Identify the Inference Methods
Brett enjoys eating ChocoChewys, a new chewy candy bar with peanuts, caramel, and chocolate that comes in 1.45 oz. sealed packaging. He begins questioning the quality control of the candy maker when he gets a bar that is much smaller than he expected. He decides to buy 10 ChocoChewys at 10 randomly chosen grocery stores. He weighs each bar with a digital scale that is accurate to a thousandth of an ounce.
1-sample t-test with 9 degrees of freedom (it is assumed he is interested in the average weight)
According to a survey conducted by the Associated Press and petside.com in 2009, 63% of dog owners and 53% of cat owners would be at least somewhat likely to give CPR to their pet in the event of a medical emergency. The survey involved a nationwide sample of 1,166 pet owners.
2-proportion z test Ho: p_dog = p_cat Ha: p_dog ≠ p_cat
Researchers Penninx et al. (Archives of General Psychiatry, 2001) looked at the relationship between depression and a person's ability to survive cardiac disease. The researchers identified 450 men and women with cardiac disease, evaluated them for depression, and followed them for four years. In those four years, of the 89 patients diagnosed with minor or major depression, 26 died of cardiac disease. Of the 361 patients with no diagnosis of depression, 67 died of cardiac disease.
2-proportion z-test Ho: p_depression= p_none Ha: p_depression ≠ p_none
A 2013 Gallup poll asked randomly selected U.S. adults whether they wanted to stay at their current body weight or change. One purpose was to investigate whether there was any difference between men and women with regard to this aspect. Of the 562 men surveyed, 242 wanted to stay at their current weight, whereas of the 477 women surveyed, 172 wanted to stay at their current weight.
2-proportion z-test Ho: p_men = p_women Ha: p_men ≠ p_women
Have you ever donated blood? If so, how often do you donate blood? Studies have looked at who is more likely to donate blood: men or women? Suppose you decide to take random samples of men and women and ask each of the participants whether they have donated blood in the past 12 months.
2-proportion z-test Ho: p_men = p_women Ha: p_men ≠ p_women
In July 2011, an Italian firm Ma-Vib was in the news for "sexual discrimination" for having chosen only its female employees to be dismissed. Before the layoffs, Ma-Vib employed 18 women and 12 men; it then fired 15 of the women and none of the men.
2-proportion z-test Ho: p_men = p_women Ha: p_men ≠ p_women Not sure if the conditions have been met...
In 1973 a lawsuit was filed against the University of California at Berkeley, alleging sex discrimination in its graduate admissions policies. Admissions data was collected on two of the graduate programs at the university. For each program, it lists the number of men accepted, the number of men denied, the number of women accepted, and the number of women denied.
2-proportion z-test or Chi-square test of homogeneity with (2-1)(2-1) = 3 degrees of freedom. (there are two groups - 2 grad programs - and gender admitted) admitted and denied are not independent...they just help get the total amount.
A survey question asked students "How many hours of TV do you watch per week?" Using their responses, we want to estimate the difference in mean hours between high school and middle school students.
2-sample t-interval (degrees of freedom obtained from technology)
An insurance company hopes to save money on repairs to autos involved in accidents. Two body shops in town seem to do most of the repairs, and the company wonders whether one of them is generally cheaper than the other. From their files of payments made during the past year they select a random sample of ten bills they paid at each repair shop. Is this evidence there is a significant difference in the costs of repairs done at these two body shops?
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_A - mu_B = 0 of mu_A = mu_B Ha: mu_A - mu_B ≠ 0 of mu_A ≠ mu_B
A supermarket chain wants to know which of two merchandise display methods is more effective. They randomly assign 15 stores to use display type A and 15 others to use display type B, then collect data about the number of items sold at each store.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_A - mu_B = 0 or mu_A = mu_B Ha: mu_A - mu_B ≠ 0 or mu_A ≠ mu_B
You need to find a new hair stylist and know that there are two terrific salons in your area, Hair by Charles and Curl Up & Dye. You want a really good haircut, but you do not want to pay too much for the cut. A random sample of costs for 10 different stylists was taken at each salon.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_hbc - mu_cud = 0 or mu_hbc = mu_cud Ha: mu_hbc - mu_cud ≠ 0 or mu_hbc ≠ mu_cud
A random sample of some of the heaviest carnivores on Earth was reviewed to determine if there is an association between the length (in meters) and weight (in kilograms) of these carnivores. Is there an association between the length and weight of these carnivores?
Linear regression t-test (degrees of freedom unknown) Ho: There is no linear association between length and weight. Ha: There is a linear association between length and weight.
A national marketing campaign is conducted to improve the name-brand recognition with potential customers. The company sells household products, primarily to the female head of the household. The current market analysis shows that 23% of their customer base recognizes the name of their product. When the marketing campaign is finished, 135 people out of 500 randomly polled recognize the brand name. Does this provide statistically significant evidence that there has been an increase in name-brand recognition?
1-proportion z-test Ho: p = 0.23 Ha: p > 0.23
A policeman believes that more than 40% of older drivers speed on highways, but a confidential survey found that 49 of 88 randomly selected older drivers admitted speeding on highways at least once. Is this strong evidence that the policeman was wrong?
1-proportion z-test Ho: p = 0.4 Ha: p > 0.4
The manager of a self-serve frozen yogurt shop, Purple Pear, knows that customers swirl different amounts of the creamy concoction into their cups. She samples cups from 10 costumers she believes to be representative of the customers and weighs the cups, finding a mean of 4.5 ounces and standard deviation of 0.25 ounces. Assuming these cups of frozen yogurt can be considered a random sample of all cups of frozen yogurt, the manager would like an estimate of the mean weight of all cups of frozen yogurt.
1-sample t-interval with 9 degrees of freedom
A wildlife biologist wants to determine the mean weight of adult red squirrels. She captures 10 squirrels she believes to be representative of the species and weighs them, finding a mean of 12.32 grams and standard deviation of 1.88gm.
1-sample t-interval with 9 degrees of freedom.
