Estimating Differences Between Populations Using Samples

_______ is a part of a group being surveyed.

• Sample

• Population

• Random Sample

• Quartile

237 fifth graders and 346 fourth graders went on a field trip. Estimate how many more fourth graders were on a field trip than fifth graders.

• 200 more

• 109 more

• 100 more

• 110 more

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?

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A 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?

• The probability is 0.90 that the difference in sample means for gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• The probability is 0.90 that the population mean difference in gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• About 90 percent of the differences in gas mileage for the two car manufacturers are between 3.5 mpg and 5.7 mpg.

• We are 90 percent confident that the difference in sample means for gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• 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.

You ask 75 randomly chosen students at a school how many movies they watch each week. If there are 1200 students in the school, estimate the number of students who watch one movie per week.

• 900

• 210

• 840

• 336

Identify the population and sample for each example.

• Population: Students

  Sample: Members of the football team

• Population: Members of the football team

  Sample: Students

• Population: Students

  Sample: Students that have had a broken bone

• Population: Members of the football team

  Sample: Members of the football team that have broken a bone.

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 one-sample t-interval for a population mean

• A one-sample t-interval for a sample mean

• A matched pairs t-interval for a mean difference

• A two-sample t-interval for a difference between sample means

• A two-sample t-interval for a difference between population means

A sociologist will conduct a two-sample t-test for a difference in means to investigate whether there is a significant difference, on average, between the salaries of people with bachelor’s degrees and people with master’s degrees. From a random sample of 32 people with a bachelor’s degree, the average salary was $55,000 with standard deviation $3,500. From a random sample of 28 people with a master’s degree, the average salary was $58,000 with a standard deviation of $4,000.

With a null hypothesis of no difference in the means, which of the following is the test statistic for the appropriate test to investigate whether there is a difference in population means (master’s degree minus bachelor’s degree) ?

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A two-sample t-test for a difference in means was conducted to investigate whether defensive players on a football team can bench-press more weight, on average, than offensive players. The conditions for inference were met, and the test produced a test statistic of t=1.083 and a p-value of 0.15. Based on the p-value and a significance level of �=0.05, which of the following is the correct conclusion?

• Reject the null hypothesis because 0.15>0.05. There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press the same amount of weight, on average, as offensive players.

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 will extend from 0.46 to 1.82.

• Approximately 90 percent of the intervals constructed will capture the difference in sample means.

• Approximately 90 percent of the intervals constructed will capture the difference in population means.

• Approximately 90 percent of the intervals constructed will capture at least one of the sample means.

• Approximately 90 percent of the intervals constructed will capture at least one of the population means.