stats
Nonnative speakers of English often have characteristics in their handwriting that differ depending on the speaker's first language. Handwriting samples from a random sample of Chinese, Malay, and Indian children in a school in Singapore were taken to investigate these differences. One characteristic of interest was a rounding at the top of the capital letter A. Data for the top-rounding characteristic for a sample of 153 children are shown in the table. If children from these countries exhibit top rounding in the same proportions, what is the expected count for Malay children who show top rounding?
((38)(51)/153)
The National Park Service writes materials for students to use while in the parks. In a study of the effectiveness of some of these materials, a random sample of students was selected to take a short quiz about oak trees after using these materials. A random sample of park professionals also took the quiz. Investigators compared classifications (low, medium, and high) of the crown shapes—the general shapes of the leafy parts of the trees—made by students in grades 6 through 12 with classifications made by professionals. Data from the study are shown in the table.
((87)(91)/200)
A survey was conducted to investigate whether there is an association between a person's age and their thoughts on what a state senate should do about state parks. Participants selected from "leave state parks the way they are," "increase funding to state parks," or "completely overhaul state parks," and their age was categorized as under 30 years old, between 30 and 50 years old, and over 50 years old. The hypothesis test statistic was calculated to be 15.01. Which of the following is closest to the p-value of the test?
0.0047
Which of the following is an appropriate description of the chi-square distribution?
A chi-square distribution will only contain positive values and will be skewed right, with the skew becoming less pronounced with increasing degrees of freedom.
A state highway commission is considering removing the lane that allows people to pay cash for a toll on a toll road and requiring all people who use the toll road to pay with an electronic transponder that is connected to their car. The commission wants to know whether the proportion of people who live in the northern part of the state and are in favor of removing the cash lane is different from the proportion of people who live in the southern part of the state and are in favor of removing the cash lane. Independent random samples are selected from the northern and southern parts of the state. The table summarizes the responses of those surveyed.
0.1342
A factory produces bags of rubber bands. A bag of rubber bands has five different sizes: extra large (XL), large (L), medium (M), small (S), and extra small (XS). A quality control specialist collects a random sample of 450 rubber bands from the bagging machine and calculates a chi-square goodness-of-fit test to see if the frequencies for each size in the sample match the hypothesized distribution. The quality control specialist will test his sample against the following null hypothesis.H0:pXL=0.10,pL=0.20,pM=0.40,pS=0.20,pXS=0.10 How many medium rubber bands are expected in the random sample of 450 rubber bands?
180
A local restaurant claims that it gets 45 percent of its customers from Monday through Thursday, 20 percent on Friday, 20 percent on Saturday, and 15 percent on Sunday. How many degrees of freedom should be used to conduct a chi-square goodness-of-fit test of the claim?
3
The ecology club at a local high school took separate random samples of sophomores, juniors, and seniors to learn whether there are differences in attitudes toward recycling by grade level. Students were asked to respond yes or no to the question, "Do you regularly recycle plastic bottles?" The results were summarized in a 3-by-2 table of counts organized by grade level and yes/no response to the recycling question. The club performed a chi-square test for homogeneity across the three grade levels. Which of the following could be an appropriate null hypothesis used by the club? H0: p1=p2=p3, where p1 is the proportion of the sample of sophomores that responded yes, p2 is the proportion of the sample of juniors that responded yes, and p3 is the proportion of the sample of seniors that responded yes. H0: There is an association between grade level and whether or not a student regularly recycles plastic bottles. H0: There is no difference in the distribution of regular recyclers across the three grade levels.
3 (III) Only
A private lake sells boating memberships and currently has 600 members. During the application process the potential members are asked which recreational activity they do the most. Their choices are fishing, skiing, boarding, swimming, or tubing. The lake manager chooses clients according to their interests to maximize the use of all areas of the lake. Every month, the lake rangers randomly sample the boats on the lake and categorize them according to the activity they are doing. The lake manager performs a chi-square goodness-of-fit test using the following null hypothesis to see whether their samples differ significantly from what the original applications claim.
46≤n≤60
Ecologists conducted a study to investigate the potential ecological impact of golf courses. Investigators monitored the reproductive success of bluebirds in birdhouses at nine golf courses and ten similar birdhouses at nongolf sites. Data on nests in birdhouses occupied only by bluebirds are shown in the table
60
Jimmy believes that the shuffle feature on his music player is malfunctioning by not playing songs that meet this distribution of music types. To test this, he listens to 100 songs randomly chosen when his player is in shuffle mode and records the number of songs in each category. Which inference procedure should he use to test whether or not the shuffle feature is working correctly?
A chi-square goodness-of-fit test
The logging company would like to use its sample to provide convincing statistical evidence that over 50 percent of the trees in the forest are spruce trees. The logging company has decided to use a chi-square goodness-of-fit test to justify its claim. Why is the chi-square goodness-of-fit test not an appropriate procedure for the logging company to use?
