Unit 8 Progress Check
Which of the following best describes the shape of the chi-square distribution when the degrees of freedom are less than 10 ?
Answer B Skewed to the right
Which of the following describes a scenario in which a chi-square goodness-of-fit test would be an appropriate procedure to justify the claim?
Answer C A campaign manager would like to show that the distribution of individuals within several social economic categories is different than what a newspaper reported. The campaign manager has a random sample of potential voters in a large city and records the number of individuals within each of the categories.
A study was conducted to investigate whether there are regional differences in peanut butter preference in the United States. The country was divided into 7 geographic regions, and in each region a random sample of peanut butter eaters were asked whether they preferred creamy or crunchy peanut butter. The responses were summarized in a 7-by-2 table of counts for each combination of geographic region and creamy or crunchy peanut butter preference. Which of the following tests is the most appropriate for the investigation?
Answer C A chi-square test of homogeneity
Jana, a high school principal, hosted a movie event at her school. Jana's assistant kept track of the number of students in each grade who attended the event. The distribution shown in the table represents the number of students in each grade that were present. Jana knows that the grade levels are equally distributed across the school of 1,200 students. She would like to use a chi-square test to see if the proportion of individuals in each class at the movie are also equally distributed. How many seniors would be expected at the event?
Answer E 59.5
Juvenile ground squirrels respond to predators by making "alarm calls" that can alert others to danger. A biologist conducted a study to investigate how squirrels respond to these alarm calls: run to burrow or freeze. The biologist played recordings of alarm calls for three samples of ground squirrels, grouped by age, and observed the squirrels' responses. The data for the three age levels are shown in the table. If ground squirrels at different ages respond to the alarm signals in the same proportions, what would be the expected count for 6- to 15-day-old squirrels that freeze?
Answer C ((40)(49)/98)
A major credit card company is investigating whether the distribution of the number of credit cards used by its customers has changed from last year to this year. Customers are classified as using 1 card, 2 cards, or more than 2 cards. The company conducts a chi-square goodness-of-fit test to investigate whether there is a change in the distribution of number of cards used from last year to this year. The value of the chi-square test statistic was χ2=7.82 with a corresponding p-value of 0.02. Assuming the conditions for inference were met, which of the following is the correct interpretation of this p-value?
Answer C If the null hypothesis were true, there is a 2 percent chance of obtaining a chi-square value of at least 7.82.
A round spinner is divided into five sections, where the sections do not have the same size. Using the measure of the interior angles, the probability of the spinner landing on any individual space on a spin is calculated and given in the table. A statistics student is asked to test the integrity of the spinner using a chi-square goodness-of-fit test. What is the minimum number of times the spinner should be spun to conduct this test?
Answer D 100
A national publication showed the following distribution of favorite class subjects for high school students. Pasquale, a student from a high school of 1,200 students, wants to see whether the distribution at his school matches that of the publication. He stands at the school entrance in the morning and asks the first 40 students he sees what their favorite class is. Pasquale records the following table of observed values. He decides to conduct a chi-square goodness-of-fit test to see whether his high school's distribution differs significantly from that of the publication. Pasquale's statistics teacher tells him that his information does not meet the conditions necessary for a goodness-of-fit test. Which condition has not been met? Data are collected using a random sample or randomized experiment n≤0.10N All expected counts are greater than 5.
Answer D I and III only
"Snoqualmie" is a name shared by a waterfall and a tribe of Native Americans. In a study of the cultural importance of the waterfall, two groups of the Snoqualmie tribe were randomly surveyed. One group consisted of Snoqualmie members living less than 25 miles from the waterfall. Another group consisted of Snoqualmie members living more than 25 miles from the waterfall. The researchers asked each member to rate the cultural importance of the waterfall as low, medium, or high. Data from the study are presented in the following table. If the distributions of ratings are the same for those Snoqualmie members living less than 25 miles from the waterfall and those living more than 25 miles from the waterfall, which of the following is equal to the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high?
Answer E ((17)(50)/88)
An amusement park keeps track of the percentage of individuals with season passes according to age category. An independent tourist company would like to show that this distribution of age category for individuals buying season passes is different from what the amusement park claims. The tourist company randomly sampled 200 individuals entering the park with a season pass and recorded the number of individuals within each age category. The tourist company will use the data to test the amusement park's claim, which is reflected in the following null hypothesis. H0:pchild=0.23, pteen=0.45, padult=0.20, and psenior=0.12. What inference procedure will the company use to investigate whether or not the distribution of age category for individuals with season passes is different from what the amusement park claims?
Answer E A chi-square goodness-of-fit test
Data was collected from a simple random sample of 200 cell phone users age 18 or older in the United States. Each user was categorized by age-group and by whether he or she uses a certain cell phone app. The table shown summarizes the data, with expected cell counts in parentheses. Which statement is true about whether the conditions for a chi-square test for independence have been met?
Answer E Not all of the expected cell counts are large enough to satisfy the conditions for the chi-square test for independence.