A team of researchers (Singer et al., 2000) used the Survey of Consumer Attitudes to investigate whether incentives would improve the response rates on telephone surveys. A national sample of 735 households was randomly selected, and all 735 of the households were sent an "advance letter" explaining that the household would be contacted shortly for a telephone survey. However, 368 households were randomly assigned to receive a monetary incentive along with the advance letter, and of these 286 responded to the telephone survey. The other 367 households were assigned to receive only the advance letter, and of these 245 responded to the telephone survey.
2-proportion z-test Ho: p_advanced = p_none Ha: p_advanced ≠ p_none
In an article published in Psychology of Music (2010), researchers reported the results of a study conducted to investigate the effects of "romantic lyrics on compliance with a courtship request." The researchers recruited undergraduate female students who were studying social and managerial science and told them that "the purpose of the study was to discuss organic products with another participant." When a participant came in for the study, she was randomly assigned to listen to either a romantic song or a neutral song. After three minutes, she was greeted by a male "confederate" (chosen and trained by the researchers to pretend to be another participant), who while discussing the organic products also asked for her phone number so that he could call her up to ask her out. The male confederate was kept unaware of the aim of the study and was trained to use the same script when talk- ing to all the females. Of the 44 women who listened to the romantic song, 23 gave their phone numbers, whereas of the 43 who listened to the neutral song, only 12 did.
2-proportion z-test Ho: p_romantic= p_neutral Ha: p_romantic ≠ p_neutral
A baby weighing more than 4,000 g at birth is considered to be large for gestational age (LGA). Gestational diabetes in the mother is believed to be a common risk factor for LGA. In an article published in the New England Journal of Medicine (October 2009), researchers Landon et al. reported a study of 958 women diagnosed with mild gestational diabetes between 24 and 31 weeks of pregnancy who volunteered to participate in the study and then were randomly assigned to one of two groups: 473 to usual prenatal care (control group) and 485 to dietary intervention, self-monitoring of blood glucose, and insulin therapy, if necessary (treatment group). Of the 473 women in the control group, 68 had babies who were LGA, and of the 485 women in the treatment group, 29 had babies who were LGA.
2-proportion z-test (easiest) OR a chi-square test of homogeneity with 1-degree of freedom (treatment vs no and positive test vs not positive) The tests are the same for 1 degree of freedom.
In a 1994 study, 164 pregnant, HIV-positive women were randomly assigned to receive the drug AZT during pregnancy and 160 such women were randomly assigned to a control group that received a placebo. It turned out that 40 of the mothers in the control group gave birth to babies who were HIV-positive, compared to only 13 in the AZT group.
2-proportion z-test (easiest) OR a chi-square test of homogeneity with 1-degree of freedom (treatment vs no and positive test vs not positive) The tests are the same for 1 degree of freedom.
Researchers Wilt et al. (New England Journal of Medicine, 2012) investigated whether surgery, compared to just observation, was (more) effective in improving men's survival chances after being diagnosed with prostate cancer. The researchers identified 731 men with localized prostate cancer who volunteered to participate. They randomly assigned 364 men to surgery and the remaining 367 to observation. All participants were followed for about 10 years. In those 10 years, 21 surgery recipients died of prostate cancer related reasons compared to 31 observation recipients.
2-proportion z-test (easiest) OR a chi-square test of homogeneity with 1-degree of freedom (treatment vs no and positive test vs not positive) The tests are the same for 1 degree of freedom.
In a study to determine whether there is a difference between the average jail time convicted bank robbers and car thieves are sentenced to, the law students randomly selected 20 cases of each type that resulted in jail sentences during the previous year.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_bank robbers - mu_car thieves = 0 or mu_bank robbers = mu_car thieves Ha: mu_bank robbers - mu_car thieves ≠ 0 or mu_bank robbers ≠ mu_car thieves
Joey, a soccer player, wants to improve his kicking accuracy so he can be the primary team member to do corner kicks. After watching David Beckham videos and practicing all summer, he tells his select soccer coach he can now get 75% of his corner kicks into the goal. He had only been 50% accurate the year before. The coach decides to let Joey take 12 practice corner kicks. If he can get at least 10 of them in the goal, the coach will make him the primary corner kicker.
1-proportion z-test (Ho: p=0.5 Ha: p>0.5)
An oral surgeon is interested in estimating how long it takes to extract all four wisdom teeth. The doctor records the times for 24 randomly chosen surgeries.
1-sample t-interval with 23 degrees of freedom.
A college alumni fund appeals for donations by phoning or emailing recent graduates. A random sample of 300 alumni shows that 40% of the 150 who were contacted by telephone actually made contributions compared to only 30% of the 150 who received email requests. The alumni fund would like to estimate the difference in the proportions of alumni who may make donations if contacted by phone or by email.
2-proportion z-interval
A New York City mayoral candidate wants to assess his constituent's opinions on the controversial "Stop and Frisk" police tactics. A sample of voters from 2 boroughs (Queens and Brooklyn) is selected and asked if they approve of this policy. Do the approval rates vary from each other?
2-proportion z-test (the answer to the survey is yes/no, so this is about proportion of yes/no) Ho: p_queens - p_brooklyn = 0 or p_queens = p_brooklyn Ha: p_queens - p_brooklyn ≠ 0 or p_queens ≠ p_brooklyn You will need a pooled proportion on this!
In a study to determine whether there is a difference between the average jail time black and white offenders of minor drug possession are sentenced to, the law students randomly selected 25 cases of each type that resulted in jail sentences during the previous year.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_black - mu_white = 0 or mu_black = mu_white Ha: mu_black - mu_white ≠ 0 or mu_black ≠ mu_white
Sixty senior account executives were classified into three groups, labeled A, B, and C. There were 26 in group A, 19 in group B and 15 in group C. At the .05 significance level, we would like to test if is it reasonable to conclude that the proportion of the population that falls into each group is the same.
Chi-square Goodness of Fit with (3-1) = 2 degrees of freedom. We are wanting to know if a single group matches a given distribution for a categorical variable. Ho: The proportion of the population in category is 0.33 Ha: The proportion in at least one of the categories is different from 0.33
A random sample of 337 college students was asked whether or not they were registered to vote. We wonder if there is an association between a student's sex and whether the student is registered to vote.
