are independent. For purposes of this question, you may assume that the captured bears are a random sample of the troublesome bear population.

The chi-squared test statistic, X², measures the extent to which the observed cell counts differ from those expected when H₀ is true.

Exhibit 11-1 In baseball, hits can be one of four types: singles, doubles, triples, or home runs. In the last baseball season, 64% of the hits were singles, 20% were doubles, 2% were triples, and 14% were home runs. Spring training games are played in March before the regular season to help the players get in shape. A sports writer thinks that perhaps the distribution of types of hits is different in these games. The sportswriter gathers a random sample of 251 hits from spring training games. Here is the distribution of types of hits for these games: -Refer to Exhibit 11-1. At the .05 level of significance, test the hypothesis that the proportions of the different types of hits in the spring is the same as for last year's season.

The authorship of ancient writings is frequently in dispute. One method for judging authorship of writings from the Classical Greek period is to analyze the proportion of sentences containing the word, γα'ρ. (γα'ρ is an article, something like "a," "an," and "the" in English.) If a particular collection of works has markedly different frequencies of use of γα'ρ, this would be considered evidence against the same author having written all the works. In the table below, data from random samples of 200 sentences each from 4 works are presented. The alleged author is one Philo Judaeus , a philosopher writing in ancient Alexandria, Egypt. ​ a)At the .05 level of significance, test the hypothesis that the frequencies of the use of γα'ρ is the same for these 4 works. b)Write a short paragraph that could be added to a history or Classics Studies textbook that explains your results. Since your audience cannot be assumed to know any statistics, you must explain your conclusions and reasoning in, so to speak, "plain English."

A random sample of 120 sixth-graders were asked to pick their preference from five different flavors of ice cream. the results are show below. A hypothesis test will be carried out to determine if there is evidence that the five flavors are not equally preferred. The test statistic will have a chi-square distribution with how many degrees of freedom (df)?

What is the approximate P-value for and df = 5? ​

Identify situation when the chi-square goodness-of-fit test is appropriate.

For the chi-squared goodness-of-fit chi-squared test, the associated P-value is the area under the appropriate chi-squared curve to the left of the calculated value of X².

Suppose that we are studying the purchasing behavior of individuals buying breakfast oatmeal. We hypothesize that half the cereal purchases will be the big name brand, "Seabiscuit Oats" and that the remaining purchases will be divided equally between two local brands, "Secretariat Oats" and "Whirlaway Oats." a)Define appropriate statistical variables, and use them to state the null and alternative hypothesis that would be used to decide if there was convincing evidence against the hypothesized distribution of purchases across the three brands. b)Suppose that each individual in a random sample of 200 purchasers provides information about his or her choice of brand. For each category, what are the expected values? c)How many degrees of freedom are associated with the chi-squared goodness-of-fit statistic? d)Suppose that X² = 6.10. What can be said about the P-value for this hypothesis test?

A chi-squared goodness-of-fit can be used to test hypotheses about the proportion of the population falling into each of the possible categories.

Black bears (Ursus americanus), and their use of National Parks can be a problem for humans. Bear travels are governed by the need for food, as well as a high degree of natural curiosity. Some humans inadvertently or purposefully feed the bears, leading them to harass the human population. When bears become troublesome, one strategy is to transplant them--that is, take them far from home and hope they don't find their way back. In a study of the success of this transplantation strategy, researchers kept track of the troublesome bears' fate over the course of a 10-year period of transplantations. One question of interest was whether the strategy worked equally well for "experienced" and "inexperienced" bears. "Experienced" bears are bears being transplanted a second time after wandering back and being troublesome again. The relevant data are presented in the table below. Test the hypothesis that the transplant outcome and the "experience level" of the population of troublesome bears are independent. For purposes of this question, you may assume that the captured bears are a random sample of the troublesome bear population.

Exhibit 11-1 In baseball, hits can be one of four types: singles, doubles, triples, or home runs. In the last baseball season, 64% of the hits were singles, 20% were doubles, 2% were triples, and 14% were home runs. Spring training games are played in March before the regular season to help the players get in shape. A sports writer thinks that perhaps the distribution of types of hits is different in these games. The sportswriter gathers a random sample of 251 hits from spring training games. Here is the distribution of types of hits for these games: -Refer to Exhibit 11-1. Write a short paragraph that could be added to a popular baseball magazine that explains your results. Since your audience cannot be assumed to know any statistics, you must explain your conclusions and reasoning in "plain English."

Suppose that we are studying the purchasing behavior of individuals buying dog food. We hypothesize that half the dog food purchases will be the big name brand, "Lassie Chow" and that the remaining purchases will be divided equally between two local brands, "Bow Wow Chow" and "Woof Woof Wafers." a)Define appropriate statistical variables, and use them to state the null and alternative hypothesis that would be used to decide if there was convincing evidence against the hypothesized distribution of purchases across the three brands. b)Suppose that each individual in a random sample of 200 purchasers provides information about his or her choice of brand. For each category, what are the expected values? c)How many degrees of freedom are associated with the chi-squared goodness-of-fit statistic? d)Suppose that X² = 6.10. What can be said about the P-value for this hypothesis test?