A bag of candy contains 5 different types of colored candies; red, green, blue, yellow, and orange. According to the manufacturer, bags should contain an equal number of each color. Students in a statistics class decided to use a chi-square procedure to test the manufacturer's claim. They opened a bag of candy and recorded the number of candies of each color. The results are shown in the following table. Which color contributes most to the chi-square test statistic?
Answer E Orange
A χ2 goodness-of-fit test where all assumptions were met yielded the chi-square test statistic χ2=1.92 and a corresponding p-value of 0.75. The researcher interpreted the p-value as a 0.75 probability of observing a test statistic of χ2=1.92 or larger. What is wrong with the researcher's interpretation?
Answer A The researcher did not state that the p-value is conditional on the null hypothesis being true.
A sports fan conducted a test to investigate whether male high school athletes are equally divided among football, soccer, swimming, tennis, and basketball. A sample of male high school athletes was selected and the resulting value of the chi-square test statistic was 10.65. Which of the following represents the p-value?
Answer B P(χ2≥10.65)=0.03
A political analyst wanted to see whether there is an association between political affiliation and where a person lives. The analyst took a random sample of 1,250 people in a state and asked them which political party they were affiliated with and what county they lived in. The following are the hypotheses the analyst tested. H0 : There is no association between party affiliation and the county in which a person lives. Ha : There is an association between party affiliation and the county in which a person lives. The chi-square test statistic and p-value of the hypothesis test were 19.78 and 0.003 respectively. Which of the following conclusions should be made about political affiliation and where a person lives?
Answer B There is convincing statistical evidence to suggest that political affiliation and where a person lives are dependent.
A college administrator wanted to know if the proportion of students who request online classes, lab classes, or lecture classes was different at two different campuses. At each campus, the administrator took a random sample of 250 students and asked each student which type of class they preferred. The conditions for the appropriate test were verified, and the chi-square test statistic for the test was calculated to be 4.01 with an associated p-value of 0.1347. If the significance level of the test was α=.05, what conclusion should the college administrator make about the proportion of students who request online classes, lab classes, or lecture classes at two different campuses?
Answer D There is not convincing statistical evidence to suggest that the proportion of students who request certain classes is different at each campus.
A job candidate at a large job fair can be classified as unacceptable, provisional, or acceptable. Based on past experience, a high-quality candidate is expected to get 80 percent acceptable ratings, 15 percent provisional ratings, and 5 percent unacceptable ratings. A high-quality candidate was evaluated by 100 companies and received 60 acceptable, 25 provisional, and 15 unacceptable ratings. A chi-square goodness-of-fit-test was conducted to investigate whether the evaluation of the candidate is consistent with past experience. What is the value of the chi-square test statistic and number of degrees of freedom for the test?
Answer A χ2=(15−5)^2/5+(25−15)^2/15+(60−80)^2/80 with 2df
A state political analyst wanted to see whether there is an association between the region where a person lives and whether the person is in favor of increasing the state gas tax. A random sample of 270 state residents was selected, and the table below shows the results of the survey. Which of the following is closest to the p-value of the appropriate test to investigate whether there is an association between the region where a person lives and whether the person is in favor of increasing the state gas tax?
Answer A 0.0644
A hypothesis test was conducted to see whether there is an association between a person's income level and his or her education level. A random sample of 225 people was selected, and the appropriate hypothesis test was conducted. The chi-square test statistic and corresponding p-value were approximately 13.36 and 0.01, respectively. Which of the following is the correct interpretation of the p-value in the context of the test?
Answer A Assuming that a person's income level and education level are independent, there is a 1 percent chance of finding a test statistic of 13.36 or greater.
A random sample of 300 United States cell phone users were asked their age and the question "Do you regularly use your cell phone to text while eating dinner?" The number responding yes was tabulated separately for young adults (aged 18-29), older adults (aged 30-64), and elderly adults (aged 65 or older). The results are summarized in the following table. Which of the following would be the appropriate hypotheses to investigate whether the survey provides convincing statistical evidence that there is an association between a person's age-group and regular use of a cell phone to text while eating dinner?
Answer B H0 : Among adults in the United States population, there is no association between age-group and cell phone usage to text while eating dinner. Ha : Among adults in the United States population, there is an association between age-group and cell phone usage to text while eating dinner.
A Labrador retriever club has 130 members: 65 black Labs, 44 golden Labs, and 21 chocolate Labs. Pablo is going to perform a chi-square goodness-of-fit test to see if the distribution of Labrador retrievers in the club is the same as the distribution nationally. Pablo is going to test his sample against the following null hypothesis, which reflects the national distribution. H0:pblack=0.53,pgolden=0.39,pchocolate=0.08 If the distribution of Labrador retrievers in the club were to match that of the national average, how many of each type of Labrador retriever would Pablo expect to see in his club?