Chi-square test of independence with (2-1)(2-1)=1 degree of freedom. (one group - college students - compared on two categorical variables - gender and registered to vote) Ho: There is no association between a student's sex and whether the student is registered to vote. Ha: There is an association between a student's sex and whether the student is registered to vote.
A random sample of 600 high school seniors reported their grade point averages and the amount of financial aid offered them by colleges. We wonder if there is an association between academic success and college aid.
Linear Regression t-test with 598 degrees of freedom Ho: there is no association between academic success and college aid. B_1 = 0 Ha: there is an association between academic success and college aid. B_1 ≠ 0
Dependent variable is: kWh No Selector R squared = 91.3% R squared (adjusted) = 90.4% s = 201.3 with 12 - 2 = 10 degrees of freedom Variable Coefficient SE of Coef t prob Constant 3898.20 269.0 14.5 0.0001 avtemp -50.9027 4.980 -10.2 0.0001
Linear Regression t-test with 10 degrees of freedom Ho: There is no association between average monthly temperature and electrical usage. B_1 = 0 Ha : There is a linear association between the variables. B_1≠0
The Comprehensive Test of Basic Skills (CTBS) is used by school district to assess student progress. Two of the areas tested are math and reading. A random sample of student results was reviewed to determine if there is an association between math and reading scores on the CTBS.
Linear Regression t-test (degrees of freedom unknown.) (are two quantitative variables - math and reading scores - related?) Ho: There is no linear association between math and reading scores. B_1 = 0 Ha: There is a linear association between math and reading scores. B_1 ≠ 0
Several volunteers engage in a special exercise program intended to lower their blood pressure. We measure each person's initial blood pressure, lead them through the exercises daily for a month, then check blood pressures again.
Matched pairs t-test (degrees of freedom unknown) Ho: mu_difference = 0 Ha: mu_difference ≠ 0
The owner of a construction company would like to know if his current work teams can build room additions quicker than the time allotted for by the contract. A random sample of 15 room additions completed recently revealed an average completion time of 0.32 days faster than contracted. Is this strong evidence that the teams can complete room additions in less than the contract times?
Matched pairs t-test with 14 degrees of freedom (this is also a 1-sample t-test on the differences) Ho: mu_difference = 0 (no faster than contracted) Ha: mu_difference < 0 (faster than contracted)
You want to see whether or not high school changes children's educational plans. You take a random sample of 6th graders and of 12th graders and ask them whether they plan to get a job right after high school, go to college, or get an advanced degree.
chi-square test of homogeneity with (2-1)(3-1)=2 degrees of freedom (We are comparing two groups on their answer to a categorical question) Ho: The proportions of those who chose high school, college, and advanced degree are the same for both groups. Ha: At least one of the proportions of those who chose high school, college, and advanced degree are different for both groups.
Which test would be the most appropriate for measuring the association between gender and favorite color based on a random sample of subjects?
chi-square test of independence (degrees of freedom unknown) (this is one group's response to two categorical questions) Ho: There is no association between gender and favorite color. Ha: There is an association between gender and favorite color.
A coffee house owner knows that customers pour different amounts of coffee into their cups. She samples cups from 10 costumers she believes to be representative of the customers and weighs the cups, finding a mean of 12.5 ounces and standard deviation of 0.5 ounces. Assuming these cups of coffee can be considered a random sample of all cups of coffee, she would like to estimate the mean weight of all cups of coffee?
1-sample t-interval with 9 degrees of freedom
From time to time police set up roadblocks to check cars to see if the safety inspection is up to date. At one such roadblock they issued tickets for expired inspection stickers to 22 of 628 cars they stopped. Based on the results at this roadblock, estimate the proportion of autos in that region whose safety inspections have expired.
1-proportion z-interval
The school newspaper wants to estimate the road test failure rate for driver's license test. In a random sample of 65 student drivers, 37 said they failed their driver's test at least once.
1-proportion z-interval
A teacher believes that no more than 10% of high school students ever cheat on an exam, but a confidential survey found that 14 of 88 randomly selected students admitted having cheated at least once. Is this strong evidence that the teacher was wrong?
1-proportion z-test Ho: p = 0.1 Ha: p > 0.1
Organizations like Gallup often track public opinion on many issues by asking the same question at different times. In October of 2013, Gallup reported that 52% of U.S. adults disapprove of the Affordable Care Act (known as Obamacare), which is up from 45% a year earlier. Assume these were both based on random samples of 450 people. Is this evidence of a real change in public opinion?
1-proportion z-test Ho: p = 0.45 Ha: p ≠ 0.52 (you could use > )
A union organization would like to represent the employees at the local market. A sample of the employees revealed 74 of 120 were in favor of the union. Does the union have the required 3 to 2 majority?
1-proportion z-test Ho: p = 0.6 (a 3 to 2 ratio means 60%) Ha: p > 0.6 You could also estimate the proportion using a 1-proportion z-interval.
A random sample of 150 teachers in an inner-city school district found that 72% of them had volunteered time to a local charitable cause within the past 12 months. Is this evidence that less than 75% of teachers volunteered?
1-proportion z-test Ho: p = 0.75 Ha: p < 0.75
A board of directors of a local homeowner's association union organization wishes to amend the bylaws. A sample of the residents revealed 310 of 430 were in favor of the amendment. Does the board of directors have the required 75% majority?
1-proportion z-test Ho: p = 0.75 Ha: p > 0.75
Forty people complaining of allergies take an antihistamine. They report that their discomfort subsided in an average of 18 minutes; the standard deviation was 4 minutes. The manufacturer wants an estimate for the "relief time".
1-sample t-interval
Workers at a large factory finish shirts with a hand sewn logo. The foreman overseeing the workers tracks the level of production. After collecting data for several months he estimates that workers complete an average of 230 shirts each day with a standard deviation of 13 shirts.
1-sample t-interval (degrees of freedom unknown)
22 people complaining of indigestion take an antacid. They report that their discomfort subsided in an average of 13 minutes; the standard deviation was 4 minutes. The manufacturer wants a 95% confidence interval for the "relief time".
1-sample t-interval with 21 degrees of freedom.