A chi-square goodness-of-fit test would be used to show that the entire distribution of trees in the forest is different than what the forester reported, not necessarily the individual proportion representing the spruce trees
A group of men and women were surveyed to investigate the association between gender and the number of friends the person has on a social media Web site. Results are shown in the table below. Which of the following procedures is the most appropriate for investigating whether an association exists between gender and the number of friends a person has on a social media Web site?
A chi-square test of independence
A polling organization uses random digit dialing of registered voters in a county to gauge the voters' opinions about a ballot initiative to increase taxes that would pay for a new county park. The polling organization will investigate whether there are differences in how voters respond based on the highest level of education completed. The phone survey asked the voters contacted whether they approve or disapprove of the tax increase. The voters were also asked to indicate their highest level of education completed: high school, bachelor's degree, or master's degree. Which of the following is the appropriate test to investigate whether there is an association between opinion about the ballot initiative and highest level of education completed?
A chi-square test of independence
A random sample of 500 people were classified by their ages into 3 age-groups: 29 years and younger, 30 to 64 years, and 65 years and older. Each person from the sample was surveyed about which of 4 major brands of cell phone they used. Their responses were compiled and displayed in a 3-by-4 contingency table. A researcher will use the data to investigate whether there is an association between cell phone brand and age-group. Which of the following is the appropriate test for the investigation?
A chi-square test of independence
In order to plan its next advertising campaign, the Trendy Motor Vehicle Company is investigating whether the type of vehicle and the color of vehicle are related. Each person in a random sample of size 275 selected from the company's mailing list was classified according to the type (car or truck) and the color of vehicle he or she drove. The data are shown in the table below. Which of the following procedures would be most appropriate to use for investigating whether there is a relationship between vehicle type and color?
A chi-square test of independence
A national health study reported that the proportion of students with elevated blood pressure is 0.15. The principal of a local high school believes that the proportion of students in the school with elevated blood pressure is greater than 0.15. If a large random sample is used, which of the following is the most appropriate test to investigate the principal's belief?
A z-test for a proportion
A simple random sample of 1,000 United States adults was collected to investigate whether gender and being a coffee drinker are independent. The results are summarized in the table with the expected counts shown in parentheses.
All necessary conditions are satisfied to apply a chi-square test for independence between gender and being a coffee drinker.
A chi-square test for homogeneity was conducted to investigate whether the four high schools in a school district have different absentee rates for each of four grade levels. The chi-square test statistic and p-value of the test were 19.02 and 0.025, respectively. Which of the following is the correct interpretation of the p-value in the context of the test?
Assuming that each high school has the same absentee rate for each grade level, there is a 2.5 percent chance of finding a test statistic 19.02 or larger.
Students in a high school statistics class wanted to see if the distribution of the colors of a popular candy was different in the bags for different types of candies the company manufactures. The students purchased several large bags of regular candies, tropical-flavored candies, and sour-flavored candies. For each type of candy, the students took a random sample of 100 candies and recorded how many of each color (red, green, yellow, or blue) were in the sample. The students verified the conditions for inference and calculated a chi-square test statistic of 12.59 with a corresponding p-value of 0.05. Which of the following is the correct interpretation of the p-value in the context of the test?
Assuming that the distribution of colors for the different types of candies is the same, there is a 5 percent chance of finding a test statistic of 12.59 or larger.
An administrator at a local high school wants to investigate whether there is an association between the grade level of a student (either ninth, tenth, eleventh, or twelfth) and how the student commutes to school (either walks, bikes, takes the bus, receives a ride, or drives). After a chi-square test for association was conducted, the results indicated that the chi-square test statistic was 14.63 with a p-value of 0.2623. Which of the following is the correct interpretation of the p-value in the context of the test?
Assuming there is no association between a student's grade level and how the student commutes to school, there is a 26.23 percent chance of finding a test statistic that is 14.63 or larger.
A fisheries biologist collected a random sample of fish from a lake and conducted a chi-square goodness-of-fit test to see if the distribution of fish changed over time. The table below shows the distribution of fish that were put into the lake when it was originally stocked. The biologist found evidence to reject the null hypothesis in favor of the alternative hypothesis. Which of the following represents the alternative hypothesis of the test?
At least one of the fish proportions is different than the corresponding proportion when the lake was originally stocked.
Which of the following chi-square distributions has the smallest number of degrees of freedom?
B (Skewed Left) (Highest point is 5) (Looks like it stops at 30)
For which of the following is a chi-square goodness-of-fit test most appropriate?
