Exam 15: Learning From Categorical Data

The formulas for the degrees of freedom for the chi-squared goodness-of-fit test and the chi-squared test of the independence of two variables arethe same.

The formulas for the expected cell counts used in the chi-squared test ofhomogeneity of proportions and the chi-squared test of the independenceof two variables are the same.

In order to decide whether the observed data is compatible with the nullhypothesis, the observed cell counts are compared to the cell counts thatwould be expected when the alternative hypothesis is true.

The popular game of golf has potential ecological consequences due to the impact ofgolf courses: habitat fragmentation, chemical pollution from pesticides, and loss ofnative vegetation. Eastern bluebirds are particularly attracted to "birdhouses" suchas those found on golf courses. Investigators monitored the reproductive success ofthese birds in birdhouses at nine golf courses and ten similar birdhouses at non-golfsites. They reasoned that if nest boxes on golf courses are less attractive to bluebirds,this would show up as a difference in the distribution of occupancy frequenciesbetween golf sites and non-golf sites. Data on nests in birdhouses occupied only bybluebirds are presented in the table below. a) Assuming no difference in the distribution of number of nests per birdhouse forgolf and non-golf locations, calculate the expected counts needed to complete thecells of table below that are marked by XXX. b) The value of Test the appropriate hypotheses using a significancelevel of 0.05. Does it appear that the birdhouses on golf courses are less attractiveto bluebirds?

As every airline passenger knows, there are never enough armrests! Over the courseof 20 flights occurring over a variety of weekdays, nights, and weekends, researchersselected a random sample of passengers who had been seated next to an individual ofthe 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 interviewedfrom a seat-mate pair, and couples were excluded from the survey. The question ofinterest was whether males and females are equally bothered by their opposite genderneighbor's use of the common armrest. The table below summarizes data from theinterviews. The indicated gender is the gender of the person being interviewed. a) What is the appropriate null hypothesis for this investigation? (You may state thishypothesis in ordinary English if you wish.) b) Using the null hypothesis from part (a), test the appropriate hypotheses using achi-square procedure.

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

The expected cell count for the row a and column b entry in a two-waytable is equal to the product of the row a and column b marginal totals.

Male horseshoe crabs (Limulus polyphemus) on a beach can become overturned bywaves approaching the beach. These "stranded" males must right themselves or theywill face death by desiccation (drying out) or fall prey to predators such as sea gulls.Over a period of many days on a South Carolina beach, investigators categorized arandom sample of crabs as either stranded or not and additionally noted their agecategory. The researchers wished to determine if there was convincing evidence ofan association between strandedness and age. a) What is the appropriate null hypothesis for this investigation? (You may state thishypothesis in ordinary English if you wish.) b) Using the null hypothesis from part (a), test the appropriate hypotheses using achi-square procedure.

The arithmetic calculations for the test of homogeneity and the test of independenceare the same. a) Explain in a few sentences what distinguishes these two hypothesis tests. b) To illustrate your answer in part (a), give an example of a research question thatwould require a test of homogeneity, and an example that would require a test ofindependence. Briefly describe the aspects of your examples that illustrate thedistinguishing characteristics you mentioned in part (a).

The National Park Service, as part of their educational outreach, writes materials forstudents to use while in the parks. In a study of the effectiveness of some of thesematerials, a random sample of students was selected to take a short quiz about oaktrees based on these materials. A random sample of professionals also took the quiz.Investigators compared classifications of the "crown shapes" - the general shapes ofthe leafy parts of the trees -- made by students in grades 6 - 12 with classificationsmade by professionals.The observed and expected counts are summarized in the table below along with thevalue of the chi-square statistic and the associated P-value. a) State the null and alternative hypotheses for this test. b) Discuss whether the conditions for a chi-square inference procedure are met forthese data. c) What should the investigators conclude based on the results of the chi-square test? d) It is possible that an error (Type I or Type II) could occur. Which error might theinvestigators make, based on your answer in part (c)? Describe this error in thecontext of the question.