Feral cats tend to be lighter than domestic cats, mainly due to a lack of regular access to food. An animal shelter in Boston weighed a random sample of 24 newly admitted stray cats over several weeks to obtain an estimate of their weights.
1-sample t-interval with 23 degrees of freedom.
A bank branch manager is interested in estimating the average wait time for customers in the teller line. The manager records the times for 40 randomly chosen customers.
1-sample t-interval with 39 degrees of freedom
An English professor at a local community college is disappointed that the average final exam score in her previous Freshman Survey courses has been 78. A friend who is a computer professor suggests that she set up a website containing tutorials created by a local actor and require students to log on for a weekly presentation. The English professor decides to give the tutorials a try for one semester for a random sample of her students.
1-sample t-test (degrees of freedom unknown) (the professor is interested in the average class score, not a before/after difference) Ho: mu=78 Ha: mu>78
A friend of yours is constantly bragging that he performs exceptionally well at a popular video game since he scores an average of 1400 points. You think he is not all that hot. Having taken a statistics class, you know that you can actually test to see whether or not your friend scores higher than other players of the game. You collect a random sample of scores from 12 friends. Is there evidence that other players average less than 1400 points per game?
1-sample t-test with 11 degrees of freedom Ho: mu = 1400 Ha: mu < 1400
A city representative claims that the policemen in the city earn an average of $52,000 per year. The local paper believes that the mean salary is less for the beat cops. A survey conducted by the paper selected a random sample of 20 beat cops and found a mean salary of $51,300 with a standard deviation of $1900.
1-sample t-test with 19 degrees of freedom (1-tail test. The survey leads to a mean, which is a t-test or interval) Ho: mu=$52,000 Ha: mu<$52,000
Tags placed on garbage cans allow the disposal of up to 30 pounds of garbage. A random sample of 22 cans averaged 33.2 pounds with a standard deviation of 3.2 pounds. Is this strong evidence that residents overload their garbage cans?
1-sample t-test with 21 degrees of freedom Ho: mu = 30 lbs Ha: mu > 30 lbs
A flower pot manufacturer is testing his clay pots to ensure that the thickness of the sides are made to proper specifications. The sides are designed to be 4 mm thick. In a random sample of 25 pots, it is found that the average thickness if 4.3 mm. Does this provide statistically significant evidence that the manufacturing process is out of alignment?
1-sample t-test with 24 degrees of freedom Ho: mu = 4 mm Ha: mu ≠ 4mm
The Centers for Disease Control report a survey of randomly chosen Americans age 45 and older, which found that 51 of 100 men and 80 of 782 women suffered from some form of arthritis. You want to estimate the difference in the proportions of men and women over 45 who have arthritis with a 95% confidence interval.
2-proportion z-interval
Are female or male students more likely to attend college outside their home state? In order to find out, random samples of male and female college-bound high school seniors were taken in the Dallas/Fort Worth metropolitan area. In September following their high school graduations, the students in the samples were contacted to see if they were attending college in Texas or outside of it. (Students who were not attending college were eliminated from the study.)
2-proportion z-interval (There are two proportions for two independent groups - male and females)
Think about the proportion of students at your school who are wearing clothing that displays their school name or logo today. Also suppose that a friend of yours attends a different school, and the two of you have a recurring discussion about which school displays more school pride. You decide to measure school pride by the proportion of students at the school who wear clothing that displays the school name or logo on a particular day. You want to investigate whether this proportion differs between your school and your friend's school.
2-proportion z-test Ho: The proportion of students who show school pride at your school = The proportion of students who show school pride at your friend's school. Ha: The proportion of students who show school pride at your school ≠ The proportion of students who show school pride at your friend's school.
A microwave manufacturing company receives large shipments of thermal shields from two suppliers. A sample from each supplier's shipment is selected and tested for the rate of defects. The microwave manufacturing company's contract with each supplier states the shipment with the smallest rate of defect will be accepted. Do the shipments' defect rates vary from each other?
2-proportion z-test Ho: p_1 - p_2 = 0 or p_1 = p_2 Ha: p_1 - p_2 ≠ 0 or p_1 ≠ p_2
Among randomly selected pets, 27% of the 188 dogs and 18% of the 167 cats had fleas. Does this indicate a significant difference in rates of flea problems for these two pets?
2-proportion z-test Ho: p_dogs - p_cats = 0 or p_dogs = p_cats Ha: p_dogs - p_cats ≠ 0 or p_dogs ≠ p_cats You will need a pooled proportion on this.
In a study on insomnia in men over the age of 65, it is found that exercise may play a role in sleep. The researchers assign a group of 100 men to exercise for a month, while another 100 volunteers are asked to abstain from most exercise. At the end of the month, the exercise group had 23 out of 100 men with difficulty in sleeping, whereas the non-exercise group had 31 out of 100. Does this provide statistically significant that exercise improves sleep for men over 65?
2-proportion z-test Ho: p_exercise - p_none = 0 or p_exercise = p_none Ho: p_exercise - p_none > 0 or p_exercise > p_none
Among a random sample of college-age students, 6% of the 473 men said they had been adopted, compared to only 4% of the 552 women. Does this indicate a significant difference between adoption rates of males and females in college-age students?
2-proportion z-test Ho: p_men - p_women = 0 of p_men = p_women Ha: p_men - p_women ≠ 0 of p_men ≠ p_women
At one SAT test site students taking the test for a second time volunteered to inhale supplemental oxygen for 10 minutes before the test. In fact, some received oxygen, but others (randomly assigned) were given just normal air. Test results showed that 42 of 66 students who breathed oxygen improved their SAT scores, compared to only 35 of 63 students who did not get the oxygen. Which procedure should we use to see if there is evidence that breathing extra oxygen can help test-takers think more clearly?
2-proportion z-test Ho: p_oxygen - p_none = 0 or p_oxygen = p_none Ha: p_oxygen - p_none > 0 or p_oxygen > p_none You will need to use a pooled proportion!
At one vehicle inspection station, 13 of 52 trucks and 11 of 88 cars failed the emissions test. Is this evidence of a difference in the percentages of all cars and trucks that are not in compliance with air quality regulations?