Determining whether a categorical variable has a significantly different distribution of proportions than the expected distribution
A veterinarian keeps track of the types of animals treated by an animal clinic. The following distribution represents the percentages of animals the clinic has historically encountered. If the animal clinic treats 230 animals in a month, how many of each animal type would be expected?
Dogs 140.3; Cats 50.6; Livestock 18.4; Birds 13.8; Other 6.9
An administrator at a local high school wanted to investigate whether there is an association between the amount of time a student studies for a test and the type of extracurricular activity the student is involved in. Three hundred students selected at random were asked how long they had studied for the last math test and how many extracurricular activities they are involved in. The times they had studied were recorded as either not at all, less than 30 minutes, or more than 30 minutes. Each student also identified which extracurricular activity (out of a total of 5 extra curricular activities) they were involved in. The calculated chi-square test statistic was 7.53 with a corresponding p-value of 0.4807. Based on this p-value, which of the following is the correct decision for the appropriate hypothesis test at the α=0.05 significance level?
Fail to reject the null hypothesis. The test is not statistically significant because a pp-value of 0.4807 is greater than a significance level of 0.05.
Polling organizations regularly collect data on the public's opinions and habits. A question on a recent survey asked, "How often do you purchase coffee from a coffeehouse?" There may be differences in how people respond to this question based on whether the person is a full-time student. Suppose a polling organization uses random digit dialing of local phone numbers to take a poll and asks respondents whether they are full-time students. In addition, they ask respondents to identify how often they purchase coffee from a coffeehouse (never, once a week, two to three times a week, daily, more than once a day). The data are collected in a 5-by-2 table of counts. Which of the following is the appropriate null hypothesis when conducting a chi-square test for the data?
H0 : How often a person purchases coffee from a coffeehouse is not associated with whether a person is a full-time student.
A large factory that builds machines has three shifts, one that starts at 4:00 A.M., one that starts at noon, and one that starts at 8:00 P.M. The manager of the factory wanted to know whether there is an association between an employee's work experience (less than five years with the company, between five and twenty years with the company, over twenty years with the company) and the time of the employee's shift. The manager selected a random sample of 125 employees and classified employees by their shift time and work experience. Which of the following is an appropriate pair of hypotheses for the manager to use?
H0 :H0 : Work experience is independent of the time of an employee's shift. Ha :Ha : Work experience is dependent on the time of an employee's shift.
A quality control specialist tests samples of the seed being packaged and uses a chi-square goodness-of-fit test to see whether the proportions in the samples match what is claimed by the company. Which of the following best describes the null hypothesis and the alternative hypothesis for the test?
H0: pF=0.55,pB=0.22,pBG=0.10,pI=0.07,pNG=0.06 Ha: At least one of the proportions is different.
A spinner made for a game of chance has 8 equally likely spaces. Alfonso records the result of a sample of 400 spins. Alfonso decides to calculate a chi-square test statistic for a goodness-of-fit test to see whether the spinner is fair. Which of the following is the appropriate null hypothesis?
H0:p1=0.125,p2=0.125,p3=0.125,p4=0.125,p5=0.125,p6=0.125,p7=0.125,p8=0.125
A statistician is conducting a chi-square goodness-of-fit test and is limited by the cost, per individual, to conduct the study. The statistician selects a sample of size 39, which is the smallest sample possible that will meet the condition for large expected counts. Which of the following could not be the null hypothesis for the study?
H0:p1=0.43,p2=0.23,p3=0.17,p4=0.09,p5=0.08
A company claims it audits its employees' transactions based on their job level. For entry-level positions, the company claims that 50 percent get a basic audit, 30 percent get an enhanced audit, and 20 percent get a complete audit. The company tests this hypothesis using a random sample and finds χ2=0.771 with a corresponding p-value of 0.68. Assuming conditions for inference were met, which of the following is the correct interpretation of the p-value?
If the null hypothesis were true, there would be a 68 percent chance of obtaining a chi-square value of at least 0.771.
A candy company claims that 10 percent of its candies are blue. A random sample of 200 of these candies is taken, and 16 are found to be blue. Which of the following tests would be most appropriate for establishing whether the candy company needs to change its claim?
One-sample proportion z-test
A newspaper article indicated that 43 percent of cars with black seats are white, 46 percent of cars with black seats are blue, 7 percent of cars with black seats are red, and 4 percent of cars with black seats are black. A test was conducted to investigate whether the color of cars with black seats was consistent with the newspaper article. A random sample of cars of these colors was selected, and the value of the chi-square test statistic was χ2=8.2. Which of the following represents the p-value for the test?
P(χ^2≥8.2)=0.04
Which of the following is a reason not to use a chi-square test of homogeneity to analyze a set of data?
The data were obtained through a simple random sample from a single population and summarized by counts on two categorical variables.
Which of the following is not a condition for a chi-square goodness-of-fit test?
The distribution of the sample should be approximately normal.
