STAT ANS
A national consumer agency selected independent random samples of 45 owners of newer cars (less than five years old) and 40 owners of older cars (more than five years old) to estimate the difference in mean dollar cost of yearly routine maintenance, such as oil changes, tire rotations, filters, and wiper blades. The agency found the mean dollar cost per year for newer cars was $195 with a standard deviation of $46. For older cars, the mean was $286 with a standard deviation of $58. Which of the following represents the 95 percent confidence interval to estimate the difference (newer minus older) in the mean dollar cost of routine maintenance between newer and older cars?
(195−286)±1.992√(46^2)/45+(58^2)/40
The weekly sales at two movie theaters were recorded for a random sample of 25 weeks. A 95 percent confidence interval for the difference in mean weekly sales for the two movie theaters was calculated as ($1,288,$2,586). With all else remaining constant, which of the following would have resulted in a confidence interval narrower than the calculated interval?
A sample size greater than 25
A researcher in sports equipment is investigating the design of racing swimsuits for women. The researcher selected a sample of 40 women swimmers from high school swim teams in the state and randomly assigned each swimmer to one of two groups: suit A or suit B. The women will wear the assigned suits for a certain race, and the mean swim times for each group will be recorded. The difference in the sample mean swim times will be calculated. Which of the following is the appropriate inference procedure for analyzing the results?
A two-sample t-interval for a difference between population means
A snowboarding competition site is using a new design for the parallel giant slalom. The designer of the slalom is investigating whether there is a difference in the mean times taken to complete a run for men and women competitors. As part of the investigation, independent random samples of men and women who will use the slalom run are selected and their times to complete a run are recorded. Which of the following is the appropriate inference procedure by which the designer can estimate the difference in the mean completion times for men and women?
A two-sample tt-interval for a difference between population means
A sports equipment researcher investigated how different types of wood used to make baseball bats might affect batting. The researcher selected a sample of 80 batters from summer baseball leagues and randomly assigned the batters to one of two groups: the ash bat group or the maple bat group. The mean number of hits for each group was recorded at the end of the season, and the difference in the sample means was calculated. Which of the following is the appropriate inference procedure for analyzing the results of the investigation?
A two-sample tt-interval for a difference between population means
To test the durability of cell phone screens, phones are dropped from a height of 1 meter until they break. A random sample of 40 phones was selected from each of two manufacturers. The phones in the samples were dropped until the screens broke. The difference in the mean number of drops was recorded and used to construct the 90 percent confidence interval (0.46,1.82) to estimate the population difference in means. Consider the sampling procedure taking place repeatedly. Each time samples are selected, the phones are dropped and the statistics are used to construct a 90 percent confidence interval for the difference in means. Which of the following statements is a correct interpretation of the intervals?
Approximately 90 percent of the intervals constructed will capture the difference in population means.
Researchers investigated if there is a difference between two headache medications, R and S. Researchers measured the mean times required to obtain relief from a headache for patients taking one of the medications. From a random sample of 75 people with chronic headaches, 38 were randomly assigned to medication R and the remaining 37 were assigned to medication S. The time, in minutes, until each person experienced relief from a headache was recorded. The sample mean times were calculated for each medication. Have the conditions been met for inference with a confidence interval for the difference in population means?
Yes, all conditions have been met.
Alicia would like to know if there is a difference in the average price between two brands of shoes. She selected and analyzed a random sample of 40 different types of Brand A shoes and 33 different types of Brand B shoes. Alicia observes that the boxplot of the sample of Brand A shoe prices shows two outliers. Alicia wants to construct a confidence interval to estimate the difference in population means. Is the sampling distribution of the difference in sample means approximately normal?
Yes, because the size of each sample is at least 30.
The management of a large hardware store is interested in estimating the difference between the mean dollar amount of purchases made by customers who use the store's credit card and the mean dollar amount of purchases made by customers who use a different credit card. A random sample of 74 customers who used the store's credit card showed a mean purchase of $107 with a standard deviation of $12. A separate random sample of 58 customers who used a different credit card showed a mean purchase of $132 with a standard deviation of $9. Technology was used to calculate that the correct number of degrees of freedom is 129.78. Which of the following represents the margin of error for a 98 percent confidence interval to estimate the difference in the mean purchase amount for the two types of credit cards?
2.355√(12^2)/74+(9^2)/58
A 99% confidence interval for a difference in means was given as 25.1±4.3. Assuming all conditions for inference were met, which of the following is a correct interpretation of the 99% confidence level?
In repeated samples of the same size, approximately 99 percent of the intervals constructed from the samples will capture the difference in population means.
Health programs study the number of days that patients stay in hospitals. In one study, a random sample of 12 men had a mean of 7.95 days and a standard deviation of 6.2 days, and a random sample of 19 women had a mean of 7.1 days and a standard deviation of 5.0 days. The sample data will be used to construct a 95 percent confidence interval to estimate the difference between men and women in the mean number of days for the length of stay at a hospital. Have the conditions been met for inference with a confidence interval?
No. The sample sizes are not large enough to assume that the sampling distribution of the difference in sample means is approximately normal.
2 99 percent confidence intervals will be constructed to estimate the difference in means of two populations, R and W. One confidence interval, I9, will be constructed using samples of size 9 from each of R and W, and the other confidence interval, I81, will be constructed using samples of size 81 from each of R and W. When all other things remain the same, which of the following describes the relationship between the two confidence intervals?
The width of I81 will be 1/3 the width of I9
A random sample of size 32 is selected from population X, and a random sample of size 43 is selected from population Y. A 90 percent confidence interval to estimate the difference in means is given as (−1.25,0.87). Consider a change in the sample sizes such that a random sample of size 52 is selected from population X and a random sample of size 63 is selected from population Y. When all other things remain the same, what effect would such a change have on the interval?
The width of the interval will decrease.
Consumer group studied two different manufacturers of cars, J and K, to investigate differences in gas mileage for cars made by the two manufacturers. For a similar type of car, a random sample of 15 cars from J and a random sample of 12 cars from K were selected, and the gas mileages, in miles per gallon (mpg), were recorded. The difference in the sample mean gas mileages was used to construct the 90 percent confidence interval (3.5,5.7). Assuming all conditions for inference were met, which of the following is a correct interpretation of the interval?
We are 90 percent confident that the population mean difference of gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.
An ecologist is examining whether the number of fish caught in a large river basin has changed since a fire burned some of the surrounding forested area and vegetation along the river. Data in the form of fishing reports was available for a five-year period before the fire. From a random sample of 195 fishing reports before the fire, the mean catch was 6.3 fish with a standard deviation of 1.6. In a random sample of 143 reports three years after the fire, the mean catch was 7.1 fish with a standard deviation of 2.1. Which of the following represents the standard error of the difference in the mean number of fish caught before and after the fire?
√(1.6^2)/195+(2.1^2)/143