2-proportion z-test Ho: p_trucks - p_cars = 0 or p_trucks = p_cars Ha: p_trucks - p_cars > 0 or p_trucks > p_cars You will need to use a pooled proportion!
A study found that 63 of 211 randomly selected men and 130 out of 651 randomly selected women prefer cats to dogs. Is this evidence that women like cats more.
2-proportion z-test Ho: p_women - p_men = 0 or p_women = p_men Ho: p_women - p_men > 0 or p_women > p_men You will need to use a pooled proportion on this test!
Among a random sample of college-age drivers 5% of the 576 men said they had been ticketed for speeding during the past year, compared to only 3% of the 552 women. Does this indicate a significant difference between college males and females in terms of being ticketed for speeding?
2-proportion z-test or a Chi-square test of independence with 1 degree of freedom (1 group - college age drivers - with two categorical variables - gender and tickets) Ho: p_men - p_women = 0 Ha: p_men - p_women ≠ 0 of Ho: There is no association between gender and whether or not they were ticketed. Ha: There is an association between gender and whether or not they were ticketed.
Every year Educational Services (ETS) selects readers for the Advanced Placement Exams. Recently the AP* Statistics exam has been graded in Lincoln, Nebraska. One objective of ETS is to achieve equity in grading by inviting teachers to be readers from all parts of the nation. However budgets are a consideration also. The accountants at ETS wonder if the flights from cities west of Lincoln are the same as flight costs from cities east of Lincoln. A random sample of the expense vouchers from last year was reviewed for the cost of airline tickets. Is there evidence of a significant difference in the costs of airline flights between the west and east coasts to Lincoln, Nebraska?
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_east - mu_west = 0 or mu_east = mu_west Ha: mu_east - mu_west ≠ 0 or mu_east ≠ mu_west
In a campus survey, a university polls its students to see how many hours they study in an average week. Females reported an average of 16.8 hours, while males reported an average of 13.8 hours. Is this evidence of a difference in average time spent studying by females compared to males.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_female - mu_males = 0 of mu_female = mu_males Ha: mu_female - mu_males ≠ 0 of mu_female ≠ mu_males
Investigators at an agricultural research facility randomly assigned equal numbers of chickens to be housed in two rooms. In group of chickens experienced normal day/night cycles, while in the other room lights were left on 24 hours a day to see if those chickens would lay more eggs. Is this evidence that chickens will lay more eggs when the lights are always on?
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_normal - mu_lights = 0 or mu_normal = mu_lights Ha: mu_normal - mu_lights < 0 or mu_normal < mu_lights
A researcher wonders if meat in the diet may be a factor in high blood pressure. She compares the blood pressures of 40 randomly selected vegetarians, to those of 40 people who eat meat.
2-sample t-test (degrees of freedom obtained from technology) Ho: mu_vegetarians - mu_meat = 0 of mu_vegetarians = mu_meat Ha: mu_vegetarians - mu_meat ≠ 0 of mu_vegetarians ≠ mu_meat
A factory recently installed new pollution control equipment ("scrubbers") on its smokestacks in hopes of reducing air pollution levels at a nearby national park. Randomly timed measurements of sulfate levels (in micrograms per cubic meter) were taken before (Set C1) and after (Set C2) the installation.
2-sample t-test (degrees of freedom obtained from technology). THIS IS NOT MATCHED PAIRS!! THERE IS NO WAY TO PAIR UP THE RANDOM MEASUREMENT TAKEN ON EACH SMOKESTACK! Ho: mu_C1 - mu_C2 = 0 or mu_C1 = mu_C2 Ha: mu_C1 - mu_C2 ≠ 0 or mu_C1 ≠ mu_C2
An orthodontist wonders if soda in the diet may be a factor in loose cement on children's braces. She checks the cement bonds (measuring the force required to loosen the bond) of 40 randomly selected patients who do not drink soda, and 40 patients who do drink soda.
2-sample t-test (degrees of freedom obtained with technology) Ho: mu_soda - mu_none = 0 or mu_soda = mu_none Ha: mu_soda - mu_none ≠ 0 or mu_soda ≠ mu_none
An independent testing lab obtained random samples of new halogen bulbs and standard incandescent bulbs made by the same company to establish the company's claim that, on average, the halogen bulb lasts longer than the incandescent one.
2-sample t-test (degrees of freedom unknown, but obtained from technology) (It is assumed the lab is interested in the average life of two bulbs) Ho: mu_halogen - mu_incandescent = 0 or mu_halogen = mu_incandescent Ha: mu_halogen - mu_incandescent ≠ 0 mu_halogen ≠ mu_incandescent
Doctors at a technology research facility randomly assigned equal numbers of people to use computer keyboards in two rooms. In one room a group of people typed a manuscript using standard keyboards, while in the other room people typed the same manuscript using ergonomic keyboards to see if those people could type more words per minute.
2-sample t-test (degrees of freedom unknown, but obtained from technology) Ho: mu_ergonomic - mu_standard = 0 or mu_ergonomic = mu_standard Ha: mu_ergonomic - mu_standard > 0 or mu_ergonomic > mu_standard
When two competing teams are equally matched, the probability that each team wins any game is 0.5. The NBA championship goes to the team that wins four games in a best-of-seven series. If the teams were equally matched, the probability that the final series ends with one of the teams sweeping four straight games would be 2(0.5)4 = 0.125 . Further probability calculations indicate that 25% of these series should last five games, 31.25% should last six games, and the other 31.25% should last the full seven games. Do you think the teams are usually equally matched?
Chi-square goodness of fit with (2-1)(4-1) = 3 degrees of freedom. Ho : The distribution of length of series is consistent with a 50-50 chance for each team to win each game. Ha : The distribution of lengths of series isn't consistent with a 50-50 chance for each team to win each game.
According to information from a college admissions office, 62% of the students there attended public high schools, 26% attended private high schools, 2% were home schooled, and the remaining students attended schools in other countries. Among this college's Honors Graduates last year there were 47 who came from public schools, 29 from private schools, 4 who had been home schooled, and 4 students from abroad. Is there any evidence that one type of high school might better equip students to attain high academic honors at this college?