How are the expected counts calculated when a chi-square goodness-of-fit test is conducted?
The expected counts are calculated by multiplying each proportion in the null hypothesis by the sample size.
The manager conducts a goodness-of-fit test to determine whether the proportions of workers of these types are identical to the population proportions of workers donating to charity, which are 50 percent for management, 30 percent for other white-collar workers, and 20 percent for blue-collar workers. Which of the following statements must be true about the sample?
The expected number of management workers donating to charity is 100.
A reporter intends to survey residents of a city to investigate whether there are differences in use of grocery delivery services by region of the city where the residents live (north, south, west, and east). City residents will be asked to respond yes or no to the question "Do you regularly use a grocery delivery service to purchase groceries?" Results will be collected in a 4-by-2 table of counts organized by region and response to the question. For a chi-square test for homogeneity, which of the conditions listed below is not necessary to investigate whether there are differences between use of grocery delivery service by region?
The number of residents sampled from each region should be greater than 30.
A chi-square goodness-of-fit test where all assumptions were met yielded the test statistic χ2=12.4. Henry claims the corresponding p-value of 0.03 means that the probability of observing a test statistic of χ2=12.4 is 0.03, assuming the null hypothesis is true. Which of the following is a valid criticism of this interpretation of the p-value?
The p-value is not the probability of observing 12.4 exactly.
A company claims they produce their mixed bag of candies so that, of the candies in the bag, 20 percent are dark chocolate, 60 percent are milk chocolate, and 20 percent are white chocolate. In a random sample of candies of size 50, the counts are as follows: 6 dark, 32 milk, and 12 white. Assuming the conditions for inference are met, what is the test statistic for a chi-square goodness-of-fit test to investigate whether the distribution of the sample is consistent with the company's claim?
χ2=(6−10)^2/10 + (32−30)^2/30 + (12−10)^2/10
A chi-square test was conducted to investigate whether there is an association between a person's favorite flavor of ice cream and their favorite toppings. Each of 200 randomly selected customers at an ice cream parlor was asked to pick their favorite flavor from vanilla, chocolate, chocolate chip, or none of these. They were also asked to pick their favorite topping from chocolate sauce, peanuts, crumbled cookies, crushed candies, or none of these. The hypothesis test had a test statistic of 24.97 with an associated p-value of 0.015. If the significance level of the test was α=0.05 , which of the following is the correct decision for this hypothesis test?
There is convincing statistical evidence to suggest an association between favorite ice cream flavor and favorite topping.
The district manager of four different restaurants wanted to investigate whether the four restaurants differed with respect to customers ordering dessert or not based on family classification (with children or without children). Independent random samples of 100 customers who ordered dessert were selected from each restaurant, and the customers were identified as either being with children or without children. After verifying the conditions for the appropriate hypothesis test, the manager calculated a chi-square test statistic of 6.45 with an associated p-value of 0.092. Based on the p-value and α=0.05, what conclusion should the manager make regarding the proportion of customers who order dessert at each restaurant and the customers' family classification?
There is not convincing statistical evidence to suggest that the proportion of customers who order dessert at each restaurant is not the same based on family classification.
A χ2 goodness-of-fit test was used to test the hypothesis that students at a local university select majors in the same proportions as other universities in the state. A chi-square test statistic of χ2=45.6 was calculated with a corresponding p-value of 0.005. Which of the following is correct?
There is sufficient evidence to conclude that students at the local university do not select majors in the same proportions as do students in the rest of the state.
In 2010, a medical research group reported the results of an experiment to evaluate the effectiveness of acupuncture to treat a chronic intestinal condition. A group of volunteers with the chronic intestinal condition agreed to participate in the experiment and be randomly assigned to either a true acupuncture treatment or a placebo treatment. The placebo treatment mimicked the application of acupuncture, but no needle penetrated the skin. Random assignment resulted in 78 subjects receiving acupuncture and 75 subjects receiving the placebo treatment. After receiving 6 treatments over the course of 3 weeks, patients were asked to report whether they had experienced a reduction in the chronic intestinal condition. The table summarizes the data from the study, with expected cell counts in parentheses.
Volunteers with the chronic intestinal condition were randomly assigned to each treatment, so the independence condition has been met.
The manager of the cafeteria at a local high school wanted to see if there was an association between a student's grade level and whether the student approves of the food choices in the cafeteria. The manager selected a random sample of students and asked if they approved of the food choices and also recorded the grade levels of the students. If a student was in ninth or tenth grade, he or she was labeled as an underclassman, and if the student was in eleventh or twelfth grade, he or she was labeled as an upperclassman. The table shows the results of the survey.
χ^2=(50−35)^2/35 + (20−35)^2/35 + (30−45)^2/45 + (60−45)^2/45 with 1 degree of freedom