Answer A Labrador Type Black Golden Chocolate Expected Counts 68.9 50.7 10.4
Horseshoe crabs on a beach can be overturned by approaching waves. These "stranded" crabs may right themselves by turning over. During a period of many days on an Atlantic Ocean beach, investigators categorized a random sample of crabs as either stranded or not stranded and additionally noted their age category. The researchers wanted to investigate whether there is convincing evidence of an association between becoming stranded and age. Data from their study are shown in the following table. If becoming stranded and age are independent in these creatures, which of the following is closest to the expected number of young stranded horseshoe crabs?
Answer B 52.54
A recent article published in Berry Weekly reported a probability distribution for the different types of jelly that individuals prefer. Editors from a competitive magazine, Jammin, conducted their own study to test the distribution. The editors from Jammin surveyed a random sample of 50 individuals and recorded the observed counts of individuals for each jelly type. They decided to test Berry Weekly's claim using a chi-square goodness-of-fit test using Jammin's observed counts compared with the number of expected counts based on the Berry Weekly data. Which of the following is the correct null hypothesis for the test?
Answer B H0:pS=0.33,pG=0.22,pWB=0.19,pp=0.15,pO=0.11H0:pS=0.33,pG=0.22,pWB=0.19,pp=0.15,pO=0.11
As a first step in developing a new drug to treat cancer in humans, an initial study of the drug is undertaken in rats. Three hundred rats with cancer are studied, and each is assigned to one of three treatments. One hundred rats are randomly assigned to a high dose of the new drug, 100 are randomly assigned to a low dose, and 100 are randomly assigned to a control group (no drug). After six months, each rat is examined and classified as having developed no tumors, one tumor, or two or more tumors. Which of the following would be an appropriate alternative hypothesis in this study for a chi-square test for homogeneity of tumor response across treatments?
Answer B Ha : There is a difference in the distribution of tumor status across the three treatments.
The national society of acupuncturists asked a random sample of 450 adults their opinion about acupuncture being a reasonable form of medicine or not a reasonable form of medicine. The table below shows the responses, along with the highest level of education achieved for each person selected. Assuming that all conditions for inference have been met, which of the following represents the correct chi-square test statistic and number of degrees of freedom to test whether there is an association between level of education and opinion about acupuncture?
Answer B χ2=(100−85.33)^2/85.33+(80−80)^2/80+(60−74.67)^2/74.67+(60−74.67)^2/74.67+(70−70)^2/70+(80−65.33)^2/65.33 with 2 degrees of freedom
A researcher is investigating the claim that the proportion of television viewers who identify one of four shows as their favorite is the same for all four shows. A χ2 goodness-of-fit test at a significance level of α=0.05 produced the test statistic χ2=8.95 with a corresponding p-value of 0.03. Which of the following is correct?
Answer C There is sufficient evidence to reject the null hypothesis at the 0.05 level since the pp-value is less than the significance level.
The table displays the distribution of the percentage of different types of home heating sources for a large mountain city, as reported by the city newspaper. A chi-square goodness-of-fit test will be performed using a simple random sample of 100 homes to investigate whether the proportion of homes heated with each source is the same as what is reported by the newspaper. Which of the following represents the alternative hypothesis of the test?
Answer D At least one of the heating source proportions is different from the proportion reported by the newspaper.
A manufacturing company with 350 employees is changing the employee health insurance plan to either plan A or plan B. The company wants to know if employees have a preference between the two plans and whether or not preference differs between those employees who have family members covered under the current plan (group 1) and those who do not (group 2). The human resources office takes a simple random sample from each of the two groups, sends information about both plans to the employees in each sample, and asks them whether they prefer plan A or plan B. The table summarizes the responses received, with expected cell counts in parentheses. Which statement is true about whether the conditions for the chi-square test for homogeneity have been met?
Answer D The total sample size is too large to apply the chi-square test for homogeneity.
A chi-square goodness-of-fit test using a significance level of α=0.05 was conducted to investigate whether the number of babies born in a town is uniformly distributed across the months of the year. The test produced a test statistic of χ2=5.6 with a corresponding p-value of 0.90. Which of the following is correct?
Answer D There is insufficient evidence to suggest that the distribution of births is not uniformly distributed across months.
The campus bookstore at a local university is interested in finding out whether the textbook preference and the class level (freshman, sophomore, junior, senior) of the student are associated. A random sample of 100 students is obtained, and each student in the sample is asked which textbook he or she prefers: new books, used books, or digital books. The students are also asked whether their class level is freshman, sophomore, junior, or senior. Which of the following is the appropriate test for the investigation?
Answer E A chi-square test of independence
A chi-square test of independence was conducted to investigate whether there is an association between the location where a person lives in a city (north, south, east, or west) and who the person planned to vote for in the upcoming mayoral election (the incumbent or the challenger). A random sample of 100 potential voters was selected, and the hypothesis test had a chi-square test statistic of χ2=9.84 with a p-value of 0.02. Which of the following statements is the correct interpretation of the p-value in context?
Answer E Assuming that the location of where a person lives and who that person plans to vote for are independent, there is a 2 percent chance of finding a test statistic that is 9.84 or greater.