Chi-square goodness of fit with 3 - 1 = 2 degrees of freedom. Ho: The distribution of secondary schools attended among Honors Graduates is the same as the rest of the college students. Ha: The distribution of secondary schools attended among Honors Graduates is different from the rest of the college students.
According to United Nations Population Division, the age distribution of the Commonwealth of Australia is: 21% less than 15 years of age, 67% between 15 and 65 years of age, and 12% are over 65 years old. A random sample of 210 residents of Canberra revealed 40 were less than 15 years of age, 145 were between 15 and 65 years of age, and 25 were over 65 years old. Are the ages of Canberra residents unusual in any way?
Chi-square goodness of fit with 3-1=2 degrees of freedom (there is one group and one categorical variable) Ho: The distribution of ages for residents of Canberra is the same as in Australia. Ha: The distribution of ages for residents of Canberra is different than Australia.
In the television series, "Mythbusters", the cast conducts studies to test urban myths and old- wives-tales; based on their findings, the cast declares the myth Busted, Plausible or Confirmed. A 2013 episode, entitled "Down and Dirty" tested the theory that in a public restroom, the bathroom stall closest to the entrance is used the least often and contains the least amount of bacteria. The cast observed 119 people enter a public restroom with 4 stalls and recorded the stall number (#1 closest to entrance and #4 farthest from entrance). The tally of the users of each stall showed that the stalls were used by 23, 38, 34 and 24 people, respectively. To ascertain if there is evidence to support the myth, test the hypothesis that the stalls are selected randomly.
Chi-square goodness of fit with 4 - 1 = 3 degrees of freedom. (This is one sample and we are seeing if the distribution of a categorical variable for this sample matches the myth) Ho: Each stall is used with the same frequency. Ha: The stalls are not used with the same frequency. One or more of the stall frequencies differs.
According to the American Red Cross, 45% of Americans have Type O blood, 40% Type A, 11% Type B, and 4% Type AB. Last week a blood drive at the high school collected 132 pints of blood. If 51 were Type O, 55 Type A, 17 Type B, and 9 were Type AB, was this yield unusual in any way?
Chi-square goodness of fit with 4 - 1 = 3 degrees of freedom. Ho: The distribution of blood types at the blood drive is the same as the distribution of blood types for all Americans. Ha: The distribution of blood types at the blood drive is different from the distribution of blood types for all Americans.
In a local school, vending machines offer a range of four types of drinks from juices to sports drinks. The purchasing agent thinks each type of drink is equally favored among the students buying drinks from the machines.
Chi-square goodness of fit with 4-1 = 3 degrees of freedom. One group and one categorical variable seeing if the distribution matches a given distribution. Ho:The types of drinks are uniformly distributed (equally favored) among the students buying drinks. Ha : The types are not uniformly distributed (equally favored) among the students buying drinks.
A biology professor reports that historically grades in her introductory biology course have been distributed as follows: 15% A's, 30% B's, 40% C's, 10% D's, and 5% F's. After a test, she wonders if the grade distribution matches the historical distribution.
Chi-square goodness of fit with 5 - 1 = 4 degrees of freedom. One group and one categorical variable seeing if the distribution matches a given distribution. Ho : The most recent grade distribution matches the historical grade distribution. Ha : The most recent grade distribution differs from the historical grade distribution.
The vast majority of states and the District of Columbia have adopted the Common Core State Standards (CCSS) for math and English language arts. Do teachers support the CCSS? In March 2003, The American Federal of Teachers (AFT) asked AFT member teachers "Based on what you know about the Common Core State Standards and the expectations they set for children, do you approve or disapprove of your state's decision to adopt them?" The following results were reported in American Educator (Volume 32, No. 2, Summer 2013, pg. 3): 27% Strongly Approve; 48% Somewhat Approve; 14% Somewhat Disapprove; 8% Strongly Approve; 3% Not Sure. A district superintendent asked the same question to the teachers in her district to assess the level of teacher support for the CCSS within the district.
Chi-square goodness of fit with 5-1=4 degrees of freedom. Comparing one group (teachers) measured on a categorical variable (CCSS approval) matches the national results. Ho: The district approval distribution matches the AFT national approval distribution. Ha: The district approval distribution differs from the AFT national approval distribution.
According to the Mars Candy Company, peanut M&M's are 12% brown, 15% yellow, 12% red, 23% blue, 23% orange, and 15% green. On a Saturday when you have run out of statistics homework, you decide to test this claim. You purchase a medium bag of peanut M&M's and find 39 browns, 44 yellows, 36 red, 78 blue, 73 orange, and 48 greens.
Chi-square goodness of fit with 6 - 1 = 5 degrees of freedom You are wanting to see the distribution of colors in your bag matches a pre-defined distribution. Ho: My bag's distribution of colors is the same as the official distribution. Ha: My bag's distribution of colors is different for at least one of the colors from the official distribution.
According to Ward's Communication, 19% of sports car enthusiasts prefer a red car, 16.2% silver, 14.7% black, 14.1% green, 14% white, and 22% other colors. A sample of 250 cars at a NASCAR raceway revealed 45 red cars, 42 silver cars, 34 black cars, 40 green cars, 39 white cars, and 50 other color cars. Are NASCAR color preferences typical of sports car enthusiasts?
Chi-square goodness of fit with 6 - 1 = 5 degrees of freedom. Ho : The distribution of the colors of cars at a NASCAR raceway is the same as the distribution of colors preferred by sports car enthusiasts. Ha : The distribution of the colors of cars at a NASCAR raceway is different than the distribution of colors preferred by sports car enthusiasts.
There are two main dog parks in Dallas, one near White Rock Lake and one near downtown. The downtown dog park is smaller and is located underneath several highway overpasses. There are many apartments, town homes, and lofts nearby. The White Rock Lake dog park is larger and provides a place for dogs to swim in the lake. The neighborhoods nearby are a mix of single family homes with some apartments. Jessica believes that since the downtown dog park is near many apartments, many of the dogs there will be smaller breeds, while the White Rock Lake park will attract larger, more active breeds. In order to test this assertion, she chooses random times during a month to visit each park. She categorizes the dogs there by size - toy, small, medium, large, and giant. Does the breed distribution for the downtown dog park differ significantly from the White Rock Lake dog park?