Suppose a researcher is studying the purchasing behavior of individuals buying dogfood. He hypothesizes that half the dog food purchases will be the big name brand,"Lassie Fritters" and that the remaining purchases will be divided equally betweentwo local brands, "Bow Wow Chow" and "Woof Woof Wafers." a) What are the null and alternative hypotheses that would be used to decide if therewas convincing evidence against the researcher's hypothesized distribution ofpurchases across the three brands? b) Suppose that each individual in a random sample of 200 purchasers providesinformation about his or her choice of brand. What are the expected counts forthe three brand categories? c) How many degrees of freedom are associated with the chi-squared goodness-of-fitstatistic? d)

Cliffs are relatively inaccessible to humans and livestock, and thus represent idealsites for birds to nest. In a study of cliff bird communities at Joshua Tree NationalPark in California, the nesting and flying behavior of cliff-dwelling birds wasrecorded by researchers. One variable of interest was the distribution of flyingdistances from a nest to a cliff faces popularly used for rock climbing. Flyingdistances from cliffs rarely used for rock climbing tend to be uniformly distributed inan interval from 0 to 50 meters. The researchers were interested to see if cliffclimbing might affect the flying distance behavior. The table below presents asummary of the observations of distances from 90 randomly selected nests to thenearest cliff used for rock climbing. a) If the flying distances from cliffs used for rock climbing are also uniformlydistributed, the expected numbers of observations for these distance categories isthe same for each category. What is this expected value? b) For these data, . Is this sufficient evidence at the 0.05 level toconclude that the distribution of distance categories is not uniform? Provideappropriate statistical evidence for your conclusion.c) Based on the data and your results in part (b), write a few sentences describingyour inferences about the distribution of distances from nests to cliffs used forclimbing.

Non-native speakers of English often have "accents" that differ, depending on thespeaker's first language. Does written English have similar "accents? " In a study ofChinese, Malay, and Indian children studying in Singapore, characteristics of theirwriting were investigated for differences. One such characteristic was a rounding atthe top of the capital letter "A." Random samples of the writings of Malay, Indian andChinese native speakers were used to create the table below. a) Assuming no difference in the proportions with and without top rounding for thethree countries, calculate the expected counts for the cells in the table below thatare indicated by XXX . b)

In the United States, professional baseball determines the winner of the year'scompetition by playing a series of games known as the World Series. The first teamto win four out of seven games is declared the winner. A sportscaster believes that bythe time the World Series comes around, it is reasonable to suppose that on averagethe competing teams are equally matched, and that the probability is 0.5 of eitherteam winning any one game. Others disagree with this theory. The table belowcontains the number of times from 1903 - 2010 that the series lasted 4, 5, 6, or 7games, as well as the probabilities associated with those outcomes if the sportscasteris correct. Do the data provide sufficient evidence at the .05 level that the sportscaster's belief isincorrect?

The row and column marginal totals provide information on the

"Snoqualmie" is a name shared by a spectacular 268-foot waterfall and a tribe ofNative Americans. The Falls provides a place for the people to worship. In a studyof the significance of the Falls to modern tribal members, surveys of two groups wereperformed. One group consisted of a random sample of members living within thetraditional tribal area, a circle with an approximate radius of 25 miles. The othergroup consisted of a random sample of members living outside this area. Theinvestigators asked questions such as whether a sense of personal renewal was felt atthe Falls, or whether personal healing/cleansing was experienced.The investigators combined responses to the various questions to make a "level ofspiritual encounter index." A chi-square test was used to determine if the distributionsof level of spiritual encounter differs for the two groups. The observed and expectedcounts are summarized in the table below along with the value of the chi-squarestatistic and the associated P-value. a) State the null and alternative hypotheses for this test. b) Discuss whether the conditions for a chi-square inference procedure are met forthese data. c) What should the investigators conclude based on the results of the chi-square test? d) It is possible that an error (Type I or Type II) could occur. Which error might theinvestigators make, based on your answer in part (c)? Describe this error in thecontext of the question.