Chi-square test for homogeneity with (2-1)(5-1)=4 degrees of freedom you are comparing two different groups (two dog parks) on one categorical variable (size of dog) Ho: There is no association between the location of the dog park and the size of the dog. or the proportion of dogs in each category are the same for both dog parks. Ha: There is an association between the location of the dog park and the size of the dog. or the proportion of dogs is different for at least one category for the two dog parks.
We randomly divide 200 volunteers with headaches into two groups who take identical- looking pills. One group gets a homeopathic remedy and the other a placebo. After 20 minutes we ask them to rate their headache pain as "no change", "somewhat better", "much better", or "gone".
Chi-square test of homogeneity with (2 - 1)(4 - 1) = 2 degrees of freedom (this is an experiment, with 3 levels of categories) Ho: The proportion of responses in each category are the same for both groups. Ha: The proportion of responses in each category are different for at least one of the categories.
A pollster collects data on gender differences in car preferences. 50 males and 50 females, selected at random, are asked if they prefer American, Asian, or European models when they purchase a car. The pollster wishes to examine if there is an association between gender and car choice when making a purchase.
Chi-square test of homogeneity with (2-1)(3-1)=2 degrees of freedom Ho: there is no association between gender and car choice when making a purchase. Ha: there is an association between gender and car choice when making a purchase.
A manufacturing plant for recreational vehicles receives shipments from three different parts vendors. There has been a defect issue with some of the electrical wiring in the recreational vehicles manufactured at the plant. The plant manager wonders if all of the vendors might be contributing equally to the defect issue. The plant manager reviews three samples of quality assurance inspections from the last six months, one from each of the vendors.
Chi-square test of homogeneity with (3-1)(3-1) = 4 degrees of freedom. We want to know whether if all of the vendors might be contributing equally to the defect issue. Ho : The type of defects in vehicles made by the three vendors has the same distribution (are homogeneous). Ha : The type of defects in vehicles made by the three vendors does not have the same distribution (are not homogeneous).
A health professional selected a random sample of 100 patients from each of four major hospital emergency rooms to see if the primary reasons for emergency room visits are similar in all four major hospitals. The primary reason were categorized as accident, illegal activity, illness, or other.
Chi-square test of homogeneity with (4-1)(3-1)=6 degrees of freedom. There are four groups (100 patients from each hospital) and 1 categorical variable (reason for emergency room visit) Ho: There is no difference in the distribution of reasons for visiting the emergency room. All the proportions are essentially the same. Ha: There is a difference in the distribution of reasons for visiting the emergency room. At least one of the proportions is different.
A sports fan selected a random sample of 100 games from each of the NBA, the NFL, the NHL, and Major League Baseball to see if overtimes (or extra innings) are equally likely to occur in all four sports.
Chi-square test of homogeneity with 4 - 1 = 3 degrees of freedom. Ho: The distribution of overtime games is uniform among the four leagues. Ha: The distribution of overtime games is not uniform among the four leagues.
In a study of how the burden of poverty varies among U. S. regions, a random sample of 1000 individuals from each of four regions of the United States recently yielded the information on poverty.
Chi-square test of homogeneity with 4-1=3 degrees of freedom. Ho: There is no association between poverty levels and region of the US. Thre poverty rates are the same in each region. Ha: There is an association between poverty levels and region of the US. The poverty rates are different in at least one of the regions.
Could eye color be a warning signal for hearing loss in patients suffering from meningitis? British researcher Helen Cullington recorded the eye color of 130 deaf patients, and noted whether the patient's deafness had developed following treatment for meningitis.
Chi-square test of independence with (2-1)(2-1) = 1 degree of freedom. One group (deaf patients) compared on two categorical variables (eye color and meningitis) Ho : Deafness and eye color are independent. Ha : There is an association between deafness and eye color.
As part of a survey, students in a large statistics class were asked whether or not they ate breakfast that morning. The data was organized by gender. Is there evidence that eating breakfast associated with the student's gender?
Chi-square test of independence with (2-1)(2-1) = 1 degrees of freedom One group (stats students) and two categorical variables. Ho : Eating breakfast and student's sex are independent. Ha : There is an association between eating breakfast and student's sex.
A random sample of 800 adults was asked the following question: "Do you think current laws concerning the use of cloning for medical research are too strict, too lenient, or about right?" The pollsters also classified the respondents with respect to highest education level attained: high school, 2-year college degree, 4-year degree, or advanced degree. We wish to know if attitudes on cloning are related to education level.
Chi-square test of independence with (3 - 1)(4 - 1) = 6 degrees of freedom. (1 group - 800 adults - compared on two categorical variables - opinion on laws and education level) Ho: There is no association between cloning opinions and education level. Ha: There is an association between cloning opinions and education level.
In 2012, the Pew Research Center asked a random sample of Facebook users about their habits with this social media tool. One of the questions asked was about the plans people had for spending time on Facebook in the coming year. They were asked if they planned to more time, less time, or about the same time as now. Respondents were also broken into three age groups. Do these data provide evidence that age is independent of plans for Facebook time?
Chi-square test of independence with (3-1)(3-1)=4 degrees of freedom Ho : Age and planned Facebook time are independent (no association). Ha : Age and planned Facebook time are not independent (there is an association).
In the study "The Role of Sports as a Social Determinant for Children," student respondents in grades 4 through 6 were asked what they would most like to do at school: make good grades, be popular or be good at sports.
Chi-square test of independence with (3-1)(3-1)=4 degrees of freedom. One group (students) compared on two categorical variables (grade and most like to do) Ho: Type of school district and personal school goal are independent. Ha: There is an association between type of school district and personal school goal.
A personal trainer would like to know if a newly designed bootcamp regimen will significantly build body mass index (BMI). In an effort to test this, the trainer recorded the BMI for 15 different clients prior to the bootcamp and after the bootcamp. The trainer assumes the BMI's are approximately normal. Does the bootcamp regimen as advertised?
Matched pairs t-test with 14 degrees of freedom Ho: mu_BMI difference = 0 Ha: mu_BMI difference > 0
Hospital nurses sampled records of 58 surgery patients. They wondered if length of surgery (in hours) might be related to the length of the hospital stay (in days).
Linear Regression t-test with 56 degrees of freedom. (are two quantitative variables - length of surgery and length of stay - related?) Ho: There is no linear association between length of surgery and length of stay. B_1 = 0 Ha: There is a linear association between length of surgery and length of stay. B_1 ≠ 0
A student in Southern California notes that the temperature is generally cooler at the beach than it is at his home, 25 miles inland. He collects data every day for month, noting the high temperature at the beach and the high temperature at his house. He then conducts a test to see if there is a significant relationship between these two variables.
Linear regression t-test (degrees of freedom unknown) (is there an association between two quantitative variables?) Ho: there is no linear association between the high temperature at the beach and the high temperature at his house. Ha: there is a linear association between the high temperature at the beach and the high temperature at his house.
It's common for a movie's ticket sales to open high for the first couple of weeks, then gradually taper off as time passes. Hoping to be able to better understand how quickly sales decline, an industry analyst keeps track of box office revenues for a new film over its first 20 weeks. What inference method might provide useful insight?
Linear regression t-test (degrees of freedom unknown) There are two quantitative variables: Sales and weeks. Ho: There is no linear association between sales and time. Ha: There is a linear association between sales and time.
Doctors offer small candies to sixty teenagers, recording the number of candies consumed by each. One hour later they test the blood sugar level for each person. Is there any evidence that high blood sugar levels in teenagers are related to the amount of candy eaten?
Linear regression t-test with 58 degrees of freedom Ho: There is no linear association between the amount of candy eaten and blood sugar levels. B_1 = 0 Ha: There is a linear association between the amount of candy eaten and blood sugar levels. B_1 ≠ 0
Researchers offer small cookies to nine nursery school children and record the number of cookies consumed by each. Forty-five minutes later they observe these children during recess, and rate each child for hyperactivity on a scale from 1 - 20. Is there any evidence that sugar contributes to hyperactivity in children?
Linear regression t-test with 7 degrees of freedom. (Is there any evidence that sugar is associated with hyperactivity in children?) Ho: there is no linear association between sugar and hyperactivity in children Ha: there is a linear association between sugar and hyperactivity in children
Many states mandate tests that have to be passed in order for the students to graduate with a high school diploma. A local school superintendent believes that after-school tutoring will improve the scores of students in his district on the state's graduation test. A tutor agrees to work with 15 students for a month before the superintendent will approach the school board about implementing an after-school tutoring program. The after-school tutoring program will be implemented if student scores increase by more than 20 points.
Matched Pairs t-test with 14 degrees of freedom or 1-sample t-test on the before/after differences Ho: mu_difference = 20 Ha: mu_difference > 20 (it must be a matched pairs. The 15 students constitute 1 group with two measurements.)
Hoping to improve the gas mileage of their cars, a car company has made an adjustment in the manufacturing process. Random samples of automobiles coming off the assembly line have been measured each week that the plant has been in operation. The data from before and after the manufacturing adjustments were collected.
Matched pairs t test. The degrees of freedom is 1 less than the number of cars in the sample. This is just a 1-sample t-test on the before/after differences. Ho: mu_diff = 0 Ha: mu_diff ≠ 0
A fitness trainer wants to know if her weight-lifting program can quickly improve upper body strength in older people. To find out, she has a group of randomly selected people over 55 years old do push-ups for 90 seconds and counts the number each can do. After these people participate in her weightlifting program for 3 weeks, she tests them again in the same way.
Matched pairs t-test (degrees of freedom unknown) (this is a 1-sample t-test on the before/after differences) Ho: The before and after difference is 0 or mu_diff = 0 Ha: The before and after difference greater than 0 or mu_diff > 0
Researchers developing new drugs must be concerned about possible side effects. They must check a new medication for arthritis to be sure that it does not cause an unsafe increase in blood pressure. They measure the blood pressures of a group of 12 subjects, then administer the drug and recheck the blood pressures one hour later. The drug will be approved for use unless there is evidence that blood pressure has increased an average of more than 20 points.
Matched pairs t-test with 11 degrees of freedom Ho: mu_difference = 20 Ha: mu_difference > 20
A packing company considers hiring a national training consultant in hopes of improving productivity on the packing line. The national consultant agrees to work with 18 employees for one week as part of a trial before the packing company makes a decision about the training program. The training program will be implemented if the average product packed increases by more than 10 cases per day per employee.
Matched pairs t-test with 17 degrees of freedom Ho: mu_difference = 10 (the mean before and after difference is 10) Ha: mu_difference > 10 (the mean before and after difference is greater than 10)
Who is paid more in New York State - teachers or policemen? We select a random sample of 25 New York cities and find the starting salaries of teachers and policemen in each.
Matched pairs t-test with 24 degrees of freedom (25 cities with two measurements for each) Ho: mu_difference = 0 Ha: mu_difference ≠ 0
Are there more broken bones in summer or winter? We get records about the number of fractures treated in January and July at a random sample of 25 emergency rooms.
Matched pairs t-test with 24 degrees of freedom (taking two month measurements on each emergency room) Ho: mu_difference = 0 Ha: mu_difference ≠ 0
Which takes less time to travel to work - car or train? We select a random sample of 45 businessmen and compare their travel time to work for both types of commute.
Matched pairs t-test with 44 degrees of freedom. Ho: mu_difference = 0 Ha: mu_difference ≠ 0
Is there more gun violence in the summer heat than the winter cold? We get records of the number of gunshot wounds in January and July in a random sample of 50 emergency rooms.
Matched pairs t-test with 49 degrees of freedom (difference = july - january) Ho: mu_difference = 0 Ha: mu_difference > 0
A farmer would like to know if a new fertilizer increases his crop yield. In an effort to decide this, the farmer recorded the yield for 10 different fields prior to adding fertilizer and after adding the fertilizer. The farmer assumes the crop yields are approximately normal. Does the fertilizer work as advertised?
Matched pairs t-test with 9 degrees of freedom (this is also a 1-sample t-test on the differences) Ho: mu_difference = 0 (no difference in before and after crop yields) Ha: mu_difference > 0 (is a positive difference in before and after crop yields)
